A monad for Epidemiologic Compartmental Modeling Workflows

Version 0.8


In this document we describe the design and demonstrate the implementation of a (software programming) monad, [Wk1], for Epidemiology Compartmental Modeling (ECM) workflows specification and execution. The design and implementation are done with Mathematica / Wolfram Language (WL). A very similar implementation is also done in R.

Monad’s name is “ECMMon”, which stands for “Epidemiology Compartmental Modeling Monad”, and its monadic implementation is based on the State monad package “StateMonadCodeGenerator.m”, [AAp1, AA1], ECMMon is implemented in the package [AAp8], which relies on the packages [AAp3-AAp6]. The original ECM workflow discussed in [AA5] was implemented in [AAp7]. An R implementation of ECMMon is provided by the package [AAr2].

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

We use real-life COVID-19 data, The New York Times COVID-19 data, see [NYT1, AA5].

The monadic programming design is used as a Software Design Pattern. The ECMMon monad can be also seen as a Domain Specific Language (DSL) for the specification and programming of epidemiological compartmental modeling workflows.

Here is an example of using the ECMMon monad over a compartmental model with two types of infected populations:

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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 ECMMon can be seen as a DSL. Because of this the monad pipelines made with ECMMon are sometimes called “specifications”.

Contents description

The document has the following structure.

  • The sections “Package load” and “Data load” obtain the needed code and data.
  • The section “Design consideration” provide motivation and design decisions rationale.
  • The section “Single site models” give brief descriptions of certain “seed” models that can be used in the monad.
  • The section “Single-site model workflow demo”, “Multi-site workflow demo” give demonstrations of how to utilize the ECMMon monad .
    • Using concrete practical scenarios and “real life” data.
  • The section “Batch simulation and calibration process” gives methodological preparation for the content of the next two sections.
  • The section “Batch simulation workflow” and “Calibration workflow” describe how to do most important monad workflows after the model is developed.
  • The section “Future plans” outlines future directions of development.

Remark: One can read only the sections “Introduction”, “Design consideration”, “Single-site models”, and “Batch simulation and calibration process”. That set of sections provide a fairly good, programming language agnostic exposition of the substance and novel ideas of this document.

Package load

In this section we load packages used in this notebook:


Remark: The import commands above would trigger some other package imports.

Data load

In this section we ingest data using the “umbrella function” MultiSiteModelReadData from [AAp5]:

Read data

AbsoluteTiming[  aData = MultiSiteModelReadData[];  ](*{38.8635, Null}*)

Data summaries

ResourceFunction["RecordsSummary"] /@ aData

Transform data

Here we transform the population related data in a form convenient for specifying the simulations with it:

aPopulations = Association@Map[{#Lon, #Lat} -> #Population &, Normal[aData["CountyRecords"]]];aInfected = Association@Map[{#Lon, #Lat} -> #Cases &, Normal[aData["CasesAndDeaths"]]];aDead = Association@Map[{#Lon, #Lat} -> #Deaths &, Normal[aData["CasesAndDeaths"]]];


Using the built-in function GeoHistogram we summarize the USA county populations, and COViD-19 infection cases and deaths:

Row@MapThread[GeoHistogram[KeyMap[Reverse, #1], Quantity[140, "Miles"], PlotLabel -> #2, PlotTheme -> "Scientific", PlotLegends -> Automatic, ImageSize -> Medium] &, {{aPopulations, aInfected, aDead}, {"Populations", "Infected", "Dead"}}]

(Note that in the plots above we have to reverse the keys of the given population associations.)

Using the function HextileHistogram from [AAp7 ] here we visualize USA county populations over a hexagonal grid with cell radius 2 degrees ((\approx 140) miles (\approx 222) kilometers):

HextileHistogram[aPopulations, 2, PlotRange -> All, PlotLegends -> Automatic, ImageSize -> Large]

In this notebook we prefer using HextileHistogram because it represents the simulation data in geometrically more faithful way.

Design considerations

The big picture

The main purpose of the designed epidemic compartmental modeling framework (i.e. software monad) is to have the ability to do multiple, systematic simulations for different scenario play-outs over large scale geographical regions. The target end-users are decision makers at government level and researchers of pandemic or other large scale epidemic effects.

Here is a diagram that shows the envisioned big picture workflow:


Large-scale modeling

The standard classical compartmental epidemiology models are not adequate over large geographical areas, like, countries. We design a software framework – the monad ECMMon – that allows large scale simulations using a simple principle workflow:

  1. Develop a single-site model for relatively densely populated geographical area for which the assumptions of the classical models (approximately) hold.
  2. Extend the single-site model into a large-scale multi-site model using statistically derived traveling patterns; see [AA4].
  3. Supply the multi-site model with appropriately prepared data.
  4. Run multiple simulations to see large scale implications of different policies.
  5. Calibrate the model to concrete observed (or faked) data. Go to 4.

Flow chart

The following flow chart visualizes the possible workflows the software monad ECMMon:


Two models in the monad

  • An ECMMon object can have one or two models. One of the models is a “seed”, single-site model from [AAp1], which, if desired, is scaled into a multi-site model, [AA3, AAp2].
  • Workflows with only the single-site model are supported.
    • Say, workflows for doing sensitivity analysis, [AA6, BC1].
  • Scaling of a single-site model into multi-site is supported and facilitated.
  • Workflows for the multi-site model include preliminary model scaling steps and simulation steps.
  • After the single-site model is scaled the monad functions use the multi-site model.
  • The workflows should be easy to specify and read.

Single-site model workflow

  1. Make a single-site model.
  2. Assign stocks initial conditions.
  3. Assign rates values.
  4. Simulate.
  5. Plot results.
  6. Go to 2.

Multi-site model workflow

  1. Make a single-site model.
  2. Assign initial conditions and rates.
  3. Scale the single-site model into a multi-site model.
    1. The single-site assigned rates become “global” when the single-site model is scaled.
    2. The scaling is based on assumptions for traveling patterns of the populations.
    3. There are few alternatives for that scaling:
      1. Using locations geo-coordinates
      2. Using regular grids covering a certain area based on in-habited locations geo-coordinates
      3. Using traveling patterns contingency matrices
      4. Using “artificial” patterns of certain regular types for qualitative analysis purposes
  4. Enhance the multi-site traveling patterns matrix and re-scale the single site model.
    1. We might want to combine traveling patterns by ground transportation with traveling patterns by airplanes.
    2. For quarantine scenarios this might a less important capability of the monad.
      1. Hence, this an optional step.
  5. Assign stocks initial conditions for each of the sites in multi-scale model.
  6. Assign rates for each of the sites.
  7. Simulate.
  8. Plot global simulation results.
  9. Plot simulation results for focus sites.

Single-site models

We have a collection of single-site models that have different properties and different modeling goals, [AAp3, AA7, AA8]. Here is as diagram of a single-site model that includes hospital beds and medical supplies as limitation resources, [AA7]:


SEI2HR model

In this sub-section we briefly describe the model SEI2HR, which is used in the examples below.

Remark: SEI2HR stands for “Susceptible, Exposed, Infected Two, Hospitalized, Recovered” (populations).

Detailed description of the SEI2HR model is given in [AA7].

