Wolfram Live-Coding Series on Quantile Regression workflows

A month or so ago I was invited to make Quantile Regression presentations at the Wolfram Research Twitch channel.

The live-coding sessions

  1. In the first
    live-streaming / live-coding session
    I demonstrated how to make
    Quantile Regression
    workflows using the software monad QRMon
    and some of the underlying software design principles. (Namely
    “monadic programming”.)
  2. In the follow up live-coding session I discussed topics like outliers removal (data cleaning), anomaly detection, and structural breaks.
  3. In the third live-coding session:
    • First, we demonstrate and explain how to do QR-based time series simulations and their applications in Operations Research.
    • Next, we discuss QR in 2D and 3D and a related application.
  4. In the fourth live-coding session we discussed the following the topics.
    • Brief review of previous sessions.
    • Proclaiming the upcoming ResourceFunction["QuantileRegression"].
    • Predict tomorrow from today’s data.
    • Using NLP techniques on time series.
    • Generation of QR workflows with natural language commands.

Notebooks

  • The notebook of the 2nd session is also attached. (I added a “References” section to it.)

Update (2019-11-13)

A few days ago the Wolfram Function Repository entry QuantileRegression was approved. The resource description page has many of the topics discussed in the live-coding sessions on Quantile regression.

Finding all structural breaks in time series

Introduction

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

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

What is a structural break?

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

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

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

Example

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

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

Load packages

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

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

Data used

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

Illustration data from Wikipedia

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

data = {{0.08, 0.34}, {0.16, 0.55}, {0.24, 0.54}, {0.32, 0.77}, {0.4, 
    0.77}, {0.48, 1.2}, {0.56, 0.57}, {0.64, 1.3}, {0.72, 1.}, {0.8, 
    1.3}, {0.88, 1.2}, {0.96, 0.88}, {1., 1.2}, {1.1, 1.3}, {1.2, 
    1.3}, {1.3, 1.4}, {1.4, 1.5}, {1.4, 1.5}, {1.5, 1.5}, {1.6, 
    1.6}, {1.7, 1.1}, {1.8, 0.98}, {1.8, 1.1}, {1.9, 1.4}, {2., 
    1.3}, {2.1, 1.5}, {2.2, 1.3}, {2.2, 1.3}, {2.3, 1.2}, {2.4, 
    1.1}, {2.5, 1.1}, {2.6, 1.2}, {2.6, 1.4}, {2.7, 1.3}, {2.8, 
    1.6}, {2.9, 1.5}, {3., 1.4}, {3., 1.8}, {3.1, 1.4}, {3.2, 
    1.4}, {3.3, 1.4}, {3.4, 2.}, {3.4, 2.}, {3.5, 1.5}, {3.6, 
    1.8}, {3.7, 2.1}, {3.8, 1.6}, {3.8, 1.8}, {3.9, 1.9}, {4., 2.1}};
ListPlot[data]
DataUsedWk2
DataUsedWk2

S&P 500 Index

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

tsSP500 = FinancialData[Entity["Financial", "^SPX"], {{2015, 1, 1}, Date[]}]
DateListPlot[tsSP500, ImageSize -> Medium]
DataUsedSP500
DataUsedSP500

Application of Chow Test

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

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

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

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

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

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

Here is an example of an automatic values call.

chowStats2 = QRMonUnit[data]⟹QRMonChowTestStatistic⟹QRMonTakeValue;
ListPlot[chowStats2, GridLines -> {
Part[
Part[chowStats2, All, 1], 
OutlierIdentifiers`OutlierPosition[
Part[chowStats2, All, 2],  OutlierIdentifiers`SPLUSQuartileIdentifierParameters]], None}, GridLinesStyle -> Directive[{Orange, Dashed}], Filling -> Axis]
ApplicationOfChowTestchowStats2
ApplicationOfChowTestchowStats2

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

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

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

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

chowStats3 = QRMonUnit[tsSP500]⟹QRMonChowTestStatistic⟹QRMonTakeValue;
DateListPlot[chowStats3, Joined -> False, Filling -> Axis]
ApplicationOfChowTestSP500
ApplicationOfChowTestSP500
OutlierPosition[chowStats3[[All, 2]], SPLUSQuartileIdentifierParameters]
(* {} *)
 
OutlierPosition[chowStats3[[All, 2]], HampelIdentifierParameters]
(* {} *)

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

Finding all structural break points

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

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

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

Computation

Chose fitting functions

fitFuncs = {1, x};

Find Chow test statistics local maxima

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

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

Find most significant structural break point

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

Plot structural breaks splits and corresponding fittings

Here we just make the plots without showing them.

