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

The main goal of this notebook is to provide some basic views and insights into the landscape of cryptocurrencies. The “landscape” we consider consists of price action and trading volume time series for cryptocurrencies found in Yahoo Finance.

Here is the work plan followed in this notebook:

Get cryptocurrency data

Do basic data analysis over suitable date ranges

Gather important cryptocurrency events

Plot together cryptocurrency prices and trading volume time series together with the events

Make observations and conjectures over the plots

Find “global” correlations between the different cryptocurrencies

Find clusters of cryptocurrencies based on time series correlations

Here are some details for the steps above:

The procedure of obtaining the cryptocurrencies data, point 1, is explained in detail in [AA1].

There is a dedicated resource object CrypocurrencyData that provides cryptocurrency data and related documentation.

The cryptocurrency events data, point 3, is taken from different news sites.

Links are provided in the corresponding dataset.

The points 6 and 7 follow similar explorations (and code) described in [AA2, AA3].

Those two articles deal with COVID-19 time series.

Remark: Note that in this notebook we do not discuss philosophical, macro-economic, and environmental issues with cryptocurrencies. We only discuss financial time series data.

Cryptocurrencies data

The cryptocurrencies data used in this notebook is obtained from found in Yahoo Finance . The procedure of obtaining the cryptocurrencies data is explained in detail in [AA1]. There is a dedicated resource object CrypocurrencyData that provides the cryptocurrency data and related documentation.

Remark:FinancialData is “aware” of 10 cryptocurrencies, but that is not documented (as far as I can tell) and only prices are provided. (For more details see the discussion in CrypocurrencyData.) Here are examples:

In this section we analyze the summaries of cryptocurrencies data in order to derive a list of the most significant ones.

We choose the phrase “significant cryptocurrency” to mean “a cryptocurrency with high market capitalization, price, or trading volume.”

Together with the summaries we look into the Pareto principle adherence of the corresponding values.

Remark: The Pareto principle adherence should be interpreted carefully here – the cryptocurrencies are not mutually exclusive when in comes to money invested and trading volumes. Nevertheless, we can interpret the corresponding value ratios as indicators of “mind share” or “significance.”

By summaries

Here is a summary of the cryptocurrencies we consider (from Yahoo Finance) ordered by “Market Cap” (largest first):

In this section we make a dataset that has the dates of certain cryptocurrency related events and links to their news announcements.

The events were taken by observing cryptocurrency board stories in the news aggregation site slashdot.org.

lsEventData ={{"Jun 18, 2021","China to shut down over 90% of its Bitcoin mining capacity after local bans","https://www.globaltimes.cn/page/202106/1226598.shtml"},{"Jun 10, 2021","Global banking regulators call for toughest rules for cryptocurrencies","https://www.theguardian.com/technology/2021/jun/10/global-banking-regulators-cryptocurrencies-bitcoin"},{"June 10, 2021","IMF sees legal, economic issues with El Salvador's bitcoin move","https://www.reuters.com/business/finance/imf-sees-legal-economic-issues-with-el-salvador-bitcoin-move-2021-06-10/"},{"June 8, 2021","El Salvador Becomes First Country To Adopt Bitcoin as Legal Tender After Passing Law","https://www.cnbc.com/2021/06/09/el-salvador-proposes-law-to-make-bitcoin-legal-tender.html"},{"June 8, 2021","US recovers millions in cryptocurrency paid to Colonial Pipeline ransomware hackers","https://edition.cnn.com/2021/06/07/politics/colonial-pipeline-ransomware-recovered/"},{"June 4, 2021","Start of Bitcoin 2021: World\[CloseCurlyQuote]s Largest Cryptocurrency Conference Coming To Wynwood","https://miami.cbslocal.com/2021/06/04/bitcoin-2021-worlds-largest-cryptocurrency-conference-coming-to-wynwood/"},{"June 6, 2021","End of Bitcoin 2021: World\[CloseCurlyQuote]s Largest Cryptocurrency Conference Coming To Wynwood","https://miami.cbslocal.com/2021/06/04/bitcoin-2021-worlds-largest-cryptocurrency-conference-coming-to-wynwood/"},{"May 28, 2021","Iran Bans Crypto Mining After Months of Blackouts","https://gizmodo.com/iran-bans-crypto-mining-after-months-of-blackouts-1846991039"},{"May 19, 2021","Bitcoin, Ethereum prices in free fall as China plans crackdown on mining and trading","https://www.cnet.com/news/bitcoin-ethereum-prices-in-freefall-as-china-plans-crackdown-on-mining-and-trading/#ftag=CAD590a51e"}};dsEventData = Dataset[lsEventData][All, AssociationThread[{"Date","Event","URL"}, #] &];dsEventData = dsEventData[All,Join[Prepend[#,"DateObject"-> DateObject[#Date]], <|"URL"-> URL[#URL]|>] &];dsEventData = dsEventData[SortBy[#DateObject &]]

Cryptocurrency time series with events

In this section we discuss possible correlation and causation effects of reported cryptocurrency events.

