Package descriptive (Maxima Manual)

39 Package descriptive

Introduction to descriptive
Functions and Variables for data manipulation
Functions and Variables for descriptive statistics
Functions and Variables for specific multivariate descriptive statistics
Functions and Variables for statistical graphs

39.1 Introduction to descriptive

Package descriptive contains a set of functions for making descriptive statistical computations and graphing. Together with the source code there are three data sets in your Maxima tree: pidigits.data, wind.data and biomed.data.

Any statistics manual can be used as a reference to the functions in package descriptive.

For comments, bugs or suggestions, please contact me at ’mario AT edu DOT xunta DOT es’.

Here is a simple example on how the descriptive functions in descriptive do they work, depending on the nature of their arguments, lists or matrices,

(%i1) load ("descriptive")$

(%i2) /* univariate sample */   mean ([a, b, c]);
                            c + b + a
(%o2)                       ---------
                                3

(%i3) matrix ([a, b], [c, d], [e, f]);
                            [ a  b ]
                            [      ]
(%o3)                       [ c  d ]
                            [      ]
                            [ e  f ]

(%i4) /* multivariate sample */ mean (%);
                      e + c + a  f + d + b
(%o4)                [---------, ---------]
                          3          3

Note that in multivariate samples the mean is calculated for each column.

In case of several samples with possible different sizes, the Maxima function map can be used to get the desired results for each sample,

(%i1) load ("descriptive")$

(%i2) map (mean, [[a, b, c], [d, e]]);
                        c + b + a  e + d
(%o2)                  [---------, -----]
                            3        2

In this case, two samples of sizes 3 and 2 were stored into a list.

Univariate samples must be stored in lists like

(%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5];
(%o1)           [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]

and multivariate samples in matrices as in

(%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
             [10.58, 6.63], [13.33, 13.25], [13.21,  8.12]);
                        [ 13.17  9.29  ]
                        [              ]
                        [ 14.71  16.88 ]
                        [              ]
                        [ 18.5   16.88 ]
(%o1)                   [              ]
                        [ 10.58  6.63  ]
                        [              ]
                        [ 13.33  13.25 ]
                        [              ]
                        [ 13.21  8.12  ]

In this case, the number of columns equals the random variable dimension and the number of rows is the sample size.

Data can be introduced by hand, but big samples are usually stored in plain text files. For example, file pidigits.data contains the first 100 digits of number %pi:

In order to load these digits in Maxima,

(%i1) s1 : read_list (file_search ("pidigits.data"))$

(%i2) length (s1);
(%o2)                          100

On the other hand, file wind.data contains daily average wind speeds at 5 meteorological stations in the Republic of Ireland (This is part of a data set taken at 12 meteorological stations. The original file is freely downloadable from the StatLib Data Repository and its analysis is discused in Haslett, J., Raftery, A. E. (1989) Space-time Modelling with Long-memory Dependence: Assessing Ireland’s Wind Power Resource, with Discussion. Applied Statistics 38, 1-50). This loads the data:

(%i1) s2 : read_matrix (file_search ("wind.data"))$

(%i2) length (s2);
(%o2)                          100

(%i3) s2 [%]; /* last record */
(%o3)            [3.58, 6.0, 4.58, 7.62, 11.25]

Some samples contain non numeric data. As an example, file biomed.data (which is part of another bigger one downloaded from the StatLib Data Repository) contains four blood measures taken from two groups of patients, A and B, of different ages,

(%i1) s3 : read_matrix (file_search ("biomed.data"))$

(%i2) length (s3);
(%o2)                          100

(%i3) s3 [1]; /* first record */
(%o3)            [A, 30, 167.0, 89.0, 25.6, 364]

The first individual belongs to group A, is 30 years old and his/her blood measures were 167.0, 89.0, 25.6 and 364.

One must take care when working with categorical data. In the next example, symbol a is asigned a value in some previous moment and then a sample with categorical value a is taken,

(%i1) a : 1$

(%i2) matrix ([a, 3], [b, 5]);
                            [ 1  3 ]
(%o2)                       [      ]
                            [ b  5 ]

39.2 Functions and Variables for data manipulation

Function: build_sample (list) ¶

Function: build_sample (matrix) ¶

Builds a sample from a table of absolute frequencies. The input table can be a matrix or a list of lists, all of them of equal size. The number of columns or the length of the lists must be greater than 1. The last element of each row or list is interpreted as the absolute frequency. The output is always a sample in matrix form.

