## Maxima – 231 – descriptive – 8

Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.726.

`cdf_empirical (list, option ...)`
`cdf_empirical (matrix, option ...)`
Empirical distribution function `F(x)`.

Data can be introduced as a `list` of numbers, or as a one column `matrix`.

The optional argument is the name of the variable in the returned expression, which is `x` by default.

Empirical distribution function.

``````(%i1) load ("descriptive")\$

(%i2) F(x):= ''(cdf_empirical([1,3,3,5,7,7,7,8,9]));
(%o2) F(x) := (charfun(x >= 9) + charfun(x >= 8) + 3 charfun(x >= 7)
+ charfun(x >= 5) + 2 charfun(x >= 3) + charfun(x >= 1))/9
(%i3) F(6);
4
(%o3)                                  -
9

(%i5) draw2d(line_width = 3, grid = true, explicit(F(z), z, -2, 12)) \$``````

`cov (matrix)`
The covariance matrix of the multivariate sample, defined as

where `Xj` is the `j`-th row of the sample matrix.

``````(%i6) s2 : read_matrix (file_search ("wind.data"))\$

(%i7) fpprintprec : 7\$ /* change precision for pretty output */
(%i8) cov (s2);
[ 17.2219   13.61811  14.37216  19.39623  15.42162 ]
[                                                  ]
[ 13.61811  14.98773  13.30448  15.15833  14.97109 ]
[                                                  ]
(%o8)        [ 14.37216  13.30448  15.47572  17.32543  16.1817  ]
[                                                  ]
[ 19.39623  15.15833  17.32543  32.17651  20.44684 ]
[                                                  ]
[ 15.42162  14.97109  16.1817   20.44684  24.42307 ]``````

`cov1 (matrix)`
The covariance matrix of the multivariate sample, defined as

where `Xj` is the `j`-th row of the sample matrix.

``````(%i9) s2 : read_matrix (file_search ("wind.data"))\$

(%i10) fpprintprec : 7\$ /* change precision for pretty output */
(%i11) cov1 (s2);
[ 17.39586  13.75567  14.51734  19.59215  15.57739 ]
[                                                  ]
[ 13.75567  15.13912  13.43886  15.31145  15.12232 ]
[                                                  ]
(%o11)       [ 14.51734  13.43886  15.63205  17.50044  16.34516 ]
[                                                  ]
[ 19.59215  15.31145  17.50044  32.50152  20.65338 ]
[                                                  ]
[ 15.57739  15.12232  16.34516  20.65338  24.66977 ]``````

`global_variances (matrix)`
`global_variances (matrix, options ...)`
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 `S1` the covariance matrix returned by `cov1`.

Option: `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

``````(%i1) load ("descriptive")\$

(%i2) s2 : read_matrix (file_search ("wind.data"))\$

(%i3) global_variances (s2);
(%o3) [105.3383420606059, 21.06766841212119, 12874.34690469686,
113.4651792608501, 6.636590811800794, 2.576158149609762]``````

Calculate the global_variances from the covariance matrix.

``````(%i4) s2 : read_matrix (file_search ("wind.data"))\$

(%i5) s : cov1 (s2)\$

(%i6) global_variances (s, data=false);
(%o6) [105.3383420606059, 21.06766841212119, 12874.34690469686,
113.4651792608501, 6.636590811800794, 2.576158149609762]``````

`cor (matrix)`
`cor (matrix, logical_value)`
The correlation matrix of the multivariate sample.

Option: `'data`, default `'true`, indicates whether the input matrix contains the sample data, in which case the covariance matrix `cov1` must be calculated, or not, and then the covariance matrix (symmetric) must be given, instead of the data.

``````(%i7) fpprintprec : 7\$

(%i8) s2 : read_matrix (file_search ("wind.data"))\$

(%i9) cor (s2);
[    1.0     0.8476338  0.8803515  0.8239623  0.7519506 ]
[                                                       ]
[ 0.8476338     1.0     0.8735834  0.6902622  0.782502  ]
[                                                       ]
(%o9)      [ 0.8803515  0.8735834     1.0     0.7764065  0.8323358 ]
[                                                       ]
[ 0.8239623  0.6902622  0.7764065     1.0     0.7293848 ]
[                                                       ]
[ 0.7519506  0.782502   0.8323358  0.7293848     1.0    ]``````

Calculate de correlation matrix from the covariance matrix.

``````(%i10) s2 : read_matrix (file_search ("wind.data"))\$

(%i11) s : cov1 (s2)\$

(%i12) cor (s, data=false); /* this is faster */
[    1.0     0.8476338  0.8803515  0.8239623  0.7519506 ]
[                                                       ]
[ 0.8476338     1.0     0.8735834  0.6902622  0.782502  ]
[                                                       ]
(%o12)     [ 0.8803515  0.8735834     1.0     0.7764065  0.8323358 ]
[                                                       ]
[ 0.8239623  0.6902622  0.7764065     1.0     0.7293848 ]
[                                                       ]
[ 0.7519506  0.782502   0.8323358  0.7293848     1.0    ]``````

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