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

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

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

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

Future plans

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

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

FuturePlansTimeSeriesStates
FuturePlansTimeSeriesStates

References

Articles

[Wk1] Wikipedia entry, Structural breaks.

[Wk2] Wikipedia entry, Chow test.

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

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

Packages

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

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

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

Videos

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

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 .

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

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

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

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

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".

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 .

ConversationalAgents Packages

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

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

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 .

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

The Great conversation in USA presidential speeches

Introduction

This document shows a way to chart in Mathematica / WL the evolution of topics in collections of texts. The making of this document (and related code) is primarily motivated by the fascinating concept of the Great Conversation, [Wk1, MA1]. In brief, all western civilization books are based on 103 great ideas; if we find the great ideas each significant book is based on we can construct a time-line (spanning centuries) of the great conversation between the authors; see [MA1, MA2, MA3].

Instead of finding the great ideas in a text collection we extract topics statistically, using dimension reduction with Non-Negative Matrix Factorization (NNMF), [AAp3, AA1, AA2].

The presented computational results are based on the text collections of State of the Union speeches of USA presidents [D2]. The code in this document can be easily configured to use the much smaller text collection [D1] available online and in Mathematica/WL. (The collection [D1] is fairly small, 51 documents; the collection [D2] is much larger, 2453 documents.)

The procedures (and code) described in this document, of course, work on other types of text collections. For example: movie reviews, podcasts, editorial articles of a magazine, etc.

A secondary objective of this document is to illustrate the use of the monadic programming pipeline as a Software design pattern, [AA3]. In order to make the code concise in this document I wrote the package MonadicLatentSemanticAnalysis.m, [AAp5]. Compare with the code given in [AA1].

The very first version of this document was written for the 2017 summer course “Data Science for the Humanities” at the University of Oxford, UK.

Outline of the procedure applied

The procedure described in this document has the following steps.

  1. Get a collection of documents with known dates of publishing.
    • Or other types of tags associated with the documents.
  2. Do preliminary analysis of the document collection.
    • Number of documents; number of unique words.

    • Number of words per document; number of documents per word.

    • (Some of the statistics of this step are done easier after the Linear vector space representation step.)

  3. Optionally perform Natural Language Processing (NLP) tasks.

    1. Obtain or derive stop words.

    2. Remove stop words from the texts.

    3. Apply stemming to the words in the texts.

  4. Linear vector space representation.

    • This means that we represent the collection with a document-word matrix.

    • Each unique word is a basis vector in that space.

    • For each document the corresponding point in that space is derived from the number of appearances of document’s words.

  5. Extract topics.

    • In this document NNMF is used.

    • In order to obtain better results with NNMF some experimentation and refinements of the topics search have to be done.

  6. Map the documents over the extracted topics.

    • The original matrix of the vector space representation is replaced with a matrix with columns representing topics (instead of words.)
  7. Order the topics according to their presence across the years (or other related tags).
    • This can be done with hierarchical clustering.

    • Alternatively,

      1. for a given topic find the weighted mean of the years of the documents that have that topic, and

      2. order the topics according to those mean values.

  8. Visualize the evolution of the documents according to their topics.

    1. This can be done by simply finding the contingency matrix year vs topic.

    2. For the president speeches we can use the president names for time-line temporal axis instead of years.

      • Because the corresponding time intervals of president office occupation do not overlap.

Remark: Some of the functions used in this document combine several steps into one function call (with corresponding parameters.)

Packages

This loads the packages [AAp1-AAp8]:

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

(Note that some of the packages that are imported automatically by [AAp5].)

The functions of the central package in this document, [AAp5], have the prefix “LSAMon”. Here is a sample of those names:

Short@Names["LSAMon*"]

(* {"LSAMon", "LSAMonAddToContext", "LSAMonApplyTermWeightFunctions", <>, "LSAMonUnit", "LSAMonUnitQ", "LSAMonWhen"} *)

Data load

In this section we load a text collection from a specified source.

