# Halloween Rorschach animations

Last weekend I made and uploaded to YouTube a presentation discussing the making of Rorschach mask animations in both 2D and 3D:

Here are Mathematica notebooks discussing the process in detail:

Here is the link to the Imgur gallery of animations: “Attempts to recreate Rorschach’s mask”.

Here is the Halloween animation I made today:

Here is the black-&-white version:

Here is a collage of the “guiding images” for the animations above:

# Random mandalas deconstruction in R, Python, and Mathematica

Today (2022-02-28) I gave a presentation Greater Boston useR Meetup titled “Random mandalas deconstruction with R, Python, and Mathematica”. (Link to the video recording.)

Here is the abstract:

In this presentation we discuss the application of different dimension reduction algorithms over collections of random mandalas. We discuss and compare the derived image bases and show how those bases explain the underlying collection structure. The presented techniques and insights (1) are applicable to any collection of images, and (2) can be included in larger, more complicated machine learning workflows. The former is demonstrated with a handwritten digits recognition
application; the latter with the generation of random Bethlehem stars. The (parallel) walk-through of the core demonstration is in all three programming languages: Mathematica, Python, and R.

Here is the related RStudio project: “RandomMandalasDeconstruction”.

Here is a link to the R-computations notebook converted to HTML: “LSA methods comparison in R”.

The Mathematica notebooks are placed in project’s folder “notebooks-WL”.

See the work plan status in the org-mode file “Random-mandalas-deconstruction-presentation-work-plan.org”.

Here is the mind-map for the presentation:

The comparison workflow implemented in the notebooks of this project is summarized in the following flow chart:

## References

### Articles

[AA1] Anton Antonov, “Comparison of dimension reduction algorithms over mandala images generation”, (2017), MathematicaForPrediction at WordPress.

[AA2] Anton Antonov, “Handwritten digits recognition by matrix factorization”, (2016), MathematicaForPrediction at WordPress.

### Mathematica packages and repository functions

[AAp1] Anton Antonov, Monadic Latent Semantic Analysis Mathematica package, (2017), MathematicaForPrediction at GitHub/antononcube.

[AAf1] Anton Antonov, NonNegativeMatrixFactorization, (2019), Wolfram Function Repository.

[AAf2] Anton Antonov, IndependentComponentAnalysis, (2019), Wolfram Function Repository.

[AAf3] Anton Antonov, RandomMandala, (2019), Wolfram Function Repository.

### Python packages

[AAp2] Anton Antonov, LatentSemanticAnalyzer Python package (2021), PyPI.org.

[AAp3] Anton Antonov, Random Mandala Python package, (2021), PyPI.org.

### R packages

[AAp4] Anton Antonov, Latent Semantic Analysis Monad R package, (2019), R-packages at GitHub/antononcube.

# Generation of Random Bethlehem Stars

## Introduction

This document/notebook is inspired by the Mathematica Stack Exchange (MSE) question “Plotting the Star of Bethlehem”, [MSE1]. That MSE question requests efficient and fast plotting of a certain mathematical function that (maybe) looks like the Star of Bethlehem, [Wk1]. Instead of doing what the author of the questions suggests, I decided to use a generative art program and workflows from three of most important Machine Learning (ML) sub-cultures: Latent Semantic Analysis, Recommendations, and Classification.

Although we discuss making of Bethlehem Star-like images, the ML workflows and corresponding code presented in this document/notebook have general applicability – in many situations we have to make classifiers based on data that has to be “feature engineered” through pipeline of several types of ML transformative workflows and that feature engineering requires multiple iterations of re-examinations and tuning in order to achieve the set goals.

The document/notebook is structured as follows:

1. Target Bethlehem Star images
2. Simplistic approach
3. Elaborated approach outline
4. Sections that follow through elaborated approach outline:
1. Data generation
2. Feature extraction
3. Recommender creation
4. Classifier creation and utilization experiments

(This document/notebook is a “raw” chapter for the book “Simplified Machine Learning Workflows”, [AAr3].)

## Target images

Here are the images taken from [MSE1] that we consider to be “Bethlehem Stars” in this document/notebook:

imgStar1 = Import["https://i.stack.imgur.com/qmmOw.png"];
imgStar2 = Import["https://i.stack.imgur.com/5gtsS.png"];
Row[{imgStar1, Spacer[5], imgStar2}]

We notice that similar images can be obtained using the Wolfram Function Repository (WFR) function RandomMandala, [AAr1]. Here are a dozen examples:

SeedRandom[5];
Multicolumn[Table[MandalaToWhiterImage@ResourceFunction["RandomMandala"]["RotationalSymmetryOrder" -> 2, "NumberOfSeedElements" -> RandomInteger[{2, 8}], "ConnectingFunction" -> FilledCurve@*BezierCurve], 12], 6, Background -> Black]

## Simplistic approach

We can just generate a large enough set of mandalas and pick the ones we like.

More precisely we have the following steps:

1. We generate, say, 200 random mandalas using BlockRandom and keeping track of the random seeds
1. The mandalas are generated with rotational symmetry order 2 and filled Bezier curve connections.
2. We pick mandalas that look, more or less, like Bethlehem Stars
3. Add picked mandalas to the results list
4. If too few mandalas are in the results list go to 1.

Here are some mandalas generated with those steps:

lsStarReferenceSeeds = DeleteDuplicates@{697734, 227488491, 296515155601, 328716690761, 25979673846, 48784395076, 61082107304, 63772596796, 128581744446, 194807926867, 254647184786, 271909611066, 296515155601, 575775702222, 595562118302, 663386458123, 664847685618, 680328164429, 859482663706};
Multicolumn[
Table[BlockRandom[ResourceFunction["RandomMandala"]["RotationalSymmetryOrder" -> 2, "NumberOfSeedElements" -> Automatic, "ConnectingFunction" -> FilledCurve@*BezierCurve, ColorFunction -> (White &), Background -> Black], RandomSeeding -> rs], {rs, lsStarReferenceSeeds}] /. GrayLevel[0.25] -> White, 6, Appearance -> "Horizontal", Background -> Black]

Remark: The plot above looks prettier in notebook converted with the resource function DarkMode.

## Elaborated approach

Assume that we want to automate the simplistic approach described in the previous section.

One way to automate is to create a Machine Learning (ML) classifier that is capable of discerning which RandomMandala objects look like Bethlehem Star target images and which do not. With such a classifier we can write a function BethlehemMandala that applies the classifier on multiple results from RandomMandala and returns those mandalas that the classifier says are good.

Here are the steps of building the proposed classifier:

• Generate a large enough Random Mandala Images Set (RMIS)
• Create a feature extractor from a subset of RMIS
• Assign features to all of RMIS
• Make a recommender with the RMIS features and other image data (like pixel values)
• Apply the RMIS recommender over the target Bethlehem Star images and determine and examine image sets that are:
• the best recommendations
• the worse recommendations
• With the best and worse recommendations sets compose training data for classifier making
• Train a classifier
• Examine classifier application to (filtering of) random mandala images (both in RMIS and not in RMIS)
• If the results are not satisfactory redo some or all of the steps above

Remark: If the results are not satisfactory we should consider using the obtained classifier at the data generation phase. (This is not done in this document/notebook.)

Remark: The elaborated approach outline and flow chart have general applicability, not just for generation of random images of a certain type.

### Flow chart

Here is a flow chart that corresponds to the outline above:

A few observations for the flow chart follow:

• The flow chart has a feature extraction block that shows that the feature extraction can be done in several ways.
• The application of LSA is a type of feature extraction which this document/notebook uses.
• If the results are not good enough the flow chart shows that the classifier can be used at the data generation phase.
• If the results are not good enough there are several alternatives to redo or tune the ML algorithms.
• Changing or tuning the recommender implies training a new classifier.
• Changing or tuning the feature extraction implies making a new recommender and a new classifier.

