# A monad for Latent Semantic Analysis workflows

## Introduction

In this document we describe the design and implementation of a (software programming) monad, [Wk1], for Latent Semantic Analysis workflows specification and execution. The design and implementation are done with Mathematica / Wolfram Language (WL).

What is Latent Semantic Analysis (LSA)? : A statistical method (or a technique) for finding relationships in natural language texts that is based on the so called Distributional hypothesis, [Wk2, Wk3]. (The Distributional hypothesis can be simply stated as “linguistic items with similar distributions have similar meanings”; for an insightful philosophical and scientific discussion see [MS1].) LSA can be seen as the application of Dimensionality reduction techniques over matrices derived with the Vector space model.

The goal of the monad design is to make the specification of LSA workflows (relatively) easy and straightforward by following a certain main scenario and specifying variations over that scenario.

The monad is named LSAMon and it is based on the State monad package “StateMonadCodeGenerator.m”, [AAp1, AA1], the document-term matrix making package “DocumentTermMatrixConstruction.m”, [AAp4, AA2], the Non-Negative Matrix Factorization (NNMF) package “NonNegativeMatrixFactorization.m”, [AAp5, AA2], and the package “SSparseMatrix.m”, [AAp2, AA5], that provides matrix objects with named rows and columns.

The data for this document is obtained from WL’s repository and it is manipulated into a certain ready-to-utilize form (and uploaded to GitHub.)

The monadic programming design is used as a Software Design Pattern. The LSAMon monad can be also seen as a Domain Specific Language (DSL) for the specification and programming of machine learning classification workflows.

Here is an example of using the LSAMon monad over a collection of documents that consists of 233 US state of union speeches.

The table above is produced with the package “MonadicTracing.m”, [AAp2, AA1], and some of the explanations below also utilize that package.

As it was mentioned above the monad LSAMon can be seen as a DSL. Because of this the monad pipelines made with LSAMon are sometimes called “specifications”.

Remark: In this document with “term” we mean “a word, a word stem, or other type of token.”

Remark: LSA and Latent Semantic Indexing (LSI) are considered more or less to be synonyms. I think that “latent semantic analysis” sounds more universal and that “latent semantic indexing” as a name refers to a specific Information Retrieval technique. Below we refer to “LSI functions” like “IDF” and “TF-IDF” that are applied within the generic LSA workflow.

### Contents description

The document has the following structure.

• The sections “Package load” and “Data load” obtain the needed code and data.
• The sections “Design consideration” and “Monad design” provide motivation and design decisions rationale.

• The sections “LSAMon overview”, “Monad elements”, and “The utilization of SSparseMatrix objects” provide technical descriptions needed to utilize the LSAMon monad .

• (Using a fair amount of examples.)
• The section “Unit tests” describes the tests used in the development of the LSAMon monad.
• (The random pipelines unit tests are especially interesting.)
• The section “Future plans” outlines future directions of development.
• The section “Implementation notes” just says that LSAMon’s development process and this document follow the ones of the classifications workflows monad ClCon, [AA6].

Remark: One can read only the sections “Introduction”, “Design consideration”, “Monad design”, and “LSAMon overview”. That set of sections provide a fairly good, programming language agnostic exposition of the substance and novel ideas of this document.

The following commands load the packages [AAp1–AAp7]:

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicLatentSemanticAnalysis.m"]
Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicTracing.m"]

In this section we load data that is used in the rest of the document. The text data was obtained through WL’s repository, transformed in a certain more convenient form, and uploaded to GitHub.

The text summarization and plots are done through LSAMon, which in turn uses the function RecordsSummary from the package “MathematicaForPredictionUtilities.m”, [AAp7].

### Hamlet

textHamlet =
ToString /@
Flatten[Import["https://raw.githubusercontent.com/antononcube/MathematicaVsR/master/Data/MathematicaVsR-Data-Hamlet.csv"]];

TakeLargestBy[
Tally[DeleteStopwords[ToLowerCase[Flatten[TextWords /@ textHamlet]]]], #[[2]] &, 20]

(* {{"ham", 358}, {"lord", 225}, {"king", 196}, {"o", 124}, {"queen", 120},
{"shall", 114}, {"good", 109}, {"hor", 109}, {"come",  107}, {"hamlet", 107},
{"thou", 105}, {"let", 96}, {"thy", 86}, {"pol", 86}, {"like", 81}, {"sir", 75},
{"'t", 75}, {"know", 74}, {"enter", 73}, {"th", 72}} *)

LSAMonUnit[textHamlet]⟹LSAMonMakeDocumentTermMatrix⟹LSAMonEchoDocumentTermMatrixStatistics;

### USA state of union speeches

url = "https://github.com/antononcube/MathematicaVsR/blob/master/Data/MathematicaVsR-Data-StateOfUnionSpeeches.JSON.zip?raw=true";
str = Import[url, "String"];
filename = First@Import[StringToStream[str], "ZIP"];
aStateOfUnionSpeeches = Association@ImportString[Import[StringToStream[str], {"ZIP", filename, "String"}], "JSON"];

lsaObj =
LSAMonUnit[aStateOfUnionSpeeches]⟹
LSAMonMakeDocumentTermMatrix⟹
LSAMonEchoDocumentTermMatrixStatistics["LogBase" -> 10];
TakeLargest[ColumnSumsAssociation[lsaObj⟹LSAMonTakeDocumentTermMatrix], 12]

(* <|"government" -> 7106, "states" -> 6502, "congress" -> 5023,
"united" -> 4847, "people" -> 4103, "year" -> 4022,
"country" -> 3469, "great" -> 3276, "public" -> 3094, "new" -> 3022,
"000" -> 2960, "time" -> 2922|> *)

### Stop words

In some of the examples below we want to explicitly specify the stop words. Here are stop words derived using the built-in functions DictionaryLookup and DeleteStopwords.

stopWords = Complement[DictionaryLookup["*"], DeleteStopwords[DictionaryLookup["*"]]];

Short[stopWords]

"you'll", "your", "you're", "yours", "yourself", "yourselves", "you've" } *)


## Design considerations

The steps of the main LSA workflow addressed in this document follow.

1. Get a collection of documents with associated ID’s.

2. Create a document-term matrix.

1. Here we apply the Bag-or-words model and Vector space model.
1. The sequential order of the words is ignored and each document is represented as a point in a multi-dimensional vector space.

2. That vector space axes correspond to the unique words found in the whole document collection.

2. Consider the application of stemming rules.

3. Consider the removal of stop words.

3. Apply matrix-entries weighting functions.

1. Those functions come from LSI.

2. Functions like “IDF”, “TF-IDF”, “GFIDF”.

4. Extract topics.

1. One possible statistical way of doing this is with Dimensionality reduction.

2. We consider using Singular Value Decomposition (SVD) and Non-Negative Matrix Factorization (NNMF).

5. Make and display the topics table.

6. Extract and display a statistical thesaurus of selected words.

7. Map search queries or unseen documents over the extracted topics.

8. Find the most important documents in the document collection. (Optional.)

The following flow-chart corresponds to the list of steps above.

• the introduction of new elements in LSA workflows,

• workflows elements variability, and

• workflows iterative changes and refining,

it is beneficial to have a DSL for LSA workflows. We choose to make such a DSL through a functional programming monad, [Wk1, AA1].

Here is a quote from [Wk1] that fairly well describes why we choose to make a classification workflow monad and hints on the desired properties of such a monad.

[…] The monad represents computations with a sequential structure: a monad defines what it means to chain operations together. This enables the programmer to build pipelines that process data in a series of steps (i.e. a series of actions applied to the data), in which each action is decorated with the additional processing rules provided by the monad. […]

Monads allow a programming style where programs are written by putting together highly composable parts, combining in flexible ways the possible actions that can work on a particular type of data. […]

Remark: Note that quote from [Wk1] refers to chained monadic operations as “pipelines”. We use the terms “monad pipeline” and “pipeline” below.

The monad we consider is designed to speed-up the programming of LSA workflows outlined in the previous section. The monad is named LSAMon for “Latent Semantic Analysis** Mon**ad”.

We want to be able to construct monad pipelines of the general form:

LSAMon is based on the State monad, [Wk1, AA1], so the monad pipeline form (1) has the following more specific form:

This means that some monad operations will not just change the pipeline value but they will also change the pipeline context.

In the monad pipelines of LSAMon we store different objects in the contexts for at least one of the following two reasons.

1. The object will be needed later on in the pipeline, or

2. The object is (relatively) hard to compute.

Such objects are document-term matrix, Dimensionality reduction factors and the related topics.

Let us list the desired properties of the monad.

• Rapid specification of non-trivial LSA workflows.

• The monad works with associations with string values, list of strings.

• The monad use the Linear vector spaces model .

• The document-term frequency matrix is can be created after removing stop words and/or word stemming.

• It is easy to specify and apply different LSI weight functions. (Like “IDF” or “GFIDF”.)

• The monad can do dimension reduction with SVD and NNMF and corresponding matrix factors are retrievable with monad functions.

• Documents (or query strings) external to the monad a easily mapped into monad’s Linear vector space of terms and the Linear vector space of topics.

• The monad allows of cursory examination and summarization of the data.

• The pipeline values can be of different types. Most monad functions modify the pipeline value; some modify the context; some just echo results.

• It is easy to obtain the pipeline value, context, and different context objects for manipulation outside of the monad.

• It is easy to tabulate extracted topics and related statistical thesauri.

• It is easy to specify and apply re-weighting functions for the entries of the document-term contingency matrices.

The LSAMon components and their interactions are fairly simple.

The main LSAMon operations implicitly put in the context or utilize from the context the following objects:

• document-term matrix,

• the factors obtained by matrix factorization algorithms,

• extracted topics.

Note the that the monadic set of types of LSAMon pipeline values is fairly heterogenous and certain awareness of “the current pipeline value” is assumed when composing LSAMon pipelines.

Obviously, we can put in the context any object through the generic operations of the State monad of the package “StateMonadGenerator.m”, [AAp1].

## LSAMon overview

When using a monad we lift certain data into the “monad space”, using monad’s operations we navigate computations in that space, and at some point we take results from it.

With the approach taken in this document the “lifting” into the LSAMon monad is done with the function LSAMonUnit. Results from the monad can be obtained with the functions LSAMonTakeValue, LSAMonContext, or with the other LSAMon functions with the prefix “LSAMonTake” (see below.)

Here is a corresponding diagram of a generic computation with the LSAMon monad:

Remark: It is a good idea to compare the diagram with formulas (1) and (2).

Let us examine a concrete LSAMon pipeline that corresponds to the diagram above. In the following table each pipeline operation is combined together with a short explanation and the context keys after its execution.

Here is the output of the pipeline:

The LSAMon functions are separated into four groups:

• operations,

• setters and droppers,

• takers,

### Monad functions interaction with the pipeline value and context

An overview of the those functions is given in the tables in next two sub-sections. The next section, “Monad elements”, gives details and examples for the usage of the LSAMon operations.

Here are the LSAMon State Monad functions (generated using the prefix “LSAMon”, [AAp1, AA1].)

Here are the usage descriptions of the main (not monad-supportive) LSAMon functions, which are explained in detail in the next section.

In this section we show that LSAMon has all of the properties listed in the previous section.

The monad head is LSAMon. Anything wrapped in LSAMon can serve as monad’s pipeline value. It is better though to use the constructor LSAMonUnit. (Which adheres to the definition in [Wk1].)

LSAMon[textHamlet, <||>]⟹LSAMonMakeDocumentTermMatrix[Automatic, Automatic]⟹LSAMonEchoFunctionContext[Short];

### Lifting data to the monad

The function lifting the data into the monad QRMon is QRMonUnit.

The lifting to the monad marks the beginning of the monadic pipeline. It can be done with data or without data. Examples follow.

LSAMonUnit[textHamlet]⟹LSAMonMakeDocumentTermMatrix⟹LSAMonTakeDocumentTermMatrix

LSAMonUnit[]⟹LSAMonSetDocuments[textHamlet]⟹LSAMonMakeDocumentTermMatrix⟹LSAMonTakeDocumentTermMatrix

(See the sub-section “Setters, droppers, and takers” for more details of setting and taking values in LSAMon contexts.)

Currently the monad can deal with data in the following forms:

• vectors of strings,

• associations with string values.

Generally, WL makes it easy to extract columns datasets order to obtain vectors or matrices, so datasets are not currently supported in LSAMon.

### Making of the document-term matrix

As it was mentioned above with “term” we mean “a word or a stemmed word”. Here is are examples of stemmed words.

WordData[#, "PorterStem"] & /@ {"consequential", "constitution", "forcing", ""}

The fundamental model of LSAMon is the so called Vector space model (or the closely related Bag-of-words model.) The document-term matrix is a linear vector space representation of the documents collection. That representation is further used in LSAMon to find topics and statistical thesauri.

Here is an example of ad hoc construction of a document-term matrix using a couple of paragraphs from “Hamlet”.

inds = {10, 19};

MatrixForm @ CrossTabulate[Flatten[KeyValueMap[Thread[{#1, #2}] &, TextWords /@ ToLowerCase[aTempText]], 1]]

When we construct the document-term matrix we (often) want to stem the words and (almost always) want to remove stop words. LSAMon’s function LSAMonMakeDocumentTermMatrix makes the document-term matrix and takes specifications for stemming and stop words.

lsaObj =
LSAMonUnit[textHamlet]⟹
LSAMonMakeDocumentTermMatrix["StemmingRules" -> Automatic, "StopWords" -> Automatic]⟹
LSAMonEchoFunctionContext[ MatrixPlot[#documentTermMatrix] &]⟹
LSAMonEchoFunctionContext[TakeLargest[ColumnSumsAssociation[#documentTermMatrix], 12] &];

We can retrieve the stop words used in a monad with the function LSAMonTakeStopWords.

Short[lsaObj⟹LSAMonTakeStopWords]

We can retrieve the stemming rules used in a monad with the function LSAMonTakeStemmingRules.

Short[lsaObj⟹LSAMonTakeStemmingRules]

The specification Automatic for stemming rules uses WordData[#,"PorterStem"]&.

Instead of the options style signature we can use positional signature.

• Options style: LSAMonMakeDocumentTermMatrix["StemmingRules" -> {}, "StopWords" -> Automatic] .

• Positional style: LSAMonMakeDocumentTermMatrix[{}, Automatic] .

### LSI weight functions

After making the document-term matrix we will most likely apply LSI weight functions, [Wk2], like “GFIDF” and “TF-IDF”. (This follows the “standard” approach used in search engines for calculating weights for document-term matrices; see [MB1].)

#### Frequency matrix

We use the following definition of the frequency document-term matrix F.

Each entry fij of the matrix F is the number of occurrences of the term j in the document i.

#### Weights

Each entry of the weighted document-term matrix M derived from the frequency document-term matrix F is expressed with the formula

where gj – global term weight; lij – local term weight; di – normalization weight.

Various formulas exist for these weights and one of the challenges is to find the right combination of them when using different document collections.

Here is a table of weight functions formulas.

#### Computation specifications

LSAMon function LSAMonApplyTermWeightFunctions delegates the LSI weight functions application to the package “DocumentTermMatrixConstruction.m”, [AAp4].

Here is an example.

lsaHamlet = LSAMonUnit[textHamlet]⟹LSAMonMakeDocumentTermMatrix;
wmat =
lsaHamlet⟹
LSAMonApplyTermWeightFunctions["IDF", "TermFrequency", "Cosine"]⟹
LSAMonTakeWeightedDocumentTermMatrix;

TakeLargest[ColumnSumsAssociation[wmat], 6]

Instead of using the positional signature of LSAMonApplyTermWeightFunctions we can specify the LSI functions using options.

wmat2 =
lsaHamlet⟹
LSAMonApplyTermWeightFunctions["GlobalWeightFunction" -> "IDF", "LocalWeightFunction" -> "TermFrequency", "NormalizerFunction" -> "Cosine"]⟹
LSAMonTakeWeightedDocumentTermMatrix;

TakeLargest[ColumnSumsAssociation[wmat2], 6]

Here we are summaries of the non-zero values of the weighted document-term matrix derived with different combinations of global, local, and normalization weight functions.

Magnify[#, 0.8] &@Multicolumn[Framed /@ #, 6] &@Flatten@
Table[
(wmat =
lsaHamlet⟹
LSAMonApplyTermWeightFunctions[gw, lw, nf]⟹
LSAMonTakeWeightedDocumentTermMatrix;
RecordsSummary[SparseArray[wmat]["NonzeroValues"],
List@StringRiffle[{gw, lw, nf}, ", "]]),
{gw, {"IDF", "GFIDF", "Binary", "None", "ColumnStochastic"}},
{lw, {"Binary", "Log", "None"}},
{nf, {"Cosine", "None", "RowStochastic"}}]
AutoCollapse[]

### Extracting topics

Streamlining topic extraction is one of the main reasons LSAMon was implemented. The topic extraction correspond to the so called “syntagmatic” relationships between the terms, [MS1].

#### Theoretical outline

The original weighed document-term matrix M is decomposed into the matrix factors W and H.

M ≈ W.H, W ∈ ℝm × k, H ∈ ℝk × n.

The i-th row of M is expressed with the i-th row of W multiplied by H.

The rows of H are the topics. SVD produces orthogonal topics; NNMF does not.

The i-the document of the collection corresponds to the i-th row W. Finding the Nearest Neighbors (NN’s) of the i-th document using the rows similarity of the matrix W gives document NN’s through topic similarity.

The terms correspond to the columns of H. Finding NN’s based on similarities of H’s columns produces statistical thesaurus entries.

The term groups provided by H’s rows correspond to “syntagmatic” relationships. Using similarities of H’s columns we can produce term clusters that correspond to “paradigmatic” relationships.

#### Computation specifications

Here is an example using the play “Hamlet” in which we specify additional stop words.

stopWords2 = {"enter", "exit", "[exit", "ham", "hor", "laer", "pol", "oph", "thy", "thee", "act", "scene"};

SeedRandom[2381]
lsaHamlet =
LSAMonUnit[textHamlet]⟹
LSAMonMakeDocumentTermMatrix["StemmingRules" -> Automatic, "StopWords" -> Join[stopWords, stopWords2]]⟹
LSAMonApplyTermWeightFunctions["GlobalWeightFunction" -> "IDF", "LocalWeightFunction" -> "None", "NormalizerFunction" -> "Cosine"]⟹
LSAMonExtractTopics["NumberOfTopics" -> 12, "MinNumberOfDocumentsPerTerm" -> 10, Method -> "NNMF", "MaxSteps" -> 20]⟹
LSAMonEchoTopicsTable["NumberOfTableColumns" -> 6, "NumberOfTerms" -> 10];

Here is an example using the USA presidents “state of union” speeches.

