45. descriptive

45.1 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:

      3
      1
      4
      1
      5
      9
      2
      6
      5
      3 ...

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 ]

45.2 Functions and Variables for data manipulation

Function: continuous_freq (list)

Function: continuous_freq (list, m)

The argument of continuous_freq must be a list of numbers, which will be then grouped in intervals and counted how many of them belong to each group. Optionally, function continuous_freq admits a second argument indicating the number of classes, 10 is default,

(%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]]

The first list contains the interval limits and the second the corresponding counts: there are 16 digits inside the interval [0, 1.8], that is 0's and 1's, 24 digits in (1.8, 3.6], that is 2's and 3's, and so on.

Categories:  Package descriptive

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.

Categories:  Package descriptive

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

Categories:  Package descriptive

45.3 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]

Categories:  Package descriptive

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

See also function var1.

Categories:  Package descriptive

Function: var1 (list)

Function: var1 (matrix)

This is the sample variance, defined as

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

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) var1 (s1), numer;
(%o3)                    8.5110101010101
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) var1 (s2);
(%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424, 
                            32.50152569696971, 24.66977392929294]

See also function var.

Categories:  Package descriptive

Function: std (list)

Function: std (matrix)

This is the the square root of function var, the variance with denominator n.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) std (s1), numer;
(%o3)                   2.902740084816414
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) std (s2);
(%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866, 
                            5.672434260526957, 4.941970881136392]

See also functions var and std1.

Categories:  Package descriptive

Function: std1 (list)

Function: std1 (matrix)

This is the the square root of function var1, the variance with denominator n-1.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) std1 (s1), numer;
(%o3)                   2.917363553109228
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) std1 (s2);
(%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555, 
                            5.701010936401517, 4.966867617451963]

See also functions var1 and std.

Categories:  Package descriptive

Function: noncentral_moment (list, k)

Function: noncentral_moment (matrix, k)

The non central moment of order k, defined as

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

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) noncentral_moment (s1, 1), numer; /* the mean */
(%o3)                         4.71
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) noncentral_moment (s2, 5);
(%o6) [319793.8724761505, 320532.1923892463,
      391249.5621381556, 2502278.205988911, 1691881.797742255]

See also function central_moment.

Categories:  Package descriptive

Function: central_moment (list, k)

Function: central_moment (matrix, k)

The central moment of order k, defined as

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

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) central_moment (s1, 2), numer; /* the variance */
(%o3)                   8.425899999999999
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) central_moment (s2, 3);
(%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                             37.5986572057918, 25.85981904394192]

See also functions central_moment and mean.

Categories:  Package descriptive

Function: cv (list)

Function: cv (matrix)

The variation coefficient is the quotient between the sample standard deviation (std) and the mean,

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) cv (s1), numer;
(%o3)                   .6193977819764815
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) cv (s2);
(%o5) [.4192426091090204, .3829365309260502, 0.363779605385983, 
                            .3627381836021478, .3346021393989506]

See also functions std and mean.

Categories:  Package descriptive

Function: mini (list)

Function: mini (matrix)

This is the minimum value of the sample list,

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) mini (s1);
(%o3)                           0
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) mini (s2);
(%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

See also function maxi.

Categories:  Package descriptive

Function: maxi (list)

Function: maxi (matrix)

This is the maximum value of the sample list,

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) maxi (s1);
(%o3)                           9
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) maxi (s2);
(%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

See also function mini.

Categories:  Package descriptive

Function: range (list)

Function: range (matrix)

The range is the difference between the extreme values.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) range (s1);
(%o3)                           9
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) range (s2);
(%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

Categories:  Package descriptive

Function: quantile (list, p)

Function: quantile (matrix, p)

This is the p-quantile, with p a number in [0, 1], of the sample list. Although there are several definitions for the sample quantile (Hyndman, R. J., Fan, Y. (1996) Sample quantiles in statistical packages. American Statistician, 50, 361-365), the one based on linear interpolation is implemented in package descriptive.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) /* 1st and 3rd quartiles */
         [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
(%o3)                      [2.0, 7.25]
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) quantile (s2, 1/4);
(%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

Categories:  Package descriptive

Function: median (list)

Function: median (matrix)

Once the sample is ordered, if the sample size is odd the median is the central value, otherwise it is the mean of the two central values.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) median (s1);
                                9
(%o3)                           -
                                2
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) median (s2);
(%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

The median is the 1/2-quantile.

See also function quantile.