## Data generation and preparation

In this section we generate random mandala graphics, transform them into images and corresponding vectors. Those image-vectors can be used to apply dimension reduction algorithms. (Other feature extraction algorithms can be applied over the images.)

### Generated data

Generate large number of mandalas:

k = 20000;
knownSeedsQ = False;
SeedRandom[343];
lsRSeeds = Union@RandomInteger[{1, 10^9}, k];
AbsoluteTiming[
aMandalas =
If[TrueQ@knownSeedsQ,
Association@Table[rs -> BlockRandom[ResourceFunction["RandomMandala"]["RotationalSymmetryOrder" -> 2, "NumberOfSeedElements" -> Automatic, "ConnectingFunction" -> FilledCurve@*BezierCurve], RandomSeeding -> rs], {rs, lsRSeeds}],
(*ELSE*)
Association@Table[i -> ResourceFunction["RandomMandala"]["RotationalSymmetryOrder" -> 2, "NumberOfSeedElements" -> Automatic, "ConnectingFunction" -> FilledCurve@*BezierCurve], {i, 1, k}]
];
]

(*{18.7549, Null}*)

Check the number of mandalas generated:

Length[aMandalas]

(*20000*)

Show a sample of the generated mandalas:

Magnify[Multicolumn[MandalaToWhiterImage /@ RandomSample[Values@aMandalas, 40], 10, Background -> Black], 0.7]

### Data preparation

Convert the mandala graphics into images using appropriately large (or appropriately small) image sizes:

AbsoluteTiming[
aMImages = ParallelMap[ImageResize[#, {120, 120}] &, aMandalas];
]

(*{248.202, Null}*)

Flatten each of the images into vectors:

AbsoluteTiming[
aMImageVecs = ParallelMap[Flatten[ImageData[Binarize@ColorNegate@ColorConvert[#, "Grayscale"]]] &, aMImages];
]

(*{16.0125, Null}*)

Remark: Below those vectors are called image-vectors.

## Feature extraction

In this section we use the software monad LSAMon, [AA1, AAp1], to do dimension reduction over a subset of random mandala images.

Remark: Other feature extraction methods can be used through the built-in functions FeatureExtraction and FeatureExtract.

### Dimension reduction

Create an LSAMon object and extract image topics using Singular Value Decomposition (SVD) or Independent Component Analysis (ICA), [AAr2]:

SeedRandom[893];
AbsoluteTiming[
lsaObj =
LSAMonUnit[]⟹
LSAMonSetDocumentTermMatrix[SparseArray[Values@RandomSample[aMImageVecs, UpTo[2000]]]]⟹
LSAMonApplyTermWeightFunctions["None", "None", "Cosine"]⟹
LSAMonExtractTopics["NumberOfTopics" -> 40, Method -> "ICA", "MaxSteps" -> 240, "MinNumberOfDocumentsPerTerm" -> 0]⟹
LSAMonNormalizeMatrixProduct[Normalized -> Left];
]

(*{16.1871, Null}*)

Show the importance coefficients of the topics (if SVD was used the plot would show the singular values):

ListPlot[Norm /@ SparseArray[lsaObj⟹LSAMonTakeH], Filling -> Axis, PlotRange -> All, PlotTheme -> "Scientific"]

Show the interpretation of the extracted image topics:

lsaObj⟹
LSAMonNormalizeMatrixProduct[Normalized -> Right]⟹
LSAMonEchoFunctionContext[ImageAdjust[Image[Partition[#, ImageDimensions[aMImages[[1]]][[1]]]]] & /@ SparseArray[#H] &];

### Approximation

Pick a test image that is a mandala image or a target image and pre-process it:

If[True,
ind = RandomChoice[Range[Length[Values[aMImages]]]];
imgTest = MandalaToWhiterImage@aMandalas[[ind]];
matImageTest = ToSSparseMatrix[SparseArray@List@ImageToVector[imgTest, ImageDimensions[aMImages[[1]]]], "RowNames" -> Automatic, "ColumnNames" -> Automatic],
(*ELSE*)
imgTest = Binarize[imgStar2, 0.5];
matImageTest = ToSSparseMatrix[SparseArray@List@ImageToVector[imgTest, ImageDimensions[aMImages[[1]]]], "RowNames" -> Automatic, "ColumnNames" -> Automatic]
];
imgTest

Find the representation of the test image with the chosen feature extractor (LSAMon object here):

matReprsentation = lsaObj⟹LSAMonRepresentByTopics[matImageTest]⟹LSAMonTakeValue;
lsCoeff = Normal@SparseArray[matReprsentation[[1, All]]];
ListPlot[lsCoeff, Filling -> Axis, PlotRange -> All]

Show the interpretation of the found representation:

H = SparseArray[lsaObj⟹LSAMonNormalizeMatrixProduct[Normalized -> Right]⟹LSAMonTakeH];
vecReprsentation = lsCoeff . H;
ImageAdjust@Image[Rescale[Partition[vecReprsentation, ImageDimensions[aMImages[[1]]][[1]]]]]

## Recommendations

In this section we utilize the software monad SMRMon, [AAp3], to create a recommender for the random mandala images.

Remark: Instead of the Sparse Matrix Recommender (SMR) object the built-in function Nearest can be used.

Create SSparseMatrix object for all image-vectors:

matImages = ToSSparseMatrix[SparseArray[Values@aMImageVecs], "RowNames" -> Automatic, "ColumnNames" -> Automatic]

Normalize the rows of the image-vectors matrix:

AbsoluteTiming[
matPixel = WeightTermsOfSSparseMatrix[matImages, "None", "None", "Cosine"]
]

Get the LSA topics matrix:

matH = (lsaObj⟹LSAMonNormalizeMatrixProduct[Normalized -> Right]⟹LSAMonTakeH)

Find the image topics representation for each image-vector (assuming matH was computed with SVD or ICA):

AbsoluteTiming[
matTopic = matPixel . Transpose[matH]
]

Here we create a recommender based on the images data (pixels) and extracted image topics (or other image features):

smrObj =
SMRMonUnit[]⟹
SMRMonCreate[<|"Pixel" -> matPixel, "Topic" -> matTopic|>]⟹
SMRMonApplyNormalizationFunction["Cosine"]⟹
SMRMonSetTagTypeWeights[<|"Pixel" -> 0.2, "Topic" -> 1|>];

Remark: Note the weights assigned to the pixels and the topics in the recommender object above. Those weights were derived by examining the recommendations results shown below.

Here is the image we want to find most similar mandala images to – the target image:

imgTarget = Binarize[imgStar2, 0.5]

Here is the profile of the target image:

aProf = MakeSMRProfile[lsaObj, imgTarget, ImageDimensions[aMImages[[1]]]];
TakeLargest[aProf, 6]

(*<|"10032-10009-4392" -> 0.298371, "3906-10506-10495" -> 0.240086, "10027-10014-4387" -> 0.156797, "8342-8339-6062" -> 0.133822, "3182-3179-11222" -> 0.131565, "8470-8451-5829" -> 0.128844|>*)

Using the target image profile here we compute the recommendation scores for all mandala images of the recommender:

aRecs =
smrObj⟹
SMRMonRecommendByProfile[aProf, All]⟹
SMRMonTakeValue;

Here is a plot of the similarity scores:

Row[{ResourceFunction["RecordsSummary"][Values[aRecs]], ListPlot[Values[aRecs], ImageSize -> Medium, PlotRange -> All, PlotTheme -> "Detailed", PlotLabel -> "Similarity scores"]}]

Here are the closest (nearest neighbor) mandala images:

Multicolumn[Values[ImageAdjust@*ColorNegate /@ aMImages[[ToExpression /@ Take[Keys[aRecs], 48]]]], 12, Background -> Black]

Here are the most distant mandala images:

Multicolumn[Values[ImageAdjust@*ColorNegate /@ aMImages[[ToExpression /@ Take[Keys[aRecs], -48]]]], 12, Background -> Black]