SeedRandom[7681]
lsaSpeeches =
LSAMonUnit[aStateOfUnionSpeeches]⟹
LSAMonMakeDocumentTermMatrix["StemmingRules" -> Automatic,  "StopWords" -> Automatic]⟹
LSAMonApplyTermWeightFunctions["GlobalWeightFunction" -> "IDF", "LocalWeightFunction" -> "None", "NormalizerFunction" -> "Cosine"]⟹
LSAMonExtractTopics["NumberOfTopics" -> 36, "MinNumberOfDocumentsPerTerm" -> 40, Method -> "NNMF", "MaxSteps" -> 12]⟹
LSAMonEchoTopicsTable["NumberOfTableColumns" -> 6, "NumberOfTerms" -> 10];

Note that in both examples:

1. stemming is used when creating the document-term matrix,

2. the default LSI re-weighting functions are used: “IDF”, “None”, “Cosine”,

3. the dimension reduction algorithm NNMF is used.

Things to keep in mind.

1. The interpretability provided by NNMF comes at a price.

2. NNMF is prone to get stuck into local minima, so several topic extractions (and corresponding evaluations) have to be done.

3. We would get different results with different NNMF runs using the same parameters. (NNMF uses random numbers initialization.)

4. The NNMF topic vectors are not orthogonal.

5. SVD is much faster than NNMF, but it topic vectors are hard to interpret.

6. Generally, the topics derived with SVD are stable, they do not change with different runs with the same parameters.

7. The SVD topics vectors are orthogonal, which provides for quick to find representations of documents not in the monad’s document collection.

The document-topic matrix W has column names that are automatically derived from the top three terms in each topic.

ColumnNames[lsaHamlet⟹LSAMonTakeW]

(* {"player-plai-welcom", "ro-lord-sir", "laert-king-attend",
"end-inde-make", "state-room-castl", "daughter-pass-love",
"hamlet-ghost-father", "father-thou-king",
"rosencrantz-guildenstern-king", "ophelia-queen-poloniu",
"answer-sir-mother", "horatio-attend-gentleman"} *)

Of course the row names of H have the same names.

RowNames[lsaHamlet⟹LSAMonTakeH]

(* {"player-plai-welcom", "ro-lord-sir", "laert-king-attend",
"end-inde-make", "state-room-castl", "daughter-pass-love",
"hamlet-ghost-father", "father-thou-king",
"rosencrantz-guildenstern-king", "ophelia-queen-poloniu",
"answer-sir-mother", "horatio-attend-gentleman"} *)

### Extracting statistical thesauri

The statistical thesaurus extraction corresponds to the “paradigmatic” relationships between the terms, [MS1].

Here is an example over the State of Union speeches.

entryWords = {"bank", "war", "economy", "school", "port", "health", "enemy", "nuclear"};

lsaSpeeches⟹
LSAMonExtractStatisticalThesaurus["Words" -> Map[WordData[#, "PorterStem"] &, entryWords], "NumberOfNearestNeighbors" -> 12]⟹
LSAMonEchoStatisticalThesaurus;

In the code above: (i) the options signature style is used, (ii) the statistical thesaurus entry words are stemmed first.

We can also call LSAMonEchoStatisticalThesaurus directly without calling LSAMonExtractStatisticalThesaurus first.

 lsaSpeeches⟹
LSAMonEchoStatisticalThesaurus["Words" -> Map[WordData[#, "PorterStem"] &, entryWords], "NumberOfNearestNeighbors" -> 12];

### Mapping queries and documents to terms

One of the most natural operations is to find the representation of an arbitrary document (or sentence or a list of words) in monad’s Linear vector space of terms. This is done with the function LSAMonRepresentByTerms.

Here is an example in which a sentence is represented as a one-row matrix (in that space.)

obj =
lsaHamlet⟹
LSAMonRepresentByTerms["Hamlet, Prince of Denmark killed the king."]⟹
LSAMonEchoValue;

Here we display only the non-zero columns of that matrix.

obj⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrixColumnSumsAssociation[#], # > 0& ]]]]& ];

#### Transformation steps

Assume that LSAMonRepresentByTerms is given a list of sentences. Then that function performs the following steps.

1. The sentence is split into a list of words.

2. If monad’s document-term matrix was made by removing stop words the same stop words are removed from the list of words.

3. If monad’s document-term matrix was made by stemming the same stemming rules are applied to the list of words.

4. The LSI global weights and the LSI local weight and normalizer functions are applied to sentence’s contingency matrix.

#### Equivalent representation

Let us look convince ourselves that documents used in the monad to built the weighted document-term matrix have the same representation as the corresponding rows of that matrix.

Here is an association of documents from monad’s document collection.

inds = {6, 10};
queries = Part[lsaHamlet⟹LSAMonTakeDocuments, inds];
queries

(* <|"id.0006" -> "Getrude, Queen of Denmark, mother to Hamlet. Ophelia, daughter to Polonius.",
"id.0010" -> "ACT I. Scene I. Elsinore. A platform before the Castle."|> *)

lsaHamlet⟹
LSAMonRepresentByTerms[queries]⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrixColumnSumsAssociation[#], # > 0& ]]]]& ];
lsaHamlet⟹
LSAMonEchoFunctionContext[MatrixForm[Part[Slot["weightedDocumentTermMatrix"], inds, Keys[Select[SSparseMatrixColumnSumsAssociation[Part[Slot["weightedDocumentTermMatrix"], inds, All]], # > 0& ]]]]& ];

### Mapping queries and documents to topics

Another natural operation is to find the representation of an arbitrary document (or a list of words) in monad’s Linear vector space of topics. This is done with the function LSAMonRepresentByTopics.

Here is an example.

inds = {6, 10};
queries = Part[lsaHamlet⟹LSAMonTakeDocuments, inds];
Short /@ queries

(* <|"id.0006" -> "Getrude, Queen of Denmark, mother to Hamlet. Ophelia, daughter to Polonius.",
"id.0010" -> "ACT I. Scene I. Elsinore. A platform before the Castle."|> *)

lsaHamlet⟹
LSAMonRepresentByTopics[queries]⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrixColumnSumsAssociation[#], # > 0& ]]]]& ];
lsaHamlet⟹
LSAMonEchoFunctionContext[MatrixForm[Part[Slot["W"], inds, Keys[Select[SSparseMatrixColumnSumsAssociation[Part[Slot["W"], inds, All]], # > 0& ]]]]& ];

#### Theory

In order to clarify what the function LSAMonRepresentByTopics is doing let us go through the formulas it is based on.

The original weighed document-term matrix M is decomposed into the matrix factors W and H.

M ≈ W.H, W ∈ ℝm × k, H ∈ ℝk × n

The i-th row of M is expressed with the i-th row of W multiplied by H.

mi ≈ wi.H.

For a query vector q0 ∈ ℝm we want to find its topics representation vector x ∈ ℝk:

q0 ≈ x.H.

Denote with H( − 1) the inverse or pseudo-inverse matrix of H. We have:

q0.H( − 1) ≈ (x.H).H( − 1) = x.(H.H( − 1)) = x.I,

x ∈ ℝk, H( − 1) ∈ ℝn × k, I ∈ ℝk × k.

In LSAMon for SVD H( − 1) = HT; for NNMF H( − 1) is the pseudo-inverse of H.

The vector x obtained with LSAMonRepresentByTopics.

### Tags representation

Sometimes we want to find the topics representation of tags associated with monad’s documents and the tag-document associations are one-to-many. See [AA3].

Let us consider a concrete example – we want to find what topics correspond to the different presidents in the collection of State of Union speeches.

Here we find the document tags (president names in this case.)

tags = StringReplace[
RowNames[
lsaSpeeches⟹LSAMonTakeDocumentTermMatrix],
RegularExpression[".\\d\\d\\d\\d-\\d\\d-\\d\\d"] -> ""];
Short[tags]

Here is the number of unique tags (president names.)

Length[Union[tags]]
(* 42 *)

Here we compute the tag-topics representation matrix using the function LSAMonRepresentDocumentTagsByTopics.

tagTopicsMat =
lsaSpeeches⟹
LSAMonRepresentDocumentTagsByTopics[tags]⟹
LSAMonTakeValue

Here is a heatmap plot of the tag-topics matrix made with the package “HeatmapPlot.m”, [AAp11].

HeatmapPlot[tagTopicsMat[[All, Ordering@ColumnSums[tagTopicsMat]]], DistanceFunction -> None, ImageSize -> Large]

### Finding the most important documents

There are several algorithms we can apply for finding the most important documents in the collection. LSAMon utilizes two types algorithms: (1) graph centrality measures based, and (2) matrix factorization based. With certain graph centrality measures the two algorithms are equivalent. In this sub-section we demonstrate the matrix factorization algorithm (that uses SVD.)

Definition: The most important sentences have the most important words and the most important words are in the most important sentences.

That definition can be used to derive an iterations-based model that can be expressed with SVD or eigenvector finding algorithms, [LE1].

Here we pick an important part of the play “Hamlet”.

focusText =
First@Pick[textHamlet, StringMatchQ[textHamlet, ___ ~~ "to be" ~~ __ ~~ "or not to be" ~~ ___, IgnoreCase -> True]];
Short[focusText]

(* "Ham. To be, or not to be- that is the question: Whether 'tis ....y.
O, woe is me T' have seen what I have seen, see what I see!" *)

LSAMonUnit[StringSplit[ToLowerCase[focusText], {",", ".", ";", "!", "?"}]]⟹
LSAMonMakeDocumentTermMatrix["StemmingRules" -> {}, "StopWords" -> Automatic]⟹
LSAMonApplyTermWeightFunctions⟹
LSAMonFindMostImportantDocuments[3]⟹
LSAMonEchoFunctionValue[GridTableForm];

### Setters, droppers, and takers

The values from the monad context can be set, obtained, or dropped with the corresponding “setter”, “dropper”, and “taker” functions as summarized in a previous section.

For example:

p = LSAMonUnit[textHamlet]⟹LSAMonMakeDocumentTermMatrix[Automatic, Automatic];

p⟹LSAMonTakeMatrix

If other values are put in the context they can be obtained through the (generic) function LSAMonTakeContext, [AAp1]:

Short@(p⟹QRMonTakeContext)["documents"]

(* <|"id.0001" -> "1604", "id.0002" -> "THE TRAGEDY OF HAMLET, PRINCE OF DENMARK", <<220>>, "id.0223" -> "THE END"|> *) 

Another generic function from [AAp1] is LSAMonTakeValue (used many times above.)

Here is an example of the “data dropper” LSAMonDropDocuments:

Keys[p⟹LSAMonDropDocuments⟹QRMonTakeContext]

(* {"documentTermMatrix", "terms", "stopWords", "stemmingRules"} *)

(The “droppers” simply use the state monad function LSAMonDropFromContext, [AAp1]. For example, LSAMonDropDocuments is equivalent to LSAMonDropFromContext[“documents”].)

## The utilization of SSparseMatrix objects

The LSAMon monad heavily relies on SSparseMatrix objects, [AAp6, AA5], for internal representation of data and computation results.

A SSparseMatrix object is a matrix with named rows and columns.

Here is an example.

n = 6;
rmat = ToSSparseMatrix[
SparseArray[{{1, 2} -> 1, {4, 5} -> 1}, {n, n}],
"RowNames" -> RandomSample[CharacterRange["A", "Z"], n],
"ColumnNames" -> RandomSample[CharacterRange["a", "z"], n]];
MatrixForm[rmat]

In this section we look into some useful SSparseMatrix idioms applied within LSAMon.

### Visualize with sorted rows and columns

In some situations it is beneficial to sort rows and columns of the (weighted) document-term matrix.

docTermMat =
lsaSpeeches⟹LSAMonTakeDocumentTermMatrix;
MatrixPlot[docTermMat[[Ordering[RowSums[docTermMat]],  Ordering[ColumnSums[docTermMat]]]], MaxPlotPoints -> 300, ImageSize -> Large]

The most popular terms in the document collection can be found through the association of the column sums of the document-term matrix.

TakeLargest[ColumnSumsAssociation[lsaSpeeches⟹LSAMonTakeDocumentTermMatrix], 10]

(* <|"state" -> 8852, "govern" -> 8147, "year" -> 6362, "nation" -> 6182,
"congress" -> 5040, "unit" -> 5040, "countri" -> 4504,
"peopl" -> 4306, "american" -> 3648, "law" -> 3496|> *)


Similarly for the lest popular terms.

TakeSmallest[
ColumnSumsAssociation[
lsaSpeeches⟹LSAMonTakeDocumentTermMatrix], 10]

(* <|"036" -> 1, "027" -> 1, "_____________________" -> 1, "0111" -> 1,
"006" -> 1, "0000" -> 1, "0001" -> 1, "______________________" -> 1,
"____" -> 1, "____________________" -> 1|> *)

### Showing only non-zero columns

In some cases we want to show only columns of the data or computation results matrices that have non-zero elements.

Here is an example (similar to other examples in the previous section.)

lsaHamlet⟹
LSAMonRepresentByTerms[{"this country is rotten",
"where is my sword my lord",
"poison in the ear should be in the play"}]⟹
LSAMonEchoFunctionValue[ MatrixForm[#1[[All, Keys[Select[ColumnSumsAssociation[#1], #1 > 0 &]]]]] &];

In the pipeline code above: (i) from the list of queries a representation matrix is made, (ii) that matrix is assigned to the pipeline value, (iii) in the pipeline echo value function the non-zero columns are selected with by using the keys of the non-zero elements of the association obtained with ColumnSumsAssociation.

### Similarities based on representation by terms

Here is way to compute the similarity matrix of different sets of documents that are not required to be in monad’s document collection.

sMat1 =
lsaSpeeches⟹
LSAMonRepresentByTerms[ aStateOfUnionSpeeches[[ Range[-5, -2] ]] ]⟹
LSAMonTakeValue

sMat2 =
lsaSpeeches⟹
LSAMonRepresentByTerms[ aStateOfUnionSpeeches[[ Range[-7, -3] ]] ]⟹
LSAMonTakeValue

MatrixForm[sMat1.Transpose[sMat2]]

### Similarities based on representation by topics

Similarly to weighted Boolean similarities matrix computation above we can compute a similarity matrix using the topics representations. Note that an additional normalization steps is required.

sMat1 =
lsaSpeeches⟹
LSAMonRepresentByTopics[ aStateOfUnionSpeeches[[ Range[-5, -2] ]] ]⟹
LSAMonTakeValue;
sMat1 = WeightTermsOfSSparseMatrix[sMat1, "None", "None", "Cosine"]

sMat2 =
lsaSpeeches⟹
LSAMonRepresentByTopics[ aStateOfUnionSpeeches[[ Range[-7, -3] ]] ]⟹
LSAMonTakeValue;
sMat2 = WeightTermsOfSSparseMatrix[sMat2, "None", "None", "Cosine"]

MatrixForm[sMat1.Transpose[sMat2]]

Note the differences with the weighted Boolean similarity matrix in the previous sub-section – the similarities that are less than 1 are noticeably larger.

## Unit tests

The development of LSAMon was done with two types of unit tests: (i) directly specified tests, [AAp7], and (ii) tests based on randomly generated pipelines, [AA8].

The unit test package should be further extended in order to provide better coverage of the functionalities and illustrate – and postulate – pipeline behavior.

### Directly specified tests

Here we run the unit tests file “MonadicLatentSemanticAnalysis-Unit-Tests.wlt”, [AAp8].

AbsoluteTiming[
]

The natural language derived test ID’s should give a fairly good idea of the functionalities covered in [AAp3].

Values[Map[#["TestID"] &, testObject["TestResults"]]]

"Make-document-term-matrix-1", "Make-document-term-matrix-2",
"Apply-term-weights-1", "Apply-term-weights-2", "Topic-extraction-1",
"Topic-extraction-2", "Topic-extraction-3", "Topic-extraction-4",
"Statistical-thesaurus-1", "Topics-representation-1",
"Take-document-term-matrix-1", "Take-weighted-document-term-matrix-1",
"Take-document-term-matrix-2", "Take-weighted-document-term-matrix-2",
"Take-terms-1", "Take-Factors-1", "Take-Factors-2", "Take-Factors-3",
"Take-Factors-4", "Take-StopWords-1", "Take-StemmingRules-1"} *)

### Random pipelines tests

Since the monad LSAMon is a DSL it is natural to test it with a large number of randomly generated “sentences” of that DSL. For the LSAMon DSL the sentences are LSAMon pipelines. The package “MonadicLatentSemanticAnalysisRandomPipelinesUnitTests.m”, [AAp9], has functions for generation of LSAMon random pipelines and running them as verification tests. A short example follows.

Generate pipelines:

SeedRandom[234]
pipelines = MakeLSAMonRandomPipelines[100];
Length[pipelines]

(* 100 *)

Here is a sample of the generated pipelines:

Here we run the pipelines as unit tests:

AbsoluteTiming[
res = TestRunLSAMonPipelines[pipelines, "Echo" -> False];
]

From the test report results we see that a dozen tests failed with messages, all of the rest passed.

rpTRObj = TestReport[res]

(The message failures, of course, have to be examined – some bugs were found in that way. Currently the actual test messages are expected.)

## Future plans

### Dimension reduction extensions

It would be nice to extend the Dimension reduction functionalities of LSAMon to include other algorithms like Independent Component Analysis (ICA), [Wk5]. Ideally with LSAMon we can do comparisons between SVD, NNMF, and ICA like the image de-nosing based comparison explained in [AA8].

Another direction is to utilize Neural Networks for the topic extraction and making of statistical thesauri.

### Conversational agent

Since LSAMon is a DSL it can be relatively easily interfaced with a natural language interface.

Here is an example of natural language commands parsed into LSA code using the package [AAp13].

## Implementation notes

The implementation methodology of the LSAMon monad packages [AAp3, AAp9] followed the methodology created for the ClCon monad package [AAp10, AA6]. Similarly, this document closely follows the structure and exposition of the ClCon monad document “A monad for classification workflows”, [AA6].

A lot of the functionalities and signatures of LSAMon were designed and programed through considerations of natural language commands specifications given to a specialized conversational agent.

## References

### Packages

[AAp1] Anton Antonov, State monad code generator Mathematica package, (2017), MathematicaForPrediction at GitHub*.

[AAp2] Anton Antonov, Monadic tracing Mathematica package, (2017), MathematicaForPrediction at GitHub*.

[AAp3] Anton Antonov, Monadic Latent Semantic Analysis Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp5] Anton Antonov, Non-Negative Matrix Factorization algorithm implementation in Mathematica, (2013), MathematicaForPrediction at GitHub.