Verbal description

We start with one infected (normally symptomatic) person, the rest of the people are susceptible. The infected people meet other people directly or get in contact with them indirectly. (Say, susceptible people touch things touched by infected.) For each susceptible person there is a probability to get the decease. The decease has an incubation period: before becoming infected the susceptible are (merely) exposed. The infected recover after a certain average infection period or die. A certain fraction of the infected become severely symptomatic. If there are enough hospital beds the severely symptomatic infected are hospitalized. The hospitalized severely infected have different death rate than the non-hospitalized ones. The number of hospital beds might change: hospitals are extended, new hospitals are build, or there are not enough medical personnel or supplies. The deaths from infection are tracked (accumulated.) Money for hospital services and money from lost productivity are tracked (accumulated.)

The equations below give mathematical interpretation of the model description above.


Here are the equations of one the epidemiology compartmental models, SEI2HR, [AA7], implemented in [AAp3]:

ModelGridTableForm[SEI2HRModel[t], "Tooltips" -> False]["Equations"] /. {eq_Equal :> TraditionalForm[eq]}

The equations for Susceptible, Exposed, Infected, Recovered populations of SEI2R are “standard” and explanations about them are found in [WK2, HH1]. For SEI2HR those equations change because of the stocks Hospitalized Population and Hospital Beds.

The equations time unit is one day. The time horizon is one year. In this document we consider COVID-19, [Wk2, AA1], hence we do not consider births.

Single-site model workflow demo

In this section we demonstrate some of the sensitivity analysis discussed in [AA6, BC1].

Make a single-site model, SEI2HR:

model1 = SEI2HRModel[t, "InitialConditions" -> True, "RateRules" -> True, "TotalPopulationRepresentation" -> "AlgebraicEquation"];

Make an association with “default” parameters:

aDefaultPars = <|    aip -> 26,     aincp -> 5,     \[Beta][ISSP] -> 0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],     \[Beta][INSP] -> 0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],     qsd -> 60,     ql -> 21,     qcrf -> 0.25,     \[Beta][HP] -> 0.01,     \[Mu][ISSP] -> 0.035/aip,     \[Mu][INSP] -> 0.01/aip,     nhbr[TP] -> 3/1000,     lpcr[ISSP, INSP] -> 1,     hscr[ISSP, INSP] -> 1    |>;

Execute the workflow multiple times with different quarantine starts:

qlVar = 56;lsRes =  Map[   ECMMonUnit[]     ECMMonSetSingleSiteModel[model1]     ECMMonAssignRateRules[Join[aDefaultPars, <|qsd -> #, ql -> qlVar|>]]     ECMMonSimulate[365]     ECMMonPlotSolutions[{"Infected Severely Symptomatic Population"}, 240,        "Together" -> True, "Derivatives" -> False,        PlotRange -> {0, 12000}, ImageSize -> 250,        Epilog -> {Orange, Dashed, Line[{{#1, 0}, {#1, 12000}}], Line[{{#1 + qlVar, 0}, {#1 + qlVar, 12000}}]},        PlotLabel -> Row[{"Quarantine start:", Spacer[5], #1, ",", Spacer[5], "end:", Spacer[5], #1 + qlVar}],        "Echo" -> False]     ECMMonTakeValue &, Range[25, 100, 5]];

Plot the simulation solutions for “Infected Severely Symptomatic Population”:

Multicolumn[#[[1, 1]] & /@ lsRes, 4]

Both theoretical and computational details of the workflow above are given [AA7, AA8].

Multi-site workflow demo

In this section we demonstrate the multi-site model workflow using COVID-19 data for USA, [WRI2, NYT1].

Here a “seed”, single-site model is created:

model1 = SEI2HRModel[t, "InitialConditions" -> True, "RateRules" -> True, "TotalPopulationRepresentation" -> "AlgebraicEquation"];

Here we specify a multi-site model workflow (the monadic steps are separated and described with purple print-outs):

ecmObj =    ECMMonUnit[]    ECMMonSetSingleSiteModel[model1]    ECMMonAssignRateRules[     <|      aip -> 26,       aincp -> 5,       \[Beta][ISSP] -> 0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],       \[Beta][INSP] -> 0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],       qsd -> 0,       ql -> 56,       qcrf -> 0.25,       \[Beta][HP] -> 0.01,       \[Mu][ISSP] -> 0.035/aip,       \[Mu][INSP] -> 0.01/aip,       nhbr[TP] -> 3/1000,       lpcr[ISSP, INSP] -> 1,       hscr[ISSP, INSP] -> 1      |>     ]    ECMMonEcho[Style["Show the single-site model tabulated form:", Bold, Purple]]    ECMMonEchoFunctionContext[Magnify[ModelGridTableForm[#singleSiteModel], 1] &]    ECMMonMakePolygonGrid[Keys[aPopulations], 1.5, "BinningFunction" -> Automatic]    ECMMonEcho[Style["Show the grid based on population coordinates:", Bold, Purple]]    ECMMonPlotGrid["CellIDs" -> True, ImageSize -> Large]    ECMMonExtendByGrid[aPopulations, 0.12]    ECMMonAssignInitialConditions[aPopulations, "Total Population", "Default" -> 0]    ECMMonAssignInitialConditions[DeriveSusceptiblePopulation[aPopulations, aInfected, aDead], "Susceptible Population", "Default" -> 0]    ECMMonAssignInitialConditions[<||>, "Exposed Population", "Default" -> 0]    ECMMonAssignInitialConditions[aInfected, "Infected Normally Symptomatic Population", "Default" -> 0]    ECMMonAssignInitialConditions[<||>, "Infected Severely Symptomatic Population", "Default" -> 0]    ECMMonEcho[Style["Show total populations initial conditions data:", Bold, Purple]]    ECMMonPlotGridHistogram[aPopulations, ImageSize -> Large, PlotLabel -> "Total populations"]    ECMMonEcho[Style["Show infected and deceased initial conditions data:", Bold,Purple]]    ECMMonPlotGridHistogram[aInfected, ColorFunction -> ColorData["RoseColors"], "ShowDataPoints" -> False, ImageSize -> Large, PlotLabel -> "Infected"]    ECMMonPlotGridHistogram[aDead, ColorFunction -> ColorData["RoseColors"], "ShowDataPoints" -> False, ImageSize -> Large, PlotLabel -> "Deceased"]    ECMMonEcho[Style["Simulate:", Bold, Purple]]    ECMMonSimulate[365]    ECMMonEcho[Style["Show global population simulation results:", Bold, Purple]]    ECMMonPlotSolutions[__ ~~ "Population", 365]    ECMMonEcho[Style["Show site simulation results for Miami and New York areas:", Bold, Purple]]    ECMMonPlotSiteSolutions[{160, 174}, __ ~~ "Population", 365]    ECMMonEcho[Style["Show deceased and hospitalzed populations results for Miami and New York areas:", Bold, Purple]]    ECMMonPlotSiteSolutions[{160, 174}, {"Deceased Infected Population", "Hospitalized Population","Hospital Beds"}, 300, "FocusTime" -> 120];
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Theoretical and computational details about the multi-site workflow can be found in [AA4, AA5].

Batch simulations and calibration processes

In this section we describe the in general terms the processes of model batch simulations and model calibration. The next two sections give more details of the corresponding software design and workflows.