## Classifier creation and utilization

In this section we:

• Prepare classifier data
• Build and examine a classifier using the software monad ClCon, [AA2, AAp2], using appropriate training, testing, and validation data ratios
• Build a classifier utilizing all training data
• Generate Bethlehem Star mandalas by filtering mandala candidates with the classifier

As it was mentioned above we prepare the data to build classifiers with by:

• Selecting top, highest scores recommendations and labeling them with True
• Selecting bad, low score recommendations and labeling them with False
AbsoluteTiming[
Block[{
lsBest = Values@aMandalas[[ToExpression /@ Take[Keys[aRecs], 120]]],
lsWorse = Values@aMandalas[[ToExpression /@ Join[Take[Keys[aRecs], -200], RandomSample[Take[Keys[aRecs], {3000, -200}], 200]]]]},
lsTrainingData =
Join[
Map[MandalaToWhiterImage[#, ImageDimensions@aMImages[[1]]] -> True &, lsBest],
Map[MandalaToWhiterImage[#, ImageDimensions@aMImages[[1]]] -> False &, lsWorse]
];
]
]

(*{27.9127, Null}*)

Using ClCon train a classifier and show its performance measures:

clObj =
ClConUnit[lsTrainingData]⟹
ClConSplitData[0.75, 0.2]⟹
ClConMakeClassifier["NearestNeighbors"]⟹
ClConClassifierMeasurements⟹
ClConEchoValue⟹
ClConClassifierMeasurements["ConfusionMatrixPlot"]⟹
ClConEchoValue;

Remark: We can re-run the ClCon workflow above several times until we obtain a classifier we want to use.

Train a classifier with all prepared data:

clObj2 =
ClConUnit[lsTrainingData]⟹
ClConSplitData[1, 0.2]⟹
ClConMakeClassifier["NearestNeighbors"];

Get the classifier function from ClCon object:

cfBStar = clObj2⟹ClConTakeClassifier

Here we generate Bethlehem Star mandalas using the classifier trained above:

SeedRandom[2020];
Multicolumn[MandalaToWhiterImage /@ BethlehemMandala[12, cfBStar, 0.87], 6, Background -> Black]

Generate Bethlehem Star mandala images utilizing the classifier (with a specified classifier probabilities threshold):

SeedRandom[32];
KeyMap[MandalaToWhiterImage, BethlehemMandala[12, cfBStar, 0.87, "Probabilities" -> True]]

Show unfiltered Bethlehem Star mandala candidates:

SeedRandom[32];
KeyMap[MandalaToWhiterImage, BethlehemMandala[12, cfBStar, 0, "Probabilities" -> True]]

Remark: Examine the probabilities in the image-probability associations above – they show that the classifier is “working.“

Here is another set generated Bethlehem Star mandalas using rotational symmetry order 4:

SeedRandom[777];
KeyMap[MandalaToWhiterImage, BethlehemMandala[12, cfBStar, 0.8, "RotationalSymmetryOrder" -> 4, "Probabilities" -> True]]

Remark: Note that although a higher rotational symmetry order is used the highly scored results still seem relevant – they have the features of the target Bethlehem Star images.

## References

[AA1] Anton Antonov, “A monad for Latent Semantic Analysis workflows”, (2019), MathematicaForPrediction at WordPress.

[AA2] Anton Antonov, “A monad for classification workflows”, (2018)), MathematicaForPrediction at WordPress.

[MSE1] “Plotting the Star of Bethlehem”, (2020),Mathematica Stack Exchange, question 236499,

[Wk1] Wikipedia entry, Star of Bethlehem.

### Packages

[AAr1] Anton Antonov, RandomMandala, (2019), Wolfram Function Repository.

[AAr2] Anton Antonov, IdependentComponentAnalysis, (2019), Wolfram Function Repository.

[AAr3] Anton Antonov, “Simplified Machine Learning Workflows” book, (2019), GitHub/antononcube.

[AAp1] Anton Antonov, Monadic Latent Semantic Analysis Mathematica package, (2017), MathematicaForPrediction at GitHub/antononcube.

[AAp2] Anton Antonov, Monadic contextual classification Mathematica package, (2017), MathematicaForPrediction at GitHub/antononcube.

[AAp3] Anton Antonov, Monadic Sparse Matrix Recommender Mathematica package, (2018), MathematicaForPrediction at GitHub/antononcube.

## Code definitions

urlPart = "https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/MonadicProgramming/";
Get[urlPart <> "/MonadicContextualClassification.m"];
Clear[MandalaToImage, MandalaToWhiterImage];
MandalaToImage[gr_Graphics, imgSize_ : {120, 120}] := ColorNegate@ImageResize[gr, imgSize];
MandalaToWhiterImage[gr_Graphics, imgSize_ : {120, 120}] := ColorNegate@ImageResize[gr /. GrayLevel[0.25] -> Black, imgSize];
Clear[ImageToVector];
ImageToVector[img_Image] := Flatten[ImageData[ColorConvert[img, "Grayscale"]]];
ImageToVector[img_Image, imgSize_] := Flatten[ImageData[ColorConvert[ImageResize[img, imgSize], "Grayscale"]]];
ImageToVector[___] := $Failed; Clear[MakeSMRProfile]; MakeSMRProfile[lsaObj_LSAMon, gr_Graphics, imgSize_] := MakeSMRProfile[lsaObj, {gr}, imgSize]; MakeSMRProfile[lsaObj_LSAMon, lsGrs : {_Graphics}, imgSize_] := MakeSMRProfile[lsaObj, MandalaToWhiterImage[#, imgSize] & /@ lsGrs, imgSize] MakeSMRProfile[lsaObj_LSAMon, img_Image, imgSize_] := MakeSMRProfile[lsaObj, {img}, imgSize]; MakeSMRProfile[lsaObj_LSAMon, lsImgs : {_Image ..}, imgSize_] := Block[{lsImgVecs, matTest, aProfPixel, aProfTopic}, lsImgVecs = ImageToVector[#, imgSize] & /@ lsImgs; matTest = ToSSparseMatrix[SparseArray[lsImgVecs], "RowNames" -> Automatic, "ColumnNames" -> Automatic]; aProfPixel = ColumnSumsAssociation[lsaObj⟹LSAMonRepresentByTerms[matTest]⟹LSAMonTakeValue]; aProfTopic = ColumnSumsAssociation[lsaObj⟹LSAMonRepresentByTopics[matTest]⟹LSAMonTakeValue]; aProfPixel = Select[aProfPixel, # > 0 &]; aProfTopic = Select[aProfTopic, # > 0 &]; Join[aProfPixel, aProfTopic] ]; MakeSMRProfile[___] :=$Failed;
Clear[BethlehemMandalaCandiate];
BethlehemMandalaCandiate[opts : OptionsPattern[]] := ResourceFunction["RandomMandala"][opts, "RotationalSymmetryOrder" -> 2, "NumberOfSeedElements" -> Automatic, "ConnectingFunction" -> FilledCurve@*BezierCurve];
Clear[BethlehemMandala];
Options[BethlehemMandala] = Join[{ImageSize -> {120, 120}, "Probabilities" -> False}, Options[ResourceFunction["RandomMandala"]]];
BethlehemMandala[n_Integer, cf_ClassifierFunction, opts : OptionsPattern[]] := BethlehemMandala[n, cf, 0.87, opts];
BethlehemMandala[n_Integer, cf_ClassifierFunction, threshold_?NumericQ, opts : OptionsPattern[]] :=
Block[{imgSize, probsQ, res, resNew, aResScores = <||>, aResScoresNew = <||>},

imgSize = OptionValue[BethlehemMandala, ImageSize];
probsQ = TrueQ[OptionValue[BethlehemMandala, "Probabilities"]];

res = {};
While[Length[res] < n,
resNew = Table[BethlehemMandalaCandiate[FilterRules[{opts}, Options[ResourceFunction["RandomMandala"]]]], 2*(n - Length[res])];
aResScoresNew = Association[# -> cf[MandalaToImage[#, imgSize], "Probabilities"][True] & /@ resNew];
aResScoresNew = Select[aResScoresNew, # >= threshold &];
aResScores = Join[aResScores, aResScoresNew];
res = Keys[aResScores]
];

aResScores = TakeLargest[ReverseSort[aResScores], UpTo[n]];
If[probsQ, aResScores, Keys[aResScores]]
] /; n > 0;
BethlehemMandala[___] := \$Failed

# Wolfram Live-Coding Series on Latent Semantic Analysis workflows

## In brief

The lectures on Latent Semantic Analysis (LSA) are to be recorded through Wolfram University (Wolfram U) in December 2019 and January-February 2020.