Categories:  Package descriptive

Function: qrange (list)

Function: qrange (matrix)

The interquartilic range is the difference between the third and first quartiles, quantile(list,3/4) - quantile(list,1/4),

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) qrange (s1);
                               21
(%o3)                          --
                               4
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) qrange (s2);
(%o5) [5.385, 5.572499999999998, 6.022500000000001, 
                            8.729999999999999, 6.649999999999999]

See also function quantile.

Categories:  Package descriptive

Function: mean_deviation (list)

Function: mean_deviation (matrix)

The mean deviation, defined as

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

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) mean_deviation (s1);
                               51
(%o3)                          --
                               20
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) mean_deviation (s2);
(%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001, 
                                               4.028546000000002]

See also function mean.

Categories:  Package descriptive

Function: median_deviation (list)

Function: median_deviation (matrix)

The median deviation, defined as

                 n
               ====
           1   \
           -    >    |x - med|
           n   /       i
               ====
               i = 1

where med is the median of list.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) median_deviation (s1);
                                5
(%o3)                           -
                                2
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) median_deviation (s2);
(%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

See also function mean.

Categories:  Package descriptive

Function: harmonic_mean (list)

Function: harmonic_mean (matrix)

The harmonic mean, defined as

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

Example:

(%i1) load (descriptive)$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i3) harmonic_mean (y), numer;
(%o3)                   3.901858027632205
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) harmonic_mean (s2);
(%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745, 
                            13.44199028193692, 13.01439145898509]

See also functions mean and geometric_mean.

Categories:  Package descriptive

Function: geometric_mean (list)

Function: geometric_mean (matrix)

The geometric mean, defined as

                 /  n      \ 1/n
                 | /===\   |
                 |  ! !    |
                 |  ! !  x |
                 |  ! !   i|
                 | i = 1   |
                 \         /

Example:

(%i1) load (descriptive)$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i3) geometric_mean (y), numer;
(%o3)                   4.454845412337012
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) geometric_mean (s2);
(%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889, 
                            14.61274126349021, 13.96184163444275]

See also functions mean and harmonic_mean.

Categories:  Package descriptive

Function: kurtosis (list)

Function: kurtosis (matrix)

The kurtosis coefficient, defined as

                    n
                  ====
            1     \          _ 4
           ----    >    (x - x)  - 3
              4   /       i
           n s    ====
                  i = 1

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) kurtosis (s1), numer;
(%o3)                  - 1.273247946514421
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) kurtosis (s2);
(%o5) [- .2715445622195385, 0.119998784429451, 
       - .4275233490482861, - .6405361979019522, - .4952382132352935]

See also functions mean, var and skewness.

Categories:  Package descriptive

Function: skewness (list)

Function: skewness (matrix)

The skewness coefficient, defined as

                    n
                  ====
            1     \          _ 3
           ----    >    (x - x)
              3   /       i
           n s    ====
                  i = 1

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) skewness (s1), numer;
(%o3)                  .009196180476450424
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) skewness (s2);
(%o5) [.1580509020000978, .2926379232061854, .09242174416107717, 
                            .2059984348148687, .2142520248890831]

See also functions mean, var and kurtosis.

Categories:  Package descriptive

Function: pearson_skewness (list)

Function: pearson_skewness (matrix)

Pearson's skewness coefficient, defined as

                _
             3 (x - med)
             -----------
                  s

where med is the median of list.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) pearson_skewness (s1), numer;
(%o3)                   .2159484029093895
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) pearson_skewness (s2);
(%o5) [- .08019976629211892, .2357036272952649, 
         .1050904062491204, .1245042340592368, .4464181795804519]

See also functions mean, var and median.

Categories:  Package descriptive

Function: quartile_skewness (list)

Function: quartile_skewness (matrix)

The quartile skewness coefficient, defined as

               c    - 2 c    + c
                3/4      1/2    1/4
               --------------------
                   c    - c
                    3/4    1/4

where c_p is the p-quantile of sample list.

Example:

(%i1) load (descriptive)$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) quartile_skewness (s1), numer;
(%o3)                  .04761904761904762
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) quartile_skewness (s2);
(%o5) [- 0.0408542246982353, .1467025572005382, 
       0.0336239103362392, .03780068728522298, .2105263157894735]

See also function quantile.

Categories:  Package descriptive

45.4 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 ]

See also function cov1.

Categories:  Package descriptive

Function: cov1 (matrix)

The covariance matrix of the multivariate sample, defined as

              n
             ====
         1   \           _        _
   S  = ---   >    (X  - X) (X  - X)'
    1   n-1  /       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) cov1 (s2);
      [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
      [                                                  ]
      [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
      [                                                  ]
(%o4) [ 14.51734  13.43887  15.63205  17.50044  16.34516 ]
      [                                                  ]
      [ 19.59216  15.31145  17.50044  32.50153  20.65338 ]
      [                                                  ]
      [ 15.5774   15.12232  16.34516  20.65338  24.66977 ]

See also function cov.