Examples:

Univariate frequency table.

(%i1) load ("descriptive")$
(%i2) sam1: build_sample([[6,1], [j,2], [2,1]]);
                       [ 6 ]
                       [   ]
                       [ j ]
(%o2)                  [   ]
                       [ j ]
                       [   ]
                       [ 2 ]
(%i3) mean(sam1);
                      2 j + 8
(%o3)                [-------]
                         4
(%i4) barsplot(sam1) $

Multivariate frequency table.

(%i1) load ("descriptive")$
(%i2) sam2: build_sample([[6,3,1], [5,6,2], [u,2,1],[6,8,2]]) ;
                           [ 6  3 ]
                           [      ]
                           [ 5  6 ]
                           [      ]
                           [ 5  6 ]
(%o2)                      [      ]
                           [ u  2 ]
                           [      ]
                           [ 6  8 ]
                           [      ]
                           [ 6  8 ]
(%i3) cov(sam2);
       [   2                 2                            ]
       [  u  + 158   (u + 28)     2 u + 174   11 (u + 28) ]
       [  -------- - ---------    --------- - ----------- ]
(%o3)  [     6          36            6           12      ]
       [                                                  ]
       [ 2 u + 174   11 (u + 28)            21            ]
       [ --------- - -----------            --            ]
       [     6           12                 4             ]
(%i4) barsplot(sam2, grouping=stacked) $

Function: continuous_freq (list) ¶

Function: continuous_freq (list, m) ¶

The argument of continuous_freq must be a list of numbers. Divides the range in intervals and counts how many values are inside them. The second argument is optional and either equals the number of classes we want, 10 by default, or equals a list containing the class limits and the number of classes we want, or a list containing only the limits. Argument list must be a list of (2 or 3) real numbers. If sample values are all equal, this function returns only one class of amplitude 2.

Examples:

Optional argument indicates the number of classes we want. The first list in the output contains the interval limits, and the second the corresponding counts: there are 16 digits inside the interval [0, 1.8], 24 digits in (1.8, 3.6], and so on.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, 5);
(%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]

Optional argument indicates we want 7 classes with limits -2 and 12:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, [-2,12,7]);
(%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]

Optional argument indicates we want the default number of classes with limits -2 and 12:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, [-2,12]);

                3  4  11  18     32  39  46  53
(%o3)  [[- 2, - -, -, --, --, 5, --, --, --, --, 12], 
                5  5  5   5      5   5   5   5
               [0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]

Function: discrete_freq (list) ¶

Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list,

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) discrete_freq (s1);

(%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 
                             [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

The first list gives the sample values and the second their absolute frequencies. Commands ? col and ? transpose should help you to understand the last input.

Function: subsample (data_matrix, predicate_function) ¶

Function: subsample (data_matrix, predicate_function, col_num1, col_num2, …) ¶

This is a sort of variant of the Maxima submatrix function. The first argument is the data matrix, the second is a predicate function and optional additional arguments are the numbers of the columns to be taken. Its behaviour is better understood with examples.

These are multivariate records in which the wind speed in the first meteorological station were greater than 18. See that in the lambda expression the i-th component is refered to as v[i].

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) subsample (s2, lambda([v], v[1] > 18));

              [ 19.38  15.37  15.12  23.09  25.25 ]
              [                                   ]
              [ 18.29  18.66  19.08  26.08  27.63 ]
(%o3)         [                                   ]
              [ 20.25  21.46  19.95  27.71  23.38 ]
              [                                   ]
              [ 18.79  18.96  14.46  26.38  21.84 ]

In the following example, we request only the first, second and fifth components of those records with wind speeds greater or equal than 16 in station number 1 and less than 25 knots in station number 4. The sample contains only data from stations 1, 2 and 5. In this case, the predicate function is defined as an ordinary Maxima function.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) g(x):= x[1] >= 16 and x[4] < 25$
(%i4) subsample (s2, g, 1, 2, 5);

                     [ 19.38  15.37  25.25 ]
                     [                     ]
                     [ 17.33  14.67  19.58 ]
(%o4)                [                     ]
                     [ 16.92  13.21  21.21 ]
                     [                     ]
                     [ 17.25  18.46  23.87 ]

Here is an example with the categorical variables of biomed.data. We want the records corresponding to those patients in group B who are older than 38 years.