## The lectures (as live-coding sessions)

1. Overview Latent Semantic Analysis (LSA) typical problems and basic workflows.
Answering preliminary anticipated questions.
Here is the recording of the first session at Twitch .

• What are the typical applications of LSA?
• Why use LSA?
• What it the fundamental philosophical or scientific assumption for LSA?
• What is the most important and/or fundamental step of LSA?
• What is the difference between LSA and Latent Semantic Indexing (LSI)?
• What are the alternatives?
• Using Neural Networks instead?
• How is LSA used to derive similarities between two given texts?
• How is LSA used to evaluate the proximity of phrases? (That have different words, but close semantic meaning.)
• How the main dimension reduction methods compare?
2. LSA for document collections.
Here is the recording of the second session at Twitch .

• Motivational example – full blown LSA workflow.
• Fundamentals, text transformation (the hard way):
• bag of words model,
• stop words,
• stemming.
• The easy way with LSAMon.
• “Eat your own dog food” example.
3. Representation of the documents – the fundamental matrix object.
Here is the recording of the third session at Twitch.

• Review: last session’s example.
• Review: the motivational example – full blown LSA workflow.
• Linear vector space representation:
• LSA’s most fundamental operation,
• matrix with named rows and columns.
• Pareto Principle adherence
• for a document,
• for a document collection, and
• (in general.)
4. Representation of unseen documents.
Here is the recording of the fourth session at Twitch.

5. LSA for image de-noising and classification. Here is the recording of the fifth session at Twitch.

• Review: last session’s image collection topics extraction.
• Let us try that two other datasets:
• Image de-noising:
• Using handwritten digits (again).
• Image classification:
• Handwritten digits.
6. Further use cases.

• LSA for time series collections.
• LSA for language translations.
• Using Dostoevsky’s novel “The Idiot”.
• LSA for phrases.

# 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 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: StateMonadCodeGenerator.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.

In order to address:

• 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,

• State Monad generic functions.

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.

### State monad functions

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"]]]

(* {"LoadPackage", "EvenOddDataset", "EvenOddDataMLRules", \
"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

[Wk1] Wikipedia entry, Monad, URL: https://en.wikipedia.org/wiki/Monad_(functional_programming) .

[Wk2] Wikipedia entry, Receiver operating characteristic, URL: https://en.wikipedia.org/wiki/Receiver_operating_characteristic .

[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 .

# Introduction

In this MathematicaVsR at GitHub project we show how to do Progressive machine learning using two types of classifiers based on:

• Tries with Frequencies, [AAp2, AAp3, AA1],

• Sparse Matrix Recommender framework [AAp4, AA2].

Progressive learning is a type of Online machine learning. For more details see [Wk1]. The Progressive learning problem is defined as follows.

Problem definition:

• Assume that the data is sequentially available.
• Meaning, at a given time only part of the data is available, and after a certain time interval new data can be obtained.

• In view of classification, it is assumed that at a given time not all class labels are presented in the data already obtained.

• Let us call this a data stream.

• Make a machine learning algorithm that updates its model continuously or sequentially in time over a given data stream.

• Let us call such an algorithm a Progressive Learning Algorithm (PLA).

In comparison, the typical (classical) machine learning algorithms assume that representative training data is available and after training that data is no longer needed to make predictions. Progressive machine learning has more general assumptions about the data and its problem formulation is closer to how humans learn to classify objects.

Below we are shown the applications of two types of classifiers as PLA’s. One is based on Tries with Frequencies (TF), [AAp2, AAp3, AA1], the other on an Item-item Recommender (IIR) framework [AAp4, AA2].

Remark: Note that both TF and IIR come from tackling Unsupervised machine learning tasks, but here they are applied in the context of Supervised machine learning.

# General workflow

The Mathematica and R notebooks follow the steps in the following flow chart.

For detailed explanations see any of the notebooks.

# Example runs

(For details see Progressive-machine-learning-examples.md.)

### Using Tries with Frequencies

Here is an example run with Tries with Frequencies, [AAp2, AA1]:

Here are the obtained ROC curves:

We can see that with the Progressive learning process does improve its success rates in time.

### Using an Item-item recommender system

Here is an example run with an Item-item recommender system, [AAp4, AA2]:

Here are the obtained ROC curves:

# References

## Packages

[AAp1] Anton Antonov, Obtain and transform Mathematica machine learning data-sets, GetMachineLearningDataset.m, (2018), MathematicaVsR at GitHub.

[AAp2] Anton Antonov, Java tries with frequencies Mathematica package, JavaTriesWithFrequencies.m, (2017), MathematicaForPrediction at GitHub.

[AAp3] Anton Antonov, Tries with frequencies R package, TriesWithFrequencies.R, (2014), MathematicaForPrediction at GitHub.

[AAp4] Anton Antonov, Sparse matrix recommender framework in Mathematica, SparseMatrixRecommenderFramework.m, (2014), MathematicaForPrediction at GitHub.

## Articles

[Wk1] Wikipedia entry, Online machine learning.

[AA1] Anton Antonov, "Tries with frequencies in Java", (2017), MathematicaForPrediction at WordPress.

[AA2] Anton Antonov, "A Fast and Agile Item-Item Recommender: Design and Implementation", (2011), Wolfram Technology Conference 2011.

# Applying Artificial Intelligence and Machine Learning to Finance and Technology

## Introduction

In this blog post I try to provide some further context for the panel discussion:

which was part of the conference "Data Science Salon Miami 2018" held in Miami on February 8 and 9.

The blog post can be read independently, but my intent is to provide a brief review of the discussion and further context to (some of) the answers. (It is probably better to see discussion’s recording first.)

Also, my comments and remarks are saturated with links for reference and further reading.

You can also read the post by panel’s discussion host, Irma Becerra.

This blog post is not written by someone particularly enamored by Artificial Intelligence (AI), Data Science (DS), or Machine Learning (ML). I do like working in those disciplines, but I do think there are more interesting and fascinating mathematical disciplines than them. Also, I think AI, DS, and ML should be seen through the lens of Operations Research, which gives the most fruitful perspectives of their utilization.

### Briefly about the event

The conference "Data Science Salon Miami 2018" included approximately 300 attendants. (I was told that there were ~60% data scientist and data analysts, and ~40% managers.)

## The panel people

Here is a list of the people participating in the panel (Irma was the host):

## The panel questions

Here are the main questions asked during the panel session:

1. What was your path to Data Science?

2. What are some misconceptions about AI? What do you see as being the future of AI?

3. How is artificial intelligence helping us engage with customers?

4. What techniques can we expect to see in terms of informing our marketing strategy so we can build predictive models to support our businesses? (backup question)

5. What can we do make advancements in Data Science more accessible to the public?

Here are some of the questions from the audience:

1. How to apply AI or Machine Learning to process automation?

2. What is going to happen if an adopted AI technology fails?

## On the main panel questions

### What was your path to Data Science?

All of the people in the panel became data scientists later in their career. Each of us at first was studying or doing something else.

Of course, that is to be expected since the term "data science" and the related occupation came into fashion relatively recently.

(Although apparently the term itself is fairly old.)

