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:

ERTMon-small-pipeline-example
ERTMon-small-pipeline-example

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].

Package load

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

Data load

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]
ERTMon-RecordsSummary-eventRecord
ERTMon-RecordsSummary-eventRecord
RecordsSummary[entityAttributes]
ERTMon-RecordsSummary-entityAttributes
ERTMon-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.

ERTMon-workflows
ERTMon-workflows

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.

ERTMon-repeated-execution-workflow
ERTMon-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]
ERTMon-eventRecords-sample
ERTMon-eventRecords-sample

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]]
ERTMon-entityAttributes-sample
ERTMon-entityAttributes-sample

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"]
ERTMon-DateListPlot-USA-Temperature
ERTMon-DateListPlot-USA-Temperature

Monad elements

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”.

Monad unit

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;
   
ERTMon-echo-data-summary
ERTMon-echo-data-summary

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 &]]
 
ERTMon-wCompSpec
ERTMon-wCompSpec

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;
ERTMon-compute-variable-statistic-1
ERTMon-compute-variable-statistic-1
p⟹ERTMonComputeVariableStatistic⟹ERTMonEchoValue;
ERTMon-compute-variable-statistic-2
ERTMon-compute-variable-statistic-2
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}]] &];
  
ERTMon-records-groups-minTime
ERTMon-records-groups-minTime

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

p =
  p⟹
   ERTMonEntityVariableGroupsToTimeSeries["MaxTime"]⟹
   ERTMonEchoFunctionContext[#timeSeries[[{1, 3, 5}]] &];
ERTMon-records-groups-maxTime
ERTMon-records-groups-maxTime

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}]] &];
   
ERTMon-restriction-and-aggregation
ERTMon-restriction-and-aggregation

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}]] &];
ERTMon-moving-average-application
ERTMon-moving-average-application

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}]] &];
   
ERTMon-normalization
ERTMon-normalization

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 /@ %
ERTMon-contingency-matrices
ERTMon-contingency-matrices
smat = p⟹ERTMonTakeContingencyMatrix;
MatrixPlot[smat, ImageSize -> 700]
ERTMon-contingency-matrix
ERTMon-contingency-matrix
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.

ERTMon-large-pipeline-example
ERTMon-large-pipeline-example
ERTMon-large-pipeline-example-output
ERTMon-large-pipeline-example-output

References

Packages

[AAp1] Anton Antonov, State monad code generator Mathematica package, (2017), MathematicaForPrediction at GitHub*. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/StateMonadCodeGenerator.m .

[AAp2] Anton Antonov, Monadic tracing Mathematica package, (2017), MathematicaForPrediction at GitHub*. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicTracing.m .

[AAp3] Anton Antonov, Monadic Event Records Transformations Mathematica package, (2018), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicEventRecordsTransformations.m .

[AAp4] Anton Antonov, MathematicaForPrediction utilities, (2014), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MathematicaForPredictionUtilities.m .

[AAp5] Anton Antonov, Cross tabulation implementation in Mathematica, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/CrossTabulate.m .

[AAp6] Anton Antonov, Implementation of one dimensional outlier identifying algorithms in Mathematica, (2013), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/OutlierIdentifiers.m.

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

[AAp8] Anton Antonov, Monadic contextual classification Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicContextualClassification.m .

[AAp9] Anton Antonov, Weather event records data Mathematica package, (2018), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/Misc/WeatherEventRecords.m .

Documents

[AA1] Anton Antonov, Monad code generation and extension, (2017), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2017/06/23/monad-code-generation-and-extension .

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

[AA2] Anton Antonov, A monad for classification workflows, (2018), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2018/05/15/a-monad-for-classification-workflows .

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QRMon for some credit risk article

Introduction

In this notebook/document we apply the monad QRMon [3] over data of the article [1]. In order to get the data we use extraction procedure described in [2].

(I saw the article [1] while browsing LinkedIn today. I met one of the authors during the event "Data Science Salon Miami Feb 2018".)