The text collection from “Presidential Nomination Acceptance Speeches”, [D1], is small and can be used for multiple code verifications and re-runnings. The “State of Union addresses of USA presidents” text collection from [D2] was converted to a Mathematica/WL object by Christopher Wolfram (and sent to me in a private communication.) The text collection [D2] provides far more interesting results (and they are shown below.)

If[True,
  speeches = ResourceData[ResourceObject["Presidential Nomination Acceptance Speeches"]];
  names = StringSplit[Normal[speeches[[All, "Person"]]][[All, 2]], "::"][[All, 1]],

  (*ELSE*)
  (*State of the union addresses provided by Christopher Wolfram. *)      
  Get["~/MathFiles/Digital humanities/Presidential speeches/speeches.mx"];
  names = Normal[speeches[[All, "Name"]]];
];

dates = Normal[speeches[[All, "Date"]]];
texts = Normal[speeches[[All, "Text"]]];

Dimensions[speeches]

(* {2453, 4} *)

Basic statistics for the texts

Using different contingency matrices we can derive basic statistical information about the document collection. (The document-word matrix is a contingency matrix.)

First we convert the text data in long-form:

docWordRecords = 
  Join @@ MapThread[
    Thread[{##}] &, {Range@Length@texts, names, 
     DateString[#, {"Year"}] & /@ dates, 
     DeleteStopwords@*TextWords /@ ToLowerCase[texts]}, 1];

Here is a sample of the rows of the long-form:

GridTableForm[RandomSample[docWordRecords, 6], 
 TableHeadings -> {"document index", "name", "year", "word"}]

Here is a summary:

Multicolumn[
 RecordsSummary[docWordRecords, {"document index", "name", "year", "word"}, "MaxTallies" -> 8], 4, Dividers -> All, Alignment -> Top]

Using the long form we can compute the document-word matrix:

ctMat = CrossTabulate[docWordRecords[[All, {1, -1}]]];
MatrixPlot[Transpose@Sort@Map[# &, Transpose[ctMat@"XTABMatrix"]], 
 MaxPlotPoints -> 300, ImageSize -> 800, 
 AspectRatio -> 1/3]

Here is the president-word matrix:

ctMat = CrossTabulate[docWordRecords[[All, {2, -1}]]];
MatrixPlot[Transpose@Sort@Map[# &, Transpose[ctMat@"XTABMatrix"]], MaxPlotPoints -> 300, ImageSize -> 800, AspectRatio -> 1/3]

Here is an alternative way to compute text collection statistics through the document-word matrix computed within the monad LSAMon:

LSAMonUnit[texts]⟹LSAMonEchoTextCollectionStatistics[];

Procedure application

Stop words

Here is one way to obtain stop words:

stopWords = Complement[DictionaryLookup["*"], DeleteStopwords[DictionaryLookup["*"]]];
Length[stopWords]
RandomSample[stopWords, 12]

(* 304 *)

(* {"has", "almost", "next", "WHO", "seeming", "together", "rather", "runners-up", "there's", "across", "cannot", "me"} *)

We can complete this list with additional stop words derived from the collection itself. (Not done here.)

Linear vector space representation and dimension reduction

Remark: In the rest of the document we use “term” to mean “word” or “stemmed word”.

The following code makes a document-term matrix from the document collection, exaggerates the representations of the terms using “TF-IDF”, and then does topic extraction through dimension reduction. The dimension reduction is done with NNMF; see [AAp3, AA1, AA2].

SeedRandom[312]

mObj =
  LSAMonUnit[texts]⟹
   LSAMonMakeDocumentTermMatrix[{}, stopWords]⟹
   LSAMonApplyTermWeightFunctions[]⟹
   LSAMonTopicExtraction[Max[5, Ceiling[Length[texts]/100]], 60, 12, "MaxSteps" -> 6, "PrintProfilingInfo" -> True];

This table shows the pipeline commands above with comments:

Detailed description

The monad object mObj has a context of named values that is an Association with the following keys:

Keys[mObj⟹LSAMonTakeContext]

(* {"texts", "docTermMat", "terms", "wDocTermMat", "W", "H", "topicColumnPositions", "automaticTopicNames"} *)

Let us clarify the values by briefly describing the computational steps.