### What are some misconceptions about AI? What do you see as being the future of AI?

This is of course a long and fruitful topic.

Here — and during the panel session — I want to concentrate of what kind of thinking processes generate those misconceptions. (Obviously this is a less laborious task than enumerating and discussing the different concrete misconceptions.)

#### Weak AI

I side with the so called "Weak AI". Here are some summarizing points.

• Weak AI has a goal to adequately do certain mental tasks by humans. In general, Weak AI does not try to replicate the human approaches, methods, and algorithms for those tasks.

• All algorithms are based on 0s and 1s, so generally, I find "AI" misleading and "Strong AI" an interesting thing to think about, but far from practical use.

• Classifiers are just complicated if-statements.

• Derived with deep learning or other algorithms.

• Also, classification problems belong to the conceptually simplest ML branch (Supervised learning.)

Generally, my personal opinions on "AI" come from living through at least one AI winter, and being a subject in the AI effect.

#### The ingredients of coming to wrong ideas about AI

Study the following mind-map, "AI misconceptions reasons".

I think the major ingredients of getting the wrong AI ideas are two.

1. Confusion of high performance with competence.
• Often, confusion of high performance in some narrow problem domain with competence in an enclosing more general problem domain.
2. Exponential growth extrapolation of AI’s advances.

My favorite analogy to point 1 above is the following.

There are two principal ways to fish: (i) with bait and a fishing rod, and (ii) by draining the lake and picking the fish you want. Any other way of fishing is something in between. (For example, using a net or a bomb.)

So, the high performance of draining the lake can be confused with the ability of bait selection and fishing-spot picking. (And yes, not knowing how the AI algorithms work is a big misconception maker too.)

As an illustration of "draining the lake" approach to interesting puzzles humans play with, review (skim over) this article: "Solving Sudoku as an integer programming problem".

• We "drain the lake" by formulating an integer optimization problem with 729 variables and 1089 constraints. By solving the integer programming problem we "pick the fish."

• Note, that this is not even considered AI now. Granted, we do need a symbolic computations system like Mathematica, which is a very advanced system. (And yes, a good symbolic manipulation system can be seen as AI, especially 4-5 decades ago.)

### How is artificial intelligence helping us engage with customers?

I specialize in making recommenders and conversational agents.

For items that require significant investment in time (movies, books) or money (houses, automobiles, freelancers) recommenders have to make very good explanations of the recommendations. This is usually based on similarities to items consumed in the past or stated preferences.

For items that can be easily tried on and discarded (songs) we are better off making prediction algorithms that predict the "survival" of the item in customer’s mind. (How long a song is going to be in a playlist?)

The recommenders I developed in the past tend to be very agile : fast, easy to tune, with clear "recommendation proofs." Very often the agility of the recommender system is more important that giving good or precise recommendations "out of the box."

Customer engagement through conversational agents is much more about envisioning of the right work-flow and mind-flow of the customer. Although clarity and simplicity are important, I like the idea of using grammar rules and agent names that are challenging or intriguing. (Like, "Will they kill me?" or "I wanna eat!".)

### What techniques can we expect to see in terms of informing our marketing strategy so we can build predictive models to support our businesses?

In many ways the application of AI to Finance is somewhat easier, since financial data tends to be well-curated.

For example,

• in healthcare different patient health data can be messy (and incomplete) and generally speaking human body is inherently complex;

• health insurance financial data, though, is well-curated (since people need to get payed) and fairly simple.

To clarify, if we have records with four columns:

   claim ID, claim total amount, transaction amount, timestamp

then we are already in a position to do a fair amount of modeling and prediction.

### What can we do make advancements in Data Science more accessible to the public?

My answer was two-fold:

1. it is important to be able to communicate AI and ML concepts, ideas, and functionalities to stakeholders in business projects and, generally, to curious people, but

2. from the other hand, some of the underlying algorithms require mastery of multiple computer science and mathematical disciplines.

In other words:

1. of course the data scientist knowing his stuff should be able to explain the AI and ML functionalities to laypersons, but

2. from the other hand, "art is for the educated."

I would like to point out that technology itself has already democratized and popularized the AI and ML advancements. Everyone is familiar with, say, content recommendations, search queries completion, and optical character recognition. Few decades ago these functionalities would have been considered science fiction or part of Strong AI.

## Some afterthoughts

• The main panel questions and the questions from the audience made me want to discuss a general classification of the AI application — study the following mind-map "Application of AI and ML".

# 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

This loads the packages [AAp1-AAp8]:

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 =
Thread[{##}] &, {Range@Length@texts, names,
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],
Thread[Tooltip[ColumnNames[m], aTopicNameToTopicTable /@ ColumnNames[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.

[MA2] Mortimer Adler, "Great Ideas".

[MA3] Mortimer Adler, "How to Think About the Great Ideas: From the Great Books of Western Civilization", 2000, Open Court.

# Monad code generation and extension

## … in Mathematica / Wolfram Language

Anton Antonov

MathematicaForPrediction at GitHub

MathematicaVsR at GitHub

June 2017

## Introduction

This document aims to introduce monadic programming in Mathematica / Wolfram Language (WL) in a concise and code-direct manner. The core of the monad codes discussed is simple, derived from the fundamental principles of Mathematica / WL.

The usefulness of the monadic programming approach manifests in multiple ways. Here are a few we are interested in:

1. easy to construct, read, and modify sequences of commands (pipelines),
2. easy to program polymorphic behaviour,
3. easy to program context utilization.

Speaking informally,

• Monad programming provides an interface that allows interactive, dynamic creation and change of sequentially structured computations with polymorphic and context-aware behavior.

The theoretical background provided in this document is given in the Wikipedia article on Monadic programming, [Wk1], and the article “The essence of functional programming” by Philip Wadler, [H3]. The code in this document is based on the primary monad definition given in [Wk1,H3]. (Based on the “Kleisli triple” and used in Haskell.)

The general monad structure can be seen as:

1. a software design pattern;
2. a fundamental programming construct (similar to class in object-oriented programming);
3. an interface for software types to have implementations of.

In this document we treat the monad structure as a design pattern, [Wk3]. (After reading [H3] point 2 becomes more obvious. A similar in spirit, minimalistic approach to Object-oriented Design Patterns is given in [AA1].)

We do not deal with types for monads explicitly, we generate code for monads instead. One reason for this is the “monad design pattern” perspective; another one is that in Mathematica / WL the notion of algebraic data type is not needed — pattern matching comes from the core “book of replacement rules” principle.

The rest of the document is organized as follows.

1. Fundamental sections The section “What is a monad?” gives the necessary definitions. The section “The basic Maybe monad” shows how to program a monad from scratch in Mathematica / WL. The section “Extensions with polymorphic behavior” shows how extensions of the basic monad functions can be made. (These three sections form a complete read on monadic programming, the rest of the document can be skipped.)

2. Monadic programming in practice The section “Monad code generation” describes packages for generating monad code. The section “Flow control in monads” describes additional, control flow functionalities. The section “General work-flow of monad code generation utilization” gives a general perspective on the use of monad code generation. The section “Software design with monadic programming” discusses (small scale) software design with monadic programming.

3. Case study sections The case study sections “Contextual monad classification” and “Tracing monad pipelines” hopefully have interesting and engaging examples of monad code generation, extension, and utilization.

## What is a monad?

### The monad definition

In this document a monad is any set of a symbol m and two operators unit and bind that adhere to the monad laws. (See the next sub-section.) The definition is taken from [Wk1] and [H3] and phrased in Mathematica / WL terms in this section. In order to be brief, we deliberately do not consider the equivalent monad definition based on unit, join, and map (also given in [H3].)

Here are operators for a monad associated with a certain symbol M:

1. monad unit function (“return” in Haskell notation) is Unit[x_] := M[x];
2. monad bind function (“>>=” in Haskell notation) is a rule like Bind[M[x_], f_] := f[x] with MatchQ[f[x],M[_]] giving True.