Extract data

I extracted the data from the image using "Recovering data points from an image".

img = Import["https://www.spglobal.com/_assets/images/marketintelligence/blog-images/demonstration-of-model-fit-comparison-visualization.png"]
enter image description here

enter image description here

extractedData
(* {{-1., 0.284894}, {-0.987395, 0.340483}, {-0.966387, 
  0.215408}, {-0.941176, 0.416918}, {-0.894958, 0.222356}, {-0.890756,
   0.215408}, {-0.878151, 0.0903323}, {-0.848739, 
  0.132024}, {-0.844538, 0.10423}, {-0.831933, 0.333535}, {-0.819328, 
  0.180665}, {-0.781513, 0.423867}, {-0.756303, 0.40997}, {-0.752101, 
  0.528097}, {-0.747899, 0.416918}, {-0.731092, 0.375227}, {-0.714286,
   0.194562}, {-0.710084, 0.340483}, {-0.651261, 
  0.555891}, {-0.647059, 0.333535}, {-0.605042, 0.496828}, {-0.57563, 
  0.}, {-0.512605, 0.354381}, {-0.491597, 0.368278}, {-0.487395, 
  0.472508}, {-0.478992, 0.479456}, {-0.453782, 0.437764}, {-0.357143,
   0.15287}, {-0.344538, 0.340483}, {-0.331933, 0.333535}, {-0.315126,
   0.500302}, {-0.285714, 0.396073}, {-0.247899, 
  0.618429}, {-0.201681, 0.541994}, {-0.159664, 0.680967}, {-0.10084, 
  1.06314}, {-0.0966387, 0.993656}, {0., 1.36193}, {0.0210084, 
  1.44532}, {0.0420168, 1.5148}, {0.0504202, 1.5148}, {0.0882353, 
  1.41405}, {0.130252, 1.70937}, {0.172269, 2.029}, {0.176471, 
  1.7858}, {0.222689, 2.20272}, {0.226891, 2.23746}, {0.231092, 
  2.23746}, {0.239496, 1.96647}, {0.268908, 1.94562}, {0.273109, 
  1.91088}, {0.277311, 1.91088}, {0.281513, 1.94562}, {0.294118, 
  2.2861}, {0.319328, 2.26526}, {0.327731, 2.3}, {0.432773, 
  1.68157}, {0.462185, 1.86918}, {0.5, 2.00121}} *)

ListPlot[extractedData, PlotRange -> All, PlotTheme -> "Detailed"]
enter image description here

enter image description here

Apply QRMon

Load packages. (For more details see [4].)

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

Apply the QRMon workflow within the TraceMonad:

TraceMonadUnit[QRMonUnit[extractedData]]⟹"lift data to the monad"⟹
  QRMonEchoDataSummary⟹"echo data summary"⟹
  QRMonQuantileRegression[12, 0.5]⟹"do Quantile Regression with\nB-spline basis with 12 knots"⟹
  QRMonPlot⟹"plot the data and regression curve"⟹
  QRMonEcho[Style["Tabulate QRMon steps and explanations:", Purple, Bold]]⟹"echo an explanation message"⟹
  TraceMonadEchoGrid;
enter image description here

enter image description here

References

[1] Moody Hadi and Danny Haydon, "A Perspective On Machine Learning In Credit Risk", (2018), S&P Global Market Intelligence.

[2] Andy Ross, answer of "Recovering data points from an image", (2012).

[3] Anton Antonov, "A monad for Quantile Regression workflows", (2018), MathematicaForPrediction at WordPress.

[4] Anton Antonov, "Monad code generation and extension", (2017), MathematicaForPrediction at GitHub*,https://github.com/antononcube/MathematicaForPrediction.

A monad for Quantile Regression workflows

Introduction

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

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

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

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

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

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

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

QRMon-introduction-monad-pipeline-example-table

QRMon-introduction-monad-pipeline-example-table

QRMon-introduction-monad-pipeline-example-echo

QRMon-introduction-monad-pipeline-example-echo

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

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

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

Contents description

The document has the following structure.

