This is the second of three blog posts about working with data sets in Maple.

In my previous post, I discussed how to use Maple to access a large number of data sets from Quandl, an online data aggregator. In this post, I’ll focus on exploring built-in data sets in Maple.

Data is being generated at an ever increasing rate. New data is generated every minute, adding to an expanding network of online information. Navigating through this information can be daunting. Simply preparing a tabular data set that collects information from several sources is often a difficult and time consuming effort. For example, even though the example in my previous post only required a couple of lines of Maple code to merge 540 different data sets from various sources, the effort to manually search for and select sources for data took significantly more time.

In an attempt to make the process of finding data easier, Maple’s built-in country data set collects information on country-specific variables including financial and economic data, as well as information on country codes, population, area, and more.

The built-in database for Country data can be accessed programmatically by creating a new DataSets Reference:

CountryData := DataSets:-Reference( "builtin", "country" );

This returns a Reference object, which can be further interrogated. There are several commands that are applicable to a DataSets Reference, including the following exports for the Reference object:

exports( CountryData, static );

The list of available countries in this data set is given using the following:

GetElementNames( CountryData );

The available data for each of these countries can be found using:

GetHeaders( CountryData );

There are many different data sets available for country data, 126 different variables to be exact. Similar to Maple’s DataFrame, the columns of information in the built-in data set can be accessed used the labelled name.

For example, the three-letter country codes for each country can be returned using:

CountryData[.., "3 Letter Country Code"];

The three-letter country code for Denmark is:

CountryData["Denmark", "3 Letter Country Code"];

Built-in data can also be queried in a similar manner to DataFrames. For example, to return the countries with a population density less than 3%:

pop_density := CountryData[ .., "Population Density" ]:

pop_density[ `Population Density` < 3 ];

At this time, Maple’s built-in country data collection contains 126 data sets for 185 countries. When I built the example from my first post, I knew exactly the data sets that I wanted to use and I built a script to collect these into a larger data container. Attempting a similar task using Maple’s built-in data left me with the difficult decision of choosing which data sets to use in my next example.

So rather than choose between these available options, I built a user interface that lets you quickly browse through all of Maple’s collection of built-in data.

Using a couple of tricks that I found in the pages for Programmatic Content Generation, I built the interface pictured above. (I’ll give more details on the method that I used to construct the interface in my next post.)

This interface allows you to select from a list of countries, and visualize up to three variables of the country data with a BubblePlot. Using the preassigned defaults, you can select several countries and then visualize how their overall number of internet users has changed along with their gross domestic product. The BubblePlot visualization also adds a third dimension of information by adjusting the bubble size according to the relative population compared with the other selected countries.

Now you may notice that the list of available data sets is longer than the list of available options in each of the selection boxes. In order to be able to generate BubblePlot animations, I made an arbitrary choice to filter out any of the built-in data sets that were not of type TimeSeries. This is something that could easily be changed in the code. The choice of a BubblePlot could also be updated to be any other type of Statistical visualization with some additional modifications.

You can download a copy of this application here: VisualizingCountryDataSets.mw

You can also interact with it via the MapleCloud: http://maplecloud.maplesoft.com/application.jsp?appId=5743882790764544

I’ll be following up this post with an in-depth post on how I authored the country selector interface using programmatic content generation.

     Example of the equidistant surface at a distance of 0.25 to the surface
x3-0.1 * (sin (4 * x1) + sin (3 * x2 + x3) + sin (2 * x2)) = 0
Constructed on the basis of universal parameterization of surfaces.

equidistant_surface.mw 

This is the first of three blog posts about working with data sets in Maple.

In 2013, I wrote a library for Maple that used the HTTP package to access the Quandl data API and import data sets into Maple. I was motivated by the fact that, when I was downloading data, I often used multiple data sources, manually updated data when updates were available, and cleaned or manipulated the data into a standardized form (which left me spending too much time on the data acquisition step).

Simply put, I needed a source for data that would provide me with a searchable, stable data API, which would also return data in a form that did not require too much post-processing.

My initial library had really just scratched the surface of what was possible.

Maple 2015 introduced the new DataSets package, which fully integrated a data set search into core library routines and made its functionality more discoverable through availability in Maple’s search bar.

Accessing online data suddenly became much easier. From within Maple, I could now search through over 12 million time series data sets provided by Quandl, and then automatically import the data into a format that I could readily work with.