[AAp6] Anton Antonov, SSparseMatrix Mathematica package, (2018), MathematicaForPrediction at GitHub.

[AAp7] Anton Antonov, MathematicaForPrediction utilities, (2014), MathematicaForPrediction at GitHub.

[AAp8] Anton Antonov, Monadic Latent Semantic Analysis unit tests, (2019), MathematicaVsR at GitHub.

[AAp9] Anton Antonov, Monadic Latent Semantic Analysis random pipelines Mathematica unit tests, (2019), MathematicaForPrediction at GitHub.

[AAp10] Anton Antonov, Monadic contextual classification Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp11] Anton Antonov, Heatmap plot Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp12] Anton Antonov,
Independent Component Analysis Mathematica package, MathematicaForPrediction at GitHub.

[AAp13] Anton Antonov, Latent semantic analysis workflows grammar in EBNF, (2018), ConverasationalAgents at GitHub.

### MathematicaForPrediction articles

[AA1] Anton Antonov, “Monad code generation and extension”, (2017), MathematicaForPrediction at GitHub.

[AA2] Anton Antonov, “Topic and thesaurus extraction from a document collection”, (2013), MathematicaForPrediction at GitHub.

[AA3] Anton Antonov, “The Great conversation in USA presidential speeches”, (2017), MathematicaForPrediction at WordPress.

[AA4] Anton Antonov, “Contingency tables creation examples”, (2016), MathematicaForPrediction at WordPress.

[AA5] Anton Antonov, “RSparseMatrix for sparse matrices with named rows and columns”, (2015), MathematicaForPrediction at WordPress.

[AA6] Anton Antonov, “A monad for classification workflows”, (2018), MathematicaForPrediction at WordPress.

[AA7] Anton Antonov, “Independent component analysis for multidimensional signals”, (2016), MathematicaForPrediction at WordPress.

[AA8] Anton Antonov, “Comparison of PCA, NNMF, and ICA over image de-noising”, (2016), MathematicaForPrediction at WordPress.

### Other

[Wk2] Wikipedia entry, Latent semantic analysis,

[Wk3] Wikipedia entry, Distributional semantics,

[Wk4] Wikipedia entry, Non-negative matrix factorization,

[LE1] Lars Elden, Matrix Methods in Data Mining and Pattern Recognition, 2007, SIAM. ISBN-13: 978-0898716269.

[MB1] Michael W. Berry & Murray Browne, Understanding Search Engines: Mathematical Modeling and Text Retrieval, 2nd. ed., 2005, SIAM. ISBN-13: 978-0898715811.

[MS1] Magnus Sahlgren, “The Distributional Hypothesis”, (2008), Rivista di Linguistica. 20 (1): 33[Dash]53.

[PS1] Patrick Scheibe, Mathematica (Wolfram Language) support for IntelliJ IDEA, (2013-2018), Mathematica-IntelliJ-Plugin at GitHub.

# Finding all structural breaks in time series

## Introduction

In this document we show how to find the so called “structural breaks”, [Wk1], in a given time series. The algorithm is based in on a systematic application of Chow Test, [Wk2], combined with an algorithm for local extrema finding in noisy time series, [AA1].

The algorithm implementation is based on the packages “MonadicQuantileRegression.m”, [AAp1], and “MonadicStructuralBreaksFinder.m”, [AAp2]. The package [AAp1] provides the software monad QRMon that allows rapid and concise specification of Quantile Regression workflows. The package [AAp2] extends QRMon with functionalities related to structural breaks finding.

### What is a structural break?

It looks like at least one type of “structural breaks” are defined through regression models, [Wk1]. Roughly speaking a structural break point of time series is a regressor point that splits the time series in such way that the obtained two parts have very different regression parameters.

One way to test such a point is to use Chow test, [Wk2]. From [Wk2] we have the definition:

The Chow test, proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known a priori (for instance, a major historical event such as a war).

### Example

Here is an example of the described algorithm application to the data from [Wk2].

QRMonUnit[data]⟹QRMonPlotStructuralBreakSplits[ImageSize -> Small];

Here we load the packages [AAp1] and [AAp2].

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicQuantileRegression.m"]
Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicStructuralBreaksFinder.m"]

## Data used

In this section we assign the data used in this document.

### Illustration data from Wikipedia

Here is the data used in the Wikipedia article “Chow test”, [Wk2].

data = {{0.08, 0.34}, {0.16, 0.55}, {0.24, 0.54}, {0.32, 0.77}, {0.4,
0.77}, {0.48, 1.2}, {0.56, 0.57}, {0.64, 1.3}, {0.72, 1.}, {0.8,
1.3}, {0.88, 1.2}, {0.96, 0.88}, {1., 1.2}, {1.1, 1.3}, {1.2,
1.3}, {1.3, 1.4}, {1.4, 1.5}, {1.4, 1.5}, {1.5, 1.5}, {1.6,
1.6}, {1.7, 1.1}, {1.8, 0.98}, {1.8, 1.1}, {1.9, 1.4}, {2.,
1.3}, {2.1, 1.5}, {2.2, 1.3}, {2.2, 1.3}, {2.3, 1.2}, {2.4,
1.1}, {2.5, 1.1}, {2.6, 1.2}, {2.6, 1.4}, {2.7, 1.3}, {2.8,
1.6}, {2.9, 1.5}, {3., 1.4}, {3., 1.8}, {3.1, 1.4}, {3.2,
1.4}, {3.3, 1.4}, {3.4, 2.}, {3.4, 2.}, {3.5, 1.5}, {3.6,
1.8}, {3.7, 2.1}, {3.8, 1.6}, {3.8, 1.8}, {3.9, 1.9}, {4., 2.1}};
ListPlot[data]

### S&P 500 Index

Here we get the time series corresponding to S&P 500 Index.

tsSP500 = FinancialData[Entity["Financial", "^SPX"], {{2015, 1, 1}, Date[]}]
DateListPlot[tsSP500, ImageSize -> Medium]

## Application of Chow Test

The Chow Test statistic is implemented in [AAp1]. In this document we rely on the relative comparison of the Chow Test statistic values: the larger the value of the Chow test statistic, the more likely we have a structural break.

Here is how we can apply the Chow Test with a QRMon pipeline to the [Wk2] data given above.

chowStats =
QRMonUnit[data]⟹
QRMonChowTestStatistic[Range[1, 3, 0.05], {1, x}]⟹
QRMonTakeValue;

We see that the regressor points \$Failed and 1.7 have the largest Chow Test statistic values.

Block[{chPoint = TakeLargestBy[chowStats, Part[#, 2]& , 1]},
ListPlot[{chowStats, chPoint}, Filling -> Axis, PlotLabel -> Row[{"Point with largest Chow Test statistic:",
Spacer[8], chPoint}]]]

The first argument of QRMonChowTestStatistic is a list of regressor points or Automatic. The second argument is a list of functions to be used for the regressions.

Here is an example of an automatic values call.

chowStats2 = QRMonUnit[data]⟹QRMonChowTestStatistic⟹QRMonTakeValue;
ListPlot[chowStats2, GridLines -> {
Part[
Part[chowStats2, All, 1],
OutlierIdentifiersOutlierPosition[
Part[chowStats2, All, 2],  OutlierIdentifiersSPLUSQuartileIdentifierParameters]], None}, GridLinesStyle -> Directive[{Orange, Dashed}], Filling -> Axis]

For the set of values displayed above we can apply simple 1D outlier identification methods, [AAp3], to automatically find the structural break point.

chowStats2[[All, 1]][[OutlierPosition[chowStats2[[All, 2]], SPLUSQuartileIdentifierParameters]]]
(* {1.7} *)

OutlierPosition[chowStats2[[All, 2]], SPLUSQuartileIdentifierParameters]
(* {20} *)

We cannot use that approach for finding all structural breaks in the general time series cases though as exemplified with the following code using the time series S&P 500 Index.

chowStats3 = QRMonUnit[tsSP500]⟹QRMonChowTestStatistic⟹QRMonTakeValue;
DateListPlot[chowStats3, Joined -> False, Filling -> Axis]
OutlierPosition[chowStats3[[All, 2]], SPLUSQuartileIdentifierParameters]
(* {} *)

OutlierPosition[chowStats3[[All, 2]], HampelIdentifierParameters]
(* {} *)

In the rest of the document we provide an algorithm that works for general time series.

## Finding all structural break points

Consider the problem of finding of all structural breaks in a given time series. That can be done (reasonably well) with the following procedure.

1. Chose functions for testing for structural breaks (usually linear.)
2. Apply Chow Test over dense enough set of regressor points.
3. Make a time series of the obtained Chow Test statistics.
4. Find the local maxima of the Chow Test statistics time series.
5. Determine the most significant break point.
6. Plot the splits corresponding to the found structural breaks.

QRMon has a function, QRMonFindLocalExtrema, for finding local extrema; see [AAp1, AA1]. For the goal of finding all structural breaks, that semi-symbolic algorithm is the crucial part in the steps above.

## Computation

### Chose fitting functions

fitFuncs = {1, x};

### Find Chow test statistics local maxima

The computation below combines steps 2,3, and 4.

qrObj =
QRMonUnit[tsSP500]⟹
QRMonFindChowTestLocalMaxima["Knots" -> 20,
"NearestWithOutliers" -> True,
"NumberOfProximityPoints" -> 5, "EchoPlots" -> True,
"DateListPlot" -> True,
ImageSize -> Medium]⟹
QRMonEchoValue;

### Find most significant structural break point

splitPoint = TakeLargestBy[qrObj⟹QRMonTakeValue, #[[2]] &, 1][[1, 1]]

### Plot structural breaks splits and corresponding fittings

Here we just make the plots without showing them.

sbPlots =
QRMonUnit[tsSP500]⟹
QRMonPlotStructuralBreakSplits[(qrObj⟹ QRMonTakeValue)[[All, 1]],
"LeftPartColor" -> Gray, "DateListPlot" -> True,
"Echo" -> False,
ImageSize -> Medium]⟹
QRMonTakeValue;


The function QRMonPlotStructuralBreakSplits returns an association that has as keys paired split points and Chow Test statistics; the plots are association’s values.

Here we tabulate the plots with plots with most significant breaks shown first.

Multicolumn[
KeyValueMap[
Show[#2, PlotLabel ->
Grid[{{"Point:", #1[[1]]}, {"Chow Test statistic:", #1[[2]]}}, Alignment -> Left]] &, KeySortBy[sbPlots, -#[[2]] &]], 2]

## Future plans

We can further apply the algorithm explained above to identifying time series states or components. The structural break points are used as knots in appropriate Quantile Regression fitting. Here is an example.

The plan is to develop such an identifier of time series states in the near future. (And present it at WTC-2019.)

## References

### Articles

[Wk1] Wikipedia entry, Structural breaks.

[Wk2] Wikipedia entry, Chow test.

[AA1] Anton Antonov, “Finding local extrema in noisy data using Quantile Regression”, (2019), MathematicaForPrediction at WordPress.

[AA2] Anton Antonov, “A monad for Quantile Regression workflows”, (2018), MathematicaForPrediction at GitHub.

### Packages

[AAp1] Anton Antonov, Monadic Quantile Regression Mathematica package, (2018), MathematicaForPrediction at GitHub.

[AAp2] Anton Antonov, Monadic Structural Breaks Finder Mathematica package, (2019), MathematicaForPrediction at GitHub.

[AAp3] Anton Antonov, Implementation of one dimensional outlier identifying algorithms in Mathematica, (2013), MathematicaForPrediction at GitHub.

### Videos

[AAv1] Anton Antonov, Structural Breaks with QRMon, (2019), YouTube.

# Call graph generation for context functions

## In brief

This document describes the package CallGraph.m for making call graphs between the functions that belong to specified contexts.

The main function is CallGraph that gives a graph with vertices that are functions names and edges that show which functions call which other functions. With the default option values the graph vertices are labeled and have tooltips with function usage messages.

## General design

The call graphs produced by the main package function CallGraph are assumed to be used for studying or refactoring of large code bases written with Mathematica / Wolfram Language.

The argument of CallGraph is a context string or a list of context strings.

With the default values of its options CallGraph produces a graph with labeled nodes and the labels have tooltips that show the usage messages of the functions from the specified contexts. It is assumed that this would be the most useful call graph type for studying the codes of different sets of packages.

We can make simple, non-label, non-tooltip call graph using CallGraph[ ... , "UsageTooltips" -> False ].

The simple call graph can be modified with the functions:

CallGraphAddUsageMessages, CallGraphAddPrintDefinitionsButtons, CallGraphBiColorCircularEmbedding

Each of those functions is decorating the simple graph in a particular way.

This loads the package CallGraph.m :

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/Misc/CallGraph.m"]

The following packages are used in the examples below.

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicQuantileRegression.m"]

Get["https://raw.githubusercontent.com/szhorvat/IGraphM/master/IGInstaller.m"];
Needs["IGraphM"]

## Usage examples

### Generate a call graph with usage tooltips

CallGraph["IGraphM", GraphLayout -> "SpringElectricalEmbedding", ImageSize -> Large]

### Generate a call graph by excluding symbols

gr = CallGraph["IGraphM", Exclusions -> Map[ToExpression, Names["IG*Q"]], ImageSize -> 900]

### Generate call graph with buttons to print definitions

gr0 = CallGraph["IGraphM", "UsageTooltips" -> False];
gr1 = CallGraphAddPrintDefinitionsButtons[gr0, GraphLayout -> "StarEmbedding", ImageSize -> 900]

### Generate circular embedding graph color

cols = RandomSample[ ColorData["Rainbow"] /@ Rescale[Range[VertexCount[gr1]]]];

CallGraphBiColorCircularEmbedding[ gr1, "VertexColors" -> cols, ImageSize -> 900 ]

(The core functions used for the implementation of CallGraphBiColorCircularEmbedding were taken from kglr’s Mathematica Stack Exchange answer: https://mathematica.stackexchange.com/a/188390/34008 . Those functions were modified to take additional arguments.)

## Options

The package functions "CallGraph*" take all of the options of the function Graph. Below are described the additional options of CallGraph.

• “PrivateContexts”
Should the functions of the private contexts be included in the call graph.

• “SelfReferencing”
Should the self referencing edges be excluded or not.

• “AtomicSymbols”
Should atomic symbols be included in the call graph.

• Exclusions
Symbols to be excluded from the call graph.

• “UsageTooltips”
Should vertex labels with the usage tooltips be added.

• “UsageTooltipsStyle”
The style of the usage tooltips.

## Possible issues

• With large context (e.g. “System”) the call graph generation might take long time. (See the TODOs below.)

• With “PrivateContexts”->False the call graph will be empty if the public functions do not depend on each other.

• For certain packages the scanning of the down values would produce (multiple) error messages or warnings.

## Future plans

The following is my TODO list for this project.

1. Special handling for the “System” context.

2. Use the symbols up-values to make the call graph.

3. Consider/implement call graph making with specified patterns and list of symbols.

• Instead of just using contexts and exclusions. (The current approach/implementation.)
4. Provide special functions for “call sequence” tracing for a specified symbol.

# Parametrized event records data transformations

## Introduction

In this document we describe transformations of events records data in order to make that data more amenable for the application of Machine Learning (ML) algorithms.

Consider the following problem formulation (done with the next five bullet points.)

• From data representing a (most likely very) diverse set of events we want to derive contingency matrices corresponding to each of the variables in that data.

• The events are observations of the values of a certain set of variables for a certain set of entities. Not all entities have events for all variables.

• The observation times do not form a regular time grid.

• Each contingency matrix has rows corresponding to the entities in the data and has columns corresponding to time.

• The software component providing the functionality should allow parametrization and repeated execution. (As in ML classifier training and testing scenarios.)

The phrase “event records data” is used instead of “time series” in order to emphasize that (i) some variables have categorical values, and (ii) the data can be given in some general database form, like transactions long-form.

The required transformations of the event records in the problem formulation above are done through the monad ERTMon, [AAp3]. (The name “ERTMon” comes from “Event Records Transformations Monad”.)

The monad code generation and utilization is explained in [AA1] and implemented with [AAp1].

It is assumed that the event records data is put in a form that makes it (relatively) easy to extract time series for the set of entity-variable pairs present in that data.

In brief ERTMon performs the following sequence of transformations.

1. The event records of each entity-variable pair are shifted to adhere to a specified start or end point,

2. The event records for each entity-variable pair are aggregated and normalized with specified functions over a specified regular grid,

3. Entity vs. time interval contingency matrices are made for each combination of variable and aggregation function.

The transformations are specified with a “computation specification” dataset.

Here is an example of an ERTMon pipeline over event records:

The rest of the document describes in detail:

• the structure, format, and interpretation of the event records data and computations specifications,

• the transformations of time series aligning, aggregation, and normalization,

• the software pattern design – a monad – that allows sequential specifications of desired transformations.

Concrete examples are given using weather data. See [AAp9].

The following commands load the packages [AAp1-AAp9].

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicEventRecordsTransformations.m"]
Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/Misc/WeatherEventRecords.m"]

The data we use is weather data from meteorological stations close to certain major cities. We retrieve the data with the function WeatherEventRecords from the package [AAp9].

?WeatherEventRecords

WeatherEventRecords[ citiesSpec_: {{_String, _String}..}, dateRange:{{_Integer, _Integer, _Integer}, {_Integer, _Integer, _Integer}}, wProps:{_String..} : {“Temperature”}, nStations_Integer : 1 ] gives an association with event records data.

citiesSpec = {{"Miami", "USA"}, {"Chicago", "USA"}, {"London",  "UK"}};
dateRange = {{2017, 7, 1}, {2018, 6, 31}};
wProps = {"Temperature", "MaxTemperature", "Pressure", "Humidity", "WindSpeed"};
res1 = WeatherEventRecords[citiesSpec, dateRange, wProps, 1];

citiesSpec = {{"Jacksonville", "USA"}, {"Peoria", "USA"}, {"Melbourne", "Australia"}};
dateRange = {{2016, 12, 1}, {2017, 12, 31}};
res2 = WeatherEventRecords[citiesSpec, dateRange, wProps, 1];

Here we assign the obtained datasets to variables we use below:

eventRecords = Join[res1["eventRecords"], res2["eventRecords"]];
entityAttributes = Join[res1["entityAttributes"], res2["entityAttributes"]];

Here are the summaries of the datasets eventRecords and entityAttributes:

RecordsSummary[eventRecords]
RecordsSummary[entityAttributes]

## Design considerations

### Workflow

The steps of the main event records transformations workflow addressed in this document follow.

1. Ingest event records and entity attributes given in the Star schema style.

2. Ingest a computation specification.

1. Specified are aggregation time intervals, aggregation functions, normalization types and functions.
3. Group event records based on unique entity ID and variable pairs.
1. Additional filtering can be applied using the entity attributes.
4. For each variable find descriptive statistics properties.
1. This is to facilitate normalization procedures.