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

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

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

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

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

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

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

Contents description

The document has the following structure.

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

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

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

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

Package load

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

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

Data load

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

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

Distribution data

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

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

(* 601 *)

QRMonUnit[distData]⟹QRMonEchoDataSummary⟹QRMonPlot;
QRMon-distData

QRMon-distData

Temperature time series

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

QRMonUnit[tsData]⟹QRMonEchoDataSummary⟹QRMonDateListPlot;
QRMon-tsData

QRMon-tsData

Financial time series

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

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

QRMonUnit[finData]⟹QRMonEchoDataSummary⟹QRMonDateListPlot;
QRMon-finData

QRMon-finData

Design considerations

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

  1. Retrieving data from a data repository.

  2. Optionally, transform the data.

    1. Delete rows with missing fields.

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

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

  3. Verify assumptions of the data.

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

    1. Quantile Regression, or

    2. Least Squares Regression.

  5. Visualize the data and regression functions.

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

  7. Utilize the found regression functions to compute:

    1. outliers,

    2. local extrema,

    3. approximation or fitting errors,

    4. conditional density distributions,

    5. time series simulations.

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

Quantile-regression-workflow-extended

Quantile-regression-workflow-extended

In order to address:

  • the introduction of new elements in regression workflows,

  • workflows elements variability, and

  • workflows iterative changes and refining,

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

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

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

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

Monad design

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

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

QRMon-formula-1

QRMon-formula-1

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

QRMon-formula-2

QRMon-formula-2

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

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

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

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

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

Let us list the desired properties of the monad.

  • Rapid specification of non-trivial quantile regression workflows.

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

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

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

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

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

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

The QRMon components and their interactions are fairly simple.

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

  • (time series) data,

  • regression functions,

  • outliers and outlier regression functions.

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

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

QRMon overview

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

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

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

QRMon-pipeline

QRMon-pipeline

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

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

Here is the output of the pipeline:

The QRMon functions are separated into four groups:

  • operations,

  • setters and droppers,

  • takers,

  • 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 QRMon operations.

Monad functions interaction with the pipeline value and context

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

QRMon-monad-functions-overview-table

QRMon-monad-functions-overview-table

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

QRMon-monad-functions-shortcuts-table

QRMon-monad-functions-shortcuts-table

State monad functions

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

QRMon-StMon-functions-overview-table

QRMon-StMon-functions-overview-table

Monad elements

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

The monad head

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

QRMon[{{1, 223}, {2, 323}}, <||>]⟹QRMonEchoDataSummary;
The-monad-head-output

The-monad-head-output

Lifting data to the monad

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

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

QRMonUnit[distData]⟹QRMonEchoDataSummary;
Lifting-data-to-the-monad-output

Lifting-data-to-the-monad-output

QRMonUnit[]⟹QRMonSetData[distData]⟹QRMonEchoDataSummary;
Lifting-data-to-the-monad-output

Lifting-data-to-the-monad-output

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

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

  • time series,

  • numerical vectors,

  • numerical matrices of rank two.

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

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

Quantile regression with B-splines

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

QRMonUnit[distData]⟹
  QRMonQuantileRegression[12]⟹
  QRMonPlot;
Quantile-regression-with-B-splines-output-1

Quantile-regression-with-B-splines-output-1

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

Options[QRMonQuantileRegression]

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

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

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

Quantile-regression-with-B-splines-output-2

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

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

ListPlot[# /@ distData[[All, 1]]] & /@ (p⟹QRMonTakeRegressionFunctions)
Quantile-regression-with-B-splines-output-3

Quantile-regression-with-B-splines-output-3

Quantile regression fit and Least squares fit

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

Here is a basis functions:

bFuncs = Table[PDF[NormalDistribution[m, 1], x], {m, Min[distData[[All, 1]]], Max[distData[[All, 1]]], 1}];
Plot[bFuncs, {x, Min[distData[[All, 1]]], Max[distData[[All, 1]]]}, 
 PlotRange -> All, PlotTheme -> "Scientific"]
Quantile-regression-fit-and-Least-squares-fit-basis