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) h(u):= u[1] = B and u[2] > 38 $
(%i4) subsample (s3, h);

                [ B  39  28.0  102.3  17.1  146 ]
                [                               ]
                [ B  39  21.0  92.4   10.3  197 ]
                [                               ]
                [ B  39  23.0  111.5  10.0  133 ]
                [                               ]
                [ B  39  26.0  92.6   12.3  196 ]
(%o4)           [                               ]
                [ B  39  25.0  98.7   10.0  174 ]
                [                               ]
                [ B  39  21.0  93.2   5.9   181 ]
                [                               ]
                [ B  39  18.0  95.0   11.3  66  ]
                [                               ]
                [ B  39  39.0  88.5   7.6   168 ]

Probably, the statistical analysis will involve only the blood measures,

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
                 3, 4, 5, 6);

                   [ 28.0  102.3  17.1  146 ]
                   [                        ]
                   [ 21.0  92.4   10.3  197 ]
                   [                        ]
                   [ 23.0  111.5  10.0  133 ]
                   [                        ]
                   [ 26.0  92.6   12.3  196 ]
(%o3)              [                        ]
                   [ 25.0  98.7   10.0  174 ]
                   [                        ]
                   [ 21.0  93.2   5.9   181 ]
                   [                        ]
                   [ 18.0  95.0   11.3  66  ]
                   [                        ]
                   [ 39.0  88.5   7.6   168 ]

This is the multivariate mean of s3,

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) mean (s3);

       65 B + 35 A  317          6 NA + 8144.999999999999
(%o3) [-----------, ---, 87.178, ------------------------, 
           100      10                     100
                                                    3 NA + 19587
                                            18.123, ------------]
                                                        100

Here, the first component is meaningless, since A and B are categorical, the second component is the mean age of individuals in rational form, and the fourth and last values exhibit some strange behaviour. This is because symbol NA is used here to indicate non available data, and the two means are nonsense. A possible solution would be to take out from the matrix those rows with NA symbols, although this deserves some loss of information.

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) g(v):= v[4] # NA and v[6] # NA $
(%i4) mean (subsample (s3, g, 3, 4, 5, 6));

(%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813, 
                                                            2514
                                                            ----]
                                                             13

39.3 Functions and Variables for descriptive statistics

Function: mean (list) ¶

Function: mean (matrix) ¶

This is the sample mean, defined as

                       n
                     ====
             _   1   \
             x = -    >    x
                 n   /      i
                     ====
                     i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$

(%i3) mean (s1);
                               471
(%o3)                          ---
                               100

(%i4) %, numer;
(%o4)                         4.71

(%i5) s2 : read_matrix (file_search ("wind.data"))$

(%i6) mean (s2);
(%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

Function: var (list) ¶

Function: var (matrix) ¶

This is the sample variance, defined as

                     n
                   ====
           2   1   \          _ 2
          s  = -    >    (x - x)
               n   /       i
                   ====
                   i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$

(%i3) var (s1), numer;
(%o3)                   8.425899999999999

39.4 Functions and Variables for specific multivariate descriptive statistics

Function: cov (matrix) ¶

The covariance matrix of the multivariate sample, defined as

              n
             ====
          1  \           _        _
      S = -   >    (X  - X) (X  - X)'
          n  /       j        j
             ====
             j = 1

where X_j is the j-th row of the sample matrix.