Batch simulation: If given a SD model (M), the set (P) of parameters of (M), and a set (B) of sets of values (P), (B\text{:=}\left{V_i\right}), then the set of multiple runs of (M) over (B) are called batch simulation.

Calibration: If given a model (M), the set (P) of parameters of (M), and a set of (k) time series (T\text{:=}\left{T_i\right}_{i=1}^k) that correspond to the set of stocks (S\text{:=}\left{S_i\right}_{i=1}^k) of (M) then the process of finding concrete the values (V) for (P) that make the stocks (S) to closely resemble the time series (T) according to some metric is called calibration of (M) over the targets (T).


  • There are three types of people dealing with the models:
    • Modeler, who develops and implements the model and prepares it for calibration.
    • Calibrator, who calibrates the model with different data for different parameters.
    • Stakeholder, who requires different features of the model and outcomes from different scenario play-outs.
  • There are two main calibration scenarios:
    • Modeler and Calibrator are the same person
    • Modeler and Calibrator are different persons


Model development and calibration is most likely going to be an iterative process.

For concreteness let us assume that the model has matured development-wise and batch simulation and model calibration is done in a (more) formal way.

Here are the steps of a well defined process between the modeling activity players described above:

  1. Stakeholder requires certain scenarios to be investigated.
  2. Modeler prepares the model for those scenarios.
  3. Stakeholder and Modeler formulate a calibration request.
  4. Calibrator uses the specifications from the calibration request to:
    1. Calibrate the model
    2. Derive model outcomes results
    3. Provide model qualitative results
    4. Provide model sensitivity analysis results
  5. Modeler (and maybe Stakeholder) review the results and decides should more calibration be done.
    1. I.e. go to 3.
  6. Modeler does batch simulations with the calibrated model for the investigation scenarios.
  7. Modeler and Stakeholder prepare report with the results.

See the documents [AA9, AA10] have questionnaires that further clarify the details of interaction between the modelers and calibrators.

Batch simulation vs calibration

In order to clarify the similarities and differences between batch simulation and calibration we list the following observations:

  • Each batch simulation or model calibration is done either for model development purposes or for scenario play-out studies.
  • Batch simulation is used for qualitative studies of the model. For example, doing sensitivity analysis; see [BC1, AA7, AA8].
  • Before starting the calibration we might want to study the “landscape” of the search space of the calibration parameters using batch simulations.
  • Batch simulation is also done after model calibration in order to evaluate different scenarios,
  • For some models with large computational complexity batch simulation – together with some evaluation metric – can be used instead of model calibration.

Batch simulations workflow

In this section we describe the specification and execution of model batch simulations.

Batch simulations can be time consuming, hence it is good idea to

In the rest of the section we go through the following steps:

  1. Make a model object
  2. Batch simulate over a few combinations of parameters and show:
    1. Plots of the simulation results for all populations
    2. Plots of the simulation results for a particular population
  3. Batch simulate over the Cartesian (outer) product of values lists of a selected pair of parameters and show the corresponding plots of all simulations

Model (object) for batch simulations

Here we make a new ECMMon object:

ecmObj2 =  ECMMonUnit[]   ECMMonSetSingleSiteModel[model1]   ECMMonAssignRateRules[aDefaultPars];

Direct specification of combinations of parameters

All populations

Here we simulate the model in the object of different parameter combinations given in a list of associations:

res1 =  ecmObj2   ECMMonBatchSimulate[___ ~~ "Population", {<|qsd -> 60, ql -> 28|>, <|qsd -> 55, ql -> 28|>, <|qsd -> 75, ql -> 21, \[Beta][ISSP] -> 0|>}, 240]   ECMMonTakeValue;

Remark: The stocks in the results are only stocks that are populations – that is specified with the string expression pattern ___~~”Population”.

Here is the shape of the result:

Short /@ res1

Here are the corresponding plots:

ListLinePlot[#, PlotTheme -> "Detailed", ImageSize -> Medium, PlotRange -> All] & /@ res1

Focus population

We might be interested in the batch simulations results for only one, focus populations. Here is an example:

res2 =    ecmObj2    ECMMonBatchSimulate["Infected Normally Symptomatic Population", {<|qsd -> 60, ql -> 28|>, <|qsd -> 55, ql -> 28|>, <|qsd -> 75, ql -> 21, \[Beta][ISSP] -> 0|>}, 240]    ECMMonTakeValue;

Here is the shape of the result:

Short /@ res2

Here are the corresponding plots:

Multicolumn[ KeyValueMap[  ListLinePlot[#2, PlotLabel -> #1, PlotTheme -> "Detailed",     Epilog -> {Directive[Orange, Dashed],       Line[{Scaled[{0, -1}, {#1[qsd], 0}], Scaled[{0, 1}, {#1[qsd], 0}]}],       Line[{Scaled[{0, -1}, {#1[qsd] + #1[ql], 0}], Scaled[{0, 1}, {#1[qsd] + #1[ql], 0}]}]},     ImageSize -> Medium] &, res2]]

Outer product of parameters

Instead of specifying an the combinations of parameters directly we can specify the values taken by each parameter using an association in which the keys are parameters and the values are list of values:

res3 =    ecmObj2    ECMMonBatchSimulate[__ ~~ "Population", <|qsd -> {60, 55, 75}, ql -> {28, 21}|>, 240]    ECMMonTakeValue;

Here is the shallow form of the results

Short /@ res3

Here are the corresponding plots:

Multicolumn[ KeyValueMap[  ListLinePlot[#2, PlotLabel -> #1, PlotTheme -> "Detailed",     Epilog -> {Directive[Gray, Dashed],       Line[{Scaled[{0, -1}, {#1[qsd], 0}], Scaled[{0, 1}, {#1[qsd], 0}]}],       Line[{Scaled[{0, -1}, {#1[qsd] + #1[ql], 0}], Scaled[{0, 1}, {#1[qsd] + #1[ql], 0}]}]},     ImageSize -> Medium] &, res3]]

Calibration workflow

In this section we go through the computation steps of the calibration of single-site SEI2HR model.

Remark: We use real data in this section, but the presented calibration results and outcome plots are for illustration purposes only. A rigorous study with discussion of the related assumptions and conclusions is beyond the scope of this notebook/document.

Calibration steps

Here are the steps performed in the rest of the sub-sections of this section:

  1. Ingest data for infected cases, deaths due to disease, etc.
  2. Choose a model to calibrate.
  3. Make the calibration targets – those a vectors corresponding to time series over regular grids.
    1. Consider using all of the data in order to evaluate model’s applicability.
    2. Consider using fractions of the data in order to evaluate model’s ability to predict the future or reconstruct data gaps.
  4. Choose calibration parameters and corresponding ranges for their values.
  5. If more than one target choose the relative weight (or importance) of the targets.
  6. Calibrate the model.
  7. Evaluate the fitting between the simulation results and data.
    1. Using statistics and plots.
  8. Make conclusions. If insufficiently good results are obtained go to 2 or 4.

Remark: When doing calibration epidemiological models a team of people it is better certain to follow (rigorously) well defined procedures. See the documents:

Remark: We plan to prepare have several notebooks dedicated to calibration of both single-site and multi-site models.