If you’re not already aware of this online service, Quandl is an online data aggregator that delivers a wide variety of high quality financial and economic data. This includes the latest data on stocks and commodities, exchange rates, and macroeconomic indicators such as population, inflation, unemployment, and so on. Quandl collects both open and proprietary data sets from many sources, such as the US Federal Reserve System, OECD, Eurostat, The World Bank, and Open Data for Africa. Best of all, Quandl's powerful API is free to use.

One of the first examples for the DataSets package that I constructed was in part based on the inspirational work of Hans Rosling. I was drawn in by his ability to use statistical visualizations to break down complex multidimensional data sets and provide insight into underlying patterns; a key example investigating the correlation between rising incomes and life expectancy.

As well as online data, the DataSets package had a database for country data. Hence it seemed fitting to add an example that explored macroeconomic indicators for several countries. Accordingly, I set out to create an example that visualized variables such as Gross Domestic Product, Life Expectancy, and Population for a collection of countries.

I’ll now describe how I constructed this application.

The three key variables are Gross Domestic Product at Power Purchasing Parity, Life Expectancy, and Population. Having browsed through Quandl’s website for available data sets, the World Bank and Open Data for Africa projects seemingly had the most available relevant data; therefore I chose these as my data sources.

Pulling data for a single country from one of these sources was pretty straight forward. For example, the DataSets Reference for the Open Data for Africa data set on GDP at PPP for Canada is:

DataSets:-Reference("quandl", "ODA/CAN_PPPPC"));

In this command, the second argument is the Quandl data set code. If you are on Quandl’s website, this is listed near the top of the data set page as well as in the last few characters of the web address itself: https://www.quandl.com/data/ODA/CAN_PPPPC . Deconstructing the code, “ODA” stands for Open Data for Africa and the rest of the string is constructed from the three letter country code for Canada, “CAN”, and the code for the GDP and PPP. Looking at a small sample of other data set codes, I theorized that both of the data sources used a standardized data set name that included the ISO-3166 3-letter country code for available data sets. Based on this theory, I created a simple script to query for available data and discovered that there was data available for many countries using this standardized code. However, not every country had available data, so I needed to filter my list somewhat in order to pick only those countries for which information was available.

The script that I had constructed required three letter country codes. In order to test all available countries, I created a table to house the country names and three-letter country codes using data from the built-in database for countries:

ccdata := DataSets:-Builtin:-Reference("country")[.., "3 Letter Country Code"];

cctable := table([seq(op(GetElementNames(ccdata[i])) = ccdata[i, "3 Letter Country Code"], 
                                i = 1 .. CountRows(ccdata))]):

My script filtered this table, returning a subset of the original table, something like:

Countries := table( [“Canada” = “CAN”, “Sweden” = “SWE”, … ] );

You can see the filtered country list in the code edit region of the application below.

With this shorter list of countries, I was now ready to download some data. I created three vectors to hold the data sets by mapping in the DataSets Reference onto the “standardized” data set names that I pulled from Quandl. Here’s the first vector for the data on GDP at PPP.

V1 := Vector( [ (x) -> Reference("quandl", cat("ODA/", x, "_PPPPC"))

                   ~([entries(Countries, nolist, indexorder)])]):

#Open Data for Africa GDP at PPP

Having created three data vectors consisting of 180 x 3 = 540 data sets, I was finally ready to visualize the large set of data that I had amassed.

In Maple’s Statistics package, BubblePlots can use the horizontal axis, vertical axis and the relative bubble size to illustrate multidimensional information. Moreover, if incoming data is stored as a TimeSeries object, BubblePlots can generate animations over a common period of time.

Putting all of this together generated the following animation for 180 available countries.

This example will be included with the next version of Maple, but for now, you can download a copy here:DataSetsBubblePlot.mw

*Note: if you try this application at home, it will download 540 data sets. This operation plus the additional BubblePlot construction can take some time, so if you just want to see the finished product, you can simply interact with the animation in the Maple worksheet using the animation toolbar.

A more advanced example that uses multiple threads for data download can be seen at the bottom of the following page: https://www.maplesoft.com/products/maple/new_features/maple19/datasets_maple2015.pdf You can also interact with this example in Maple by searching for: ?updates,Maple2015,DataSets

In my next post, I’ll discuss how I used programmatic content generation to construct an interactive application for data retrieval.