Example:

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec : 7$  /* change precision for pretty output */

(%i4) cov (s2);
      [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
      [                                                  ]
      [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
      [                                                  ]
(%o4) [ 14.37217  13.30448  15.47573  17.32544  16.18171 ]
      [                                                  ]
      [ 19.39624  15.15834  17.32544  32.17651  20.44685 ]
      [                                                  ]
      [ 15.42162  14.9711   16.18171  20.44685  24.42308 ]

39.5 Functions and Variables for statistical graphs

Function: barsplot (data1, data2, …, option_1, option_2, …) ¶

Function: barsplot_description (…) ¶

Plots bars diagrams for discrete statistical variables, both for one or multiple samples.

data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

Available options are:

box_width (default, 3/4): relative width of rectangles. This value must be in the range [0,1].
grouping (default, clustered): indicates how multiple samples are shown. Valid values are: clustered and stacked.
groups_gap (default, 1): a positive integer number representing the gap between two consecutive groups of bars.
bars_colors (default, []): a list of colors for multiple samples. When there are more samples than specified colors, the extra necesary colors are chosen at random. See color to learn more about them.
frequency (default, absolute): indicates the scale of the ordinates. Possible values are: absolute, relative, and percent.
ordering (default, orderlessp): possible values are orderlessp or ordergreatp, indicating how statistical outcomes should be ordered on the x-axis.
sample_keys (default, []): a list with the strings to be used in the legend. When the list length is other than 0 or the number of samples, an error message is returned.
start_at (default, 0): indicates where the plot begins to be plotted on the x-axis.
All global draw options, except xtics, which is internally assigned by barsplot. If you want to set your own values for this option or want to build complex scenes, make use of barsplot_description. See example below.
The following local draw options: key, color, fill_color, fill_density and line_width. See also bars.

Function barsplot_description creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function wxbarsplot for creating embedded histograms in interfaces wxMaxima and iMaxima.

Examples:

Univariate sample in matrix form. Absolute frequencies.

(%i1) load ("descriptive")$
(%i2) m : read_matrix (file_search ("biomed.data"))$

(%i3) barsplot(
        col(m,2),
        title        = "Ages",
        xlabel       = "years",
        box_width    = 1/2,
        fill_density = 3/4)$

Two samples of different sizes, with relative frequencies and user declared colors.

(%i1) load ("descriptive")$
(%i2) l1:makelist(random(10),k,1,50)$
(%i3) l2:makelist(random(10),k,1,100)$

(%i4) barsplot(
        l1,l2,
        box_width    = 1,
        fill_density = 1,
        bars_colors  = [black, grey],
        frequency = relative,
        sample_keys = ["A", "B"])$

Four non numeric samples of equal size.

(%i1) load ("descriptive")$

(%i2) barsplot(
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        title  = "Asking for something to four groups",
        ylabel = "# of individuals",
        groups_gap   = 3,
        fill_density = 0.5,
        ordering     = ordergreatp)$

Stacked bars.

(%i1) load ("descriptive")$

(%i2) barsplot(
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        title  = "Asking for something to four groups",
        ylabel = "# of individuals",
        grouping     = stacked,
        fill_density = 0.5,
        ordering     = ordergreatp)$

barsplot in a multiplot context.

(%i1) load ("descriptive")$
(%i2) l1:makelist(random(10),k,1,50)$
(%i3) l2:makelist(random(10),k,1,100)$
(%i4) bp1 : 
        barsplot_description(
         l1,
         box_width = 1,
         fill_density = 0.5,
         bars_colors = [blue],
         frequency = relative)$

(%i5) bp2 : 
        barsplot_description(
         l2,
         box_width = 1,
         fill_density = 0.5,
         bars_colors = [red],
         frequency = relative)$

(%i6) draw(gr2d(bp1), gr2d(bp2))$

For bars diagrams related options, see bars of package draw. See also functions histogram and piechart.

Function: boxplot (data) ¶

Function: boxplot (data, option_1, option_2, …) ¶

Function: boxplot_description (…) ¶

This function plots box-and-whishker diagrams. Argument data can be a list, which is not of great interest, since these diagrams are mainly used for comparing different samples, or a matrix, so it is possible to compare two or more components of a multivariate statistical variable. But it is also allowed data to be a list of samples with possible different sample sizes, in fact this is the only function in package descriptive that admits this type of data structure.

Available options are:

box_width (default, 3/4): relative width of boxes. This value must be in the range [0,1].
box_orientation (default, vertical): possible values: vertical and horizontal.
All draw options, except points_joined, point_size, point_type, xtics, ytics, xrange, and yrange, which are internally assigned by boxplot. If you want to set your own values for this options or want to build complex scenes, make use of boxplot_description.
The following local draw options: key, color, and line_width.