2. Optionally, for each variable find outlier boundaries.

5. Align each group of records to start or finish at some specified point.

1. For each variable we want to impose a regular time grid.
6. From each group of records produce a time series.

7. For each time series do prescribed aggregation and normalization.

1. The variable that corresponds to each group of records has at least one (possibly several) computation specifications.
8. Make a contingency matrix for each time series obtained in the previous step.
1. The contingency matrices have entity ID’s as rows, and time intervals enumerating values of time intervals.

The following flow-chart corresponds to the list of steps above.

A corresponding monadic pipeline is given in the section “Larger example pipeline”.

### Feature engineering perspective

The workflow above describes a way to do feature engineering over a collection of event records data. For a given entity ID and a variable we derive several different time series.

Couple of examples follow.

• One possible derived feature (times series) is for each entity-variable pair we make time series of the hourly mean value in each of the eight most recent hours for that entity. The mean values are normalized by the average values of the records corresponding to that entity-variable pair.

• Another possible derived feature (time series) is for each entity-variable pair to make a time series with the number of outliers in the each half-hour interval, considering the most recent 20 half-hour intervals. The outliers are found by using outlier boundaries derived by analyzing all values of the corresponding variable, across all entities.

From the examples above – and some others – we conclude that for each feature we want to be able to specify:

• maximum history length (say from the most recent observation),

• aggregation interval length,

• aggregation function (to be applied in each interval),

• normalization function (per entity, per cohort of entities, per variable),

• conversion of categorical values into numerical ones.

### Repeated execution

We want to be able to do repeated executions of the specified workflow steps.

Consider the following scenario. After the event records data is converted to a entity-vs-feature contingency matrix, we use that matrix to train and test a classifier. We want to find the combination of features that gives the best classifier results. For that reason we want to be able to easily and systematically change the computation specifications (interval size, aggregation and normalization functions, etc.) With different computation specifications we obtain different entity-vs-feature contingency matrices, that would have different performance with different classifiers.

Using the classifier training and testing scenario we see that there is another repeated execution perspective: after the feature engineering is done over the training data, we want to be able to execute exactly the same steps over the test data. Note that with the training data we find certain global or cohort normalization values and outlier boundaries that have to be used over the test data. (Not derived from the test data.)

The following diagram further describes the repeated execution workflow.

Further discussion of making and using ML classification workflows through the monad software design pattern can be found in [AA2].

## Event records data design

The data is structured to follow the style of Star schema. We have event records dataset (table) and entity attributes dataset (table).

The structure datasets (tables) proposed satisfy a wide range of modeling data requirements. (Medical and financial modeling included.)

### Entity event data

The entity event data has the columns “EntityID”, “LocationID”, “ObservationTime”, “Variable”, “Value”.

RandomSample[eventRecords, 6]

Most events can be described through “Entity event data”. The entities can be anything that produces a set of event data: financial transactions, vital sign monitors, wind speed sensors, chemical concentrations sensors.

The locations can be anything that gives the events certain “spatial” attributes: medical units in hospitals, sensors geo-locations, tiers of financial transactions.

### Entity attributes data

The entity attributes dataset (table) has attributes (immutable properties) of the entities. (Like, gender and race for people, longitude and latitude for wind speed sensors.)

entityAttributes[[1 ;; 6]]

### Example

For example, here we take all weather stations in USA:

ws = Normal[entityAttributes[Select[#Attribute == "Country" && #Value == "USA" &], "EntityID"]]

(* {"KMFL", "KMDW", "KNIP", "KGEU"} *)

Here we take all temperature event records for those weather stations:

srecs = eventRecords[Select[#Variable == "Temperature" && MemberQ[ws, #EntityID] &]];

And here plot the corresponding time series obtained by grouping the records by station (entity ID’s) and taking the columns “ObservationTime” and “Value”:

grecs = Normal @ GroupBy[srecs, #EntityID &][All, All, {"ObservationTime", "Value"}];
DateListPlot[grecs, ImageSize -> Large, PlotTheme -> "Detailed"]

This section goes through the steps of the general ERTMon workflow. For didactic purposes each sub-section changes the pipeline assigned to the variable p. Of course all functions can be chained into one big pipeline as shown in the section “Larger example pipeline”.

The monad is initialized with ERTMonUnit.

ERTMonUnit[]

(* ERTMon[None, <||>] *)

### Ingesting event records and entity attributes

The event records dataset (table) and entity attributes dataset (table) are set with corresponding setter functions. Alternatively, they can be read from files in a specified directory.

p =
ERTMonUnit[]⟹
ERTMonSetEventRecords[eventRecords]⟹
ERTMonSetEntityAttributes[entityAttributes]⟹
ERTMonEchoDataSummary;


### Computation specification

Using the package [AAp3] we can create computation specification dataset. Below is given an example of constructing a fairly complicated computation specification.

The package function EmptyComputationSpecificationRow can be used to construct the rows of the specification.

EmptyComputationSpecificationRow[]

(* <|"Variable" -> Missing[], "Explanation" -> "",
"MaxHistoryLength" -> 3600, "AggregationIntervalLength" -> 60,
"AggregationFunction" -> "Mean", "NormalizationScope" -> "Entity",
"NormalizationFunction" -> "None"|> *)

compSpecRows =
Join[EmptyComputationSpecificationRow[], <|"Variable" -> #,
"MaxHistoryLength" -> 60*24*3600,
"AggregationIntervalLength" -> 2*24*3600,
"AggregationFunction" -> "Mean",
"NormalizationScope" -> "Entity",
"NormalizationFunction" -> "Mean"|>] & /@
Union[Normal[eventRecords[All, "Variable"]]];
compSpecRows =
Join[
compSpecRows,
Join[EmptyComputationSpecificationRow[], <|"Variable" -> #,
"MaxHistoryLength" -> 60*24*3600,
"AggregationIntervalLength" -> 2*24*3600,
"AggregationFunction" -> "Range",
"NormalizationScope" -> "Country",
"NormalizationFunction" -> "Mean"|>] & /@
Union[Normal[eventRecords[All, "Variable"]]],
Join[EmptyComputationSpecificationRow[], <|"Variable" -> #,
"MaxHistoryLength" -> 60*24*3600,
"AggregationIntervalLength" -> 2*24*3600,
"AggregationFunction" -> "OutliersCount",
"NormalizationScope" -> "Variable"|>] & /@
Union[Normal[eventRecords[All, "Variable"]]]
];

The constructed rows are assembled into a dataset (with Dataset). The function ProcessComputationSpecification is used to convert a user-made specification dataset into a form used by ERTMon.

wCompSpec =
ProcessComputationSpecification[Dataset[compSpecRows]][SortBy[#Variable &]]


The computation specification is set to the monad with the function ERTMonSetComputationSpecification.

Alternatively, a computation specification can be created and filled-in as a CSV file and read into the monad. (Not described here.)

### Grouping event records by entity-variable pairs

With the function ERTMonGroupEntityVariableRecords we group the event records by the found unique entity-variable pairs. Note that in the pipeline below we set the computation specification first.

p =
p⟹
ERTMonSetComputationSpecification[wCompSpec]⟹
ERTMonGroupEntityVariableRecords;

### Descriptive statistics (per variable)

After the data is ingested into the monad and the event records are grouped per entity-variable pairs we can find certain descriptive statistics for the data. This is done with the general function ERTMonComputeVariableStatistic and the specialized function ERTMonFindVariableOutlierBoundaries.

p⟹ERTMonComputeVariableStatistic[RecordsSummary]⟹ERTMonEchoValue;
p⟹ERTMonComputeVariableStatistic⟹ERTMonEchoValue;
p⟹ERTMonComputeVariableStatistic[TakeLargest[#, 3] &]⟹ERTMonEchoValue;

(* value: <|Humidity->{1.,1.,0.993}, MaxTemperature->{48,48,48},
Pressure->{1043.1,1042.8,1041.1}, Temperature->{42.28,41.94,41.89},
WindSpeed->{54.82,44.63,44.08}|> *)



### Finding the variables outlier boundaries

The finding of outliers counts and fractions can be specified in the computation specification. Because of this there is a specialized function for outlier finding ERTMonFindVariableOutlierBoundaries. That function makes the association of the found variable outlier boundaries (i) to be the pipeline value and (ii) to be the value of context key “variableOutlierBoundaries”. The outlier boundaries are found using the functions of the package [AAp6].

If no argument is specified ERTMonFindVariableOutlierBoundaries uses the Hampel identifier (HampelIdentifierParameters).

p⟹ERTMonFindVariableOutlierBoundaries⟹ERTMonEchoValue;

(* value: <|Humidity->{0.522536,0.869464}, MaxTemperature->{14.2106,31.3494},
Pressure->{1012.36,1022.44}, Temperature->{9.88823,28.3318},
WindSpeed->{5.96141,19.4086}|> *)

Keys[p⟹ERTMonFindVariableOutlierBoundaries⟹ERTMonTakeContext]

(* {"eventRecords", "entityAttributes", "computationSpecification",
"entityVariableRecordGroups", "variableOutlierBoundaries"} *)


In the rest of document we use the outlier boundaries found with the more conservative identifier SPLUSQuartileIdentifierParameters.

p =
p⟹
ERTMonFindVariableOutlierBoundaries[SPLUSQuartileIdentifierParameters]⟹
ERTMonEchoValue;

(* value: <|Humidity->{0.176,1.168}, MaxTemperature->{-1.67,45.45},
Pressure->{1003.75,1031.35}, Temperature->{-5.805,43.755},
WindSpeed->{-5.005,30.555}|> *)

### Conversion of event records to time series

The grouped event records are converted into time series with the function ERTMonEntityVariableGroupsToTimeSeries. The time series are aligned to a time point specification given as an argument. The argument can be: a date object, “MinTime”, “MaxTime”, or “None”. (“MaxTime” is the default.)

p⟹
ERTMonEntityVariableGroupsToTimeSeries["MinTime"]⟹
ERTMonEchoFunctionContext[#timeSeries[[{1, 3, 5}]] &];


Compare the last output with the output of the following command.

p =
p⟹
ERTMonEntityVariableGroupsToTimeSeries["MaxTime"]⟹
ERTMonEchoFunctionContext[#timeSeries[[{1, 3, 5}]] &];

### Time series restriction and aggregation.

The main goal of ERTMon is to convert a diverse, general collection of event records into a collection of aligned time series over specified regular time grids.

The regular time grids are specified with the columns “MaxHistoryLength” and “AggregationIntervalLength” of the computation specification. The time series of the variables in the computation specification are restricted to the corresponding maximum history lengths and are aggregated using the corresponding aggregation lengths and functions.

p =
p⟹
ERTMonAggregateTimeSeries⟹
ERTMonEchoFunctionContext[DateListPlot /@ #timeSeries[[{1, 3, 5}]] &];


### Application of time series functions

At this point we can apply time series modifying functions. An often used such function is moving average.

p⟹
ERTMonApplyTimeSeriesFunction[MovingAverage[#, 6] &]⟹
ERTMonEchoFunctionValue[DateListPlot /@ #[[{1, 3, 5}]] &];

Note that the result is given as a pipeline value, the value of the context key “timeSeries” is not changed.

(In the future, the computation specification and its handling might be extended to handle moving average or other time series function specifications.)

### Normalization

With “normalization” we mean that the values of a given time series values are divided (normalized) with a descriptive statistic derived from a specified set of values. The specified set of values is given with the parameter “NormalizationScope” in computation specification.

At the normalization stage each time series is associated with an entity ID and a variable.

Normalization is done at three different scopes: “entity”, “attribute”, and “variable”.

Given a time series $T(i,var)$ corresponding to entity ID $i$ and a variable $var$ we define the normalization values for the different scopes in the following ways.

• Normalization with scope “entity” means that the descriptive statistic is derived from the values of $T(i,var)$ only.

• Normalization with scope attribute means that

• from the entity attributes dataset we find attribute value that corresponds to $i$,

• next we find all entity ID’s that are associated with the same attribute value,

• next we find value of normalization descriptive statistic using the time series that correspond to the variable $var$ and the entity ID’s found in the previous step.

• Normalization with scope “variable” means that the descriptive statistic is derived from the values of all time series corresponding to $var$.

Note that the scope “entity” is the most granular, and the scope “variable” is the coarsest.

The following command demonstrates the normalization effect – compare the $y$-axes scales of the time series corresponding to the same entity-variable pair.

p =
p⟹
ERTMonEchoFunctionContext[DateListPlot /@ #timeSeries[[{1, 3, 5}]] &]⟹
ERTMonNormalize⟹
ERTMonEchoFunctionContext[DateListPlot /@ #timeSeries[[{1, 3, 5}]] &];


Here are the normalization values that should be used when normalizing “unseen data.”

p⟹ERTMonTakeNormalizationValues

(* <|{"Humidity.Range", "Country", "USA"} -> 0.0864597,
{"Humidity.Range", "Country", "UK"} -> 0.066,
{"Humidity.Range", "Country", "Australia"} -> 0.145968,
{"MaxTemperature.Range", "Country", "USA"} -> 2.85468,
{"MaxTemperature.Range", "Country", "UK"} -> 78/31,
{"MaxTemperature.Range", "Country", "Australia"} -> 3.28871,
{"Pressure.Range", "Country", "USA"} -> 2.08222,
{"Pressure.Range", "Country", "Australia"} -> 3.33871,
{"Temperature.Range", "Country", "USA"} -> 2.14411,
{"Temperature.Range", "Country", "UK"} -> 1.25806,
{"Temperature.Range", "Country", "Australia"} -> 2.73032,
{"WindSpeed.Range", "Country", "USA"} -> 4.13532,
{"WindSpeed.Range", "Country", "UK"} -> 3.62097,
{"WindSpeed.Range", "Country", "Australia"} -> 3.17226|> *)

### Making contingency matrices

One of the main goals of ERTMon is to produce contingency matrices corresponding to the event records data.

The contingency matrices are created and stored as SSparseMatrix objects, [AAp7].

p =
p⟹ERTMonMakeContingencyMatrices;

We can obtain an association of the contingency matrices for each variable-and-aggregation-function pair, or obtain the overall contingency matrix.

p⟹ERTMonTakeContingencyMatrices
Dimensions /@ %
smat = p⟹ERTMonTakeContingencyMatrix;
MatrixPlot[smat, ImageSize -> 700]
RowNames[smat]

(* {"EGLC", "KGEU", "KMDW", "KMFL", "KNIP", "WMO95866"} *)

## Larger example pipeline

The pipeline shown in this section utilizes all main workflow functions of ERTMon. The used weather data and computation specification are described above.

## References

### Packages

[AAp7] Anton Antonov, SSparseMatrix Mathematica package, (2018), MathematicaForPrediction at GitHub*. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/SSparseMatrix.m .

### Documents

[AA1a] Anton Antonov, Monad code generation and extension, (2017), MathematicaForPrediction at GitHub.

# A monad for Quantile Regression workflows

## Introduction

In this document we describe the design and implementation of a (software programming) monad for Quantile Regression workflows specification and execution. The design and implementation are done with Mathematica / Wolfram Language (WL).

What is Quantile Regression? : Assume we have a set of two dimensional points each point being a pair of an independent variable value and a dependent variable value. We want to find a curve that is a function of the independent variable that splits the points in such a way that, say, 30% of the points are above that curve. This is done with Quantile Regression, see [Wk2, CN1, AA2, AA3]. Quantile Regression is a method to estimate the variable relations for all parts of the distribution. (Not just, say, the mean of the relationships found with Least Squares Regression.)

The goal of the monad design is to make the specification of Quantile Regression workflows (relatively) easy, straightforward, by following a certain main scenario and specifying variations over that scenario. Since Quantile Regression is often compared with Least Squares Regression and some type of filtering (like, Moving Average) those functionalities should be included in the monad design scenarios.

The monad is named QRMon and it is based on the State monad package "StateMonadCodeGenerator.m", [AAp1, AA1] and the Quantile Regression package "QuantileRegression.m", [AAp4, AA2].

The data for this document is read from WL’s repository or created ad-hoc.

The monadic programming design is used as a Software Design Pattern. The QRMon monad can be also seen as a Domain Specific Language (DSL) for the specification and programming of machine learning classification workflows.

Here is an example of using the QRMon monad over heteroscedastic data::

The table above is produced with the package "MonadicTracing.m", [AAp2, AA1], and some of the explanations below also utilize that package.

As it was mentioned above the monad QRMon can be seen as a DSL. Because of this the monad pipelines made with QRMon are sometimes called "specifications".

Remark: With "regression quantile" we mean "a curve or function that is computed with Quantile Regression".

### Contents description

The document has the following structure.

• The sections "Package load" and "Data load" obtain the needed code and data.
• The sections "Design consideration" and "Monad design" provide motivation and design decisions rationale.

• The sections "QRMon overview" and "Monad elements" provide technical description of the QRMon monad needed to utilize it.

• (Using a fair amount of examples.)
• The section "Unit tests" describes the tests used in the development of the QRMon monad.
• (The random pipelines unit tests are especially interesting.)
• The section "Future plans" outlines future directions of development.
• The section "Implementation notes" just says that QRMon’s development process and this document follow the ones of the classifications workflows monad ClCon, [AA6].

Remark: One can read only the sections "Introduction", "Design consideration", "Monad design", and "QRMon overview". That set of sections provide a fairly good, programming language agnostic exposition of the substance and novel ideas of this document.

The table above is produced with the package "MonadicTracing.m", [AAp2, AA1], and some of the explanations below also utilize that package.

As it was mentioned above the monad QRMon can be seen as a DSL. Because of this the monad pipelines made with QRMon are sometimes called "specifications".

Remark: With "regression quantile" we mean "a curve or function that is computed with Quantile Regression".

The following commands load the packages [AAp1–AAp6]:

Import["https://raw.githubusercontent.com/antononcube/\
Import["https://raw.githubusercontent.com/antononcube/\
MathematicaForPrediction/master/MonadicProgramming/MonadicTracing.m"]

In this section we load data that is used in the rest of the document. The time series data is obtained through WL’s repository.

The data summarization and plots are done through QRMon, which in turn uses the function RecordsSummary from the package "MathematicaForPredictionUtilities.m", [AAp6].