Remark: The discussion is based on time series and events only, without considering other operational properties of the cryptocurrencies.

Here is a date range:

dateRange ={"May 15 2021","Jun 21 2021"};

Here get time series for the daily opening and closing prices for the selected date range:

Here we plot the cryptocurrency events with together with the BTC trading volume time series:

CryptocurrencyPlot[{aCCVolume, dsEventData},PlotLabel->"BTC and ETH trading volumes",ImageSize->1200]

Observations

Going down

We can see that opening prices and volume going down correlate with:

The news announcement that China plans to crackdown on mining and trading

The news announcement Iran bans crypto mining

The Sichuan Provincial Development and Reform Commission and the Sichuan Energy Bureau issue of a joint notice, ordering local electricity companies to “screen, clean up and terminate” mining operations

The start of the “Bitcoin 2021” conference

Related conjectures:

We can easily conjecture that 1 and 2 made cryptocurrencies (Bitcoin) less attractive to miners or traders in China and Iran, hence the price and the volume went down.

The most active Bitcoin traders were attending the “Bitcoin 2021” conference, hence the price and volume went down.

Going up

We can see the prices and volume going up correlate with:

The news announcement of El Salvador adopting BTC as legal tender currency

The news announcement that US Justice Department recovered most of the ransom paid to the Colonial Pipeline hackers

The end of the “Bitcoin 2021” conference

Related conjectures:

Of course, a country deciding to use BTC as legal tender would make (some) traders willing to invest in BTC.

The announcement that USA Justice Department, have made (some) traders to more confidently invest in BTC.

Although, the opposite could also happen – for some people if BTC can be recovered by law enforcement, then BTC is less attractive for financial transactions.

After the end of “Bitcoin 2021” conference the attending traders resumed their usual activity.

That conjecture and the “start of Bitcoin 2021” conjecture above support each other.

The same pattern is observed for both BTC and ETH trading volumes.

Time series correlations

In this section we compute and visualize correlations between the time series of a set of cryptocurrencies.

I investigated the data with several other methods:

Clustering with different methods and distance functions

Clustering after the application of Independent Component Analysis (ICA), [AAw5]

Time series analysis with Quantile Regression (QR), [AAw6]

None of the outcomes provided some “immediate”, notable insight. The analyses with ICA and QR, though, seem to provide some interesting and fruitful future explorations.

In this article we proclaim the preparation and availability of interactive interfaces to two Time Series Search Engines (TSSEs) over COVID-19 data. One TSSE is based on Apple Mobility Trends data, [APPL1]; the other on The New York Times COVID-19 data, [NYT1].

Here are links to interactive interfaces of the TSSEs hosted (and publicly available) at shinyapps.io by RStudio:

Motivation: The primary motivation for making the TSSEs and their interactive interfaces is to use them as exploratory tools. Combined with relevant data analysis (e.g. [AA1, AA2]) the TSSEs should help to form better intuition and feel of the spread of COVID-19 and related data aggregation, public reactions, and government polices.

The rest of the article is structured as follows:

Brief descriptions the overall process, the data

Brief descriptions the search engines structure and implementation

Discussions of a few search examples and their (possible) interpretations

The overall process

For both search engines the overall process has the same steps:

Ingest the data

Do basic (and advanced) data analysis

Make (and publish) reports detailing the data ingestion and transformation steps

Enhance the data with transformed versions of it or with additional related data

Make a Time Series Sparse Matrix Recommender (TSSMR)

Make a Time Series Search Engine Interactive Interface (TSSEII)

Make the interactive interface easily accessible over the World Wide Web

Here is a flow chart that corresponds to the steps listed above:

Data

The Apple data

The Apple Mobility Trends data is taken from Apple’s site, see [APPL1]. The data ingestion, basic data analysis, time series seasonality demonstration, (graph) clusterings are given in [AA1]. (Here is a link to the corresponding R-notebook .)