Categories:  Package descriptive

Function: global_variances (matrix)

Function: global_variances (matrix, logical_value)

Function global_variances returns a list of global variance measures:

• total variance: trace(S_1),
• mean variance: trace(S_1)/p,
• generalized variance: determinant(S_1),
• generalized standard deviation: sqrt(determinant(S_1)),
• efective variance determinant(S_1)^(1/p), (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
• efective standard deviation: determinant(S_1)^(1/(2*p)).

where p is the dimension of the multivariate random variable and S_1 the covariance matrix returned by cov1.

Example:

(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) global_variances (s2);
(%o3) [105.338342060606, 21.06766841212119, 12874.34690469686, 
         113.4651792608501, 6.636590811800795, 2.576158149609762]

Function global_variances has an optional logical argument: global_variances(x,true) tells Maxima that x is the data matrix, making the same as global_variances(x). On the other hand, global_variances(x,false) means that x is not the data matrix, but the covariance matrix, avoiding its recalculation,

(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) s : cov1 (s2)$
(%i4) global_variances (s, false);
(%o4) [105.338342060606, 21.06766841212119, 12874.34690469686, 
         113.4651792608501, 6.636590811800795, 2.576158149609762]

See also cov and cov1.

Categories:  Package descriptive

Function: cor (matrix)

Function: cor (matrix, logical_value)

The correlation matrix of the multivariate sample.

Example:

(%i1) load (descriptive)$
(%i2) fpprintprec : 7 $
(%i3) s2 : read_matrix (file_search ("wind.data"))$
(%i4) cor (s2);
      [   1.0     .8476339  .8803515  .8239624  .7519506 ]
      [                                                  ]
      [ .8476339    1.0     .8735834  .6902622  0.782502 ]
      [                                                  ]
(%o4) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
      [                                                  ]
      [ .8239624  .6902622  .7764065    1.0     .7293848 ]
      [                                                  ]
      [ .7519506  0.782502  .8323358  .7293848    1.0    ]

Function cor has an optional logical argument: cor(x,true) tells Maxima that x is the data matrix, making the same as cor(x). On the other hand, cor(x,false) means that x is not the data matrix, but the covariance matrix, avoiding its recalculation,

(%i1) load (descriptive)$
(%i2) fpprintprec : 7 $
(%i3) s2 : read_matrix (file_search ("wind.data"))$
(%i4) s : cov1 (s2)$
(%i5) cor (s, false); /* this is faster */
      [   1.0     .8476339  .8803515  .8239624  .7519506 ]
      [                                                  ]
      [ .8476339    1.0     .8735834  .6902622  0.782502 ]
      [                                                  ]
(%o5) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
      [                                                  ]
      [ .8239624  .6902622  .7764065    1.0     .7293848 ]
      [                                                  ]
      [ .7519506  0.782502  .8323358  .7293848    1.0    ]

See also cov and cov1.

Categories:  Package descriptive

Function: list_correlations (matrix)

Function: list_correlations (matrix, logical_value)

Function list_correlations returns a list of correlation measures:

• precision matrix: the inverse of the covariance matrix S_1,

         -1     ij
        S   = (s  )             
         1         i,j = 1,2,...,p
  

• multiple correlation vector: (R_1^2, R_2^2, ..., R_p^2), with

         2          1
        R  = 1 - -------
         i        ii
                 s   s
                      ii
  

  being an indicator of the goodness of fit of the linear multivariate regression model on X_i when the rest of variables are used as regressors.

• partial correlation matrix: with element (i, j) being

                           ij
                          s
        r        = - ------------
         ij.rest     / ii  jj\ 1/2
                     |s   s  |
                     \       /
  

Example:

(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) z : list_correlations (s2)$
(%i4) fpprintprec : 5$ /* for pretty output */
(%i5) z[1];  /* precision matrix */
      [  .38486   - .13856   - .15626   - .10239    .031179  ]
      [                                                      ]
      [ - .13856   .34107    - .15233    .038447   - .052842 ]
      [                                                      ]
(%o5) [ - .15626  - .15233    .47296    - .024816  - .10054  ]
      [                                                      ]
      [ - .10239   .038447   - .024816   .10937    - .034033 ]
      [                                                      ]
      [ .031179   - .052842  - .10054   - .034033   .14834   ]
(%i6) z[2];  /* multiple correlation vector */
(%o6)      [.85063, .80634, .86474, .71867, .72675]
(%i7) z[3];  /* partial correlation matrix */
      [  - 1.0     .38244   .36627   .49908   - .13049 ]
      [                                                ]
      [  .38244    - 1.0    .37927  - .19907   .23492  ]
      [                                                ]
(%o7) [  .36627    .37927   - 1.0    .10911    .37956  ]
      [                                                ]
      [  .49908   - .19907  .10911   - 1.0     .26719  ]
      [                                                ]
      [ - .13049   .23492   .37956   .26719    - 1.0   ]

Function list_correlations also has an optional logical argument: list_correlations(x,true) tells Maxima that x is the data matrix, making the same as list_correlations(x). On the other hand, list_correlations(x,false) means that x is not the data matrix, but the covariance matrix, avoiding its recalculation.