Quantile-regression-fit-and-Least-squares-fit-basis

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

p =
  QRMonUnit[distData]⟹
   QRMonQuantileRegressionFit[bFuncs]⟹
   QRMonLeastSquaresFit[bFuncs]⟹
   QRMonPlot;
   
Quantile-regression-fit-and-Least-squares-fit-output-1

Quantile-regression-fit-and-Least-squares-fit-output-1

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

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

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

ListPlot[# /@ distData[[All, 1]]] & /@ (p⟹QRMonTakeRegressionFunctions)
Quantile-regression-fit-and-Least-squares-fit-output-2

Quantile-regression-fit-and-Least-squares-fit-output-2

Default basis to fit (using Chebyshev polynomials)

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

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

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

QRMonUnit[distData]⟹
  QRMonQuantileRegressionFit[12]⟹
  QRMonLeastSquaresFit[12]⟹
  QRMonPlot;
Default-basis-to-fit-output-1-and-2

Default-basis-to-fit-output-1-and-2

The code above is equivalent to the following code:

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

p =
  QRMonUnit[distData]⟹
   QRMonQuantileRegressionFit[bfuncs]⟹
   QRMonLeastSquaresFit[bfuncs]⟹
   QRMonPlot;
Default-basis-to-fit-output-1-and-2

Default-basis-to-fit-output-1-and-2

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

QRMonUnit[distData]⟹
  QRMonQuantileRegressionFit[{4, {-1, 1}}]⟹
  QRMonPlot;
Default-basis-to-fit-output-3

Default-basis-to-fit-output-3

Regression functions evaluation

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

Evaluation for a given value of the independent variable:

p⟹QRMonEvaluate[0.12]⟹QRMonTakeValue

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

Evaluation for a vector of values:

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

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

Evaluation for complicated lists of numbers:

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

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

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

Errors and error plots

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

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

QRMonUnit[finData]⟹
  QRMonQuantileRegression[10, {0.5, 0.1}]⟹
  QRMonDateListPlot[Joined -> False]⟹
  QRMonErrorPlots["DateListPlot" -> True, Joined -> False]⟹
  QRMonErrors⟹
  QRMonEchoFunctionValue["Errors summary:", RecordsSummary[#[[All, 2]]] & /@ # &];
Errors-and-error-plots-output-1

Errors-and-error-plots-output-1

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

Finding outliers

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

Here is an example:

p =
  QRMonUnit[distData]⟹
   QRMonQuantileRegression[6, {0.02, 0.98}]⟹
   QRMonOutliers⟹
   QRMonEchoValue⟹
   QRMonOutliersPlot;
Finding-outliers-output-1

Finding-outliers-output-1

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

Keys[p⟹QRMonTakeContext]

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

Here are the corresponding quantiles of the plot above:

Keys[p⟹QRMonTakeOutlierRegressionFunctions]

(* {0.02, 0.98} *)

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

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

Here is an example for finding only the top outliers:

QRMonUnit[finData]⟹
  QRMonQuantileRegression[5, 0.8]⟹
  QRMonOutliers⟹
  QRMonEchoFunctionContext["outlier quantiles:", Keys[#outlierRegressionFunctions] &]⟹
  QRMonOutliersPlot["DateListPlot" -> True];
  
Finding-outliers-output-2

Finding-outliers-output-2

Plotting outliers

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

Here are the options of QRMonOutliersPlot:

Options[QRMonOutliersPlot]

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

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

QRMonOutliersPlot utilizes combines with Show two plots:

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

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

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

QRMonUnit[tsData]⟹
 QRMonQuantileRegression[12, {0.01, 0.99}]⟹
 QRMonOutliersPlot[
  "Echo" -> False,
  "DateListPlot" -> True,
  ListPlot -> {PlotStyle -> {Green, {PointSize[0.02], 
       Red}, {PointSize[0.02], Blue}}, Joined -> False, 
    PlotTheme -> "Grid"},
  Plot -> {PlotStyle -> Orange}]⟹
 QRMonTakeValue
 
Plotting-outliers-output-2

Plotting-outliers-output-2

Estimating conditional distributions

Consider the following problem:

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

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

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

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

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

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

Estimating-conditional-distributions-output-1

Moving average, moving median, and moving map

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

  • visualize the obtained transformed data,

  • do regression over the transformed data,

  • compare with regression curves over the original data.