(It was too much work to get the weather data using some of the well known weather data R packages.)

The New York Times data

The New York Times COVID-19 data is taken from GitHub, see [NYT1]. The data ingestion, basic data analysis, and visualizations are given in [AA2]. (Here is a link to the corresponding R-notebook .)

The search engines

The following sub-sections have screenshots of the TSSE interactive interfaces.

I did experiment with combining the data of the two engines, but did not turn out to be particularly useful. It seems that is more interesting and useful to enhance the Apple data engine with temperature data, and to enhance The New Your Times engine with the (consecutive) differences of the time series.

Structure

The interactive interfaces have three panels:

Nearest Neighbors

Gives the time series nearest neighbors for the time series of selected entity.

Has interactive controls for entity selection and filtering.

Trend Finding

Gives the time series that adhere to a specified named trend.

Has interactive controls for trend curves selection and entity filtering.

Notes

Gives references and data objects summary.

Implementation

Both TSSEs are implemented using the R packages “SparseMatrixRecommender”, [AAp1], and “SparseMatrixRecommenderInterfaces”, [AAp2].

The package “SparseMatrixRecommender” provides functions to create and use Sparse Matrix Recommender (SMR) objects. Both TSSEs use underlying SMR objects.

The Apple data TSSE has four types of time series (“entities”). The first three are normalized volumes of Apple maps requests while driving, transit transport use, and walking. (See [AA1] for more details.) The fourth is daily mean temperature at different geo-locations.

Here are screenshots of the panels “Nearest Neighbors” and “Trend Finding” (at interface launch):

The New York Times TSSE has four types of time series (aggregated) cases and deaths, and their corresponding time series differences.

Here are screenshots of the panels “Nearest Neighbors” and “Trend Finding” (at interface launch):

Examples

In this section we discuss in some detail several examples of using each of the TSSEs.

Apple data search engine examples

Here are a few observations from [AA1]:

The COVID-19 lockdowns are clearly reflected in the time series.

The time series from the Apple Mobility Trends data shows strong weekly seasonality. Roughly speaking, people go to places they are not familiar with on Fridays and Saturdays. Other work week days people are more familiar with their trips. Since much lesser number of requests are made on Sundays, we can conjecture that many people stay at home or visit very familiar locations.

Here are a few assumptions:

Where people frequently go (work, school, groceries shopping, etc.) they do not need directions that much.

People request directions when they have more free time and will for “leisure trips.”

During vacations people are more likely to be in places they are less familiar with.

People are more likely to take leisure trips when the weather is good. (Warm, not raining, etc.)

Nice, France vs Florida, USA

Consider the results of the Nearest Neighbors panel for Nice, France.

Since French tend to go on vacation in July and August ([SS1, INSEE1]) we can see that driving, transit, and walking in Nice have pronounced peaks during that time:

Of course, we also observe the lockdown period in that geographical area.

Compare those time series with the time series from driving in Florida, USA:

We can see that people in Florida, USA have driving patterns unrelated to the typical weather seasons and vacation periods.

(Further TSSE queries show that there is a negative correlation with the temperature in south Florida and the volumes of Apple Maps directions requests.)

Italy and Balkan countries driving

We can see that according to the data people who have access to both iPhones and cars in Italy and the Balkan countries Bulgaria, Greece, and Romania have similar directions requests patterns:

(The similarities can be explained with at least a few “obvious” facts, but we are going to restrain ourselves.)

The New York Times data search engine examples

In Broward county, Florida, USA and Cook county, Illinois, USA we can see two waves of infections in the difference time series:

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:

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.

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:

(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):

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:

Develop a single-site model for relatively densely populated geographical area for which the assumptions of the classical models (approximately) hold.

Extend the single-site model into a large-scale multi-site model using statistically derived traveling patterns; see [AA4].

Supply the multi-site model with appropriately prepared data.

Run multiple simulations to see large scale implications of different policies.

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

Make a single-site model.

Assign stocks initial conditions.

Assign rates values.

Simulate.

Plot results.

Go to 2.

Multi-site model workflow

Make a single-site model.

Assign initial conditions and rates.

Scale the single-site model into a multi-site model.

The single-site assigned rates become “global” when the single-site model is scaled.

The scaling is based on assumptions for traveling patterns of the populations.