  1. From texts we derive the document-term matrix \text{docTermMat}\in \mathbb{R}^{m \times n}, where n is the number of documents and m is the number of terms.
    • The terms are words or stemmed words.

    • This is done with LSAMonMakeDocumentTermMatrix.

  2. From docTermMat is derived the (weighted) matrix wDocTermMat using “TF-IDF”.

    • This is done with LSAMonApplyTermWeightFunctions.
  3. Using docTermMat we find the terms that are present in sufficiently large number of documents and their column indices are assigned to topicColumnPositions.

  4. Matrix factorization.

    1. Assign to \text{wDocTermMat}[[\text{All},\text{topicsColumnPositions}]], \text{wDocTermMat}[[\text{All},\text{topicsColumnPositions}]]\in \mathbb{R}^{m_1 \times n}, where m_1 = |topicsColumnPositions|.

    2. Compute using NNMF the factorization \text{wDocTermMat}[[\text{All},\text{topicsColumnPositions}]]\approx H W, where W\in \mathbb{R}^{k \times n}, H\in \mathbb{R}^{k \times m_1}, and k is the number of topics.

    3. The values for the keys “W, “H”, and “topicColumnPositions” are computed and assigned by LSAMonTopicExtraction.

  5. From the top terms of each topic are derived automatic topic names and assigned to the key automaticTopicNames in the monad context.

    • Also done by LSAMonTopicExtraction.

Statistical thesaurus

At this point in the object mObj we have the factors of NNMF. Using those factors we can find a statistical thesaurus for a given set of words. The following code calculates such a thesaurus, and echoes it in a tabulated form.

queryWords = {"arms", "banking", "economy", "education", "freedom", 
   "tariff", "welfare", "disarmament", "health", "police"};

mObj⟹
  LSAMonStatisticalThesaurus[queryWords, 12]⟹
  LSAMonEchoStatisticalThesaurus[];

By observing the thesaurus entries we can see that the words in each entry are semantically related.

Note, that the word “welfare” strongly associates with “[applause]”. The rest of the query words do not, which can be seen by examining larger thesaurus entries:

thRes =
  mObj⟹
   LSAMonStatisticalThesaurus[queryWords, 100]⟹
   LSAMonTakeValue;
Cases[thRes, "[applause]", Infinity]

(* {"[applause]", "[applause]"} *)

The second “[applause]” associated word is “education”.

Detailed description

The statistical thesaurus is computed by using the NNMF’s right factor H.

For a given term, its corresponding column in H is found and the nearest neighbors of that column are found in the space \mathbb{R}^{m_1} using Euclidean norm.

Extracted topics

The topics are the rows of the right factor H of the factorization obtained with NNMF .

Let us tabulate the topics found above with LSAMonTopicExtraction :

mObj⟹ LSAMonEchoTopicsTable["NumberOfTerms" -> 6, "MagnificationFactor" -> 0.8, Appearance -> "Horizontal"];

Map documents over the topics

The function LSAMonTopicsRepresentation finds the top outliers for each row of NNMF’s left factor W. (The outliers are found using the package [AAp4].) The obtained list of indices gives the topic representation of the collection of texts.

Short@(mObj⟹LSAMonTopicsRepresentation[]⟹LSAMonTakeContext)["docTopicIndices"]

{{53}, {47, 53}, {25}, {46}, {44}, {15, 42}, {18}, <>, {30}, {33}, {7, 60}, {22, 25}, {12, 13, 25, 30, 49, 59}, {48, 57}, {14, 41}}

Further we can see that if the documents have tags associated with them — like author names or dates — we can make a contingency matrix of tags vs topics. (See [AAp8, AA4].) This is also done by the function LSAMonTopicsRepresentation that takes tags as an argument. If the tags argument is Automatic, then the tags are simply the document indices.

Here is a an example:

rsmat = mObj⟹LSAMonTopicsRepresentation[Automatic]⟹LSAMonTakeValue;
MatrixPlot[rsmat]

Here is an example of calling the function LSAMonTopicsRepresentation with arbitrary tags.

rsmat = mObj⟹LSAMonTopicsRepresentation[DateString[#, "MonthName"] & /@ dates]⟹LSAMonTakeValue;
MatrixPlot[rsmat]

Note that the matrix plots above are very close to the charting of the Great conversation that we are looking for. This can be made more obvious by observing the row names and columns names in the tabulation of the transposed matrix rsmat:

Magnify[#, 0.6] &@MatrixForm[Transpose[rsmat]]

Charting the great conversation

In this section we show several ways to chart the Great Conversation in the collection of speeches.