USA COVID-19 data

Here data for the USA COVID-19 infection cases and deaths from [NYT1] (see [AA6] data ingestion details):

lsCases = {1, 1, 1, 2, 3, 5, 5, 5, 5, 6, 7, 8, 11, 11, 11, 12, 12, 12,12, 12, 13, 13, 14, 15, 15, 15, 15, 25, 25, 25, 27, 30, 30, 30, 43, 45, 60, 60, 65, 70, 85, 101, 121, 157, 222, 303, 413, 530, 725,976, 1206, 1566, 2045, 2603, 3240, 4009, 5222, 6947, 9824, 13434, 17918, 23448, 30082, 37696, 46791, 59678, 73970, 88796, 103318, 119676, 139165, 160159, 184764, 212033, 241127, 262275, 288195, 314991, 341540, 370689, 398491, 423424, 445213, 467106, 490170, 512972, 539600, 566777, 590997, 613302, 637812, 660549, 685165, 714907, 747741, 777098, 800341, 820764, 844225, 868644, 895924, 927372, 953923, 977395, 998136, 1020622, 1043873, 1069587, 1095405,1118643, 1137145, 1155671, 1176913, 1196485, 1222057, 1245777, 1267911, 1285105, 1306316, 1326559, 1349019, 1373255, 1395981, 1416682, 1436260, 1455183, 1473813, 1491974, 1513223, 1536848, 1559020, 1578876, 1600414, 1620096, 1639677, 1660303, 1688335, 1709852, 1727711, 1745546, 1763803, 1784049, 1806724, 1831494, 1855870, 1874023, 1894074, 1918373, 1943743, 1970066, 2001470, 2031613, 2057493, 2088420, 2123068, 2159633, 2199841, 2244876, 2286401, 2324563, 2362875, 2411709, 2461341, 2514500, 2573030, 2622980, 2667278, 2713656, 2767129, 2825865, 2885325, 2952393, 3012349, 3069369, 3129738, 3194944, 3263077, 3338308, 3407962, 3469137, 3529938, 3588229, 3653114, 3721574, 3790356, 3862588, 3928575, 3981476, 4039440, 4101329, 4167741, 4235717, 4303663, 4359188, 4408708, 4455340, 4507370, 4560539, 4617036, 4676822, 4730639, 4777548, 4823529, 4876038, 4929115, 4981066, 5038637, 5089258, 5130147, 5166032, 5206970, 5251824, 5297150, 5344322, 5388034, 5419494, 5458726, 5497530, 5541830, 5586297, 5631403, 5674714, 5707327, 5742814, 5786178, 5817338, 5862014, 5917466, 5958619, 5988001, 6012054, 6040456, 6073671, 6110645, 6157050, 6195893, 6228601, 6264192, 6301923, 6341145, 6385411, 6432677, 6472474, 6507345, 6560827, 6597281, 6638806, 6682079, 6734971, 6776512, 6812354, 6847745, 6889421, 6930523, 6975693, 7027692, 7073962, 7107992, 7168298, 7210171, 7261433, 7315687, 7373073, 7422798, 7466501, 7513020, 7565839, 7625285, 7688761, 7757326, 7808123, 7853753, 7916879, 7976530, 8039653, 8113165, 8193658, 8270925, 8328369, 8401001, 8473618, 8555199, 8642599, 8737995, 8814233, 8892933, 8983153, 9074711, 9182627, 9301455, 9436244, 9558668, 9659817, 9784920, 9923082, 10065150, 10222441, 10405550, 10560047, 10691686, 10852769, 11011484, 11183982, 11367840, 11561152, 11727724, 11864571, 12039323, 12213742, 12397014, 12495699, 12693598, 12838076, 12972986, 13135728, 13315143, 13516558, 13728192, 13958512, 14158135, 14325555, 14519697, 14731424, 14954596, 15174109, 15447371, 15647963, 15826415, 16020169, 16218331, 16465552, 16697862, 16941306, 17132902, 17305013, 17498382, 17694678, 17918870, 18106293, 18200349, 18410644, 18559596, 18740591, 18932346, 19157710};
lsDeaths = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 10, 12, 12, 15, 19, 22, 26, 31, 37, 43, 50, 59, 63, 84, 106, 137, 181, 223, 284, 335, 419, 535, 694, 880, 1181, 1444, 1598, 1955, 2490, 3117, 3904, 4601, 5864, 6408, 7376, 8850, 10159, 11415,12924, 14229, 15185, 16320, 18257, 20168, 21941, 23382, 24617, 26160, 27535, 29821, 31633, 33410, 35104, 36780, 37660, 38805, 40801, 42976, 44959, 46552, 48064, 49122, 50012, 52079, 54509, 56277, 57766, 59083, 59903, 60840, 62299, 63961, 65623, 67143, 68260, 68959, 69633, 71042, 72474, 73718, 74907, 75891, 76416, 76888, 77579, 79183, 80329, 81452, 82360, 82904, 83582, 84604, 85545, 86487, 87559, 88256, 88624, 89235, 90220, 91070, 91900, 92621, 93282, 93578, 93988, 94710, 95444, 96123, 96760, 97297, 97514, 97832, 98638, 99372, 101807, 102525, 103018, 103258, 103594,104257, 104886, 105558, 106148, 106388, 106614, 106998, 107921, 108806, 109705, 110522, 111177, 111567, 111950, 112882, 113856, 114797, 115695, 116458, 116854, 117367, 118475, 119606, 120684, 121814, 122673, 123105, 124782, 126089, 127465, 128720, 130131, 131172, 131584, 132174, 133517, 134756, 135271, 136584, 137547, 138076, 138601, 140037, 141484, 142656, 143793, 144819, 145315, 145831, 147148, 148517, 149550, 150707, 151641, 152065, 152551, 153741, 154908, 156026, 157006, 157869, 158235, 158714, 159781, 160851, 161916, 162869, 163573, 163970, 164213, 164654, 165811, 166713, 167913, 168594, 168990, 169423, 170674, 171663, 172493, 173408, 174075, 174281, 174689, 175626, 176698, 177559, 178390, 179145, 179398, 179736, 180641, 181607, 182455, 183282, 183975, 184298, 184698, 185414, 186359, 187280, 188168, 188741, 189165, 189501, 190312, 191267, 192077, 192940, 193603, 193976, 194481, 195405, 196563, 197386, 198271, 199126, 199462, 199998, 200965, 201969, 202958, 203878, 204691, 205119, 205623, 206757, 208331, 209417, 210829, 211838, 212277, 212989, 214442, 215872, 217029, 218595, 219791, 220402, 221165, 222750, 224641, 226589, 228452, 229868, 230695, 231669, 233856, 236127, 237284, 238636, 239809, 240607, 241834, 244430, 247258, 250102, 252637, 254784, 255857, 257282, 260038, 263240, 266144, 268940, 271172, 272499, 274082, 277088, 280637, 283899, 286640, 289168, 290552, 292346, 295549, 298953, 301713, 302802, 304403, 305581, 307396, 310994, 314654};

Remark: The COVID-19 data was ingested from [NYT1] on 2020-12-31,

Calibration targets

From the data we make the calibration targets association:

aTargets = <|{ISSP -> 0.2 lsCases, INSP -> 0.8 lsCases, DIP -> lsDeaths}|>;

Remark: Note that we split the infection cases into 20% severely symptomatic cases and 80% normally symptomatic cases.