There are few alternatives for that scaling:

Using locations geo-coordinates

Using regular grids covering a certain area based on in-habited locations geo-coordinates

Using traveling patterns contingency matrices

Using “artificial” patterns of certain regular types for qualitative analysis purposes

Enhance the multi-site traveling patterns matrix and re-scale the single site model.

We might want to combine traveling patterns by ground transportation with traveling patterns by airplanes.

For quarantine scenarios this might a less important capability of the monad.

Hence, this an optional step.

Assign stocks initial conditions for each of the sites in multi-scale model.

Assign rates for each of the sites.

Simulate.

Plot global simulation results.

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.

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.

Equations

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

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].

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.

Definitions

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).

Roles

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

Process

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:

Stakeholder requires certain scenarios to be investigated.

Modeler prepares the model for those scenarios.

Stakeholder and Modeler formulate a calibration request.

Calibrator uses the specifications from the calibration request to:

Calibrate the model

Derive model outcomes results

Provide model qualitative results

Provide model sensitivity analysis results

Modeler (and maybe Stakeholder) review the results and decides should more calibration be done.

I.e. go to 3.

Modeler does batch simulations with the calibrated model for the investigation scenarios.

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:

Make a model object

Batch simulate over a few combinations of parameters and show:

Plots of the simulation results for all populations

Plots of the simulation results for a particular population

Batch simulate over the Cartesian (outer) product of values lists of a selected pair of parameters and show the corresponding plots of all simulations

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:

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:

Ingest data for infected cases, deaths due to disease, etc.

Choose a model to calibrate.

Make the calibration targets – those a vectors corresponding to time series over regular grids.

Consider using all of the data in order to evaluate model’s applicability.

Consider using fractions of the data in order to evaluate model’s ability to predict the future or reconstruct data gaps.

Choose calibration parameters and corresponding ranges for their values.

If more than one target choose the relative weight (or importance) of the targets.

Calibrate the model.

Evaluate the fitting between the simulation results and data.

Using statistics and plots.

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):

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.)

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.

(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 for240 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:

[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.

(Note that mind-map’s PDF has hyperlinks. Also, see the folder Presentation-aids. )

The organizers and I did a poll for what people want to hear. After discussing the results of the 15 votes from that poll we decided the presentation to be a methodological one instead of a know-how one.

Approximately 30% of the presentation was based on the R-project “COVID-19-modeling-in-R”, [AA1].

Approximately 30% of the presentation was based on an R-programmed software monad for epidemiology compartmental models, ECMMon-R, [AAr2].

For the rest were used frameworks, simulations, and graphics made with Mathematica, [AAr1].

The presentation was given online (because of COVID-19) using Zoom. 90 people registered. Nearly 40 showed up (and maybe 20 stayed throughout.)

(Note that mind-map’s PDF has hyperlinks. Also, see the folder Presentation-aids. )

Approximately 70% of the presentation was based on an R-programmed software monad for epidemiology compartmental models, ECMMon-R, [AAr2]. For the rest were used frameworks, simulations, and graphics made with Mathematica, [AAr1], and Wolfram System Modeler .

The presentation was given online (because of COVID-19) using Zoom. 190 people registered. Nearly 70 showed up (and maybe 60 stayed throughout.)

“SEI2HR” stands for “Susceptible, Exposed, Infected two, Hospitalized, Recovered” (populations.) “Econ” stands for “Economic”.

In this notebook we also deal with both quarantine scenarios and medical supplies scenarios. In the notebook [AA4] we deal with quarantine scenarios over a simpler model, SEI2HR.

Remark: We consider the contagious disease propagation models as instances of the more general System Dynamics (SD) models. We use SD terminology in this notebook.

The models

SEI2R

The model SEI2R is introduced and explained in the notebook [AA2]. SEI2R differs from the classical SEIR model, [Wk1, HH1], with the following elements:

Two separate infected populations: one is “severely symptomatic”, the other is “normally symptomatic”

The monetary equivalent of lost productivity due to infected or died people is tracked

SEI2HR

For the formulation of SEI2HR we use a system of Differential Algebraic Equations (DAE’s). The package [AAp1] allows the use of a formulation that has just Ordinary Differential Equations (ODE’s).

Here are the unique features of SEI2HR:

People stocks

There are two types of infected populations: normally symptomatic and severely symptomatic.

There is a hospitalized population.

There is a deceased from infection population.