Note that:

• the function Bind unwraps the content of M[_] and gives it to the function f;
• the functions fi are responsible to return results wrapped with the monad symbol M.

Here is an illustration formula showing a monad pipeline:

From the definition and formula it should be clear that if for the result of Bind[_M,f[x]] the test MatchQ[f[x],_M] is True then the result is ready to be fed to the next binding operation in monad’s pipeline. Also, it is clear that it is easy to program the pipeline functionality with Fold:

Fold[Bind, M[x], {f1, f2, f3}]

(* Bind[Bind[Bind[M[x], f1], f2], f3] *)

### The monad laws

The monad laws definitions are taken from [H1] and [H3].In the monad laws given below the symbol “⟹” is for monad’s binding operation and ↦ is for a function in anonymous form.

Here is a table with the laws:

Remark: The monad laws are satisfied for every symbol in Mathematica / WL with List being the unit operation and Apply being the binding operation.

### Expected monadic programming features

Looking at formula (1) — and having certain programming experiences — we can expect the following features when using monadic programming.

• Computations that can be expressed with monad pipelines are easy to construct and read.
• By programming the binding function we can tuck-in a variety of monad behaviours — this is the so called “programmable semicolon” feature of monads.
• Monad pipelines can be constructed with Fold, but with suitable definitions of infix operators like DoubleLongRightArrow (⟹) we can produce code that resembles the pipeline in formula (1).
• A monad pipeline can have polymorphic behaviour by overloading the signatures of fi (and if we have to, Bind.)

These points are clarified below. For more complete discussions see [Wk1] or [H3].

## The basic Maybe monad

It is fairly easy to program the basic monad Maybe discussed in [Wk1].

The goal of the Maybe monad is to provide easy exception handling in a sequence of chained computational steps. If one of the computation steps fails then the whole pipeline returns a designated failure symbol, say None otherwise the result after the last step is wrapped in another designated symbol, say Maybe.

Here is the special version of the generic pipeline formula (1) for the Maybe monad:

Here is the minimal code to get a functional Maybe monad (for a more detailed exposition of code and explanations see [AA7]):

MaybeUnitQ[x_] := MatchQ[x, None] || MatchQ[x, Maybe[___]];

MaybeUnit[None] := None;
MaybeUnit[x_] := Maybe[x];

MaybeBind[None, f_] := None;
MaybeBind[Maybe[x_], f_] :=
Block[{res = f[x]}, If[FreeQ[res, None], res, None]];

MaybeEcho[x_] := Maybe@Echo[x];
MaybeEchoFunction[f___][x_] := Maybe@EchoFunction[f][x];

MaybeOption[f_][xs_] :=
Block[{res = f[xs]}, If[FreeQ[res, None], res, Maybe@xs]];

In order to make the pipeline form of the code we write let us give definitions to a suitable infix operator (like “⟹”) to use MaybeBind:

DoubleLongRightArrow[x_?MaybeUnitQ, f_] := MaybeBind[x, f];
DoubleLongRightArrow[x_, y_, z__] :=
DoubleLongRightArrow[DoubleLongRightArrow[x, y], z];

Here is an example of a Maybe monad pipeline using the definitions so far:

data = {0.61, 0.48, 0.92, 0.90, 0.32, 0.11};

MaybeUnit[data]⟹(* lift data into the monad *)
(Maybe@ Join[#, RandomInteger[8, 3]] &)⟹(* add more values *)
MaybeEcho⟹(* display current value *)
(Maybe @ Map[If[# < 0.4, None, #] &, #] &)(* map values that are too small to None *)

(* {0.61,0.48,0.92,0.9,0.32,0.11,4,4,0}
None *)

The result is None because:

1. the data has a number that is too small, and
2. the definition of MaybeBind stops the pipeline aggressively using a FreeQ[_,None] test.

### Monad laws verification

Let us convince ourselves that the current definition of MaybeBind gives a monad.

The verification is straightforward to program and shows that the implemented Maybe monad adheres to the monad laws.

## Extensions with polymorphic behavior

We can see from formulas (1) and (2) that the monad codes can be easily extended through overloading the pipeline functions.

For example the extension of the Maybe monad to handle of Dataset objects is fairly easy and straightforward.

Here is the formula of the Maybe monad pipeline extended with Dataset objects:

Here is an example of a polymorphic function definition for the Maybe monad:

MaybeFilter[filterFunc_][xs_] := Maybe@Select[xs, filterFunc[#] &];

MaybeFilter[critFunc_][xs_Dataset] := Maybe@xs[Select[critFunc]];

See [AA7] for more detailed examples of polymorphism in monadic programming with Mathematica / WL.

A complete discussion can be found in [H3]. (The main message of [H3] is the poly-functional and polymorphic properties of monad implementations.)

### Polymorphic monads in R’s dplyr

The R package dplyr, [R1], has implementations centered around monadic polymorphic behavior. The command pipelines based on dplyrcan work on R data frames, SQL tables, and Spark data frames without changes.

Here is a diagram of a typical work-flow with dplyr:

The diagram shows how a pipeline made with dplyr can be re-run (or reused) for data stored in different data structures.

## Monad code generation

We can see monad code definitions like the ones for Maybe as some sort of initial templates for monads that can be extended in specific ways depending on their applications. Mathematica / WL can easily provide code generation for such templates; (see [WL1]). As it was mentioned in the introduction, we do not deal with types for monads explicitly, we generate code for monads instead.

In this section are given examples with packages that generate monad codes. The case study sections have examples of packages that utilize generated monad codes.

### Maybe monads code generation

The package [AA2] provides a Maybe code generator that takes as an argument a prefix for the generated functions. (Monad code generation is discussed further in the section “General work-flow of monad code generation utilization”.)

Here is an example:

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

data = {0.61, 0.48, 0.92, 0.90, 0.32, 0.11};

AnotherMaybeUnit[data]⟹(* lift data into the monad *)
(AnotherMaybe@Join[#, RandomInteger[8, 3]] &)⟹(* add more values *)
AnotherMaybeEcho⟹(* display current value *)
(AnotherMaybe @ Map[If[# < 0.4, None, #] &, #] &)(* map values that are too small to None *)

(* {0.61,0.48,0.92,0.9,0.32,0.11,8,7,6}
AnotherMaybeBind: Failure when applying: Function[AnotherMaybe[Map[Function[If[Less[Slot[1], 0.4], None, Slot[1]]], Slot[1]]]]
None *)

We see that we get the same result as above (None) and a message prompting failure.

### State monads code generation

The State monad is also basic and its programming in Mathematica / WL is not that difficult. (See [AA3].)

Here is the special version of the generic pipeline formula (1) for the State monad:

Note that since the State monad pipeline caries both a value and a state, it is a good idea to have functions that manipulate them separately. For example, we can have functions for context modification and context retrieval. (These are done in [AA3].)

This loads the package [AA3]:

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

This generates the State monad for the prefix “StMon”:

GenerateStateMonadCode["StMon"]

The following StMon pipeline code starts with a random matrix and then replaces numbers in the current pipeline value according to a threshold parameter kept in the context. Several times are invoked functions for context deposit and retrieval.

SeedRandom[34]
StMonUnit[RandomReal[{0, 1}, {3, 2}], <|"mark" -> "TooSmall", "threshold" -> 0.5|>]⟹
StMonEchoValue⟹
StMonEchoContext⟹
StMonEchoContext⟹
(StMon[#1 /. (x_ /; x < #2["threshold"] :> #2["mark"]), #2] &)⟹
StMonEchoValue⟹
StMonRetrieveFromContext["data"]⟹
StMonEchoValue⟹
StMonRetrieveFromContext["mark"]⟹
StMonEchoValue;

(* value: {{0.789884,0.831468},{0.421298,0.50537},{0.0375957,0.289442}}
context: <|mark->TooSmall,threshold->0.5|>
context: <|mark->TooSmall,threshold->0.5,data->{{0.789884,0.831468},{0.421298,0.50537},{0.0375957,0.289442}}|>
value: {{0.789884,0.831468},{TooSmall,0.50537},{TooSmall,TooSmall}}
value: {{0.789884,0.831468},{0.421298,0.50537},{0.0375957,0.289442}}
value: TooSmall *)

## Flow control in monads

We can implement dedicated functions for governing the pipeline flow in a monad.