Here is the corresponding plot:

ListLogPlot[aTargets, PlotTheme -> "Detailed", PlotLabel -> "Calibration targets", ImageSize -> Medium]

Here we prepare a smaller set of the targets data for the calibration experiments below:

aTargetsShort = Take[#, 170] & /@ aTargets;

Model creation

modelSEI2HR = SEI2HRModel[t, "TotalPopulationRepresentation" -> "AlgebraicEquation"];

Here are the parameters we want to experiment with (or do calibration with):

lsFocusParams = {aincp, aip, sspf[SP], \[Beta][HP], qsd, ql, qcrf, nhbcr[ISSP, INSP], nhbr[TP]};

Here we set custom rates and initial conditions:

aDefaultPars = <|    \[Beta][ISSP] -> 0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],     \[Beta][INSP] -> 0.5*Piecewise[{{1, t < qsd}, {qcrf, qsd <= t <= qsd + ql}}, 1],     qsd -> 60,     ql -> 8*7,     qcrf -> 0.25,     \[Beta][HP] -> 0.01,     \[Mu][ISSP] -> 0.035/aip,     \[Mu][INSP] -> 0.01/aip,     nhbr[TP] -> 3/1000,     lpcr[ISSP, INSP] -> 1,     hscr[ISSP, INSP] -> 1    |>;

Remark: Note the piecewise functions for (\beta [\text{ISSP}]) and (\beta [\text{INSP}]).


Here is the USA population number we use for calibration:

usaPopulation = QuantityMagnitude@CountryData["UnitedStates", "Population"](*329064917*)

Here is we create a ECMMon object that has default parameters and initial conditions assigned above:

AbsoluteTiming[  ecmObj3 =     ECMMonUnit[]     ECMMonSetSingleSiteModel[modelSEI2HR]     ECMMonAssignInitialConditions[<|TP[0] -> usaPopulation, SP[0] -> usaPopulation - 1, ISSP[0] -> 1|>]     ECMMonAssignRateRules[KeyDrop[aDefaultPars, {aip, aincp, qsd, ql, qcrf}]]     ECMMonCalibrate[      "Target" -> KeyTake[aTargetsShort, {ISSP, DIP}],       "StockWeights" -> <|ISSP -> 0.8, DIP -> 0.2|>,       "Parameters" -> <|aip -> {10, 35}, aincp -> {2, 16}, qsd -> {60, 120}, ql -> {20, 160}, qcrf -> {0.1, 0.9}|>,       DistanceFunction -> EuclideanDistance,       Method -> {"NelderMead", "PostProcess" -> False},       MaxIterations -> 1000      ];  ](*{28.0993, Null}*)

Here are the found parameters:

calRes = ecmObj3ECMMonTakeValue(*{152516., {aip -> 10., aincp -> 3.67018, qsd -> 81.7067, ql -> 111.422, qcrf -> 0.312499}}*)

Using different minimization methods and distance functions

In the monad the calibration of the models is done with NMinimize. Hence, the monad function ECMMonCalibrate takes all options of NMinimize and can do calibrations with the same data and parameter search space using different global minima finding methods and distance functions.

Remark: EuclideanDistance is an obvious distance function, but use others like infinity norm and sum norm. Also, we can use a distance function that takes parts of the data. (E.g. between days 50 and 150 because the rest of the data is, say, unreliable.)

Verification of the fit

maxTime = Length[aTargets[[1]]];
ecmObj4 =    ECMMonUnit[]    ECMMonSetSingleSiteModel[modelSEI2HR]    ECMMonAssignInitialConditions[<|TP[0] -> usaPopulation, SP[0] -> usaPopulation - 1, ISSP[0] -> 1|>]    ECMMonAssignRateRules[Join[aDefaultPars, Association[calRes[[2]]]]]    ECMMonSimulate[maxTime]    ECMMonPlotSolutions[___ ~~ "Population" ~~ ___, maxTime, ImageSize -> Large, LogPlot -> False];
aSol4 = ecmObj4ECMMonGetSolutionValues[Keys[aTargets], maxTime]ECMMonTakeValue;
Map[ListLogPlot[{aSol4[#], aTargets[#]}, PlotLabel -> #, PlotRange -> All, ImageSize -> Medium, PlotLegends -> {"Calibrated model", "Target"}] &, Keys[aTargets]]

Conclusions from the calibration results

We see the that with the calibration found parameter values the model can fit the data for the first 200 days, after that it overestimates the evolution of the infected and deceased popupulations.

We can conjecture that:

  • The model is too simple, hence inadequate
  • That more complicated quarantine policy functions have to be used
  • That the calibration process got stuck in some local minima

Future plans

In this section we outline some of the directions in which the presented work on ECMMon can be extended.

More unit tests and random unit tests

We consider the preparation and systematic utilization of unit tests to be a very important component of any software development. Unit tests are especially important when complicated software package like ECMMon are developed.

For the presented software monad (and its separately developed, underlying packages) have implemented a few collections of tests, see [AAp10, AAp11].

We plan to extend and add more complicated unit tests that test for both quantitative and qualitative behavior. Here are some examples for such tests:

  • Stock-vs-stock orbits produced by simulations of certain epidemic models
  • Expected theoretical relationships between populations (or other stocks) for certain initial conditions and rates
  • Wave-like propagation of the proportions of the infected populations in multi-site models over artificial countries and traveling patterns
  • Finding of correct parameter values with model calibration over different data (both artificial and real life)
  • Expected number of equations for different model set-ups
  • Expected (relative) speed of simulations with respect to model sizes

Further for the monad ECMMon we plant to develop random pipeline unit tests as the ones in [AAp12] for the classification monad ClCon, [AA11].

More comprehensive calibration guides and documentation

We plan to produce more comprehensive guides for doing calibration with ECMMon and in general with Mathematica’s NDSolve and NMinimize functions.

Full correspondence between the Mathematica and R implementations

The ingredients of the software monad ECMMon and ECMMon itself were designed and implemented in Mathematica first. The corresponding design and implementation was done in R, [AAr2]. To distinguish the two implementations we call the R one ECMMon-R and Mathematica (Wolfram Language) one ECMMon-WL.

At this point the calibration is not implemented in ECMMon-R, but we plan to do that soon.

Using ECMMon-R (and the RStudio’s Shiny ecosystem) allows for highly shareable interactive interfaces to be programed. Here is an example: https://antononcube.shinyapps.io/SEI2HR-flexdashboard/ .

(With Mathematica similar interactive interfaces are presented in [AA7, AA8].)

Model transfer between Mathematica and R

We are very interested in transferring epidemiological models from Mathematica to R (or Python, or Julia.)

This can be done in two principle ways: (i) using Mathematica expressions parsers, or (ii) using matrix representations. We plan to investigate the usage of both approaches.

Conversational agent

Consider the making of a conversational agent for epidemiology modeling workflows building. Initial design and implementation is given in [AA13, AA14].