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

Here is an example:

QRMonUnit[tsData]⟹
  QRMonDateListPlot[ImageSize -> Small]⟹
  QRMonMovingAverage[20]⟹
  QRMonEchoFunctionValue["Moving avg: ", DateListPlot[#, ImageSize -> Small] &]⟹
  QRMonMovingMap[Mean, Quantity[20, "Days"]]⟹
  QRMonEchoFunctionValue["Moving map: ", DateListPlot[#, ImageSize -> Small] &];
Moving-average-moving-median-and-moving-map-output-1

Moving-average-moving-median-and-moving-map-output-1

Dependent variable simulation

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

More formally,

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

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

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

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

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

tsNew =
  p⟹
   QRMonSimulate[1000]⟹
   QRMonTakeValue;

opts = {ImageSize -> Medium, PlotTheme -> "Detailed"};
GraphicsGrid[{{DateListPlot[tsData, PlotLabel -> "Actual", opts],
    DateListPlot[tsNew, PlotLabel -> "Simulated", opts]}}]
Dependent-variable-simulation-output-1

Dependent-variable-simulation-output-1

Finding local extrema in noisy data

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

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

  1. Fit a polynomial through the data.

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

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

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

An example of finding local extrema follows.

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

Finding-local-extrema-in-noisy-data-output-1

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

Setters, droppers, and takers

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

For example:

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

p⟹QRMonTakeRegressionFunctions

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

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

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

(p⟹QRMonTakeContext)["data"]

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

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

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

p⟹QRMonDropData⟹QRMonTakeContext

(* <||> *)

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

Unit tests

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

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

Directly specified tests

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

AbsoluteTiming[
 testObject = TestReport["~/MathematicaForPrediction/UnitTests/MonadicQuantileRegression-Unit-Tests.wlt"]
]
Unit-tests-output-1

Unit-tests-output-1

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

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

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

Random pipelines tests

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

Generate pipelines:

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

(* 100 *)

Here is a sample of the generated pipelines:

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

Unit-tests-random-pipelines-sample

Here we run the pipelines as unit tests:

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

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

rpTRObj = TestReport[res]

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

Future plans

Workflow operations

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

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

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

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

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

Conversational agent

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

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

QRMonUnit[distData]⟹
  ToQRMonPipelineFunction["show data summary"]⟹
  ToQRMonPipelineFunction["calculate quantile regression for quantiles 0.2, 0.8 and with 40 knots"]⟹
  ToQRMonPipelineFunction["plot"];
Future-plans-conversational-agent-output-1

Future-plans-conversational-agent-output-1

Implementation notes

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

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

References

Packages

[AAp1] Anton Antonov, State monad code generator Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/StateMonadCodeGenerator.m .

[AAp2] Anton Antonov, Monadic tracing Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicTracing.m .

[AAp3] Anton Antonov, Monadic Quantile Regression Mathematica package, (2018), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicQuantileRegression.m.

[AAp4] Anton Antonov, Quantile regression Mathematica package, (2014), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/QuantileRegression.m .

[AAp5] Anton Antonov, Monadic contextual classification Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicContextualClassification.m .

[AAp6] Anton Antonov, MathematicaForPrediction utilities, (2014), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MathematicaForPredictionUtilities.m .

[AAp7] Anton Antonov, Monadic Quantile Regression unit tests, (2018), MathematicaVsR at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/UnitTests/MonadicQuantileRegression-Unit-Tests.wlt .