Let us look at a breakdown of these kind of functions using the State monad StMon generated above.

### Optional acceptance of a function result

A basic and simple pipeline control function is for optional acceptance of result — if failure is obtained applying f then we ignore its result (and keep the current pipeline value.)

Here is an example with StMonOption :

SeedRandom[34]
StMonUnit[RandomReal[{0, 1}, 5]]⟹
StMonEchoValue⟹
StMonOption[If[# < 0.3, None] & /@ # &]⟹
StMonEchoValue

(* value: {0.789884,0.831468,0.421298,0.50537,0.0375957}
value: {0.789884,0.831468,0.421298,0.50537,0.0375957}
StMon[{0.789884, 0.831468, 0.421298, 0.50537, 0.0375957}, <||>] *)

Without StMonOption we get failure:

SeedRandom[34]
StMonUnit[RandomReal[{0, 1}, 5]]⟹
StMonEchoValue⟹
(If[# < 0.3, None] & /@ # &)⟹
StMonEchoValue

(* value: {0.789884,0.831468,0.421298,0.50537,0.0375957}
StMonBind: Failure when applying: Function[Map[Function[If[Less[Slot[1], 0.3], None]], Slot[1]]]
None *)

### Conditional execution of functions

It is natural to want to have the ability to chose a pipeline function application based on a condition.

This can be done with the functions StMonIfElse and StMonWhen.

SeedRandom[34]
StMonUnit[RandomReal[{0, 1}, 5]]⟹
StMonEchoValue⟹
StMonIfElse[
Or @@ (# < 0.4 & /@ #) &,
(Echo["A too small value is present.", "warning:"];
StMon[Style[#1, Red], #2]) &,
StMon[Style[#1, Blue], #2] &]⟹
StMonEchoValue

(* value: {0.789884,0.831468,0.421298,0.50537,0.0375957}
warning: A too small value is present.
value: {0.789884,0.831468,0.421298,0.50537,0.0375957}
StMon[{0.789884,0.831468,0.421298,0.50537,0.0375957},<||>] *)

Remark: Using flow control functions like StMonIfElse and StMonWhen with appropriate messages is a better way of handling computations that might fail. The silent failures handling of the basic Maybe monad is convenient only in a small number of use cases.

### Iterative functions

The last group of pipeline flow control functions we consider comprises iterative functions that provide the functionalities of Nest, NestWhile, FoldList, etc.

In [AA3] these functionalities are provided through the function StMonIterate.

Here is a basic example using Nest that corresponds to Nest[#+1&,1,3]:

StMonUnit[1]⟹StMonIterate[Nest, (StMon[#1 + 1, #2]) &, 3]

(* StMon[4, <||>] *)

Consider this command that uses the full signature of NestWhileList:

NestWhileList[# + 1 &, 1, # < 10 &, 1, 4]

(* {1, 2, 3, 4, 5} *)

Here is the corresponding StMon iteration code:

StMonUnit[1]⟹StMonIterate[NestWhileList, (StMon[#1 + 1, #2]) &, (#[[1]] < 10) &, 1, 4]

(* StMon[{1, 2, 3, 4, 5}, <||>] *)

Here is another results accumulation example with FixedPointList :

StMonUnit[1.]⟹
StMonIterate[FixedPointList, (StMon[(#1 + 2/#1)/2, #2]) &]

(* StMon[{1., 1.5, 1.41667, 1.41422, 1.41421, 1.41421, 1.41421}, <||>] *)

When the functions NestList, NestWhileList, FixedPointList are used with StMonIterate their results can be stored in the context. Here is an example:

StMonUnit[1.]⟹
StMonIterate[FixedPointList, (StMon[(#1 + 2/#1)/2, #2]) &, "fpData"]

(* StMon[{1., 1.5, 1.41667, 1.41422, 1.41421, 1.41421, 1.41421}, <|"fpData" -> {StMon[1., <||>],
StMon[1.5, <||>], StMon[1.41667, <||>], StMon[1.41422, <||>], StMon[1.41421, <||>],
StMon[1.41421, <||>], StMon[1.41421, <||>]} |>] *)

More elaborate tests can be found in [AA8].

### Partial pipelines

Because of the associativity law we can design pipeline flows based on functions made of “sub-pipelines.”

fEcho = Function[{x, ct}, StMonUnit[x, ct]⟹StMonEchoValue⟹StMonEchoContext];

fDIter = Function[{x, ct},
StMonUnit[y^x, ct]⟹StMonIterate[FixedPointList, StMonUnit@D[#, y] &, 20]];

StMonUnit[7]⟹fEcho⟹fDIter⟹fEcho;

(*
value: 7
context: <||>
value: {y^7,7 y^6,42 y^5,210 y^4,840 y^3,2520 y^2,5040 y,5040,0,0}
context: <||> *)

## General work-flow of monad code generation utilization

With the abilities to generate and utilize monad codes it is natural to consider the following work flow. (Also shown in the diagram below.)

1. Come up with an idea that can be expressed with monadic programming.
2. Look for suitable monad implementation.
3. If there is no such implementation, make one (or two, or five.)
4. Having a suitable monad implementation, generate the monad code.
5. Implement additional pipeline functions addressing envisioned use cases.
6. Start making pipelines for the problem domain of interest.
7. Are the pipelines are satisfactory? If not go to 5. (Or 2.)

The template nature of the general monads can be exemplified with the group of functions in the package StateMonadCodeGenerator.m, [4].

They are in five groups:

1. base monad functions (unit testing, binding),
2. display of the value and context,
3. context manipulation (deposit, retrieval, modification),
4. flow governing (optional new value, conditional function application, iteration),
5. other convenience functions.

We can say that all monad implementations will have their own versions of these groups of functions. The more specialized monads will have functions specific to their intended use. Such special monads are discussed in the case study sections.

## Software design with monadic programming

The application of monadic programming to a particular problem domain is very similar to designing a software framework or designing and implementing a Domain Specific Language (DSL).

The answers of the question “When to use monadic programming?” can form a large list. This section provides only a couple of general, personal viewpoints on monadic programming in software design and architecture. The principles of monadic programming can be used to build systems from scratch (like Haskell and Scala.) Here we discuss making specialized software with or within already existing systems.

#### Framework design

Software framework design is about architectural solutions that capture the commonality and variability in a problem domain in such a way that: 1) significant speed-up can be achieved when making new applications, and 2) a set of policies can be imposed on the new applications.

The rigidness of the framework provides and supports its flexibility — the framework has a backbone of rigid parts and a set of “hot spots” where new functionalities are plugged-in.

Usually Object-Oriented Programming (OOP) frameworks provide inversion of control — the general work-flow is already established, only parts of it are changed. (This is characterized with “leave the driving to us” and “don’t call us we will call you.”)

The point of utilizing monadic programming is to be able to easily create different new work-flows that share certain features. (The end user is the driver, on certain rail paths.)

In my opinion making a software framework of small to moderate size with monadic programming principles would produce a library of functions each with polymorphic behaviour that can be easily sequenced in monadic pipelines. This can be contrasted with OOP framework design in which we are more likely to end up with backbone structures that (i) are static and tree-like, and (ii) are extended or specialized by plugging-in relevant objects. (Those plugged-in objects themselves can be trees, but hopefully short ones.)

#### DSL development

Given a problem domain the general monad structure can be used to shape and guide the development of DSLs for that problem domain.