Consider the following epidemiology modeling workflow specification:

lsCommands = "create with SEI2HR;assign 100000 to total population;set infected normally symptomatic population to be 0;set infected severely symptomatic population to be 1;assign 0.56 to contact rate of infected normally symptomatic population;assign 0.58 to contact rate of infected severely symptomatic population;assign 0.1 to contact rate of the hospitalized population;simulate for 240 days;plot populations results;calibrate for target DIPt -> tsDeaths, over parameters contactRateISSP in from 0.1 to 0.7;echo pipeline value";

Here is the ECMMon code generated using the workflow specification:

ToEpidemiologyModelingWorkflowCode[lsCommands, "Execute" -> False, "StringResult" -> True](*"ECMMonUnit[SEI2HRModel[t]] ECMMonAssignInitialConditions[<|TP[0] -> 100000|>] ECMMonAssignInitialConditions[<|INSP[0] -> 0|>] ECMMonAssignInitialConditions[<|ISSP[0] -> 1|>] ECMMonAssignRateRules[<|\\[Beta][INSP] -> 0.56|>] ECMMonAssignRateRules[<|\\[Beta][ISSP] -> 0.58|>] ECMMonAssignRateRules[<|\\[Beta][HP] -> 0.1|>] ECMMonSimulate[\"MaxTime\" -> 240] ECMMonPlotSolutions[ \"Stocks\" -> __ ~~ \"Population\"] ECMMonCalibrate[ \"Target\" -> <|DIP -> tsDeaths|>, \"Parameters\" -> <|\\[Beta][ISSP] -> {0.1, 0.7}|> ] ECMMonEchoValue[]"*)

Here is the execution of the code above:

Block[{tsDeaths = Take[lsDeaths, 150]}, ToEpidemiologyModelingWorkflowCode[lsCommands]];
0cs73r03ki1mi 0227wiq3locdf

Different target languages

Using the natural commands workflow specification we can generate code to other languages, like, Python or R:

ToEpidemiologyModelingWorkflowCode[lsCommands, "Target" -> "Python"](*"obj = ECMMonUnit( model = SEI2HRModel())obj = ECMMonAssignInitialConditions( ecmObj = obj, initConds = [TPt = 100000])obj = ECMMonAssignInitialConditions( ecmObj = obj, initConds = [INSPt = 0])obj = ECMMonAssignInitialConditions( ecmObj = obj, initConds = [ISSPt = 1])obj = ECMMonAssignRateValues( ecmObj = obj, rateValues = [contactRateINSP = 0.56])obj = ECMMonAssignRateValues( ecmObj = obj, rateValues = [contactRateISSP = 0.58])obj = ECMMonAssignRateValues( ecmObj = obj, rateValues = [contactRateHP = 0.1])obj = ECMMonSimulate( ecmObj = obj, maxTime = 240)obj = ECMMonPlotSolutions( ecmObj = obj, stocksSpec = \".*Population\")obj = ECMMonCalibrate( ecmObj = obj,  target = [DIPt = tsDeaths], parameters = [contactRateISSP = [0.1, 0.7]] )"*)



[Wk1] Wikipedia entry, Monad.

[Wk2] Wikipedia entry, “Compartmental models in epidemiology”.

[Wk3] Wikipedia entry, “Coronavirus disease 2019”.

[BC1] Lucia Breierova, Mark Choudhari, An Introduction to Sensitivity Analysis, (1996), Massachusetts Institute of Technology.

[JS1] John D.Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World. (2000), New York: McGraw.

[HH1] Herbert W. Hethcote (2000). “The Mathematics of Infectious Diseases”. SIAM Review. 42 (4): 599–653. Bibcode:2000SIAMR..42..599H. doi:10.1137/s0036144500371907.

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

[AA2] Anton Antonov, “Coronavirus propagation modeling considerations”, (2020), SystemModeling at GitHub/antononcube.

[AA3] Anton Antonov, “Basic experiments workflow for simple epidemiological models”, (2020), SystemModeling at GitHub/antononcube.

[AA4] Anton Antonov, “Scaling of Epidemiology Models with Multi-site Compartments”, (2020), SystemModeling at GitHub/antononcube.

[AA5] Anton Antonov, “WirVsVirus hackathon multi-site SEI2R over a hexagonal grid graph”, (2020), SystemModeling at GitHub/antononcube.

[AA6] Anton Antonov, “NY Times COVID-19 data visualization”, (2020), SystemModeling at GitHub/antononcube.

[AA7] Anton Antonov, “SEI2HR model with quarantine scenarios”, (2020), SystemModeling at GitHub/antononcube.

[AA8] Anton Antonov, “SEI2HR-Econ model with quarantine and supplies scenarios”, (2020), SystemModeling at GitHub/antononcube.

[AA9] Anton Antonov, Modelers questionnaire, (2020), SystemModeling at GitHub/antononcube.

[AA10] Anton Antonov, Calibrators questionnaire, (2020), SystemModeling at GitHub/antononcube.

[AA11] Anton Antonov, A monad for classification workflows, (2018), MathematicaForPrediction at WordPress.

Repositories, packages

[WRI1] Wolfram Research, Inc., “Epidemic Data for Novel Coronavirus COVID-19”, WolframCloud.

[WRI2] Wolfram Research Inc., USA county records, (2020), System Modeling at GitHub.

[NYT1] The New York Times, Coronavirus (Covid-19) Data in the United States, (2020), GitHub.

[AAr1] Anton Antonov, Coronavirus propagation dynamics project, (2020), SystemModeling at GitHub/antononcube.

[AAr2] Anton Antonov, Epidemiology Compartmental Modeling Monad R package, (2020), ECMMon-R at GitHu/antononcube.

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

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

[AAp3] Anton Antonov, Epidemiology models Mathematica package, (2020), SystemModeling at GitHub/antononcube.

[AAp4] Anton Antonov, Epidemiology models modifications Mathematica package, (2020), SystemModeling at GitHub/antononcube.

[AAp5] Anton Antonov, Epidemiology modeling visualization functions Mathematica package, (2020), SystemModeling at GitHub/antononcube.

[AAp6] Anton Antonov, System dynamics interactive interfaces functions Mathematica package, (2020), SystemModeling at GitHub/antononcube.

[AAp7] Anton Antonov, Multi-site model simulation Mathematica package, (2020), SystemModeling at GitHub/antononcube.

[AAp8] Anton Antonov, Monadic Epidemiology Compartmental Modeling Mathematica package, (2020), SystemModeling at GitHub/antononcube.

[AAp9] Anton Antonov, Hextile bins Mathematica package, (2020), MathematicaForPrediction at GitHub/antononcube.

[AAp10] Anton Antonov, Monadic Epidemiology Compartmental Modeling Mathematica unit tests, (2020), SystemModeling at GitHub/antononcube.

[AAp11] Anton Antonov, Epidemiology Models NDSolve Mathematica unit tests, (2020), SystemModeling at GitHub/antononcube.

[AAp12] Anton Antonov, Monadic contextual classification random pipelines Mathematica unit tests, (2018), MathematicaForPrediction at GitHub/antononcube.

[AAp13] Anton Antonov, Epidemiology Modeling Workflows Raku package, (2020), Raku-DSL-English-EpidemiologyModelingWorkflows at GitHu/antononcube.

[AAp14] Anton Antonov, External Parsers Hookup Mathematica package, (2019), ConversationalAgents at GitHub.