Below is the worksheet with the whole material presented yesterday in the webinar, “Applying the power of computer algebra to theoretical physics”, broadcasted by the “Institute of Physics” (IOP, England). The material was very well received, rated 4.5 out of 5 (around 30 voters among the more than 300 attendants), and generated a lot of feedback. The webinar was recorded so that it is possible to watch it (for free, of course, click the link above, it will ask you for registration, though, that’s how IOP works).

Anyway, you can reproduce the presentation with the worksheet below (mw file linked at the end, or the corresponding pdf also linked with all the input lines executed). As usual, to reproduce the input/output you need to have installed the latest version of Physics, available in the Maplesoft R&D Physics webpage.

Physics_2016_IOP_webinar.mw     Physics_2016_IOP_webinar.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Maple 2016.1a now available, fixes issues reported here http://www.mapleprimes.com/questions/211561-Why-Does-Maple-20161-Improperly-Treat-X2

Thank you Maplesoft for your quick response to this issue.

An update to Maple 2016 is now available. Maple 2016.1 provides:

• Updated translations for Simplified and Traditional Chinese,  French, Greek, Japanese, Brazilian Portuguese, and Spanish
• Updates to the new Maple Workbook
• Enhancements to Maple’s context-sensitive menus
• A variety of improvements to the math engine and interface

To get this update, use Tools>Check for Updates from within Maple, or visit the Maple 2016.1 downloads page.

eithne

MapleSim 2016 is here!

MapleSim 2016 provides variety of improvements to streamline the user experience, expand modeling scope, and enhance connectivity with other tools. Here are some highlights:

• Collapsible task panes provide a larger model workspace, so you can see more of your model at once.
• Improved layout ensures the tools you need for your current task are available at your fingertips.
• The expanded Multibody component library now supports contact modeling.
• A new add-on library, the MapleSim Pneumatics Library from Modelon, supports the modeling and simulation of pneumatic systems.
• The MapleSim CAD Toolbox has been extended to support the latest versions of Inventor®, NX®, SOLIDWORKS®, CATIA® V5, Solid Edge®, PTC® Creo Parametric™, and more.
• The MapleSim Connector, which provides connectivity to Simulink®, now supports single precision export of S-functions so you can run your MapleSim models on hardware that only supports single precision.

See What’s New in MapleSim 2016 for more information about these and other improvements.

eithne

MapleSim 2016 is here!

MapleSim 2016 provides variety of improvements to streamline the user experience, expand modeling scope, and enhance connectivity with other tools. Here are some highlights:

• Collapsible task panes provide a larger model workspace, so you can see more of your model at once.
• Improved layout ensures the tools you need for your current task are available at your fingertips.
• The expanded Multibody component library now supports contact modeling.
• A new add-on library, the MapleSim Pneumatics Library from Modelon, supports the modeling and simulation of pneumatic systems.
• The MapleSim CAD Toolbox has been extended to support the latest versions of Inventor®, NX®, SOLIDWORKS®, CATIA® V5, Solid Edge®, PTC® Creo Parametric™, and more.
• The MapleSim Connector, which provides connectivity to Simulink®, now supports single precision export of S-functions so you can run your MapleSim models on hardware that only supports single precision.

See What’s New in MapleSim 2016 for more information about these and other improvements.

eithne

New color schemes for plotting have been added to Maple 2016. They are summarized here. You can also see more details on the plot/colorscheme help pages, which have been reworked (hopefully in a way that makes them more useful to you).

We released a new video a few weeks ago describing one of these features: coloring by value. The worksheet I used for the video is available here: ColorByValueWebinar.mw

The mechanism of transport of the material of the sewing machine M 1022 class: mathematical animation.   BELORUS.mw 

I thought I would share some code for computing sparse matrix products in Maple.  For floating point matrices this is done quickly, but for algebraic datatypes there is a performance problem with the builtin routines, as noted in http://www.mapleprimes.com/questions/205739-How-Do-I-Improve-The-Performance-Of

Download spm.txt

The code is fairly straightforward in that it uses op(1,A) to extract the dimensions and op(2,A) to extract the non-zero elements of a Matrix or Vector, and then loops over those elements.  I included a sparse map function for cases where you want to map a function (like expand) over non-zero elements only.