[AAp8] Anton Antonov, Monadic Quantile Regression random pipelines Mathematica unit tests, (2018), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/UnitTests/MonadicQuantileRegressionRandomPipelinesUnitTests.m .

[AAp9] Anton Antonov, Monadic contextual classification Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicContextualClassification.m .

ConverationalAgents Packages

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

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

MathematicaForPrediction articles

[AA1] Anton Antonov, "Monad code generation and extension", (2017), MathematicaForPrediction at GitHub, https://github.com/antononcube/MathematicaForPrediction.

[AA2] Anton Antonov, "Quantile regression through linear programming", (2013), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2013/12/16/quantile-regression-through-linear-programming/ .

[AA3] Anton Antonov, "Quantile regression with B-splines", (2014), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2014/01/01/quantile-regression-with-b-splines/ .

[AA4] Anton Antonov, "Estimation of conditional density distributions", (2014), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2014/01/13/estimation-of-conditional-density-distributions/ .

[AA5] Anton Antonov, "Finding local extrema in noisy data using Quantile Regression", (2015), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2015/09/27/finding-local-extrema-in-noisy-data-using-quantile-regression/ .

[AA6] Anton Antonov, "A monad for classification workflows", (2018), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2018/05/15/a-monad-for-classification-workflows/ .

Other

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

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

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

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

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

[RG1] Roger Koenker, Quantile Regression, ‪Cambridge University Press, 2005‬.

Mathematica-vs-R: Deep learning examples

Introduction

This MathematicaVsR at GitHub project is for the comparison of the Deep Learning functionalities in R/RStudio and Mathematica/Wolfram Language (WL).

The project is aimed to mirror and aid the talk "Deep Learning series (session 2)" of the meetup Orlando Machine Learning and Data Science.

The focus of the talk is R and Keras, so the project structure is strongly influenced by the content of the book Deep learning with R, [1], and the corresponding Rmd notebooks, [2].

Some of Mathematica’s notebooks repeat the material in [2]. Some are original versions.

WL’s Neural Nets framework and abilities are fairly well described in the reference page "Neural Networks in the Wolfram Language overview", [4], and the webinar talks [5].

The corresponding documentation pages [3] (R) and [6] (WL) can be used for a very fruitful comparison of features and abilities.

Remark: With "deep learning with R" here we mean "Keras with R".

Remark: An alternative to R/Keras and Mathematica/MXNet is the library H2O (that has interfaces to Java, Python, R, Scala.) See project’s directory R.H2O for examples.

The presentation

The big picture

Deep learning can be used for both supervised and unsupervised learning. In this project we concentrate on supervised learning.

The following diagram outlines the general, simple classification workflow we have in mind.

simple_classification_workflow

Here is a corresponding classification monadic pipeline in Mathematica:

monadic_pipeline

monadic_pipeline

Code samples

R-Keras uses monadic pipelines through the library magrittr. For example:

model <- keras_model_sequential() 
model %>% 
  layer_dense(units = 256, activation = 'relu', input_shape = c(784)) %>% 
  layer_dropout(rate = 0.4) %>% 
  layer_dense(units = 128, activation = 'relu') %>%
  layer_dropout(rate = 0.3) %>%
  layer_dense(units = 10, activation = 'softmax')

The corresponding Mathematica command is:

model =
 NetChain[{
   LinearLayer[256, "Input" -> 784],
   ElementwiseLayer[Ramp],            
   DropoutLayer[0.4],
   LinearLayer[128],
   ElementwiseLayer[Ramp],            
   DropoutLayer[0.3],
   LinearLayer[10]
 }]

Comparison

Installation

  • Mathematica

  • The neural networks framework comes with Mathematica. (No additional installation required.)

  • R

  • Pretty straightforward using the directions in [3]. (A short list.)

  • Some additional Python installation is required.

Simple neural network classifier over MNIST data

Vector classification

TBD…

Categorical classification

TBD…

Regression

Encoders and decoders

The Mathematica encoders (for neural networks and generally for machine learning tasks) are very well designed and with a very advanced development.