There are several possible ways to make the chart: using a time-line plot, using heat-map plot, and using appropriate tabulation (with MatrixForm or Grid).

In order to make the code in this section more concise the package RSparseMatrix.m, [AAp7, AA5], is used.

Topic name to topic words

This command makes an Association between the topic names and the top topic words.

aTopicNameToTopicTable = 
  AssociationThread[(mObj⟹LSAMonTakeContext)["automaticTopicNames"], 
   mObj⟹LSAMonTopicsTable["NumberOfTerms" -> 12]⟹LSAMonTakeValue];

Here is a sample:

Magnify[#, 0.7] &@ aTopicNameToTopicTable[[1 ;; 3]]

Time-line plot

This command makes a contingency matrix between the documents and the topics (as described above):

rsmat = ToRSparseMatrix[mObj⟹LSAMonTopicsRepresentation[Automatic]⟹LSAMonTakeValue]

This time-plot shows great conversation in the USA presidents state of union speeches:

TimelinePlot[
 Association@
  MapThread[
   Tooltip[#2, aTopicNameToTopicTable[#2]] -> dates[[ToExpression@#1]] &, 
   Transpose[RSparseMatrixToTriplets[rsmat]]], 
 PlotTheme -> "Detailed", ImageSize -> 1000, AspectRatio -> 1/2, PlotLayout -> "Stacked"]

The plot is too cluttered, so it is a good idea to investigate other visualizations.

Topic vs president heatmap

We can use the USA president names instead of years in the Great Conversation chart because the USA presidents terms do not overlap.

This makes a contingency matrix presidents vs topics:

rsmat2 = ToRSparseMatrix[
   mObj⟹LSAMonTopicsRepresentation[
     names]⟹LSAMonTakeValue];

Here we compute the chronological order of the presidents based on the dates of their speeches:

nameToMeanYearRules = 
  Map[#[[1, 1]] -> Mean[N@#[[All, 2]]] &, 
   GatherBy[MapThread[List, {names, ToExpression[DateString[#, "Year"]] & /@ dates}], First]];
ordRowInds = Ordering[RowNames[rsmat2] /. nameToMeanYearRules];

This heat-map plot uses the (experimental) package HeatmapPlot.m, [AAp6]:

Block[{m = rsmat2[[ordRowInds, All]]},
 HeatmapPlot[SparseArray[m], RowNames[m], 
  Thread[Tooltip[ColumnNames[m], aTopicNameToTopicTable /@ ColumnNames[m]]],
  DistanceFunction -> {None, Sort}, ImageSize -> 1000, 
  AspectRatio -> 1/2]
 ]

Note the value of the option DistanceFunction: there is not re-ordering of the rows and columns are reordered by sorting. Also, the topics on the horizontal names have tool-tips.

References

Text data

[D1] Wolfram Data Repository, "Presidential Nomination Acceptance Speeches".

[D2] US Presidents, State of the Union Addresses, Trajectory, 2016. ‪ISBN‬1681240009, 9781681240008‬.

[D3] Gerhard Peters, "Presidential Nomination Acceptance Speeches and Letters, 1880-2016", The American Presidency Project.

[D4] Gerhard Peters, "State of the Union Addresses and Messages", The American Presidency Project.

Packages

[AAp1] Anton Antonov, MathematicaForPrediction utilities, (2014), MathematicaForPrediction at GitHub.

[AAp2] Anton Antonov, Implementation of document-term matrix construction and re-weighting functions in Mathematica(2013), MathematicaForPrediction at GitHub.

[AAp3] Anton Antonov, Implementation of the Non-Negative Matrix Factorization algorithm in Mathematica, (2013), MathematicaForPrediction at GitHub.

[AAp4] Anton Antonov, Implementation of one dimensional outlier identifying algorithms in Mathematica, (2013), MathematicaForPrediction at GitHub.

[AAp5] Anton Antonov, Monadic latent semantic analysis Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp6] Anton Antonov, Heatmap plot Mathematica package, (2017), MathematicaForPrediction at GitHub.