Generally, in order to make a DSL we have to choose the language syntax and grammar. Using monadic programming the syntax and grammar commands are clear. (The monad pipelines are the commands.) What is left is “just” the choice of particular functions and their implementations.

Another way to develop such a DSL is through a grammar of natural language commands. Generally speaking, just designing the grammar — without developing the corresponding interpreters — would be very helpful in figuring out the components at play. Monadic programming meshes very well with this approach and applying the two approaches together can be very fruitful.

## Contextual monad classification (case study)

In this section we show an extension of the State monad into a monad aimed at machine learning classification work-flows.

### Motivation

We want to provide a DSL for doing machine learning classification tasks that allows us:

1. to do basic summarization and visualization of the data,
2. to control splitting of the data into training and testing sets;
3. to apply the built-in classifiers;
4. to apply classifier ensembles (see [AA9] and [AA10]);
5. to evaluate classifier performances with standard measures and
6. ROC plots.

Also, we want the DSL design to provide clear directions how to add (hook-up or plug-in) new functionalities.

The package [AA4] discussed below provides such a DSL through monadic programming.

This loads the package [AA4]:

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

This gets some test data (the Titanic dataset):

dataName = "Titanic";
ds = Dataset[Flatten@*List @@@ ExampleData[{"MachineLearning", dataName}, "Data"]];
varNames = Flatten[List @@ ExampleData[{"MachineLearning", dataName}, "VariableDescriptions"]];
varNames = StringReplace[varNames, "passenger" ~~ (WhitespaceCharacter ..) -> ""];
If[dataName == "FisherIris", varNames = Most[varNames]];
ds = ds[All, AssociationThread[varNames -> #] &];

The package [AA4] provides functions for the monad ClCon — the functions implemented in [AA4] have the prefix “ClCon”.

The classifier contexts are Association objects. The pipeline values can have the form:

ClCon[ val, context:(_String|_Association) ]

The ClCon specific monad functions deposit or retrieve values from the context with the keys: “trainData”, “testData”, “classifier”. The general idea is that if the current value of the pipeline cannot provide all arguments for a ClCon function, then the needed arguments are taken from the context. If that fails, then an message is issued. This is illustrated with the following pipeline with comments example.

The pipeline and results above demonstrate polymorphic behaviour over the classifier variable in the context: different functions are used if that variable is a ClassifierFunction object or an association of named ClassifierFunction objects.

Note the demonstrated granularity and sequentiality of the operations coming from using a monad structure. With those kind of operations it would be easy to make interpreters for natural language DSLs.

### Another usage example

This monadic pipeline in this example goes through several stages: data summary, classifier training, evaluation, acceptance test, and if the results are rejected a new classifier is made with a different algorithm using the same data splitting. The context keeps track of the data and its splitting. That allows the conditional classifier switch to be concisely specified.

First let us define a function that takes a Classify method as an argument and makes a classifier and calculates performance measures.

ClSubPipe[method_String] :=
Function[{x, ct},
ClConUnit[x, ct]⟹
ClConMakeClassifier[method]⟹
ClConEchoFunctionContext["classifier:",
ClassifierInformation[#["classifier"], Method] &]⟹
ClConEchoFunctionContext["training time:", ClassifierInformation[#["classifier"], "TrainingTime"] &]⟹
ClConClassifierMeasurements[{"Accuracy", "Precision", "Recall"}]⟹
ClConEchoValue⟹
ClConEchoFunctionContext[
ClassifierMeasurements[#["classifier"],
ClConToNormalClassifierData[#["testData"]], "ROCCurve"] &]
];

Using the sub-pipeline function ClSubPipe we make the outlined pipeline.

SeedRandom[12]
res =
ClConUnit[ds]⟹
ClConSplitData[0.7]⟹
ClConEchoFunctionValue["summaries:", ColumnForm[Normal[RecordsSummary /@ #]] &]⟹
ClConEchoFunctionValue["xtabs:",
MatrixForm[CrossTensorate[Count == varNames[[1]] + varNames[[-1]], #]] & /@ # &]⟹
ClSubPipe["LogisticRegression"]⟹
(If[#1["Accuracy"] > 0.8,
Echo["Good accuracy!", "Success:"]; ClConFail,
Echo["Make a new classifier", "Inaccurate:"];
ClConUnit[#1, #2]] &)⟹
ClSubPipe["RandomForest"];

## Tracing monad pipelines (case study)

The monadic implementations in the package MonadicTracing.m, [AA5] allow tracking of the pipeline execution of functions within other monads.

The primary reason for developing the package was the desire to have the ability to print a tabulated trace of code and comments using the usual monad pipeline notation. (I.e. without conversion to strings etc.)

It turned out that by programming MonadicTracing.m I came up with a monad transformer; see [Wk2], [H2].

This loads the package [AA5]:

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

### Usage example

This generates a Maybe monad to be used in the example (for the prefix “Perhaps”):

GenerateMaybeMonadCode["Perhaps"]
GenerateMaybeMonadSpecialCode["Perhaps"]

In following example we can see that pipeline functions of the Perhaps monad are interleaved with comment strings. Producing the grid of functions and comments happens “naturally” with the monad function TraceMonadEchoGrid.

data = RandomInteger[10, 15];

PerhapsFilter[# > 3 &]⟹"filter current value"⟹
PerhapsEcho⟹"display current value"⟹
PerhapsWhen[#[[3]] > 3 &,
PerhapsEchoFunction[Style[#, Red] &]]⟹
(Perhaps[#/4] &)⟹
PerhapsEcho⟹"display current value again"⟹
TraceMonadEchoGrid[Grid[#, Alignment -> Left] &];

Note that :

1. the tracing is initiated by just using TraceMonadUnit;
2. pipeline functions (actual code) and comments are interleaved;
3. putting a comment string after a pipeline function is optional.

Another example is the ClCon pipeline in the sub-section “Monad design” in the previous section.

## Summary

This document presents a style of using monadic programming in Wolfram Language (Mathematica). The style has some shortcomings, but it definitely provides convenient features for day-to-day programming and in coming up with architectural designs.

The style is based on WL’s basic language features. As a consequence it is fairly concise and produces light overhead.

Ideally, the packages for the code generation of the basic Maybe and State monads would serve as starting points for other more general or more specialized monadic programs.

## References

[Wk1] Wikipedia entry: Monad (functional programming), URL: https://en.wikipedia.org/wiki/Monad_(functional_programming) .

[Wk2] Wikipedia entry: Monad transformer, URL: https://en.wikipedia.org/wiki/Monad_transformer .

[Wk3] Wikipedia entry: Software Design Pattern, URL: https://en.wikipedia.org/wiki/Software_design_pattern .

[H2] Sheng Liang, Paul Hudak, Mark Jones, “Monad transformers and modular interpreters”, (1995), Proceedings of the 22nd ACM SIGPLAN-SIGACT symposium on Principles of programming languages. New York, NY: ACM. pp. 333[Dash]343. doi:10.1145/199448.199528.

[H3] Philip Wadler, “The essence of functional programming”, (1992), 19’th Annual Symposium on Principles of Programming Languages, Albuquerque, New Mexico, January 1992.

### R

[R1] Hadley Wickham et al., dplyr: A Grammar of Data Manipulation, (2014), tidyverse at GitHub, URL: https://github.com/tidyverse/dplyr . (See also, http://dplyr.tidyverse.org .)

### Mathematica / Wolfram Language

[WL1] Leonid Shifrin, “Metaprogramming in Wolfram Language”, (2012), Mathematica StackExchange. (Also posted at Wolfram Community in 2017.) URL of the Mathematica StackExchange answer: https://mathematica.stackexchange.com/a/2352/34008 . URL of the Wolfram Community post: http://community.wolfram.com/groups/-/m/t/1121273 .

### MathematicaForPrediction

[AA7] Anton Antonov, Simple monadic programming, (2017), MathematicaForPrediction at GitHub. (Preliminary version, 40% done.) URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/Documentation/Simple-monadic-programming.pdf .