Hospital beds

Hospital beds are a limited resource that determines the number of hospitalized people.

Only severely symptomatic people are hospitalized according to the available hospital beds.

The hospital beds stock is not assumed constant, it has its own change rate.

Money stocks

The money from lost productivity is tracked.

The money for hospital services is tracked.

SEI2HR-Econ

SEI2HR-Econ adds the following features to SEI2HR:

Medical supplies

Medical supplies production is part of the model.

Medical supplies delivery is part of the model..

Medical supplies accumulation at hospitals is taken into account.

Medical supplies demand tracking.

Hospitalization

Severely symptomatic people are hospitalized according to two limited resources: hospital beds and medical supplies.

Money stocks

Money for medical supplies production is tracked.

SEI2HR-Econ’s place a development plan

This graph shows the “big picture” of the model development plan undertaken in [AAr1] and SEI2HR (discussed in this notebook) is in that graph:

Notebook structure

The rest of notebook has the following sequence of sections:

Package load section

SEI2HR-Econ structure in comparison of SEI2HR

Explanations of the equations of SEI2HR-Econ

Quarantine scenario modeling preparation

Medical supplies production and delivery scenario modeling preparation

Parameters and initial conditions setup

Populations, hospital beds, quarantine scenarios, medical supplies scenarios

Simulation solutions

Interactive interface

Sensitivity analysis

Load packages

The epidemiological models framework used in this notebook is implemented with the packages [AAp1-AAp4, AA3]; many of the plot functions are from the package [AAp5].

In this section we provide rationale for the equations of SEI2HR-Econ.

The equations for Susceptible, Exposed, Infected, Recovered populations of SEI2R are “standard” and explanations about them are found in [WK1, HH1]. For SEI2HR those equations change because of the stocks Hospitalized Population and Hospital Beds. For SEI2HR-Econ the SEI2HR equations change because of the stocks Medical Supplies, Medical Supplies Demand, and Hospital Medical Supplies.

The equations time unit is one day. The time horizon is one year. Since we target COVID-19, [Wk2, AA1], we do not consider births.

Remark: For convenient reading the equations in this section have tooltips for the involved stocks and rates.

Verbalization description of the model

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. The severely symptomatic infected are hospitalized if there are enough hospital beds and enough medical supplies. 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 different types of populations (infected, hospitalized, recovered, etc.) have their own consumption rates of medical supplies. The medical supplies are produced with a certain rate (units per day) and delivered after a certain delay period. The hospitals have their own storage for medical supplies. Medical supplies are delivered to the hospitals only, non-hospitalized people go to the medical supplies producer to buy supplies. The hospitals have precedence for the medical supplies: if the medical supplies are not enough for everyone, the hospital needs are covered first (as much as possible.)

The medical supplies producer has a certain storage capacity (for supplies.) The medical supplies delivery vehicles have a certain – generally speaking, smaller – capacity. The hospitals have a certain capacity to store medical supplies. It is assumed that both producer and hospitals have initial stocks of medical supplies. (Following a certain normal, general preparedness protocol.)

The combined demand from all populations for medical supplies is tracked (accumulated.) The deaths from infection are tracked (accumulated.) Money for medical supplies production, money for hospital services, and money from lost productivity are tracked (accumulated.)

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

Code for the equations

Each equation in this section are derived with code like this:

and then the output cell is edited to be “DisplayFormula” and have CellLabel value corresponding to the stock of interest.

The infected and hospitalized populations

SEI2HR has two types of infected populations: a normally symptomatic one and a severely symptomatic one. A major assumption for SEI2HR is that only the severely symptomatic people are hospitalized. (That assumption is also reflected in the diagram in the introduction.)

Each of those three populations have their own contact rates and mortality rates.

Here are the contact rates from the SEI2HR-Econ dictionary

Remark: Below with “Infected Population” we mean both stocks Infected Normally Symptomatic Population (INSP) and Infected Severely Symptomatic Population (ISSP).

Total Population

In this notebook we consider a DAE’s formulation of SEI2HR-Econ. The stock Total Population has the following (obvious) algebraic equation:

Note that with Max we specified that the total population cannot be less than .

Remark: As mentioned in the introduction, the package [AAp1] allows for the use of non-algebraic formulation, without an equation for TP.

Susceptible Population

The stock Susceptible Population (SP) is decreased by (1) infections derived from stocks Infected Populations and Hospitalized Population (HP), and (2) morality cases derived with the typical mortality rate.