[AAp7] Anton Antonov, RSparseMatrix Mathematica package, (2015), MathematicaForPrediction at GitHub.

[AAp8] Anton Antonov, Cross tabulation implementation in Mathematica, (2017), MathematicaForPrediction at GitHub.

Books and articles

[AA1] Anton Antonov, "Topic and thesaurus extraction from a document collection", (2013), MathematicaForPrediction at GitHub.

[AA2] Anton Antonov, "Statistical thesaurus from NPR podcasts", (2013), MathematicaForPrediction at WordPress blog.

[AA3] Anton Antonov, "Monad code generation and extension", (2017), MathematicaForPrediction at GitHub.

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

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

[Wk1] Wikipedia entry, Great Conversation.

[MA1] Mortimer Adler, "The Great Conversation Revisited," in The Great Conversation: A Peoples Guide to Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago,1990, p. 28.

[MA2] Mortimer Adler, "Great Ideas".

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

Tries with frequencies in Java

Introduction

This blog post describes the installation and use in Mathematica of Tries with frequencies [1] implemented in Java [2] through a corresponding Mathematica package [3].

Prefix tree or Trie, [6], is a tree data structure that stores a set of "words" that consist of "characters" — each element can be seen as a key to itself. The article [1] and packages [2,3,4] extend that data structure to have additional data (frequencies) associated with each key.

The packages [2,3] work with lists of strings only. The package [4] can work with more general data but it is much slower.

The main motivation to create the package [3] was to bring the fast Trie functions implementations of [2] into Mathematica in order to prototype, implement, and experiment with different text processing algorithms. (Like, inductive grammar parsers generation and entity name recognition.) The speed of combining [2] and [3] is evaluated in the section "Performance tests" below.

Set-up

This following directory path has to have the jar file "TriesWithFrequencies.jar".

$JavaTriesWithFrequenciesPath = 
  "/Users/antonov/MathFiles/MathematicaForPrediction/Java/TriesWithFrequencies";
FileExistsQ[
 FileNameJoin[{$JavaTriesWithFrequenciesPath, "TriesWithFrequencies.jar"}]]

(* True *)

For more details see the explanations in the README file in the GitHub directory of [2].

The following directory is expected to have the Mathematica package [3].

dirName = "/Users/antonov/MathFiles/MathematicaForPrediction";
FileExistsQ[FileNameJoin[{dirName, "JavaTriesWithFrequencies.m"}]]

(* True *)

AppendTo[$Path, dirName];
Needs["JavaTriesWithFrequencies`"]

This commands installs Java (via JLink`) and loads the necessary Java libraries.

JavaTrieInstall[$JavaTriesWithFrequenciesPath]

Basic examples

For brevity the basic examples are not included in this blog post. Here is album of images that shows the "JavaTrie.*" commands with their effects:

"JavaTrieExample" .

More detailed explanations can be found in the Markdown document, [7]:

Next, we are going to look into performance evaluation examples (also given in [7].)

Membership of words

Assume we want find the words of "Hamlet" that are not in the book "Origin of Species". This section shows that the Java trie creation and query times for this task are quite small.

Read words

The following code reads the words in the texts. We get 33000 words from "Hamlet" and 151000 words from "Origin of Species".

hWords =
  Block[{words},
   words = 
    StringSplit[
     ExampleData[{"Text", "Hamlet"}], {Whitespace, 
      PunctuationCharacter}];
   words = Select[ToLowerCase[words], StringLength[#] > 0 &]
   ];
Length[hWords]

(* 32832 *)

osWords =
  Block[{words},
   words = 
    StringSplit[
     ExampleData[{"Text", "OriginOfSpecies"}], {Whitespace, 
      PunctuationCharacter}];
   words = Select[ToLowerCase[words], StringLength[#] > 0 &]
   ];
Length[osWords]

(* 151205 *)

Membership

First we create trie with "Origin of species" words:

AbsoluteTiming[
 jOStr = JavaTrieCreateBySplit[osWords];
]

(* {0.682531, Null} *)

Sanity check — the "Origin of species" words are in the trie:

AbsoluteTiming[
 And @@ JavaObjectToExpression[
   JavaTrieContains[jOStr, Characters /@ osWords]]
]

(* {1.32224, True} *)

Membership of "Hamlet" words into "Origin of Species":