Phone dialing conversational agent


This blog post proclaims the first committed project in the repository ConversationalAgents at GitHub. The project has designs and implementations of a phone calling conversational agent that aims at providing the following functionalities:

  • contacts retrieval (querying, filtering, selection),
  • contacts prioritization, and
  • phone call (work flow) handling.
  • The design is based on a Finite State Machine (FSM) and context free grammar(s) for commands that switch between the states of the FSM. The grammar is designed as a context free grammar rules of a Domain Specific Language (DSL) in Extended Backus-Naur Form (EBNF). (For more details on DSLs design and programming see [1].)

    The (current) implementation is with Wolfram Language (WL) / Mathematica using the functional parsers package [2, 3].

    This movie gives an overview from an end user perspective.

    General design

    The design of the Phone Conversational Agent (PhCA) is derived in a straightforward manner from the typical work flow of calling a contact (using, say, a mobile phone.)

    The main goals for the conversational agent are the following:

    1. contacts retrieval — search, filtering, selection — using both natural language commands and manual interaction,
    2. intuitive integration with the usual work flow of phone calling.

    An additional goal is to facilitate contacts retrieval by determining the most appropriate contacts in query responses. For example, while driving to work by pressing the dial button we might prefer the contacts of an up-coming meeting to be placed on top of the prompting contacts list.

    In this project we assume that the voice to text conversion is done with an external (reliable) component.

    It is assumed that an user of PhCA can react to both visual and spoken query results.

    The main algorithm is the following.

    1) Parse and interpret a natural language command.

    2) If the command is a contacts query that returns a single contact then call that contact.

    3) If the command is a contacts query that returns multiple contacts then :

    3.1) use natural language commands to refine and filter the query results,

    3.2) until a single contact is obtained. Call that single contact.

    4) If other type of command is given act accordingly.

    PhCA has commands for system usage help and for canceling the current contact search and starting over.

    The following FSM diagram gives the basic structure of PhCA:


    This movie demonstrates how different natural language commands switch the FSM states.

    Grammar design

    The derived grammar describes sentences that: 1. fit end user expectations, and 2. are used to switch between the FSM states.

    Because of the simplicity of the FSM and the natural language commands only few iterations were done with the Parser-generation-by-grammars work flow.

    The base grammar is given in the file "./Mathematica/PhoneCallingDialogsGrammarRules.m" in EBNF used by [2].

    Here are parsing results of a set of test natural language commands:


    using the WL command:

    ParsingTestTable[ParseJust[pCALLCONTACT\[CirclePlus]pCALLFILTER], ToLowerCase /@ queries]

    (Note that according to PhCA’s FSM diagram the parsing of pCALLCONTACT is separated from pCALLFILTER, hence the need to combine the two parsers in the code line above.)

    PhCA’s FSM implementation provides interpretation and context of the functional programming expressions obtained by the parser.

    In the running script "./Mathematica/PhoneDialingAgentRunScript.m" the grammar parsers are modified to do successful parsing using data elements of the provided fake address book.

    The base grammar can be extended with the "Time specifications grammar" in order to include queries based on temporal commands.


    In order to experiment with the agent just run in Mathematica the command:


    The imported Wolfram Language file, "./Mathematica/PhoneDialingAgentRunScript.m", uses a fake address book based on movie creators metadata. The code structure of "./Mathematica/PhoneDialingAgentRunScript.m" allows easy experimentation and modification of the running steps.

    Here are several screen-shots illustrating a particular usage path (scan left-to-right):

    "PhCA-1-call-someone-from-x-men"" "PhCA-2-a-producer" "PhCA-3-the-third-one

    See this movie demonstrating a PhCA run.


    [1] Anton Antonov, "Creating and programming domain specific languages", (2016), MathematicaForPrediction at WordPress blog.

    [2] Anton Antonov, Functional parsers, Mathematica package, MathematicaForPrediction at GitHub, 2014.

    [3] Anton Antonov, "Natural language processing with functional parsers", (2014), MathematicaForPrediction at WordPress blog.

    Simple time series conversational engine


    In this blog post I am going to discuss the creation (design and programming) of a simple conversational engine for time series analysis. The conversational engine responds to commands for:
    1. loading time series data (weather data, stock data, or data files),
    2. finding outliers,
    3. analysis by curve fitting,
    4. plotting, and
    5. help and state changes.

    This blog post is related to the blog post [1] and uses the package with functional parsers [2] and the Quantile regression package [3].

    I programmed this engine ten months ago and here is link to a movie that demonstrates it, [4]: https://www.youtube.com/watch?v=wlZ5ANglVI4 .

    A package with all of the code of the conversational engine can be downloaded from GitHub, [12] and executed with Import:


    More about functional parsers, which were essential for the building of the conversational engine, can be read in “Functional parsers” by Fokker, [5].

    Design of the conversational engine

    The conversational engine is a finite state machine. It has a command recognizer state and command application state. The switching between the states is done upon completion states functions. The recognized commands are English sentences. Some of the sentences include Mathematica code.

    Let use describe the structure of the conversational engine.


    1. Acceptor state
    The Acceptor state parses and recognizes commands. If the given command is successfully parsed then the parsing result is given to the Transducer state. Otherwise a message “Unknown command” is emitted and no state change is made (i.e.we are still in the Acceptor state).

    2. Transducer state
    Interprets the parsed a command by the Acceptor state. This state emits different messages according to the loaded data and accumulated graphics objects.

    The states are named according to the classification given in the Wikipedia article about finite-state machines, [5]. According to that classification the conversational engine is a Mealy machine.


    The results of the applied time series analysis are graphical. The Transducer state accumulates the graphs of the commands and displays them with the data. The conversational engine responses have a spoken part and a graphics part. The responses have at least a spoken part. The graphics part can omitted.


    The weather data commands can be for temperature, wind speed, and pressure over a specified city. For example, “load temperature of Atlanta” or “load the Chicago wind speeds”.

    The financial data commands can be for stock price or trading volume. For example, “load trade volume of NYSE:GOOG” or “load stock price of NYSE:APPL”.

    The weather and stocks data loading commands do not take time period, they take only city and company specifications for WeatherData and FinancialData respectively.

    The loading of data files is done with commands like “load file ‘~/LogDataWithSkewedNoise.csv’ “.

    The time series analysis commands we consider are: curve fitting by a list specified functions, finding regression quantiles, and finding outliers. For example, “compute least squares fit of {1,x,Sin[x/10^6]}”, “compute 5 regression quantiles”, “find top outliers”.

    The graphics commands are “plot data”, “plot(s) joined”, “plot(s) not joined”, “clear graphics”.

    In addition, the conversational engine takes global, service commands like “help”, “all commands”, “start over’, etc.

    Grammar specification

    In order to respond to the commands that are English sentences we have to be able to parse these sentences into concrete state or message specifications. This means that we have to define the grammatical structures of the command sentences and the words in them.

    This section describes the grammar of the commands using Extended Backus-Naur Form (EBNF). The package FunctionalParsers.m, [2], has Mathematica functions that generate parsers from EBNF of a grammar (see [9] and the next section). The resulting parsers are used in the conversational engine described above.