Because we hospitalize the severely infected people only instead of the term

we have the terms

The first term is for the infections derived from the hospitalized population. The second term for the infections derived from people who are infected severely symptomatic and not hospitalized.

Births term

Note that we do not consider in this notebook births, but the births term can be included in SP’s equation:

The births rate is the same as the death rate, but it can be programmatically changed. (See [AAp2].)

Exposed Population

The stock Exposed Population (EP) is increased by (1) infections derived from the stocks Infected Populations and Hospitalized Population, and (2) mortality cases derived with the typical mortality rate. EP is decreased by (1) the people who after a certain average incubation period (aincp) become ill, and (2) mortality cases derived with the typical mortality rate.

Infected Normally Symptomatic Population

INSP is increased by a fraction of the people who have been exposed. That fraction is derived with the parameter severely symptomatic population fraction (sspf). INSP is decreased by (1) the people who recover after a certain average infection period (aip), and (2) the normally symptomatic people who die from the disease.

Infected Severely Symptomatic Population

ISSP is increased by a fraction of the people who have been exposed. That fraction is corresponds to the parameter severely symptomatic population fraction (sspf). ISSP is decreased by (1) the people who recover after a certain average infection period (aip), (2) the hospitalized severely symptomatic people who die from the disease, and (3) the non-hospitalized severely symptomatic people who die from the disease.

Note that we do not assume that severely symptomatic people recover faster if they are hospitalized, only that they have a different death rate.

Hospitalized Population

The amount of people that can be hospitalized is determined by the available Hospital Beds (HB) – the stock Hospitalized Population (HP) is subject to a resource limitation by the stock HB.

The equation of the stock HP can be easily understood from the following dynamics description points:

If the number of hospitalized people is less that the number of hospital beds we hospitalize the new ISSP people.

The Available Hospital Beds (AHB) are determined by the minimum of (i) the non-occupied hospital beds, and (ii) the hospital medical supplies divided by the ISSP consumption rate.

If the new ISSP people are more than AHB the hospital takes as many as AHB.

Hospitalized people have the same average infection period (aip).

Hospitalized (severely symptomatic) people have their own mortality rate.

Here is the HP equation:

Note that although we know that in a given day some hospital beds are going to be freed they are not considered in the hospitalization plans for that day. Similarly, we know that new medical supplies are coming but we do not include them into AHB.

Recovered Population

The stock Recovered Population (RP) is increased by the recovered infected people and decreased by mortality cases derived with the typical mortality rate.

Deceased Infected Population

The stock Deceased Infected Population (DIP) accumulates the deaths of the people who are infected. Note that we utilize the different death rates for HP and ISSP.

Hospital Beds

The stock Hospital Beds (HB) can change with a rate that reflects the number of hospital beds change rate (nhbcr) per day. Generally speaking, using nhbcr we can capture scenarios, like, extending hospitals, building new hospitals, recruitment of new medical personnel, loss of medical personnel (due to infections.)

Hospital Medical Supplies

The Hospital Medical Supplies (HMS) are decreased according to the medical supplies consumption rate (mscr) of HP and increased by a Medical Supplies (MS) delivery term (to be described next.)

The MS delivery term is build with the following assumptions / postulates:

Every day the hospital attempts to order MS that correspond to HB multiplied by mscr.

The hospital has limited capacity of MS storage, .

The MS producer has limited capacity for delivery, .

The hospital demand for MS has precedence over the demands for the non-hospitalized populations.

Hence, if the MS producer has less stock of MS than the demand of the hospital then MS producer’s whole amount of MS goes to the hospital.

The supplies are delivered with some delay period: the medical supplies delivery period (msdp).

Here is the MS delivery term:

Here is the corresponding HMS equation:

Medical Supplies

The equation of the Medical Supplies (MS) stock is based on the following assumptions / postulates:

The non-hospitalized people go to the MS producer to buy supplies. (I.e. MS delivery is to the hospital only.)

The MS producer vehicles have certain capacity, .

The MS producer has a certain storage capacity (for MS stock.)

Each of the populations INSP, ISSP, and HP has its own specific medical supplies consumption rate (mscr). EP, RP, and TP have the same mscr.

The hospital has precedence in its MS order. I.e. the demand from the hospital is satisfied first, and then the demand of the rest of the populations.