AbsoluteTiming[
 res = JavaObjectToExpression[
    JavaTrieContains[jOStr, Characters /@ hWords]];
]

(* {0.265307, Null} *)

Tallies of belonging:

Tally[res]

(* {{True, 24924}, {False, 7908}} *)

Sample of words from "Hamlet" that do not belong to "Origin of Species":

RandomSample[Pick[hWords, Not /@ res], 30]

(* {"rosencrantz", "your", "mar", "airy", "rub", "honesty", \
"ambassadors", "oph", "returns", "pale", "virtue", "laertes", \
"villain", "ham", "earnest", "trail", "unhand", "quit", "your", \
"your", "fishmonger", "groaning", "your", "wake", "thou", "liest", \
"polonius", "upshot", "drowned", "grosser"} *)

Common words sample:

RandomSample[Pick[hWords, res], 30]

(* {"as", "indeed", "it", "with", "wild", "will", "to", "good", "so", \
"dirt", "the", "come", "not", "or", "but", "the", "why", "my", "to", \
"he", "and", "you", "it", "to", "potent", "said", "the", "are", \
"question", "soft"} *)

Statistics

The node counts statistics calculation is fast:

AbsoluteTiming[
 JavaTrieNodeCounts[jOStr]
]

(* {0.002344, <|"total" -> 20723, "internal" -> 15484, "leaves" -> 5239|>} *)

The node counts statistics computation after shrinking is comparably fast :

AbsoluteTiming[
 JavaTrieNodeCounts[JavaTrieShrink[jOStr]]
]

(* {0.00539, <|"total" -> 8918,  "internal" -> 3679, "leaves" -> 5239|>} *)

The conversion of a large trie to JSON and computing statistics over the obtained tree is reasonably fast:

AbsoluteTiming[
 res = JavaTrieToJSON[jOStr];
]

(* {0.557221, Null} *)

AbsoluteTiming[
 Quantile[
  Cases[res, ("value" -> v_) :> v, \[Infinity]], 
  Range[0, 1, 0.1]]
]

(* {0.019644, {1., 1., 1., 1., 2., 3., 5., 9., 17., 42., 151205.}} *)

Dictionary infixes

Get all words from a dictionary:

allWords =  DictionaryLookup["*"];
allWords // Length

(* 92518 *)

Trie creation and shrinking:

AbsoluteTiming[
 jDTrie = JavaTrieCreateBySplit[allWords];
 jDShTrie = JavaTrieShrink[jDTrie];
]

(* {0.30508, Null} *)

JSON form extraction:

AbsoluteTiming[
 jsonRes = JavaTrieToJSON[jDShTrie];
]

(* {3.85955, Null} *)

Here are the node statistics of the original and shrunk tries:

"Orginal-trie-vs-Shrunk-trie-Node-Counts"

Find the infixes that have more than three characters and appear more than 10 times:

Multicolumn[#, 4] &@
 Select[SortBy[
   Tally[Cases[
     jsonRes, ("key" -> v_) :> v, Infinity]], -#[[-1]] &], StringLength[#[[1]]] > 3 && #[[2]] > 10 &]
"Long-infixes-in-shrunk-dictionary-trie"

Unit tests

Many of example shown in this document have corresponding tests in the file JavaTriesWithFrequencies-Unit-Tests.wlt hosted at GitHub.

tr = TestReport[
  dirName <> "/UnitTests/JavaTriesWithFrequencies-Unit-Tests.wlt"]
"TestReport"

References

[1] Anton Antonov, "Tries with frequencies for data mining", (2013), MathematicaForPrediction at WordPress blog. URL: https://mathematicaforprediction.wordpress.com/2013/12/06/tries-with-frequencies-for-data-mining/ .

[2] Anton Antonov, Tries with frequencies in Java, (2017), source code at MathematicaForPrediction at GitHub, project Java/TriesWithFrequencies.

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

[4] Anton Antonov, Tries with frequencies Mathematica package, (2013), source code at MathematicaForPrediction at GitHub, package TriesWithFrequencies.m .

[5] Anton Antonov, Java tries with frequencies Mathematica unit tests, (2017), source code at MathematicaForPrediction at GitHub, unit tests file JavaTriesWithFrequencies-Unit-Tests.wlt .

[6] Wikipedia, Trie, https://en.wikipedia.org/wiki/Trie .

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