Function boxplot_description creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function wxboxplot for creating embedded histograms in interfaces wxMaxima and iMaxima.

Examples:

Box-and-whishker diagram from a multivariate sample.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix(file_search("wind.data"))$

(%i3) boxplot(s2,
        box_width  = 0.2,
        title      = "Windspeed in knots",
        xlabel     = "Stations",
        color      = red,
        line_width = 2)$

Box-and-whishker diagram from three samples of different sizes.

(%i1) load ("descriptive")$

(%i2) A :
       [[6, 4, 6, 2, 4, 8, 6, 4, 6, 4, 3, 2],
        [8, 10, 7, 9, 12, 8, 10],
        [16, 13, 17, 12, 11, 18, 13, 18, 14, 12]]$

(%i3) boxplot (A, box_orientation = horizontal)$

Function: histogram (list) ¶

Function: histogram (list, option_1, option_2, …) ¶

Function: histogram (one_column_matrix) ¶

Function: histogram (one_column_matrix, option_1, option_2, …) ¶

Function: histogram (one_row_matrix) ¶

Function: histogram (one_row_matrix, option_1, option_2, …) ¶

Function: histogram_description (…) ¶

This function plots an histogram from a continuous sample. Sample data must be stored in a list of numbers or an one dimensional matrix.

Available options are:

nclasses (default, 10): number of classes of the histogram, or a list indicating the limits of the classes and the number of them, or only the limits.
frequency (default, absolute): indicates the scale of the ordinates. Possible values are: absolute, relative, and percent.
htics (default, auto): format of the histogram tics. Possible values are: auto, endpoints, intervals, or a list of labels.
All global draw options, except xrange, yrange, and xtics, which are internally assigned by histogram. If you want to set your own values for these options, make use of histogram_description. See examples bellow.
The following local draw options: key, color, fill_color, fill_density and line_width. See also bars.

Function histogram_description creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function wxhistogram for creating embedded histograms in interfaces wxMaxima and iMaxima.

Examples:

A simple with eight classes:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (
           s1,
           nclasses     = 8,
           title        = "pi digits",
           xlabel       = "digits",
           ylabel       = "Absolute frequency",
           fill_color   = grey,
           fill_density = 0.6)$

Setting the limits of the histogram to -2 and 12, with 3 classes. Also, we introduce predefined tics:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (
           s1,
           nclasses     = [-2,12,3],
           htics        = ["A", "B", "C"],
           terminal     = png,
           fill_color   = "#23afa0",
           fill_density = 0.6)$

We make use of histogram_description for setting the xrange and adding an explicit curve into the scene:

(%i1) load ("descriptive")$
(%i2) ( load("distrib"),
        m: 14, s: 2,
        s2: random_normal(m, s, 1000) ) $
(%i3) draw2d(
        grid   = true,
        xrange = [5, 25],
        histogram_description(
          s2,
          nclasses     = 9,
          frequency    = relative,
          fill_density = 0.5),
        explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$

Function: piechart (list) ¶

Function: piechart (list, option_1, option_2, …) ¶

Function: piechart (one_column_matrix) ¶

Function: piechart (one_column_matrix, option_1, option_2, …) ¶

Function: piechart (one_row_matrix) ¶

Function: piechart (one_row_matrix, option_1, option_2, …) ¶

Function: piechart_description (…) ¶

Similar to barsplot, but plots sectors instead of rectangles.

Available options are:

sector_colors (default, []): a list of colors for sectors. When there are more sectors than specified colors, the extra necesary colors are chosen at random. See color to learn more about them.
pie_center (default, [0,0]): diagram’s center.
pie_radius (default, 1): diagram’s radius.
All global draw options, except key, which is internally assigned by piechart. If you want to set your own values for this option or want to build complex scenes, make use of piechart_description.
The following local draw options: key, color, fill_density and line_width. See also ellipse.

Function piechart_description creates a graphic object suitable for creating complex scenes, together with other graphic objects. There is also a function wxpiechart for creating embedded histograms in interfaces wxMaxima and iMaxima.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) piechart(
        s1,
        xrange  = [-1.1, 1.3],
        yrange  = [-1.1, 1.1],
        title   = "Digit frequencies in pi")$