Here is the MS delivery term described in the previous section:

Here is the MS formula with the MS delivery term replaced with “Dlvr”:

We can see from that equation that MS is increased by medical supplies production rate (mspr) with measuring dimension number of units per day. The production is restricted by the storage capacity, :

(*Min[mspr[HB], -MS[t] + \[Kappa][MS]]*)

MS is decreased by the MS delivery term and the demand from the non-hospitalized populations. Because the hospital has precedence, we use this term form in the equation:

(*Min[-Dlvr + MS[t], "non-hospital demand"]*)

Here is the full MS equation:

Medical Supplies Demand

The stock Medical Supplies Demand (MSD) simply accumulates the MS demand derived from population stocks and their corresponding mscr:

Money for Hospital Services

The stock Money for Hospital Services (MHS) simply tracks expenses for hospitalized people. The parameter hospital services cost rate (hscr) with unit money per bed per day simply multiplies HP.

Money from Lost Productivity

The stock Money from Lost Productivity (MLP) simply tracks the work non-availability of the infected and died from infection people. The parameter lost productivity cost rate (lpcr) with unit money per person per day multiplies the total count of the infected and dead from infection.

Quarantine scenarios

In order to model quarantine scenarios we use piecewise constant functions for the contact rates and .

Remark: Other functions can be used, like, functions derived through some statistical fitting.

Remark: We use the code in this section to do the computations in the section “Sensitivity Analysis”.

Interactive interface

Using the interface in this section we can interactively see the effects of changing parameters. (This interface is programmed without using parametrized NDSolve solutions in order to be have code that corresponds to the interface implementations in [AAr2].)

When making and using this kind of dynamics models it is important to see how the solutions react to changes of different parameters. For example, we should try to find answers to questions like “What ranges of which parameters bring dramatic changes into important stocks?”

Sensitivity Analysis (SA) is used to determine how sensitive is a SD model to changes of the parameters and to changes of model’s equations, [BC1]. More specifically, parameter sensitivity, which we apply below, allows us to see the changes of stocks dynamic behaviour for different sequences (and combinations) of parameter values.

Remark: This section to mirrors to a point the section with same name in [AA4], except in this notebook we are more interested in medical supplies related SA because quarantine related SA is done in [AA4].

Remark: SA shown below should be done for other stocks and rates. In order to keep this exposition short we focus on ISSP, DIP, and HP. Also, it is interesting to think in terms of “3D parameter sensitivity plots.” We also do such plots.

Evaluations by Area under the curve

For certain stocks we might be not just interested in their evolution in time but also in their cumulative values. I.e. we are interested in the so called Area Under the Curve (AUC) metric for those stocks.

There are three ways to calculate AUC for stocks of interest:

Add aggregation equations in the system of ODE’s. (Similar to the stock DIP in SEI2HR.)

For example, in order to compute AUC for ISSP we can add to SEI2HR the equation:

(*aucISSP'[t] = ISSP[t]*)

- More details for such equation addition are given in [AA2].

Apply NIntegrate over stocks solution functions.

Apply Trapezoidal rule to stock solution function values over a certain time grid.

Below we use 1 and 3.

Variation of medical supplies delivery period

Here we calculate the solutions for a certain combination of capacities and rates:

In order to demonstrate the effect of medical supplies production rate (mspr) it is beneficial to eliminate the hospital beds availability restriction – we assume that we have enough hospital beds for all infected severely symptomatic people.

Here we calculate the solutions for a certain combination of capacities and rates:

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

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

The lectures (as live-coding sessions)

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

What are the typical applications of LSA?

Why use LSA?

What it the fundamental philosophical or scientific assumption for LSA?

What is the most important and/or fundamental step of LSA?

What is the difference between LSA and Latent Semantic Indexing (LSI)?

What are the alternatives?

Using Neural Networks instead?

How is LSA used to derive similarities between two given texts?

How is LSA used to evaluate the proximity of phrases? (That have different words, but close semantic meaning.)

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

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

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

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

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

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

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

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

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

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

Contents description

The document has the following structure.

The sections “Package load” and “Data load” obtain the needed code and data.

The sections “Design consideration” and “Monad design” provide motivation and design decisions rationale.

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

(Using a fair amount of examples.)

The section “Unit tests” describes the tests used in the development of the LSAMon monad.

(The random pipelines unit tests are especially interesting.)

The section “Future plans” outlines future directions of development.

The section “Implementation notes” just says that LSAMon’s development process and this document follow the ones of the classifications workflows monad ClCon, [AA6].