# sparse matrix vector product
spmv := proc(A::Matrix,V::Vector)
local m,n,Ae,Ve,Vi,R,e;
  n, m := op(1,A);
  if op(1,V) <> m then error "incompatible dimensions"; end if;
  Ae := op(2,A);
  Ve := op(2,V);
  Vi := map2(op,1,Ve);
  R := Vector(n, storage=sparse);
  for e in Ae do
    n, m := op(1,e);
    if member(m, Vi) then R[n] := R[n] + A[n,m]*V[m]; end if;
  end do;
  return R;
end proc:

# sparse matrix product
spmm := proc(A::Matrix, B::Matrix)
local m,n,Ae,Be,Bi,R,l,e,i;
  n, m := op(1,A);
  i, l := op(1,B);
  if i <> m then error "incompatible dimensions"; end if;
  Ae := op(2,A);
  Be := op(2,B);
  R := Matrix(n,l,storage=sparse);
  for i from 1 to l do
    Bi, Be := selectremove(type, Be, (anything,i)=anything);
    Bi := map2(op,[1,1],Bi);
    for e in Ae do
      n, m := op(1,e);
      if member(m, Bi) then R[n,i] := R[n,i] + A[n,m]*B[m,i]; end if;
    end do;
  end do;
  return R;
end proc:

# sparse map
smap := proc(f, A::{Matrix,Vector})
local B, Ae, e;
  if A::Vector then
    B := Vector(op(1,A),storage=sparse):
  else
    B := Matrix(op(1,A),storage=sparse):
  end if;
  Ae := op(2,A);
  for e in Ae do
    B[op(1,e)] := f(op(2,e),args[3..nargs]);
  end do;
  return B;
end proc:

As for how it performs, here is a demo inspired by the original post.

n := 674;
k := 6;

A := Matrix(n,n,storage=sparse):
for i to n do
  for j to k do
    A[i,irem(rand(),n)+1] := randpoly(x):
  end do:
end do:

V := Vector(n):
for i to k do
  V[irem(rand(),n)+1] := randpoly(x):
end do:

C := CodeTools:-Usage( spmv(A,V) ):  # 7ms, 25x faster
CodeTools:-Usage( A.V ):  # 174 ms

B := Matrix(n,n,storage=sparse):
for i to n do
  for j to k do
    B[i,irem(rand(),n)+1] := randpoly(x):
  end do:
end do:

C := CodeTools:-Usage( spmm(A,B) ):  # 2.74 sec, 50x faster
CodeTools:-Usage( A.B ):  # 2.44 min

# expand and collect like terms
C := CodeTools:-Usage( smap(expand, C) ):

eulermac(1/(n*ln(n)^2),n=2..N,1);  #Error
Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation


eulermac(1/(n*ln(n)^2+1),n=2..N,1);  #nonsense

polygon_2_color.mw

Imitation coloring both sides of the polygon in 3d.  We  build a new polygon in parallel with our polygon on a very short distance t. (We need any three points on the polygon plane, do not lie on a straight line.) This place in the program is highlighted in blue.

Paint the polygons are in different colors.

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations. 
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw 
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf

The method of solving underdetermined systems of equations, and universal method for calculating link mechanisms. It is based on the Draghilev’s method for solving systems of nonlinear equations. 
When calculating link mechanisms we can use geometrical relationships to produce their mathematical models without specifying the “input link”. The new method allows us to specify the “input link”, any link of mechanism.

Example.
Three-bar mechanism.  The system of equations linkages in this mechanism is as follows:

f1 := x1^2+(x2+1)^2+(x3-.5)^2-R^2;
f2 := x1-.5*x2+.5*x3;
f3 := (x1-x4)^2+(x2-x5)^2+(x3-x6)^2-19;
f4 := sin(x4)-x5;
f5 := sin(2*x4)-x6;

Coordinates green point x'i', i = 1..3, the coordinates of red point x'i', i = 4..6.
Set of x0'i', i = 1..6 searched arbitrarily, is the solution of the system of equations and is the initial point for the solution of the ODE system. The solution of ODE system is the solution of system of equations linkages for concrete assembly linkage.
Two texts of the program for one mechanism. In one case, the “input link” is the red-green, other case the “input link” is the green-blue.
After the calculation trajectories of points, we can always find the values of other variables, for example, the angles.
Animation displays the kinematics of the mechanism.
MECAN_3_GR_P_bar.mw 
MECAN_3_Red_P_bar.mw

(if to use another color instead of color = "Niagara Dark Orchid", the version of Maple <17)

Method_Mechan_PDF.pdf