The encoders in R-Keras are fairly useful but not was advanced as those in Mathematica.

[TBD: Encoder correspondence…]

Dealing with over-fitting

Repositories of pre-trained models

Documentation

References

[1] F. Chollet, J. J. Allaire, Deep learning with R, (2018).

[2] J. J. Allaire, Deep Learing with R notebooks, (2018), GitHub.

[3] RStudio, Keras reference.

[4] Wolfram Research, "Neural Networks in the Wolfram Language overview".

[5] Wolfram Research, "Machine Learning Webinar Series".

[6] Wolfram Research, "Neural Networks guide".

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:

"ClCon-simple-dsTitanic-pipeline"

"ClCon-simple-dsTitanic-pipeline"

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.

Package load

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/MathematicaForPrediction/master/MonadicProgramming/MonadicTracing.m"]
Import["https://raw.githubusercontent.com/antononcube/MathematicaVsR/master/Projects/ProgressiveMachineLearning/Mathematica/GetMachineLearningDataset.m"]
Import["https://raw.githubusercontent.com/antononcube/MathematicaForPrediction/master/UnitTests/MonadicContextualClassificationRandomPipelinesUnitTests.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
*)

Data load

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"}, #] &]]
"ClCon-datasets-dimensions"

"ClCon-datasets-dimensions"

Here is the summary of dsTitanic:

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

"ClCon-dsTitanic-summary"

Here is the summary of dsMushroom in long form:

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

"ClCon-dsMushroom-summary"

Here is the summary of dsWineQuality in long form:

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

"ClCon-dsWineQuality-summary"

"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]
"ClCon-quick-data-sample"

"ClCon-quick-data-sample"

Here is a summary of the data:

ClConUnit[dsData]⟹ClConSummarizeData;
"ClCon-quick-data-summary-ds"

"ClCon-quick-data-summary-ds"

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

mlrData = ClConToNormalClassifierData[dsData];
ClConUnit[mlrData]⟹ClConSummarizeData;
"ClCon-quick-data-summary-mlr"

"ClCon-quick-data-summary-mlr"

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.

"Classification-workflow-horizontal-layout"

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.

"Making-competitions-classifiers-mind-map.png"

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.

"Classification-workflow-extended.jpg"

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.

Monad design

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-generic-monad-formula"

"ClCon-generic-monad-formula"

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

"ClCon-State-monad-formula"

"ClCon-State-monad-formula"

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.)

"ClCon-components-interaction.jpg"

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:

"ClCon-pipeline"

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.

"ClCon-pipeline-TraceMonad-table"

"ClCon-pipeline-TraceMonad-table"

Here is the output of the pipeline:

"ClCon-pipeline-TraceMonad-Echo-output"

"ClCon-pipeline-TraceMonad-Echo-output"

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.

"ClCon-table-of-operations-setters-takers"

"ClCon-table-of-operations-setters-takers"

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]):

"ClCon-StateMonad-functions-table"

"ClCon-StateMonad-functions-table"

Monad elements

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

The monad head

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;
"ClCon-monad-head-example"

"ClCon-monad-head-example"

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;
"ClCon-lifting-data-example-1"

"ClCon-lifting-data-example-1"

ClConUnit[]⟹ClConSetTrainingData[dsData]⟹ClConSummarizeData;
"ClCon-lifting-data-example-2"

"ClCon-lifting-data-example-2"

(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
"ClCon-ensemble-classifier-example-1"

"ClCon-ensemble-classifier-example-1"

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
"ClCon-ensemble-classifier-example-2"

"ClCon-ensemble-classifier-example-2"

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;
"ClCon-classifier-testing-ConfusionMatrixPlot-echo"

"ClCon-classifier-testing-ConfusionMatrixPlot-echo"

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];
"ClCon-classifier-testing-ROCPlot-echo"

"ClCon-classifier-testing-ROCPlot-echo"

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"}];
ClCon-classifier-testing-ROCListLinePlot-echo