    Commands for loading data explanations

    Let us explain the derivation and rationale of the grammar rules for the data loading commands. (I am not going to describe all the language considerations and decisions I have made for the commands grammar. These decisions can be traced in the EBNF grammar specification.)

    We want to parse and interpret commands like:
    1. load data file ‘timeTable.csv’ ,
    2. load temperature of Atlanta ,
    3. load the wind speeds of Chicago ,
    4. load the stock price of NYSE:APPL ,
    5. load trading volume of NYSE:GOOG .

    We can see that we have three different types of data loading commands: for files, for weather data, and for financial data. Each of these commands when parsed would be easy to interpret using Mathematica’s functions Import, WeatherData, and FinancialData respectively. Note that — because we want to keep the grammar simple — for the weather and financial data there is no time period specification. For the weather data we are going to load data for the last 60 days; for the financial data for the last 365 days. Also, for simplicity, we are not going to consider data loading commands that start with the phrases “get”, “get me”, “please load”, etc.

    To this end the general load data command structure is:


    Note that in the grammar rules above some of the words are optional; the optional words are put within square brackets. With the first rule we have separated the file loading from the weather and financial data loading. The file loading is further specified with rule 2. Rule 3 separates the preamble of the weather and financial data loading commands from the data specification. The latter is partitioned in rule 4. Rule 5 gives all of the possible data type requests for the weather data: temperature, pressure, and wind speed. Rule 6 gives the two possible types for financial data requests: stock price and trade volume. The file name, city, and company specifications are left out, but the rules for them are be easy to formulate.

    The command grammar used in the conversational engine uses additional data loading definitions and allows “reverse” data load commands like: “load the Chicago wind speeds”.

    Full grammar specification

    In this sub-section the full grammar specification is given. Note that some parts of the commands are specified to be dropped from the parsing results using the symbols “\[RightTriangle]” and “\[LeftTriangle]”. (For example ” ‘trade’ \[RightTriangle] ‘volume’ ” means “parse the sequence of words ‘trade’ ‘volume’ and return ‘volume’ as a result”.) The specification ‘_LetterString’ is an implementation shortcut for matching words and identifiers with the package FunctionalParsers.m, [2].


    The rules were simplified for clarity, the actual conversational engine code can be retrieved, [8].

    The grammar specified above takes complicated commands like “find regression quantiles over temperature of Atlanta”.

    (In certain cases I prefer reading grammars in EBNF using Emacs and ebnf-mode.el, [7].)

    Parsers for the grammar

    Using the grammar specification [8] and the package FunctionalParsers.m, [2], we can generate the parsers for that grammar. A detailed description of the parser generation process is given in [9].

    Here is the generation step:

    tokens = ToTokens[timeSeriesEBNFCode];
    res = GenerateParsersFromEBNF[tokens];
    res // LeafCount

    Out[4]= 3069

    The generation step creates a set of parsers:

    In[16]:= Magnify[Shallow[res[[1, 1, 2]], 6], 0.8]


    Here are the parsing results returned by the “master” rule pTSCOMMAND:

    Here is another list for the engine’s “service” commands:

    And see their parsing outcome using the function ParsingTestTable provided by FunctionalParsers.m:

    questions = {"load data file '~/example.csv'",
    "load file '~/anotherExample.csv'"};
    ParsingTestTable[ParseShortest[pLOADFILECOMMAND], questions]


    Note that the parser takes commands with Mathematica expressions parts if those expressions are written without spaces:

    questions = {"least squares fit with x+Sin[x]"};
    ParsingTestTable[ParseShortest[pTSCOMMAND], questions]


    This type of commands are specified with the grammar rule:

    = ( ‘least’ , ‘squares’ , [ ‘fit’ ] , [ ‘with’ | ‘of’ ] ) \[RightTriangle] ‘_String’ ;

    Interpreters (for the parsers)

    Next we have to write interpreters for the parsing results. Note that the parsing results are Mathematica expressions with heads that hint to the semantic meaning behind the parsed commands. Since we are viewing the conversational engine as a finite state machine we can say that for each parsing result we have to write a function that “reacts” to result’s head and content. This is a fairly straightforward process very similar to the one described by the Interpreter design pattern, [10].

    A package with all of the code of the conversational engine can be downloaded from GitHub, [12]. The interpreters code can be found in the section “Interpreters”.

    The conversational engine programming

    The conversational engine was programmed using Mathematica’s dynamic evaluation functions and mechanisms. The implementation follows the Absorber and Transductor state design described above in the section “Design of the conversational engine”.

    The engine has three panels: for command input, for spoken output, and for graphical output. It is fully demonstrated with the movie “Time series conversational engine” posted on YouTube, [4].

    As it was mentioned in the introduction the engine can be run using Import:



    Future plans

    The engine can be extended in many directions depending on the final goals for it.

    One must have extension is the ability to take time period specifications for the weather and financial data.

    The time series analysis can be extended with more detailed analysis using functions like TimeSeriesModelFit or heteroscedasticity analysis using multiple regression quantiles, [11].

    A third set of extensions is the ability to load additional kinds of data, like, the evolutions of countries populations or gross domestic products.


    [1] Anton Antonov, Natural language processing with functional parsers, blog post at WordPress, (2014).

    [2] Anton Antonov, Functional parsers Mathematica package, source code at GitHub, https://github.com/antononcube/MathematicaForPrediction, package FunctionalParsers.m, (2014).

    [3] Anton Antonov, Quantile regression Mathematica package, source code at GitHub, https://github.com/antononcube/MathematicaForPrediction, package QuantileRegression.m, (2013).

    [4] Anton Antonov, Time series conversational engine, a YouTube movie.

    [5] Wikipedia entry: Finite-state Machine, http://en.wikipedia.org/wiki/Finite-state_machine .

    [6] Jeroen Fokker, Functional parsers, In: Advanced Functional Programming, Tutorial text of the First international spring school on advanced functional programming techniques in Båstad, Sweden, may 1995. (Johan Jeuring and Erik Meijer, eds). Berlin, Springer 1995 (LNCS 925), pp. 1-23. http://www.staff.science.uu.nl/~fokke101/article/parsers/index.html .

    [7] Jeramey Crawford, Extended Backus-Naur Form Mode, ebnf-mode.el, emacs LISP at Github, https://github.com/jeramey/ebnf-mode .

    [8] Anton Antonov, Time series conversational engine grammar in EBNF, TimeSeriesConversationalEngineGrammar.ebnf at the MathematicaForPrediction project at GitHub.

    [9] Anton Antonov, Functional parsers for an integration requests language, PDF document at GitHub, https://github.com/antononcube/MathematicaForPrediction/tree/master/Documentation , (2014).

    [10] Erich GammaTaligent, Richard Helm, Ralph Johnson, John Vlissides, Design patterns: elements of reusable object-oriented softwareAddison-Wesley Longman Publishing Co., Inc. Boston, MA, USA ©1995, ISBN:0-201-63361-2, http://en.wikipedia.org/wiki/Design_Patterns .

    [11] Anton Antonov, Estimation of conditional density distributions, “Mathematica for prediction algorithms” blog at WordPress, (2014).

    [12] Anton Antonov, Simple Time Series Conversational Engine Mathematica package, SimpleTimeSeriesConversationalEngine.m at the MathematicaForPrediction project at GitHub.