See also cov and cov1.

Categories:  Package descriptive

45.5 Functions and Variables for statistical graphs

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, ...)

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

Available options are:

• Those defined in the draw package. See also bars and barsplot.
• nclasses: number of classes of the histogram (10 by default).

See also discrete_freq and continuous_freq to count data, and bars and barsplot to display bar graphs.

Examples:

A simple histogram 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)$

Categories:  Package descriptive · Plotting

Function: scatterplot (list)

Function: scatterplot (list, option_1, option_2, ...)

Function: scatterplot (matrix)

Function: scatterplot (matrix, option_1, option_2, ...)

Plots scatter diagrams both for univariate (list) and multivariate (matrix) samples.

Available options are:

• Those defined in the draw package.
• nclasses: number of classes of the histogram (10 by default).

Examples:

Univariate scatter diagram from a simulated Gaussian sample.

(%i1) load (descriptive)$
(%i2) load (distrib)$
(%i3) scatterplot(
        random_normal(0,1,200),
        xaxis      = true,
        point_size = 2,
        terminal   = eps,
        eps_width  = 10,
        eps_height = 2)$

Two dimensional scatter plot.

(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(
       submatrix(s2, 1,2,3),
       title      = "Data from stations #4 and #5",
       point_type = diamant,
       point_size = 2,
       color      = blue)$

Three dimensional scatter plot.

(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(submatrix (s2, 1,2))$

Five dimensional scatter plot, with five classes histograms.

(%i1) load (descriptive)$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(
        s2,
        nclasses     = 5,
        fill_color   = blue,
        fill_density = 0.3,
        xtics        = 5)$

For plotting isolated or line-joined points in two and three dimensions, see points. For histogram related options, see bars.

See also histogram.

Categories:  Package descriptive · Plotting

Function: barsplot (data1, data2, ..., option_1, option_2, ...)

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:

• Those defined in the draw package.
• box_width: relative width of rectangles (3/4 by default). This value must be in the range [0,1].
• groups_gap: a positive integer number representing the gap between two consecutive groups of bars. Default value is 1.
• bars_colors: a list of colors for multiple samples. Defaults to the empty list []. When there are more samples than specified colors, the extra necesary colors are chosen at random. See color to learn more about them.
• relative_frequencies: if false, absolute frequencies are used; if true, tics on the y-axis are relative frequencies. Default value is false.
• ordering: possible values are orderlessp or ordergreatp, indicating how statistical outcomes should be ordered on the x-axis. Default value is orderlessp.
• sample_keys: a list with the strings to be used in the legend. Defaults to the empty list []. When the list length is other than 0 or the number of samples, an error message is returned.

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],
        relative_frequencies = true,
        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)$

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) draw_compound : false $
(%i5) bp1 : 
        barsplot(
         l1,
         box_width = 1,
         fill_density = 0.5,
         bars_colors = [blue],
         relative_frequencies = true)$
(%i6) bp2 : 
        barsplot(
         l2,
         box_width = 1,
         fill_density = 0.5,
         bars_colors = [red],
         relative_frequencies = true)$
(%i7) draw(apply(gr2d,bp1), 
           apply(gr2d,bp2)) $

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

Categories:  Package descriptive · Plotting

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, ...)

Similar to barsplot, but plots sectors instead of rectangles.

Available options are:

• Those defined in the draw package.
• pie_center: diagram's center ([0,0] by default).
• pie_radius: diagram's radius (1 by default).

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],
        axis_top    = false,
        axis_right  = false,
        axis_left   = false,
        axis_bottom = false,
        xtics       = none,
        ytics       = none,
        title       = "Digit frequencies in pi")$

See also function barsplot.

Categories:  Package descriptive · Plotting

Function: boxplot (data)

Function: boxplot (data, option_1, option_2, ...)

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:

• Those defined in the draw package.
• box_width: relative width of boxes (3/4 by default). This value must be in the range [0,1].

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)$

Categories:  Package descriptive · Plotting

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