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

Package load

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

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

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

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

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 LSA workflows outlined in the previous section. The monad is named LSAMon for “Latent Semantic Analysis** Mon**ad”.

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

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

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

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

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

The object is (relatively) hard to compute.

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

Let us list the desired properties of the monad.

Rapid specification of non-trivial LSA workflows.

The text data supplied to the monad can be: (i) a list of strings, or (ii) an association with string values.

The monad uses the Linear vector space model .

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

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

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

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

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

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

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

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

The LSAMon components and their interactions are fairly simple.

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

document-term matrix,

the factors obtained by matrix factorization algorithms,

LSI weight functions specifications,

extracted topics.

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

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

LSAMon overview

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

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

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

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

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

Here is the output of the pipeline:

The LSAMon functions are separated into four groups:

operations,

setters and droppers,

takers,

State Monad generic functions.

Monad functions interaction with the pipeline value and context

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

State monad functions

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

Main monad functions

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

Monad elements

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

The monad head

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

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

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

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

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

Frequency matrix

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

Each entry f_{ij} of the matrix F is the number of occurrences of the term j in the document i.

Weights

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

where g_{j} – global term weight; l_{ij} – local term weight; d_{i} – normalization weight.

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

Here is a table of weight functions formulas.

Computation specifications

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

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

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

Theoretical outline

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

M ≈ W.H, W ∈ ℝ^{m × k}, H ∈ ℝ^{k × n}.

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

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

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

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

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

Computation specifications

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

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

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

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

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

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

Transformation steps

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

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

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

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

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

Equivalent representation

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

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

inds = {6, 10};
queries = Part[lsaHamlet⟹LSAMonTakeDocuments, inds];
queries
(* <|"id.0006" -> "Getrude, Queen of Denmark, mother to Hamlet. Ophelia, daughter to Polonius.",
"id.0010" -> "ACT I. Scene I. Elsinore. A platform before the Castle."|> *)
lsaHamlet⟹
LSAMonRepresentByTerms[queries]⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrix`ColumnSumsAssociation[#], # > 0& ]]]]& ];

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

Here is an example.

inds = {6, 10};
queries = Part[lsaHamlet⟹LSAMonTakeDocuments, inds];
Short /@ queries
(* <|"id.0006" -> "Getrude, Queen of Denmark, mother to Hamlet. Ophelia, daughter to Polonius.",
"id.0010" -> "ACT I. Scene I. Elsinore. A platform before the Castle."|> *)
lsaHamlet⟹
LSAMonRepresentByTopics[queries]⟹
LSAMonEchoFunctionValue[MatrixForm[Part[#, All, Keys[Select[SSparseMatrix`ColumnSumsAssociation[#], # > 0& ]]]]& ];

In LSAMon for SVD H^{( − 1)} = H^{T}; for NNMF H^{( − 1)} is the pseudo-inverse of H.

The vector x obtained with LSAMonRepresentByTopics.

Tags representation

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

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

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

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

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

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

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

focusText =
First@Pick[textHamlet, StringMatchQ[textHamlet, ___ ~~ "to be" ~~ __ ~~ "or not to be" ~~ ___, IgnoreCase -> True]];
Short[focusText]
(* "Ham. To be, or not to be- that is the question: Whether 'tis ....y.
O, woe is me T' have seen what I have seen, see what I see!" *)
LSAMonUnit[StringSplit[ToLowerCase[focusText], {",", ".", ";", "!", "?"}]]⟹
LSAMonMakeDocumentTermMatrix["StemmingRules" -> {}, "StopWords" -> Automatic]⟹
LSAMonApplyTermWeightFunctions⟹
LSAMonFindMostImportantDocuments[3]⟹
LSAMonEchoFunctionValue[GridTableForm];

Setters, droppers, and takers

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

For example:

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

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

Short@(p⟹QRMonTakeContext)["documents"]
(* <|"id.0001" -> "1604", "id.0002" -> "THE TRAGEDY OF HAMLET, PRINCE OF DENMARK", <<220>>, "id.0223" -> "THE END"|> *)

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

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

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

The utilization of SSparseMatrix objects

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

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

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

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

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

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

Similarities based on representation by terms

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

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

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

Unit tests

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

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

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

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

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

rpTRObj = TestReport[res]

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

Future plans

Dimension reduction extensions

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

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

Conversational agent

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

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

Implementation notes

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

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

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].

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;