### Distribution data

The following data is generated to have [heteroscedasticity(https://en.wikipedia.org/wiki/Heteroscedasticity).

distData =
Table[{x,
Exp[-x^2] +
RandomVariate[
NormalDistribution[0, .15 Sqrt[Abs[1.5 - x]/1.5]]]}, {x, -3,
3, .01}];
Length[distData]

(* 601 *)

QRMonUnit[distData]⟹QRMonEchoDataSummary⟹QRMonPlot;

### Temperature time series

tsData = WeatherData[{"Orlando", "USA"}, "Temperature", {{2015, 1, 1}, {2018, 1, 1}, "Day"}]

QRMonUnit[tsData]⟹QRMonEchoDataSummary⟹QRMonDateListPlot;

### Financial time series

The following data is typical for financial time series. (Note the differences with the temperature time series.)

finData = TimeSeries[FinancialData["NYSE:GE", {{2014, 1, 1}, {2018, 1, 1}, "Day"}]];

QRMonUnit[finData]⟹QRMonEchoDataSummary⟹QRMonDateListPlot;

## Design considerations

The steps of the main regression workflow addressed in this document follow.

1. Retrieving data from a data repository.

2. Optionally, transform the data.

1. Delete rows with missing fields.

2. Rescale data along one or both of the axes.

3. Apply moving average (or median, or map.)

3. Verify assumptions of the data.

4. Run a regression algorithm with a certain basis of functions using:

1. Quantile Regression, or

2. Least Squares Regression.

5. Visualize the data and regression functions.

6. If the regression functions fit is not satisfactory go to step 4.

7. Utilize the found regression functions to compute:

1. outliers,

2. local extrema,

3. approximation or fitting errors,

4. conditional density distributions,

5. time series simulations.

The following flow-chart corresponds to the list of steps above.

• the introduction of new elements in regression workflows,

• workflows elements variability, and

• workflows iterative changes and refining,

it is beneficial to have a DSL for regression workflows. We choose to make such a DSL through a functional programming monad, [Wk1, AA1].

Here is a quote from [Wk1] that fairly well describes why we choose to make a classification workflow monad and hints on the desired properties of such a monad.

[…] The monad represents computations with a sequential structure: a monad defines what it means to chain operations together. This enables the programmer to build pipelines that process data in a series of steps (i.e. a series of actions applied to the data), in which each action is decorated with the additional processing rules provided by the monad. […] Monads allow a programming style where programs are written by putting together highly composable parts, combining in flexible ways the possible actions that can work on a particular type of data. […]

Remark: Note that quote from [Wk1] refers to chained monadic operations as "pipelines". We use the terms "monad pipeline" and "pipeline" below.

The monad we consider is designed to speed-up the programming of quantile regression workflows outlined in the previous section. The monad is named QRMon for "Quantile Regression Monad".

We want to be able to construct monad pipelines of the general form:

QRMon is based on the State monad, [Wk1, AA1], so the monad pipeline form (1) has the following more specific form:

This means that some monad operations will not just change the pipeline value but they will also change the pipeline context.

In the monad pipelines of QRMon we store different objects in the contexts for at least one of the following two reasons.

1. The object will be needed later on in the pipeline, or

2. The object is (relatively) hard to compute.

Such objects are transformed data, regression functions, and outliers.

Let us list the desired properties of the monad.

• Rapid specification of non-trivial quantile regression workflows.

• The monad works with time series, numerical matrices, and numerical vectors.

• The pipeline values can be of different types. Most monad functions modify the pipeline value; some modify the context; some just echo results.

• The monad can do quantile regression with B-Splines bases, quantile regression fit and least squares fit with specified bases of functions.

• The monad allows of cursory examination and summarization of the data.

• It is easy to obtain the pipeline value, context, and different context objects for manipulation outside of the monad.

• It is easy to plot different combinations of data, regression functions, outliers, approximation errors, etc.

The QRMon components and their interactions are fairly simple.

The main QRMon operations implicitly put in the context or utilize from the context the following objects:

• (time series) data,

• regression functions,

• outliers and outlier regression functions.

Note the that the monadic set of types of QRMon pipeline values is fairly heterogenous and certain awareness of "the current pipeline value" is assumed when composing QRMon pipelines.

Obviously, we can put in the context any object through the generic operations of the State monad of the package "StateMonadGenerator.m", [AAp1].

## QRMon overview

When using a monad we lift certain data into the "monad space", using monad’s operations we navigate computations in that space, and at some point we take results from it.

With the approach taken in this document the "lifting" into the QRMon monad is done with the function QRMonUnit. Results from the monad can be obtained with the functions QRMonTakeValue, QRMonContext, or with the other QRMon functions with the prefix "QRMonTake" (see below.)

Here is a corresponding diagram of a generic computation with the QRMon monad:

Remark: It is a good idea to compare the diagram with formulas (1) and (2).

Let us examine a concrete QRMon pipeline that corresponds to the diagram above. In the following table each pipeline operation is combined together with a short explanation and the context keys after its execution.

Here is the output of the pipeline:

The QRMon functions are separated into four groups:

• operations,

• setters and droppers,

• takers,

An overview of the those functions is given in the tables in next two sub-sections. The next section, "Monad elements", gives details and examples for the usage of the QRMon operations.

### Monad functions interaction with the pipeline value and context

The following table gives an overview the interaction of the QRMon monad functions with the pipeline value and context.

The following table shows the functions that are function synonyms or short-cuts.

Here are the QRMon State Monad functions (generated using the prefix "QRMon", [AAp1, AA1]):

In this section we show that QRMon has all of the properties listed in the previous section.

The monad head is QRMon. Anything wrapped in QRMon can serve as monad’s pipeline value. It is better though to use the constructor QRMonUnit. (Which adheres to the definition in [Wk1].)

QRMon[{{1, 223}, {2, 323}}, <||>]⟹QRMonEchoDataSummary;

### Lifting data to the monad

The function lifting the data into the monad QRMon is QRMonUnit.

The lifting to the monad marks the beginning of the monadic pipeline. It can be done with data or without data. Examples follow.

QRMonUnit[distData]⟹QRMonEchoDataSummary;
QRMonUnit[]⟹QRMonSetData[distData]⟹QRMonEchoDataSummary;

(See the sub-section "Setters, droppers, and takers" for more details of setting and taking values in QRMon contexts.)

Currently the monad can deal with data in the following forms:

• time series,

• numerical vectors,

• numerical matrices of rank two.

When the data lifted to the monad is a numerical vector vec it is assumed that vec has to become the second column of a "time series" matrix; the first column is derived with Range[Length[vec]] .

Generally, WL makes it easy to extract columns datasets order to obtain numerical matrices, so datasets are not currently supported in QRMon.

### Quantile regression with B-splines

This computes quantile regression with B-spline basis over 12 regularly spaced knots. (Using Linear Programming algorithms; see [AA2] for details.)

QRMonUnit[distData]⟹
QRMonQuantileRegression[12]⟹
QRMonPlot;

The monad function QRMonQuantileRegression has the same options as QuantileRegression. (The default value for option Method is different, since using "CLP" is generally faster.)

Options[QRMonQuantileRegression]

(* {InterpolationOrder -> 3, Method -> {LinearProgramming, Method -> "CLP"}} *)

Let us compute regression using a list of particular knots, specified quantiles, and the method "InteriorPoint" (instead of the Linear Programming library CLP):

p =
QRMonUnit[distData]⟹
QRMonQuantileRegression[{-3, -2, 1, 0, 1, 1.5, 2.5, 3}, Range[0.1, 0.9, 0.2], Method -> {LinearProgramming, Method -> "InteriorPoint"}]⟹
QRMonPlot;

Remark: As it was mentioned above the function QRMonRegression is a synonym of QRMonQuantileRegression.

The fit functions can be extracted from the monad with QRMonTakeRegressionFunctions, which gives an association of quantiles and pure functions.

ListPlot[# /@ distData[[All, 1]]] & /@ (p⟹QRMonTakeRegressionFunctions)

### Quantile regression fit and Least squares fit

Instead of using a B-spline basis of functions we can compute a fit with our own basis of functions.

Here is a basis functions:

bFuncs = Table[PDF[NormalDistribution[m, 1], x], {m, Min[distData[[All, 1]]], Max[distData[[All, 1]]], 1}];
Plot[bFuncs, {x, Min[distData[[All, 1]]], Max[distData[[All, 1]]]},
PlotRange -> All, PlotTheme -> "Scientific"]

Here we do a Quantile Regression fit, a Least Squares fit, and plot the results:

p =
QRMonUnit[distData]⟹
QRMonQuantileRegressionFit[bFuncs]⟹
QRMonLeastSquaresFit[bFuncs]⟹
QRMonPlot;


Remark: The functions "QRMon*Fit" should generally have a second argument for the symbol of the basis functions independent variable. Often that symbol can be omitted and implied. (Which can be seen in the pipeline above.)

Remark: As it was mentioned above the function QRMonRegressionFit is a synonym of QRMonQuantileRegressionFit and QRMonFit is a synonym of QRMonLeastSquaresFit.

As it was pointed out in the previous sub-section, the fit functions can be extracted from the monad with QRMonTakeRegressionFunctions. Here the keys of the returned/taken association consist of quantiles and "mean" since we applied both Quantile Regression and Least Squares Regression.

ListPlot[# /@ distData[[All, 1]]] & /@ (p⟹QRMonTakeRegressionFunctions)

### Default basis to fit (using Chebyshev polynomials)

One of the main advantages of using the function QuanileRegression of the package [AAp4] is that the functions used to do the regression with are specified with a few numeric parameters. (Most often only the number of knots is sufficient.) This is achieved by using a basis of B-spline functions of a certain interpolation order.

We want similar behaviour for Quantile Regression fitting we need to select a certain well known basis with certain desirable properties. Such basis is given by Chebyshev polynomials of first kind [Wk3]. Chebyshev polynomials bases can be easily generated in Mathematica with the functions ChebyshevT or ChebyshevU.

Here is an application of fitting with a basis of 12 Chebyshev polynomials of first kind:

QRMonUnit[distData]⟹
QRMonQuantileRegressionFit[12]⟹
QRMonLeastSquaresFit[12]⟹
QRMonPlot;

The code above is equivalent to the following code:

bfuncs = Table[ChebyshevT[i, Rescale[x, MinMax[distData[[All, 1]]], {-0.95, 0.95}]], {i, 0, 12}];

p =
QRMonUnit[distData]⟹
QRMonQuantileRegressionFit[bfuncs]⟹
QRMonLeastSquaresFit[bfuncs]⟹
QRMonPlot;

The shrinking of the ChebyshevT domain seen in the definitions of bfuncs is done in order to prevent approximation error effects at the ends of the data domain. The following code uses the ChebyshevT domain { − 1, 1} instead of the domain { − 0.95, 0.95} used above.

QRMonUnit[distData]⟹
QRMonQuantileRegressionFit[{4, {-1, 1}}]⟹
QRMonPlot;

### Regression functions evaluation

The computed quantile and least squares regression functions can be evaluated with the monad function QRMonEvaluate.

Evaluation for a given value of the independent variable:

p⟹QRMonEvaluate[0.12]⟹QRMonTakeValue

(* <|0.25 -> 0.930402, 0.5 -> 1.01411, 0.75 -> 1.08075, "mean" -> 0.996963|> *)

Evaluation for a vector of values:

p⟹QRMonEvaluate[Range[-1, 1, 0.5]]⟹QRMonTakeValue

(* <|0.25 -> {0.258241, 0.677461, 0.943299, 0.703812, 0.293741},
0.5 -> {0.350025, 0.768617, 1.02311, 0.807879, 0.374545},
0.75 -> {0.499338, 0.912183, 1.10325, 0.856729, 0.431227},
"mean" -> {0.355353, 0.776006, 1.01118, 0.783304, 0.363172}|> *)

Evaluation for complicated lists of numbers:

p⟹QRMonEvaluate[{0, 1, {1.5, 1.4}}]⟹QRMonTakeValue

(* <|0.25 -> {0.943299, 0.293741, {0.0762883, 0.10759}},
0.5 -> {1.02311, 0.374545, {0.103386, 0.139142}},
0.75 -> {1.10325, 0.431227, {0.133755, 0.177161}},
"mean" -> {1.01118, 0.363172, {0.107989, 0.142021}}|> *)


The obtained values can be used to compute estimates of the distributions of the dependent variable. See the sub-sections "Estimating conditional distributions" and "Dependent variable simulation".

### Errors and error plots

Here with "errors" we mean the differences between data’s dependent variable values and the corresponding values calculated with the fitted regression curves.

In the pipeline below we compute couple of regression quantiles, plot them together with the data, we plot the errors, compute the errors, and summarize them.

QRMonUnit[finData]⟹
QRMonQuantileRegression[10, {0.5, 0.1}]⟹
QRMonDateListPlot[Joined -> False]⟹
QRMonErrorPlots["DateListPlot" -> True, Joined -> False]⟹
QRMonErrors⟹
QRMonEchoFunctionValue["Errors summary:", RecordsSummary[#[[All, 2]]] & /@ # &];

Each of the functions QRMonErrors and QRMonErrorPlots computes the errors. (That computation is considered cheap.)

### Finding outliers

Finding outliers can be done with the function QRMonOultiers. The outliers found by QRMonOutliers are simply points that below or above certain regression quantile curves, for example, the ones corresponding to 0.02 and 0.98.

Here is an example:

p =
QRMonUnit[distData]⟹
QRMonQuantileRegression[6, {0.02, 0.98}]⟹
QRMonOutliers⟹
QRMonEchoValue⟹
QRMonOutliersPlot;

The function QRMonOutliers puts in the context values for the keys "outliers" and "outlierRegressionFunctions". The former is for the found outliers, the latter is for the functions corresponding to the used regression quantiles.

Keys[p⟹QRMonTakeContext]

(* {"data", "regressionFunctions", "outliers", "outlierRegressionFunctions"} *)

Here are the corresponding quantiles of the plot above:

Keys[p⟹QRMonTakeOutlierRegressionFunctions]

(* {0.02, 0.98} *)

The control of the outliers computation is done though the arguments and options of QRMonQuantileRegression (or the rest of the regression calculation functions.)

If only one regression quantile is found in the context and the corresponding quantile is less than 0.5 then QRMonOutliers finds only bottom outliers. If only one regression quantile is found in the context and the corresponding quantile is greater than 0.5 then QRMonOutliers finds only top outliers.

Here is an example for finding only the top outliers:

QRMonUnit[finData]⟹
QRMonQuantileRegression[5, 0.8]⟹
QRMonOutliers⟹
QRMonEchoFunctionContext["outlier quantiles:", Keys[#outlierRegressionFunctions] &]⟹
QRMonOutliersPlot["DateListPlot" -> True];


### Plotting outliers

The function QRMonOutliersPlot makes an outliers plot. If the outliers are not in the context then QRMonOutliersPlot calls QRMonOutliers first.

Here are the options of QRMonOutliersPlot:

Options[QRMonOutliersPlot]

(* {"Echo" -> True, "DateListPlot" -> False, ListPlot -> {Joined -> False}, Plot -> {}} *)

The default behavior is to echo the plot. That can be suppressed with the option "Echo".

QRMonOutliersPlot utilizes combines with Show two plots:

• one with ListPlot (or DateListPlot) for the data and the outliers,

• the other with Plot for the regression quantiles used to find the outliers.

That is why separate lists of options can be given to manipulate those two plots. The option DateListPlot can be used make plots with date or time axes.

QRMonUnit[tsData]⟹
QRMonQuantileRegression[12, {0.01, 0.99}]⟹
QRMonOutliersPlot[
"Echo" -> False,
"DateListPlot" -> True,
ListPlot -> {PlotStyle -> {Green, {PointSize[0.02],
Red}, {PointSize[0.02], Blue}}, Joined -> False,
PlotTheme -> "Grid"},
Plot -> {PlotStyle -> Orange}]⟹
QRMonTakeValue


### Estimating conditional distributions

Consider the following problem:

How to estimate the conditional density of the dependent variable given a value of the conditioning independent variable?

(In other words, find the distribution of the y-values for a given, fixed x-value.)

The solution of this problem using Quantile Regression is discussed in detail in [PG1] and [AA4].

Finding a solution for this problem can be seen as a primary motivation to develop Quantile Regression algorithms.

The following pipeline (i) computes and plots a set of five regression quantiles and (ii) then using the found regression quantiles computes and plots the conditional distributions for two focus points (−2 and 1.)

QRMonUnit[distData]⟹
QRMonQuantileRegression[6,
Range[0.1, 0.9, 0.2]]⟹
QRMonPlot[GridLines -> {{-2, 1}, None}]⟹
QRMonConditionalCDF[{-2, 1}]⟹
QRMonConditionalCDFPlot;

### Moving average, moving median, and moving map

Fairly often it is a good idea for a given time series to apply filter functions like Moving Average or Moving Median. We might want to:

• visualize the obtained transformed data,

• do regression over the transformed data,

• compare with regression curves over the original data.

For these reasons QRMon has the functions QRMonMovingAverage, QRMonMovingMedian, and QRMonMovingMap that correspond to the built-in functions MovingAverage, MovingMedian, and MovingMap.

Here is an example:

QRMonUnit[tsData]⟹
QRMonDateListPlot[ImageSize -> Small]⟹
QRMonMovingAverage[20]⟹
QRMonEchoFunctionValue["Moving avg: ", DateListPlot[#, ImageSize -> Small] &]⟹
QRMonMovingMap[Mean, Quantity[20, "Days"]]⟹
QRMonEchoFunctionValue["Moving map: ", DateListPlot[#, ImageSize -> Small] &];

### Dependent variable simulation

Consider the problem of making a time series that is a simulation of a process given with a known time series.

More formally,

• we are given a time-axis grid (regular or irregular),

• we consider each grid node to correspond to a random variable,

• we want to generate time series based on the empirical CDF’s of the random variables that correspond to the grid nodes.

The formulation of the problem hints to an (almost) straightforward implementation using Quantile Regression.

p = QRMonUnit[tsData]⟹QRMonQuantileRegression[30, Join[{0.01}, Range[0.1, 0.9, 0.1], {0.99}]];

tsNew =
p⟹
QRMonSimulate[1000]⟹
QRMonTakeValue;

opts = {ImageSize -> Medium, PlotTheme -> "Detailed"};
GraphicsGrid[{{DateListPlot[tsData, PlotLabel -> "Actual", opts],
DateListPlot[tsNew, PlotLabel -> "Simulated", opts]}}]

### Finding local extrema in noisy data

Using regression fitting — and Quantile Regression in particular — we can easily construct semi-symbolic algorithms for finding local extrema in noisy time series data; see [AA5]. The QRMon function with such an algorithm is QRMonLocalExtrema.

In brief, the algorithm steps are as follows. (For more details see [AA5].)

1. Fit a polynomial through the data.

2. Find the local extrema of the fitted polynomial. (We will call them fit estimated extrema.)

3. Around each of the fit estimated extrema find the most extreme point in the data by a nearest neighbors search (by using Nearest).