ClCon-classifier-testing-ROCListLinePlot-echo

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;
"ClCon-classifier-testing-ROCListLinePlot-survived-echo"

"ClCon-classifier-testing-ROCListLinePlot-survived-echo"

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;
"ClCon-MNIST-example-output"

"ClCon-MNIST-example-output"

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 &];
"ClCon-conditional-continuation-example-output"

"ClCon-conditional-continuation-example-output"

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 *);
"ClCon-best-thresholds-example-output"

"ClCon-best-thresholds-example-output"

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[
 testObject = TestReport["~/MathematicaForPrediction/UnitTests/MonadicContextualClassification-Unit-Tests.wlt"]
]
"ClCon-direct-unit-tests-TestReport-icon"

"ClCon-direct-unit-tests-TestReport-icon"

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], 
  TableHeadings -> {"pipeline"}]
]
AutoCollapse[]
"ClCon-random-pipelines-tests-sample-table"

"ClCon-random-pipelines-tests-sample-table"

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]
"ClCon-random-pipelines-TestReport-icon"

"ClCon-random-pipelines-TestReport-icon"

(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;
"ClCon-dimension-reduction-example-echo"

"ClCon-dimension-reduction-example-echo"

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.

"ClCon-development-cycle"

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

[AAp1] Anton Antonov, State monad code generator Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/StateMonadCodeGenerator.m .

[AAp2] Anton Antonov, Monadic tracing Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicTracing.m .

[AAp3] Anton Antonov, Monadic contextual classification Mathematica package, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MonadicProgramming/MonadicContextualClassification.m .

[AAp4] Anton Antonov, Classifier ensembles functions Mathematica package, (2016), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/ClassifierEnsembles.m .

[AAp5] Anton Antonov, Receiver operating characteristic functions Mathematica package, (2016), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/ROCFunctions.m .

[AAp6] Anton Antonov, Variable importance determination by classifiers implementation in Mathematica,(2015), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/VariableImportanceByClassifiers.m .

[AAp7] Anton Antonov, MathematicaForPrediction utilities, (2014), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MathematicaForPredictionUtilities.m .

[AAp8] Anton Antonov, Cross tabulation implementation in Mathematica, (2017), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/CrossTabulate.m .

[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.

[AAp11] Anton Antonov, Monadic contextual classification Mathematica unit tests, (2018), MathematicaVsR at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/UnitTests/MonadicContextualClassification-Unit-Tests.wlt .

[AAp12] Anton Antonov, Monadic contextual classification random pipelines Mathematica unit tests, (2018), MathematicaVsR at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/UnitTests/MonadicContextualClassificationRandomPipelinesUnitTests.m .

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.

[AA2] Anton Antonov, "ROC for classifier ensembles, bootstrapping, damaging, and interpolation", (2016), MathematicaForPrediction at WordPress. URL: https://mathematicaforprediction.wordpress.com/2016/10/15/roc-for-classifier-ensembles-bootstrapping-damaging-and-interpolation/ .

[AA3] Anton Antonov, "Importance of variables investigation guide", (2016), MathematicaForPrediction at GitHub. URL: https://github.com/antononcube/MathematicaForPrediction/blob/master/MarkdownDocuments/Importance-of-variables-investigation-guide.md .

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 .

Progressive Machine Learning Examples

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.

"Progressive-machine-learning-with-Tries"

For detailed explanations see any of the notebooks.

Project organization

Mathematica files

R files

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]:

"PLA-Trie-run"

Here are the obtained ROC curves:

"PLA-Trie-ROCs-thresholds"

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]:

"PLA-SMR-run"

Here are the obtained ROC curves:

"PLA-SMR-ROCs-thresholds"

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:

"Artificial Intelligence and Machine Learning application in finance and technology",

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.

"AI-winters-and-winter-episodes"

The ingredients of coming to wrong ideas about AI

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

"AI-misconseptions-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.)

"Two-ways-to-fish"

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".

"AI and Machine Learning application in technology"