The function QRMonLocalExtrema uses the regression quantiles previously found in the monad pipeline (and stored in the context.) The bottom regression quantile is used for finding local minima, the top regression quantile is used for finding the local maxima.

An example of finding local extrema follows.

QRMonUnit[TimeSeriesWindow[tsData, {{2015, 1, 1}, {2018, 12, 31}}]]⟹
QRMonQuantileRegression[10, {0.05, 0.95}]⟹
QRMonDateListPlot[Joined -> False, PlotTheme -> "Scientific"]⟹
QRMonLocalExtrema["NumberOfProximityPoints" -> 100]⟹
QRMonEchoValue⟹
QRMonEchoFunctionContext[
DateListPlot[{#localMinima, #localMaxima, #data},
PlotStyle -> {PointSize[0.015], PointSize[0.015], Gray},
Joined -> False,
PlotLegends -> {"localMinima", "localMaxima", "data"},
PlotTheme -> "Scientific"] &];

Note that in the pipeline above in order to plot the data and local extrema together some additional steps are needed. The result of QRMonLocalExtrema becomes the pipeline value; that pipeline value is displayed with QRMonEchoValue, and stored in the context with QRMonAddToContext. If the pipeline value is an association — which is the case here — the monad function QRMonAddToContext joins that association with the context association. In this case this means that we will have key-value elements in the context for "localMinima" and "localMaxima". The date list plot at the end of the pipeline uses values of those context keys (together with the value for "data".)

### Setters, droppers, and takers

The values from the monad context can be set, obtained, or dropped with the corresponding "setter", "dropper", and "taker" functions as summarized in a previous section.

For example:

p = QRMonUnit[distData]⟹QRMonQuantileRegressionFit[2];

p⟹QRMonTakeRegressionFunctions

(* <|0.25 -> (0.0191185 + 0.00669159 #1 + 3.05509*10^-14 #1^2 &),
0.5 -> (0.191408 + 9.4728*10^-14 #1 + 3.02272*10^-14 #1^2 &),
0.75 -> (0.563422 + 3.8079*10^-11 #1 + 7.63637*10^-14 #1^2 &)|> *)


If other values are put in the context they can be obtained through the (generic) function QRMonTakeContext, [AAp1]:

p = QRMonUnit[RandomReal[1, {2, 2}]]⟹QRMonAddToContext["data"];

(p⟹QRMonTakeContext)["data"]

(* {{0.608789, 0.741599}, {0.877074, 0.861554}} *)

Another generic function from [AAp1] is QRMonTakeValue (used many times above.)

Here is an example of the "data dropper" QRMonDropData:

p⟹QRMonDropData⟹QRMonTakeContext

(* <||> *)

(The "droppers" simply use the state monad function QRMonDropFromContext, [AAp1]. For example, QRMonDropData is equivalent to QRMonDropFromContext["data"].)

## Unit tests

The development of QRMon was done with two types of unit tests: (i) directly specified tests, [AAp7], and (ii) tests based on randomly generated pipelines, [AA8].

The unit test package should be further extended in order to provide better coverage of the functionalities and illustrate — and postulate — pipeline behavior.

### Directly specified tests

Here we run the unit tests file "MonadicQuantileRegression-Unit-Tests.wlt", [AAp7]:

AbsoluteTiming[
]

The natural language derived test ID’s should give a fairly good idea of the functionalities covered in [AAp3].

Values[Map[#["TestID"] &, testObject["TestResults"]]]

"QuantileRegression-2", "QuantileRegression-3", \
"QuantileRegression-and-Fit-1", "Fit-and-QuantileRegression-1", \
"QuantileRegressionFit-and-Fit-1", "Fit-and-QuantileRegressionFit-1", \
"Outliers-1", "Outliers-2", "GridSequence-1", "BandsSequence-1", \
"ConditionalCDF-1", "Evaluate-1", "Evaluate-2", "Evaluate-3", \
"Simulate-1", "Simulate-2", "Simulate-3"} *)

### Random pipelines tests

Since the monad QRMon is a DSL it is natural to test it with a large number of randomly generated "sentences" of that DSL. For the QRMon DSL the sentences are QRMon pipelines. The package "MonadicQuantileRegressionRandomPipelinesUnitTests.m", [AAp8], has functions for generation of QRMon random pipelines and running them as verification tests. A short example follows.

Generate pipelines:

SeedRandom[234]
pipelines = MakeQRMonRandomPipelines[100];
Length[pipelines]

(* 100 *)

Here is a sample of the generated pipelines:

(*
Block[{DoubleLongRightArrow, pipelines = RandomSample[pipelines, 6]},
Clear[DoubleLongRightArrow];
pipelines = pipelines /. {_TemporalData -> "tsData", _?MatrixQ -> "distData"};
GridTableForm[Map[List@ToString[DoubleLongRightArrow @@ #, FormatType -> StandardForm] &, pipelines], TableHeadings -> {"pipeline"}]
]
AutoCollapse[] *)

Here we run the pipelines as unit tests:

AbsoluteTiming[
res = TestRunQRMonPipelines[pipelines, "Echo" -> False];
]

From the test report results we see that a dozen tests failed with messages, all of the rest passed.

rpTRObj = TestReport[res]

(The message failures, of course, have to be examined — some bugs were found in that way. Currently the actual test messages are expected.)

## Future plans

### Workflow operations

A list of possible, additional workflow operations and improvements follows.

• Certain improvements can be done over the specification of the different plot options.

• It will be useful to develop a function for automatic finding of over-fitting parameters.

• The time series simulation should be done by aggregation of similar time intervals.

• For example, for time series with span several years, for each month name is made Quantile Regression simulation and the results are spliced to obtain a one year simulation.
• If the time series is represented as a sequence of categorical values, then the time series simulation can use Bayesian probabilities derived from sub-sequences.
• QRMon already has functions that facilitate that, QRMonGridSequence and QRMonBandsSequence.

### Conversational agent

Using the packages [AAp10, AAp11] we can generate QRMon pipelines with natural commands. The plan is to develop and document those functionalities further.

Here is an example of a pipeline constructed with natural language commands:

QRMonUnit[distData]⟹
ToQRMonPipelineFunction["show data summary"]⟹
ToQRMonPipelineFunction["calculate quantile regression for quantiles 0.2, 0.8 and with 40 knots"]⟹
ToQRMonPipelineFunction["plot"];

## Implementation notes

The implementation methodology of the QRMon monad packages [AAp3, AAp8] followed the methodology created for the ClCon monad package [AAp9, AA6]. Similarly, this document closely follows the structure and exposition of the ClCon monad document "A monad for classification workflows", [AA6].

A lot of the functionalities and signatures of QRMon were designed and programed through considerations of natural language commands specifications given to a specialized conversational agent. (As discussed in the previous section.)

## References

### ConversationalAgents Packages

[AAp10] Anton Antonov, Time series workflows grammar in EBNF, (2018), ConversationalAgents at GitHub, https://github.com/antononcube/ConversationalAgents.

[AAp11] Anton Antonov, QRMon translator Mathematica package,(2018), ConversationalAgents at GitHub, https://github.com/antononcube/ConversationalAgents.

### Other

[Wk2] Wikipedia entry, Quantile Regression, URL: https://en.wikipedia.org/wiki/Quantile_regression .

[Wk3] Wikipedia entry, Chebyshev polynomials, URL: https://en.wikipedia.org/wiki/Chebyshev_polynomials .

[CN1] Brian S. Code and Barry R. Noon, "A gentle introduction to quantile regression for ecologists", (2003). Frontiers in Ecology and the Environment. 1 (8): 412[Dash]420. doi:10.2307/3868138. URL: http://www.econ.uiuc.edu/~roger/research/rq/QReco.pdf .

[PS1] Patrick Scheibe, Mathematica (Wolfram Language) support for IntelliJ IDEA, (2013-2018), Mathematica-IntelliJ-Plugin at GitHub. URL: https://github.com/halirutan/Mathematica-IntelliJ-Plugin .

[RK1] Roger Koenker, Quantile Regression, ‪Cambridge University Press, 2005‬.

# A monad for classification workflows

## Introduction

In this document we describe the design and implementation of a (software programming) monad for classification workflows specification and execution. The design and implementation are done with Mathematica / Wolfram Language (WL).

The goal of the monad design is to make the specification of classification workflows (relatively) easy, straightforward, by following a certain main scenario and specifying variations over that scenario.

The monad is named ClCon and it is based on the State monad package "StateMonadCodeGenerator.m", [AAp1, AA1], the classifier ensembles package "ClassifierEnsembles.m", [AAp4, AA2], and the package for Receiver Operating Characteristic (ROC) functions calculation and plotting "ROCFunctions.m", [AAp5, AA2, Wk2].

The data for this document is read from WL’s repository using the package "GetMachineLearningDataset.m", [AAp10].

The monadic programming design is used as a Software Design Pattern. The ClCon monad can be also seen as a Domain Specific Language (DSL) for the specification and programming of machine learning classification workflows.

Here is an example of using the ClCon monad over the Titanic data:

The table above is produced with the package "MonadicTracing.m", [AAp2, AA1], and some of the explanations below also utilize that package.

As it was mentioned above the monad ClCon can be seen as a DSL. Because of this the monad pipelines made with ClCon are sometimes called "specifications".

### Contents description

The document has the following structure.

• The sections "Package load" and "Data load" obtain the needed code and data.
(Needed and put upfront from the "Reproducible research" point of view.)

• The sections "Design consideration" and "Monad design" provide motivation and design decisions rationale.

• The sections "ClCon overview" and "Monad elements" provide technical description of the ClCon monad needed to utilize it.
(Using a fair amount of examples.)

• The section "Example use cases" gives several more elaborated examples of ClCon that have "real life" flavor.
(But still didactic and concise enough.)

• The section "Unit test" describes the tests used in the development of the ClCon monad.
(The random pipelines unit tests are especially interesting.)

• The section "Future plans" outlines future directions of development.
(The most interesting and important one is the "conversational agent" direction.)

• The section "Implementation notes" has (i) a diagram outlining the ClCon development process, and (ii) a list of observations and morals.
(Some fairly obvious, but deemed fairly significant and hence stated explicitly.)

Remark: One can read only the sections "Introduction", "Design consideration", "Monad design", and "ClCon overview". That set of sections provide a fairly good, programming language agnostic exposition of the substance and novel ideas of this document.

The following commands load the packages [AAp1–AAp10, AAp12]:

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicContextualClassification.m"]
Import["https://raw.githubusercontent.com/antononcube/MathematicaVsR/master/Projects/ProgressiveMachineLearning/Mathematica/GetMachineLearningDataset.m"]

(*
Importing from GitHub: MathematicaForPredictionUtilities.m
Importing from GitHub: MosaicPlot.m
Importing from GitHub: CrossTabulate.m
Importing from GitHub: ClassifierEnsembles.m
Importing from GitHub: ROCFunctions.m
Importing from GitHub: VariableImportanceByClassifiers.m
Importing from GitHub: SSparseMatrix.m
Importing from GitHub: OutlierIdentifiers.m
*)

In this section we load data that is used in the rest of the document. The "quick" data is created in order to specify quick, illustrative computations.

Remark: In all datasets the classification labels are in the last column.

The summarization of the data is done through ClCon, which in turn uses the function RecordsSummary from the package "MathematicaForPredictionUtilities.m", [AAp7].

### WL resources data

The following commands produce datasets using the package [AAp10] (that utilizes ExampleData):

dsTitanic = GetMachineLearningDataset["Titanic"];
dsMushroom = GetMachineLearningDataset["Mushroom"];
dsWineQuality = GetMachineLearningDataset["WineQuality"];

Here is are the dimensions of the datasets:

Dataset[Dataset[Map[Prepend[Dimensions[ToExpression[#]], #] &, {"dsTitanic", "dsMushroom", "dsWineQuality"}]][All, AssociationThread[{"name", "rows", "columns"}, #] &]]

Here is the summary of dsTitanic:

ClConUnit[dsTitanic]⟹ClConSummarizeData["MaxTallies" -> 12];

Here is the summary of dsMushroom in long form:

ClConUnit[dsMushroom]⟹ClConSummarizeDataLongForm["MaxTallies" -> 12];

Here is the summary of dsWineQuality in long form:

ClConUnit[dsWineQuality]⟹ClConSummarizeDataLongForm["MaxTallies" -> 12];

### "Quick" data

In this subsection we make up some data that is used for illustrative purposes.

SeedRandom[212]
dsData = RandomInteger[{0, 1000}, {100}];
dsData = Dataset[
Transpose[{dsData, Mod[dsData, 3], Last@*IntegerDigits /@ dsData, ToString[Mod[#, 3]] & /@ dsData}]];
dsData = Dataset[dsData[All, AssociationThread[{"number", "feature1", "feature2", "label"}, #] &]];
Dimensions[dsData]

(* {100, 4} *)

Here is a sample of the data:

RandomSample[dsData, 6]

Here is a summary of the data:

ClConUnit[dsData]⟹ClConSummarizeData;

Here we convert the data into a list of record-label rules (and show the summary):

mlrData = ClConToNormalClassifierData[dsData];
ClConUnit[mlrData]⟹ClConSummarizeData;

Finally, we make the array version of the dataset:

arrData = Normal[dsData[All, Values]];

## Design considerations

The steps of the main classification workflow addressed in this document follow.

1. Retrieving data from a data repository.

2. Optionally, transform the data.

3. Split data into training and test parts.

• Optionally, split training data into training and validation parts.
4. Make a classifier with the training data.

5. Test the classifier over the test data.

• Computation of different measures including ROC.

The following diagram shows the steps.

Very often the workflow above is too simple in real situations. Often when making "real world" classifiers we have to experiment with different transformations, different classifier algorithms, and parameters for both transformations and classifiers. Examine the following mind-map that outlines the activities in making competition classifiers.

In view of the mind-map above we can come up with the following flow-chart that is an elaboration on the main, simple workflow flow-chart.

• the introduction of new elements in classification workflows,

• workflows elements variability, and

• workflows iterative changes and refining,

it is beneficial to have a DSL for classification workflows. We choose to make such a DSL through a functional programming monad, [Wk1, AA1].

Here is a quote from [Wk1] that fairly well describes why we choose to make a classification workflow monad and hints on the desired properties of such a monad.

[…] The monad represents computations with a sequential structure: a monad defines what it means to chain operations together. This enables the programmer to build pipelines that process data in a series of steps (i.e. a series of actions applied to the data), in which each action is decorated with the additional processing rules provided by the monad. […]

Monads allow a programming style where programs are written by putting together highly composable parts, combining in flexible ways the possible actions that can work on a particular type of data. […]

Remark: Note that quote from [Wk1] refers to chained monadic operations as "pipelines". We use the terms "monad pipeline" and "pipeline" below.

The monad we consider is designed to speed-up the programming of classification workflows outlined in the previous section. The monad is named ClCon for "Classification with Context".

We want to be able to construct monad pipelines of the general form:

ClCon is based on the State monad, [Wk1, AA1], so the monad pipeline form (1) has the following more specific form:

This means that some monad operations will not just change the pipeline value but they will also change the pipeline context.

In the monad pipelines of ClCon we store different objects in the contexts for at least one of the following two reasons.

1. The object will be needed later on in the pipeline.

2. The object is hard to compute.

Such objects are training data, ROC data, and classifiers.

Let us list the desired properties of the monad.

• Rapid specification of non-trivial classification workflows.

• The monad works with different data types: Dataset, lists of machine learning rules, full arrays.

• The pipeline values can be of different types. Most monad functions modify the pipeline value; some modify the context; some just echo results.

• The monad works with single classifier objects and with classifier ensembles.

• This means support of different classifier measures and ROC plots for both single classifiers and classifier ensembles.
• The monad allows of cursory examination and summarization of the data.
• For insight and in order to verify assumptions.
• The monad has operations to compute importance of variables.

• We can easily obtain the pipeline value, context, and different context objects for manipulation outside of the monad.

• We can calculate classification measures using a specified ROC parameter and a class label.

• We can easily plot different combinations of ROC functions.

The ClCon components and their interaction are given in the following diagram. (The components correspond to the main workflow given in the previous section.)

In the diagram above the operations are given in rectangles. Data objects are given in round corner rectangles and classifier objects are given in round corner squares.

The main ClCon operations implicitly put in the context or utilize from the context the following objects:

• training data,

• test data,

• validation data,

• classifier (a classifier function or an association of classifier functions),

• ROC data,

• variable names list.

Note the that the monadic set of types of ClCon pipeline values is fairly heterogenous and certain awareness of "the current pipeline value" is assumed when composing ClCon pipelines.

Obviously, we can put in the context any object through the generic operations of the State monad of the package "StateMonadGenerator.m", [AAp1].

## ClCon overview

When using a monad we lift certain data into the "monad space", using monad’s operations we navigate computations in that space, and at some point we take results from it.

With the approach taken in this document the "lifting" into the ClCon monad is done with the function ClConUnit. Results from the monad can be obtained with the functions ClConTakeValue, ClConContext, or with the other ClCon functions with the prefix "ClConTake" (see below.)

Here is a corresponding diagram of a generic computation with the ClCon monad:

Remark: It is a good idea to compare the diagram with formulas (1) and (2).

Let us examine a concrete ClCon pipeline that corresponds to the diagram above. In the following table each pipeline operation is combined together with a short explanation and the context keys after its execution.

Here is the output of the pipeline:

In the specified pipeline computation the last column of the dataset is assumed to be the one with the class labels.

The ClCon functions are separated into four groups:

• operations,

• setters,

• takers,

An overview of the those functions is given in the tables in next two sub-sections. The next section, "Monad elements", gives details and examples for the usage of the ClCon operations.

### Monad functions interaction with the pipeline value and context

The following table gives an overview the interaction of the ClCon monad functions with the pipeline value and context.

Several functions that use ROC data have two rows in the table because they calculate the needed ROC data if it is not available in the monad context.

Here are the ClCon State Monad functions (generated using the prefix "ClCon", [AAp1, AA1]):

In this section we show that ClCon has all of the properties listed in the previous section.

The monad head is ClCon. Anything wrapped in ClCon can serve as monad’s pipeline value. It is better though to use the constructor ClConUnit. (Which adheres to the definition in [Wk1].)

ClCon[{{1, "a"}, {2, "b"}}, <||>]⟹ClConSummarizeData;

### Lifting data to the monad

The function lifting the data into the monad ClCon is ClConUnit.

The lifting to the monad marks the beginning of the monadic pipeline. It can be done with data or without data. Examples follow.

ClConUnit[dsData]⟹ClConSummarizeData;
ClConUnit[]⟹ClConSetTrainingData[dsData]⟹ClConSummarizeData;

(See the sub-section "Setters and takers" for more details of setting and taking values in ClCon contexts.)

Currently the monad can deal with data in the following forms:

• datasets,

• matrices,

• lists of example->label rules.

The ClCon monad also has the non-monadic function ClConToNormalClassifierData which can be used to convert datasets and matrices to lists of example->label rules. Here is an example:

Short[ClConToNormalClassifierData[dsData], 3]

(*
{{639, 0, 9} -> "0", {121, 1, 1} -> "1", {309, 0, 9} ->  "0", {648, 0, 8} -> "0", {995, 2, 5} -> "2", {127, 1, 7} -> "1", {908, 2, 8} -> "2", {564, 0, 4} -> "0", {380, 2, 0} -> "2", {860, 2, 0} -> "2",
<<80>>,
{464, 2, 4} -> "2", {449, 2, 9} -> "2", {522, 0, 2} -> "0", {288, 0, 8} -> "0", {51, 0, 1} -> "0", {108, 0, 8} -> "0", {76, 1, 6} -> "1", {706, 1, 6} -> "1", {765, 0, 5} -> "0", {195, 0, 5} -> "0"}
*)

When the data lifted to the monad is a dataset or a matrix it is assumed that the last column has the class labels. WL makes it easy to rearrange columns in such a way the any column of dataset or a matrix to be the last.

### Data splitting

The splitting is made with ClConSplitData, which takes up to two arguments and options. The first argument specifies the fraction of training data. The second argument — if given — specifies the fraction of the validation part of the training data. If the value of option Method is "LabelsProportional", then the splitting is done in correspondence of the class labels tallies. ("LabelsProportional" is the default value.) Data splitting demonstration examples follow.

Here are the dimensions of the dataset dsData:

Dimensions[dsData]

(* {100, 4} *)

Here we split the data into 70% for training and 30% for testing and then we verify that the corresponding number of rows add to the number of rows of dsData:

val = ClConUnit[dsData]⟹ClConSplitData[0.7]⟹ClConTakeValue;
Map[Dimensions, val]
Total[First /@ %]

(*
<|"trainingData" -> {69, 4}, "testData" -> {31, 4}|>
100
*)

Note that if Method is not "LabelsProportional" we get slightly different results.

val = ClConUnit[dsData]⟹ClConSplitData[0.7, Method -> "Random"]⟹ClConTakeValue;
Map[Dimensions, val]
Total[First /@ %]

(*
<|"trainingData" -> {70, 4}, "testData" -> {30, 4}|>
100
*)

In the following code we split the data into 70% for training and 30% for testing, then the training data is further split into 90% for training and 10% for classifier training validation; then we verify that the number of rows add up.

val = ClConUnit[dsData]⟹ClConSplitData[0.7, 0.1]⟹ClConTakeValue;
Map[Dimensions, val]
Total[First /@ %]

(*
<|"trainingData" -> {61, 4}, "testData" -> {31, 4}, "validationData" -> {8, 4}|>
100
*)

### Classifier training

The monad ClCon supports both single classifiers obtained with Classify and classifier ensembles obtained with Classify and managed with the package "ClassifierEnsembles.m", [AAp4].

#### Single classifier training

With the following pipeline we take the Titanic data, split it into 75/25 % parts, train a Logistic Regression classifier, and finally take that classifier from the monad.

cf =
ClConUnit[dsTitanic]⟹
ClConSplitData[0.75]⟹
ClConMakeClassifier["LogisticRegression"]⟹
ClConTakeClassifier;

Here is information about the obtained classifier:

ClassifierInformation[cf, "TrainingTime"]

(* Quantity[3.84008, "Seconds"] *)

If we want to pass parameters to the classifier training we can use the Method option. Here we train a Random Forest classifier with 400 trees:

cf =
ClConUnit[dsTitanic]⟹
ClConSplitData[0.75]⟹
ClConMakeClassifier[Method -> {"RandomForest", "TreeNumber" -> 400}]⟹
ClConTakeClassifier;

ClassifierInformation[cf, "TreeNumber"]

(* 400 *)

#### Classifier ensemble training

With the following pipeline we take the Titanic data, split it into 75/25 % parts, train a classifier ensemble of three Logistic Regression classifiers and two Nearest Neighbors classifiers using random sampling of 90% of the training data, and finally take that classifier ensemble from the monad.

ensemble =
ClConUnit[dsTitanic]⟹
ClConSplitData[0.75]⟹
ClConMakeClassifier[{{"LogisticRegression", 0.9, 3}, {"NearestNeighbors", 0.9, 2}}]⟹
ClConTakeClassifier;

The classifier ensemble is simply an association with keys that are automatically assigned names and corresponding values that are classifiers.

ensemble

Here are the training times of the classifiers in the obtained ensemble:

ClassifierInformation[#, "TrainingTime"] & /@ ensemble

(*
<|"LogisticRegression[1,0.9]" -> Quantity[3.47836, "Seconds"],
"LogisticRegression[2,0.9]" -> Quantity[3.47681, "Seconds"],
"LogisticRegression[3,0.9]" -> Quantity[3.4808, "Seconds"],
"NearestNeighbors[1,0.9]" -> Quantity[1.82454, "Seconds"],
"NearestNeighbors[2,0.9]" -> Quantity[1.83804, "Seconds"]|>
*)

A more precise specification can be given using associations. The specification

<|"method" -> "LogisticRegression", "sampleFraction" -> 0.9, "numberOfClassifiers" -> 3, "samplingFunction" -> RandomChoice|>

says "make three Logistic Regression classifiers, for each taking 90% of the training data using the function RandomChoice."

Here is a pipeline specification equivalent to the pipeline specification above:

ensemble2 =
ClConUnit[dsTitanic]⟹
ClConSplitData[0.75]⟹
ClConMakeClassifier[{
<|"method" -> "LogisticRegression",
"sampleFraction" -> 0.9,
"numberOfClassifiers" -> 3,
"samplingFunction" -> RandomSample|>,
<|"method" -> "NearestNeighbors",
"sampleFraction" -> 0.9,
"numberOfClassifiers" -> 2,
"samplingFunction" -> RandomSample|>}]⟹
ClConTakeClassifier;

ensemble2

### Classifier testing

Classifier testing is done with the testing data in the context.

Here is a pipeline that takes the Titanic data, splits it, and trains a classifier:

p =
ClConUnit[dsTitanic]⟹
ClConSplitData[0.75]⟹
ClConMakeClassifier["DecisionTree"];

Here is how we compute selected classifier measures:

p⟹
ClConClassifierMeasurements[{"Accuracy", "Precision", "Recall", "FalsePositiveRate"}]⟹
ClConTakeValue

(*
<|"Accuracy" -> 0.792683,
"Precision" -> <|"died" -> 0.802691, "survived" -> 0.771429|>,
"Recall" -> <|"died" -> 0.881773, "survived" -> 0.648|>,
"FalsePositiveRate" -> <|"died" -> 0.352, "survived" -> 0.118227|>|>
*)

(The measures are listed in the function page of ClassifierMeasurements.)

Here we show the confusion matrix plot:

p⟹ClConClassifierMeasurements["ConfusionMatrixPlot"]⟹ClConEchoValue;

Here is how we plot ROC curves by specifying the ROC parameter range and the image size:

p⟹ClConROCPlot["FPR", "TPR", "ROCRange" -> Range[0, 1, 0.1], ImageSize -> 200];

Remark: ClCon uses the package ROCFunctions.m, [AAp5], which implements all functions defined in [Wk2].

Here we plot ROC functions values (y-axis) over the ROC parameter (x-axis):

p⟹ClConROCListLinePlot[{"ACC", "TPR", "FPR", "SPC"}];

Note of the "ClConROC*Plot" functions automatically echo the plots. The plots are also made to be the pipeline value. Using the option specification "Echo"->False the automatic echoing of plots can be suppressed. With the option "ClassLabels" we can focus on specific class labels.

p⟹
ClConROCListLinePlot[{"ACC", "TPR", "FPR", "SPC"}, "Echo" -> False, "ClassLabels" -> "survived", ImageSize -> Medium]⟹
ClConEchoValue;

### Variable importance finding

Using the pipeline constructed above let us find the most decisive variables using systematic random shuffling (as explained in [AA3]):

p⟹
ClConAccuracyByVariableShuffling⟹
ClConTakeValue

(*
<|None -> 0.792683, "id" -> 0.664634, "passengerClass" -> 0.75, "passengerAge" -> 0.777439, "passengerSex" -> 0.612805|>
*)

We deduce that "passengerSex" is the most decisive variable because its corresponding classification success rate is the smallest. (See [AA3] for more details.)

Using the option "ClassLabels" we can focus on specific class labels:

p⟹ClConAccuracyByVariableShuffling["ClassLabels" -> "survived"]⟹ClConTakeValue

(*
<|None -> {0.771429}, "id" -> {0.595506}, "passengerClass" -> {0.731959}, "passengerAge" -> {0.71028}, "passengerSex" -> {0.414414}|>
*)

### Setters and takers

The values from the monad context can be set or obtained with the corresponding "setters" and "takers" functions as summarized in previous section.

For example:

p⟹ClConTakeClassifier

(* ClassifierFunction[__] *)

Short[Normal[p⟹ClConTakeTrainingData]]

(*
{<|"id" -> 858, "passengerClass" -> "3rd", "passengerAge" -> 30, "passengerSex" -> "male", "passengerSurvival" -> "survived"|>, <<979>> }
*)

Short[Normal[p⟹ClConTakeTestData]]

(* {<|"id" -> 285, "passengerClass" -> "1st", "passengerAge" -> 60, "passengerSex" -> "female", "passengerSurvival" -> "survived"|> , <<327>> }
*)

p⟹ClConTakeVariableNames

(* {"id", "passengerClass", "passengerAge", "passengerSex", "passengerSurvival"} *)

If other values are put in the context they can be obtained through the (generic) function ClConTakeContext, [AAp1]:

p = ClConUnit[RandomReal[1, {2, 2}]]⟹ClConAddToContext["data"];

(p⟹ClConTakeContext)["data"]

(* {{0.815836, 0.191562}, {0.396868, 0.284587}} *)

Another generic function from [AAp1] is ClConTakeValue (used many times above.)

## Example use cases

### Classification with MNIST data

Here we show an example of using ClCon with the reasonably large dataset of images MNIST, [YL1].

mnistData = ExampleData[{"MachineLearning", "MNIST"}, "Data"];

SeedRandom[3423]
p =
ClConUnit[RandomSample[mnistData, 20000]]⟹
ClConSplitData[0.7]⟹
ClConSummarizeData⟹
ClConMakeClassifier["NearestNeighbors"]⟹
ClConClassifierMeasurements[{"Accuracy", "ConfusionMatrixPlot"}]⟹
ClConEchoValue;

Here we plot the ROC curve for a specified digit:

p⟹ClConROCPlot["ClassLabels" -> 5];

### Conditional continuation

In this sub-section we show how the computations in a ClCon pipeline can be stopped or continued based on a certain condition.

The pipeline below makes a simple classifier ("LogisticRegression") for the WineQuality data, and if the recall for the important label ("high") is not large enough makes a more complicated classifier ("RandomForest"). The pipeline marks intermediate steps by echoing outcomes and messages.

SeedRandom[267]
res =
ClConUnit[dsWineQuality[All, Join[#, <|"wineQuality" -> If[#wineQuality >= 7, "high", "low"]|>] &]]⟹
ClConSplitData[0.75, 0.2]⟹
ClConSummarizeData(* summarize the data *)⟹
ClConMakeClassifier[Method -> "LogisticRegression"](* training a simple classifier *)⟹
ClConROCPlot["FPR", "TPR", "ROCPointCallouts" -> False]⟹
ClConClassifierMeasurements[{"Accuracy", "Precision", "Recall", "FalsePositiveRate"}]⟹
ClConEchoValue⟹
ClConIfElse[#["Recall", "high"] > 0.70 & (* criteria based on the recall for "high" *),
ClConEcho["Good recall for \"high\"!", "Success:"],
ClConUnit[##]⟹
ClConEcho[Style["Recall for \"high\" not good enough... making a large random forest.", Darker[Red]], "Info:"]⟹
ClConMakeClassifier[Method -> {"RandomForest", "TreeNumber" -> 400}](* training a complicated classifier *)⟹
ClConROCPlot["FPR", "TPR", "ROCPointCallouts" -> False]⟹
ClConClassifierMeasurements[{"Accuracy", "Precision", "Recall", "FalsePositiveRate"}]⟹
ClConEchoValue &];

We can see that the recall with the more complicated is classifier is higher. Also the ROC plots of the second classifier are visibly closer to the ideal one. Still, the recall is not good enough, we have to find a threshold that is better that the default one. (See the next sub-section.)

### Classification with custom thresholds

(In this sub-section we use the monad from the previous sub-section.)

Here we compute classification measures using the threshold 0.3 for the important class label ("high"):

res⟹
ClConClassifierMeasurementsByThreshold[{"Accuracy", "Precision", "Recall", "FalsePositiveRate"}, "high" -> 0.3]⟹
ClConTakeValue

(* <|"Accuracy" -> 0.782857,  "Precision" -> <|"high" -> 0.498871, "low" -> 0.943734|>,
"Recall" -> <|"high" -> 0.833962, "low" -> 0.76875|>,
"FalsePositiveRate" -> <|"high" -> 0.23125, "low" -> 0.166038|>|> *)

We can see that the recall for "high" is fairly large and the rest of the measures have satisfactory values. (The accuracy did not drop that much, and the false positive rate is not that large.)

Here we compute suggestions for the best thresholds:

res (* start with a previous monad *)⟹
ClConROCPlot[ImageSize -> 300] (* make ROC plots *)⟹
ClConSuggestROCThresholds[3] (* find the best 3 thresholds per class label *)⟹
ClConEchoValue (* echo the result *);

The suggestions are the ROC points that closest to the point {0, 1} (which corresponds to the ideal classifier.)

Here is a way to use threshold suggestions within the monad pipeline:

res⟹
ClConSuggestROCThresholds⟹
ClConEchoValue⟹
(ClConUnit[##]⟹
ClConClassifierMeasurementsByThreshold[{"Accuracy", "Precision", "Recall"}, "high" -> First[#1["high"]]] &)⟹
ClConEchoValue;

(*
value: <|high->{0.35},low->{0.65}|>
value: <|Accuracy->0.825306,Precision-><|high->0.571831,low->0.928736|>,Recall-><|high->0.766038,low->0.841667|>|>
*)

## Unit tests

The development of ClCon was done with two types of unit tests: (1) directly specified tests, [AAp11], and (2) tests based on randomly generated pipelines, [AAp12].

Both unit test packages should be further extended in order to provide better coverage of the functionalities and illustrate — and postulate — pipeline behavior.

### Directly specified tests

Here we run the unit tests file "MonadicContextualClassification-Unit-Tests.wlt", [AAp11]:

AbsoluteTiming[
]

The natural language derived test ID’s should give a fairly good idea of the functionalities covered in [AAp11].

Values[Map[#["TestID"] &, testObject["TestResults"]]]

"DataToContext-no-[]", "DataToContext-with-[]", \
"ClassifierMaking-with-Dataset-1", "ClassifierMaking-with-MLRules-1", \
"AccuracyByVariableShuffling-1", "ROCData-1", \
"ClassifierEnsemble-different-methods-1", \
"ClassifierEnsemble-different-methods-2-cont", \
"ClassifierEnsemble-different-methods-3-cont", \
"ClassifierEnsemble-one-method-1", "ClassifierEnsemble-one-method-2", \
"ClassifierEnsemble-one-method-3-cont", \
"ClassifierEnsemble-one-method-4-cont", "AssignVariableNames-1", \
"AssignVariableNames-2", "AssignVariableNames-3", "SplitData-1", \
"Set-and-take-training-data", "Set-and-take-test-data", \
"Set-and-take-validation-data", "Partial-data-summaries-1", \
"Assign-variable-names-1", "Split-data-100-pct", \
"MakeClassifier-with-empty-unit-1", \
"No-rocData-after-second-MakeClassifier-1"} *)

### Random pipelines tests

Since the monad ClCon is a DSL it is natural to test it with a large number of randomly generated "sentences" of that DSL. For the ClCon DSL the sentences are ClCon pipelines. The package "MonadicContextualClassificationRandomPipelinesUnitTests.m", [AAp12], has functions for generation of ClCon random pipelines and running them as verification tests. A short example follows.

Generate pipelines:

SeedRandom[234]
pipelines = MakeClConRandomPipelines[300];
Length[pipelines]

(* 300 *)

Here is a sample of the generated pipelines:

Block[{DoubleLongRightArrow, pipelines = RandomSample[pipelines, 6]},
Clear[DoubleLongRightArrow];
pipelines = pipelines /. {_Dataset -> "ds", _?DataRulesForClassifyQ -> "mlrData"};
GridTableForm[
Map[List@ToString[DoubleLongRightArrow @@ #, FormatType -> StandardForm] &, pipelines],
]
AutoCollapse[]

Here we run the pipelines as unit tests:

AbsoluteTiming[
res = TestRunClConPipelines[pipelines, "Echo" -> True];
]

(* {350.083, Null} *)

From the test report results we see that a dozen tests failed with messages, all of the rest passed.

rpTRObj = TestReport[res]

(The message failures, of course, have to be examined — some bugs were found in that way. Currently the actual test messages are expected.)

## Future plans

### Workflow operations

#### Outliers

Better outliers finding and manipulation incorporation in ClCon. Currently only outlier finding is surfaced in [AAp3]. (The package internally has other related functions.)

ClConUnit[dsTitanic[Select[#passengerSex == "female" &]]]⟹
ClConOutlierPosition⟹
ClConTakeValue

(* {4, 17, 21, 22, 25, 29, 38, 39, 41, 59} *)

#### Dimension reduction

Support of dimension reduction application — quick construction of pipelines that allow the applying different dimension reduction methods.

Currently with ClCon dimension reduction is applied only to data the non-label parts of which can be easily converted into numerical matrices.

ClConUnit[dsWineQuality]⟹
ClConSplitData[0.7]⟹
ClConReduceDimension[2, "Echo" -> True]⟹
ClConRetrieveFromContext["svdRes"]⟹
ClConEchoFunctionValue["SVD dimensions:", Dimensions /@ # &]⟹
ClConSummarizeData;

### Conversational agent

Using the packages [AAp13, AAp15] we can generate ClCon pipelines with natural commands. The plan is to develop and document those functionalities further.

## Implementation notes

The ClCon package, MonadicContextualClassification.m, [AAp3], is based on the packages [AAp1, AAp4-AAp9]. It was developed using Mathematica and the Mathematica plug-in for IntelliJ IDEA, by Patrick Scheibe , [PS1]. The following diagram shows the development workflow.

Some observations and morals follow.

• Making the unit tests [AAp11] made the final implementation stage much more comfortable.
• Of course, in retrospect that is obvious.
• Initially "MonadicContextualClassification.m" was not real a package, just a collection of global context functions with the prefix "ClCon". This made some programming design decisions harder, slower, and more cumbersome. By making a proper package the development became much easier because of the "peace of mind" brought by the context feature encapsulation.
• The making of random pipeline tests, [AAp12], helped catch a fair amount of inconvenient "features" and bugs.
• (Both tests sets [AAp11, AAp12] can be made to be more comprehensive.)
• The design of a conversational agent for producing ClCon pipelines with natural language commands brought a very fruitful viewpoint on the overall functionalities and the determination and limits of the ClCon development goals. See [AAp13, AAp14, AAp15].

• "Eat your own dog food", or in this case: "use ClCon functionalities to implement ClCon functionalities."

• Since we are developing a DSL it is natural to use that DSL for its own advancement.

• Again, in retrospect that is obvious. Also probably should be seen as a consequence of practicing a certain code refactoring discipline.

• The reason to list that moral is that often it is somewhat "easier" to implement functionalities thinking locally, ad-hoc, forgetting or not reviewing other, already implemented functions.

• In order come be better design and find inconsistencies: write many pipelines and discuss with co-workers.

• This is obvious. I would like to mention that a somewhat good alternative to discussions is (i) writing this document and related ones and (ii) making, running, and examining of the random pipelines tests.

## References

### Packages

[AAp9] Anton Antonov, SSparseMatrix Mathematica package, (2018), MathematicaForPrediction at GitHub.

[AAp10] Anton Antonov, Obtain and transform Mathematica machine learning data-sets, (2018), MathematicaVsR at GitHub.

### ConverationalAgents Packages

[AAp13] Anton Antonov, Classifier workflows grammar in EBNF, (2018), ConversationalAgents at GitHub, https://github.com/antononcube/ConversationalAgents.

[AAp14] Anton Antonov, Classifier workflows grammar Mathematica unit tests, (2018), ConversationalAgents at GitHub, https://github.com/antononcube/ConversationalAgents.

[AAp15] Anton Antonov, ClCon translator Mathematica package, (2018), ConversationalAgents at GitHub, https://github.com/antononcube/ConversationalAgents.

### MathematicaForPrediction articles

[AA1] Anton Antonov, Monad code generation and extension, (2017), MathematicaForPrediction at GitHub, https://github.com/antononcube/MathematicaForPrediction.

### Other

[YL1] Yann LeCun et al., MNIST database site. URL: http://yann.lecun.com/exdb/mnist/ .

[PS1] Patrick Scheibe, Mathematica (Wolfram Language) support for IntelliJ IDEA, (2013-2018), Mathematica-IntelliJ-Plugin at GitHub. URL: https://github.com/halirutan/Mathematica-IntelliJ-Plugin .

# The Great conversation in USA presidential speeches

## Introduction

This document shows a way to chart in Mathematica / WL the evolution of topics in collections of texts. The making of this document (and related code) is primarily motivated by the fascinating concept of the Great Conversation, [Wk1, MA1]. In brief, all western civilization books are based on great ideas; if we find the great ideas each significant book is based on we can construct a time-line (spanning centuries) of the great conversation between the authors; see [MA1, MA2, MA3].

Instead of finding the great ideas in a text collection we extract topics statistically, using dimension reduction with Non-Negative Matrix Factorization (NNMF), [AAp3, AA1, AA2].

The presented computational results are based on the text collections of State of the Union speeches of USA presidents [D2]. The code in this document can be easily configured to use the much smaller text collection [D1] available online and in Mathematica/WL. (The collection [D1] is fairly small, documents; the collection [D2] is much larger, documents.)

The procedures (and code) described in this document, of course, work on other types of text collections. For example: movie reviews, podcasts, editorial articles of a magazine, etc.

A secondary objective of this document is to illustrate the use of the monadic programming pipeline as a Software design pattern, [AA3]. In order to make the code concise in this document I wrote the package MonadicLatentSemanticAnalysis.m, [AAp5]. Compare with the code given in [AA1].

The very first version of this document was written for the 2017 summer course “Data Science for the Humanities” at the University of Oxford, UK.

## Outline of the procedure applied

The procedure described in this document has the following steps.

1. Get a collection of documents with known dates of publishing.
• Or other types of tags associated with the documents.
2. Do preliminary analysis of the document collection.
• Number of documents; number of unique words.

• Number of words per document; number of documents per word.

• (Some of the statistics of this step are done easier after the Linear vector space representation step.)

3. Optionally perform Natural Language Processing (NLP) tasks.

1. Obtain or derive stop words.

2. Remove stop words from the texts.

3. Apply stemming to the words in the texts.

4. Linear vector space representation.

• This means that we represent the collection with a document-word matrix.

• Each unique word is a basis vector in that space.

• For each document the corresponding point in that space is derived from the number of appearances of document’s words.

5. Extract topics.

• In this document NNMF is used.

• In order to obtain better results with NNMF some experimentation and refinements of the topics search have to be done.

6. Map the documents over the extracted topics.

• The original matrix of the vector space representation is replaced with a matrix with columns representing topics (instead of words.)
7. Order the topics according to their presence across the years (or other related tags).
• This can be done with hierarchical clustering.

• Alternatively,

1. for a given topic find the weighted mean of the years of the documents that have that topic, and

2. order the topics according to those mean values.

8. Visualize the evolution of the documents according to their topics.

1. This can be done by simply finding the contingency matrix year vs topic.

2. For the president speeches we can use the president names for time-line temporal axis instead of years.

• Because the corresponding time intervals of president office occupation do not overlap.

Remark: Some of the functions used in this document combine several steps into one function call (with corresponding parameters.)

## Packages

Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/MonadicLatentSemanticAnalysis.m"];
Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/Misc/HeatmapPlot.m"];
Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/Misc/RSparseMatrix.m"];

(Note that some of the packages that are imported automatically by [AAp5].)

The functions of the central package in this document, [AAp5], have the prefix “LSAMon”. Here is a sample of those names:

Short@Names["LSAMon*"]

(* {"LSAMon", "LSAMonAddToContext", "LSAMonApplyTermWeightFunctions", <>, "LSAMonUnit", "LSAMonUnitQ", "LSAMonWhen"} *)

In this section we load a text collection from a specified source.

The text collection from “Presidential Nomination Acceptance Speeches”, [D1], is small and can be used for multiple code verifications and re-runnings. The “State of Union addresses of USA presidents” text collection from [D2] was converted to a Mathematica/WL object by Christopher Wolfram (and sent to me in a private communication.) The text collection [D2] provides far more interesting results (and they are shown below.)

If[True,
speeches = ResourceData[ResourceObject["Presidential Nomination Acceptance Speeches"]];
names = StringSplit[Normal[speeches[[All, "Person"]]][[All, 2]], "::"][[All, 1]],

(*ELSE*)
(*State of the union addresses provided by Christopher Wolfram. *)
Get["~/MathFiles/Digital humanities/Presidential speeches/speeches.mx"];
names = Normal[speeches[[All, "Name"]]];
];

dates = Normal[speeches[[All, "Date"]]];
texts = Normal[speeches[[All, "Text"]]];

Dimensions[speeches]

(* {2453, 4} *)

## Basic statistics for the texts

Using different contingency matrices we can derive basic statistical information about the document collection. (The document-word matrix is a contingency matrix.)

First we convert the text data in long-form:

docWordRecords =
DateString[#, {"Year"}] & /@ dates,
DeleteStopwords@*TextWords /@ ToLowerCase[texts]}, 1];

Here is a sample of the rows of the long-form:

GridTableForm[RandomSample[docWordRecords, 6],
TableHeadings -> {"document index", "name", "year", "word"}]

Here is a summary:

Multicolumn[
RecordsSummary[docWordRecords, {"document index", "name", "year", "word"}, "MaxTallies" -> 8], 4, Dividers -> All, Alignment -> Top]

Using the long form we can compute the document-word matrix:

ctMat = CrossTabulate[docWordRecords[[All, {1, -1}]]];
MatrixPlot[Transpose@Sort@Map[# &, Transpose[ctMat@"XTABMatrix"]],
MaxPlotPoints -> 300, ImageSize -> 800,
AspectRatio -> 1/3]

Here is the president-word matrix:

ctMat = CrossTabulate[docWordRecords[[All, {2, -1}]]];
MatrixPlot[Transpose@Sort@Map[# &, Transpose[ctMat@"XTABMatrix"]], MaxPlotPoints -> 300, ImageSize -> 800, AspectRatio -> 1/3]

Here is an alternative way to compute text collection statistics through the document-word matrix computed within the monad LSAMon:

LSAMonUnit[texts]⟹LSAMonEchoTextCollectionStatistics[];

## Procedure application

### Stop words

Here is one way to obtain stop words:

stopWords = Complement[DictionaryLookup["*"], DeleteStopwords[DictionaryLookup["*"]]];
Length[stopWords]
RandomSample[stopWords, 12]

(* 304 *)

(* {"has", "almost", "next", "WHO", "seeming", "together", "rather", "runners-up", "there's", "across", "cannot", "me"} *)

We can complete this list with additional stop words derived from the collection itself. (Not done here.)

### Linear vector space representation and dimension reduction

Remark: In the rest of the document we use “term” to mean “word” or “stemmed word”.

The following code makes a document-term matrix from the document collection, exaggerates the representations of the terms using “TF-IDF”, and then does topic extraction through dimension reduction. The dimension reduction is done with NNMF; see [AAp3, AA1, AA2].

SeedRandom[312]

mObj =
LSAMonUnit[texts]⟹
LSAMonMakeDocumentTermMatrix[{}, stopWords]⟹
LSAMonApplyTermWeightFunctions[]⟹
LSAMonTopicExtraction[Max[5, Ceiling[Length[texts]/100]], 60, 12, "MaxSteps" -> 6, "PrintProfilingInfo" -> True];

This table shows the pipeline commands above with comments:

#### Detailed description

The monad object mObj has a context of named values that is an Association with the following keys:

Keys[mObj⟹LSAMonTakeContext]

(* {"texts", "docTermMat", "terms", "wDocTermMat", "W", "H", "topicColumnPositions", "automaticTopicNames"} *)

Let us clarify the values by briefly describing the computational steps.

1. From texts we derive the document-term matrix , where is the number of documents and is the number of terms.
• The terms are words or stemmed words.

• This is done with LSAMonMakeDocumentTermMatrix.

2. From docTermMat is derived the (weighted) matrix wDocTermMat using “TF-IDF”.

• This is done with LSAMonApplyTermWeightFunctions.
3. Using docTermMat we find the terms that are present in sufficiently large number of documents and their column indices are assigned to topicColumnPositions.

4. Matrix factorization.

1. Assign to , , where .

2. Compute using NNMF the factorization , where , , and is the number of topics.

3. The values for the keys “W, “H”, and “topicColumnPositions” are computed and assigned by LSAMonTopicExtraction.

5. From the top terms of each topic are derived automatic topic names and assigned to the key automaticTopicNames in the monad context.

• Also done by LSAMonTopicExtraction.

### Statistical thesaurus

At this point in the object mObj we have the factors of NNMF. Using those factors we can find a statistical thesaurus for a given set of words. The following code calculates such a thesaurus, and echoes it in a tabulated form.

queryWords = {"arms", "banking", "economy", "education", "freedom",
"tariff", "welfare", "disarmament", "health", "police"};

mObj⟹
LSAMonStatisticalThesaurus[queryWords, 12]⟹
LSAMonEchoStatisticalThesaurus[];

By observing the thesaurus entries we can see that the words in each entry are semantically related.

Note, that the word “welfare” strongly associates with “[applause]”. The rest of the query words do not, which can be seen by examining larger thesaurus entries:

thRes =
mObj⟹
LSAMonStatisticalThesaurus[queryWords, 100]⟹
LSAMonTakeValue;
Cases[thRes, "[applause]", Infinity]

(* {"[applause]", "[applause]"} *)

The second “[applause]” associated word is “education”.

#### Detailed description

The statistical thesaurus is computed by using the NNMF’s right factor .

For a given term, its corresponding column in is found and the nearest neighbors of that column are found in the space using Euclidean norm.

### Extracted topics

The topics are the rows of the right factor of the factorization obtained with NNMF .

Let us tabulate the topics found above with LSAMonTopicExtraction :

mObj⟹ LSAMonEchoTopicsTable["NumberOfTerms" -> 6, "MagnificationFactor" -> 0.8, Appearance -> "Horizontal"];

### Map documents over the topics

The function LSAMonTopicsRepresentation finds the top outliers for each row of NNMF’s left factor . (The outliers are found using the package [AAp4].) The obtained list of indices gives the topic representation of the collection of texts.

Short@(mObj⟹LSAMonTopicsRepresentation[]⟹LSAMonTakeContext)["docTopicIndices"]

{{53}, {47, 53}, {25}, {46}, {44}, {15, 42}, {18}, <>, {30}, {33}, {7, 60}, {22, 25}, {12, 13, 25, 30, 49, 59}, {48, 57}, {14, 41}}

Further we can see that if the documents have tags associated with them — like author names or dates — we can make a contingency matrix of tags vs topics. (See [AAp8, AA4].) This is also done by the function LSAMonTopicsRepresentation that takes tags as an argument. If the tags argument is Automatic, then the tags are simply the document indices.

Here is a an example:

rsmat = mObj⟹LSAMonTopicsRepresentation[Automatic]⟹LSAMonTakeValue;
MatrixPlot[rsmat]

Here is an example of calling the function LSAMonTopicsRepresentation with arbitrary tags.

rsmat = mObj⟹LSAMonTopicsRepresentation[DateString[#, "MonthName"] & /@ dates]⟹LSAMonTakeValue;
MatrixPlot[rsmat]

Note that the matrix plots above are very close to the charting of the Great conversation that we are looking for. This can be made more obvious by observing the row names and columns names in the tabulation of the transposed matrix rsmat:

Magnify[#, 0.6] &@MatrixForm[Transpose[rsmat]]

## Charting the great conversation

In this section we show several ways to chart the Great Conversation in the collection of speeches.

There are several possible ways to make the chart: using a time-line plot, using heat-map plot, and using appropriate tabulation (with MatrixForm or Grid).

In order to make the code in this section more concise the package RSparseMatrix.m, [AAp7, AA5], is used.

### Topic name to topic words

This command makes an Association between the topic names and the top topic words.

aTopicNameToTopicTable =
mObj⟹LSAMonTopicsTable["NumberOfTerms" -> 12]⟹LSAMonTakeValue];

Here is a sample:

Magnify[#, 0.7] &@ aTopicNameToTopicTable[[1 ;; 3]]

### Time-line plot

This command makes a contingency matrix between the documents and the topics (as described above):

rsmat = ToRSparseMatrix[mObj⟹LSAMonTopicsRepresentation[Automatic]⟹LSAMonTakeValue]

This time-plot shows great conversation in the USA presidents state of union speeches:

TimelinePlot[
Association@
Tooltip[#2, aTopicNameToTopicTable[#2]] -> dates[[ToExpression@#1]] &,
Transpose[RSparseMatrixToTriplets[rsmat]]],
PlotTheme -> "Detailed", ImageSize -> 1000, AspectRatio -> 1/2, PlotLayout -> "Stacked"]

The plot is too cluttered, so it is a good idea to investigate other visualizations.

### Topic vs president heatmap

We can use the USA president names instead of years in the Great Conversation chart because the USA presidents terms do not overlap.

This makes a contingency matrix presidents vs topics:

rsmat2 = ToRSparseMatrix[
mObj⟹LSAMonTopicsRepresentation[
names]⟹LSAMonTakeValue];

Here we compute the chronological order of the presidents based on the dates of their speeches:

nameToMeanYearRules =
Map[#[[1, 1]] -> Mean[N@#[[All, 2]]] &,
GatherBy[MapThread[List, {names, ToExpression[DateString[#, "Year"]] & /@ dates}], First]];
ordRowInds = Ordering[RowNames[rsmat2] /. nameToMeanYearRules];

This heat-map plot uses the (experimental) package HeatmapPlot.m, [AAp6]:

Block[{m = rsmat2[[ordRowInds, All]]},
HeatmapPlot[SparseArray[m], RowNames[m],
DistanceFunction -> {None, Sort}, ImageSize -> 1000,
AspectRatio -> 1/2]
]

Note the value of the option DistanceFunction: there is not re-ordering of the rows and columns are reordered by sorting. Also, the topics on the horizontal names have tool-tips.

## References

### Text data

[D1] Wolfram Data Repository, "Presidential Nomination Acceptance Speeches".

[D2] US Presidents, State of the Union Addresses, Trajectory, 2016. ‪ISBN‬1681240009, 9781681240008‬.

[D4] Gerhard Peters, "State of the Union Addresses and Messages", The American Presidency Project.

### Packages

[AAp1] Anton Antonov, MathematicaForPrediction utilities, (2014), MathematicaForPrediction at GitHub.

[AAp3] Anton Antonov, Implementation of the Non-Negative Matrix Factorization algorithm in Mathematica, (2013), MathematicaForPrediction at GitHub.

[AAp4] Anton Antonov, Implementation of one dimensional outlier identifying algorithms in Mathematica, (2013), MathematicaForPrediction at GitHub.

[AAp5] Anton Antonov, Monadic latent semantic analysis Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp6] Anton Antonov, Heatmap plot Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp7] Anton Antonov, RSparseMatrix Mathematica package, (2015), MathematicaForPrediction at GitHub.

[AAp8] Anton Antonov, Cross tabulation implementation in Mathematica, (2017), MathematicaForPrediction at GitHub.

### Books and articles

[AA1] Anton Antonov, "Topic and thesaurus extraction from a document collection", (2013), MathematicaForPrediction at GitHub.

[AA2] Anton Antonov, "Statistical thesaurus from NPR podcasts", (2013), MathematicaForPrediction at WordPress blog.

[AA3] Anton Antonov, "Monad code generation and extension", (2017), MathematicaForPrediction at GitHub.

[AA4] Anton Antonov, "Contingency tables creation examples", (2016), MathematicaForPrediction at WordPress blog.

[AA5] Anton Antonov, "RSparseMatrix for sparse matrices with named rows and columns", (2015), MathematicaForPrediction at WordPress blog.

[Wk1] Wikipedia entry, Great Conversation.

[MA1] Mortimer Adler, "The Great Conversation Revisited," in The Great Conversation: A Peoples Guide to Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago,1990, p. 28.