[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
. They can be also downloaded from the web site www.biomates.net
.
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) load (numericalio)$ (%i2) s1 : read_list (file_search ("pidigits.data"))$ (%i3) length (s1); (%o3) 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) load (numericalio)$ (%i2) s2 : read_matrix (file_search ("wind.data"))$ (%i3) length (s2); (%o3) 100 (%i4) s2 [%]; /* last record */ (%o4) [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) load (numericalio)$ (%i2) s3 : read_matrix (file_search ("biomed.data"))$ (%i3) length (s3); (%o3) 100 (%i4) s3 [1]; /* first record */ (%o4) [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 ] |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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 (numericalio)$ (%i2) load (descriptive)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) continuous_freq (s1, 5); (%o4) [[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.
Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data")); (%o3) [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7] (%i4) discrete_freq (s1); (%o4) [[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.
This is a sort of variation of the Maxima submatrix
function. The first argument is the name of the data matrix, the second is a quoted logical expression and optional additional arguments are the numbers of the columns to be taken. Its behaviour is better understood with examples,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) subsample (s2, '(%c[1] > 18)); [ 19.38 15.37 15.12 23.09 25.25 ] [ ] [ 18.29 18.66 19.08 26.08 27.63 ] (%o4) [ ] [ 20.25 21.46 19.95 27.71 23.38 ] [ ] [ 18.79 18.96 14.46 26.38 21.84 ] |
These are multivariate records in which the wind speeds in the first meteorological station were greater than 18. See that in the quoted logical expression the i-th component is refered to as %c[i]
. Symbol %c[i]
is used inside function subsample
, therefore when used as a categorical variable, Maxima gets confused. 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 lesser than 25 knots in station number 4,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) subsample (s2, '(%c[1] >= 16 and %c[4] < 25), 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) load (numericalio)$ (%i3) s3 : read_matrix (file_search ("biomed.data"))$ (%i4) subsample (s3, '(%c[1] = B and %c[2] > 38)); [ 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) load (numericalio)$ (%i3) s3 : read_matrix (file_search ("biomed.data"))$ (%i4) subsample (s3, '(%c[1] = B and %c[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 ] (%o4) [ ] [ 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) load (numericalio)$ (%i3) s3 : read_matrix (file_search ("biomed.data"))$ (%i4) mean (s3); 65 B + 35 A 317 6 NA + 8145.0 (%o4) [-----------, ---, 87.178, -------------, 18.123, 100 10 100 3 NA + 19587 ------------] 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 of course 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) load (numericalio)$ (%i3) s3 : read_matrix (file_search ("biomed.data"))$ (%i4) mean (subsample (s3, '(%c[4] # NA and %c[6] # NA), 3, 4, 5, 6)); (%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813, 2514 ----] 13 |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This is the sample mean, defined as
n ==== _ 1 \ x = - > x n / i ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) mean (s1); 471 (%o4) --- 100 (%i5) %, numer; (%o5) 4.71 (%i6) s2 : read_matrix (file_search ("wind.data"))$ (%i7) mean (s2); (%o7) [9.9485, 10.1607, 10.8685, 15.7166, 14.8441] |
This is the sample variance, defined as
n ==== 2 1 \ _ 2 s = - > (x - x) n / i ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) var (s1), numer; (%o4) 8.425899999999999 |
See also function var1
.
This is the sample variance, defined as
n ==== 1 \ _ 2 --- > (x - x) n-1 / i ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) var1 (s1), numer; (%o4) 8.5110101010101 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) var1 (s2); (%o6) [17.39586540404041, 15.13912778787879, 15.63204924242424, 32.50152569696971, 24.66977392929294] |
See also function var
.
This is the the square root of function var
, the variance with denominator n.
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) std (s1), numer; (%o4) 2.902740084816414 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) std (s2); (%o6) [4.149928523480858, 3.871399812729241, 3.933920277534866, 5.672434260526957, 4.941970881136392] |
See also functions var
and std1
.
This is the the square root of function var1
, the variance with denominator n-1.
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) std1 (s1), numer; (%o4) 2.917363553109228 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) std1 (s2); (%o6) [4.17083509672109, 3.89090320978032, 3.953738641137555, 5.701010936401517, 4.966867617451963] |
See also functions var1
and std
.
The non central moment of order k, defined as
n ==== 1 \ k - > x n / i ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) noncentral_moment (s1, 1), numer; /* the mean */ (%o4) 4.71 (%i6) s2 : read_matrix (file_search ("wind.data"))$ (%i7) noncentral_moment (s2, 5); (%o7) [319793.8724761506, 320532.1923892463, 391249.5621381556, 2502278.205988911, 1691881.797742255] |
See also function central_moment
.
The central moment of order k, defined as
n ==== 1 \ _ k - > (x - x) n / i ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) central_moment (s1, 2), numer; /* the variance */ (%o4) 8.425899999999999 (%i6) s2 : read_matrix (file_search ("wind.data"))$ (%i7) central_moment (s2, 3); (%o7) [11.29584771375004, 16.97988248298583, 5.626661952750102, 37.5986572057918, 25.85981904394192] |
See also functions central_moment
and mean
.
The variation coefficient is the quotient between the sample standard deviation (std
) and the mean
,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) cv (s1), numer; (%o4) .6193977819764815 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) cv (s2); (%o6) [.4192426091090204, .3829365309260502, 0.363779605385983, .3627381836021478, .3346021393989506] |
See also functions std
and mean
.
This is the minimum value of the sample list,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) mini (s1); (%o4) 0 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) mini (s2); (%o6) [0.58, 0.5, 2.67, 5.25, 5.17] |
See also function maxi
.
This is the maximum value of the sample list,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) maxi (s1); (%o4) 9 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) maxi (s2); (%o6) [20.25, 21.46, 20.04, 29.63, 27.63] |
See also function mini
.
The range is the difference between the extreme values.
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) range (s1); (%o4) 9 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) range (s2); (%o6) [19.67, 20.96, 17.37, 24.38, 22.46] |
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) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) /* 1st and 3rd quartiles */ [quantile (s1, 1/4), quantile (s1, 3/4)], numer; (%o4) [2.0, 7.25] (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) quantile (s2, 1/4); (%o6) [7.2575, 7.477500000000001, 7.82, 11.28, 11.48] |
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) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) median (s1); 9 (%o4) - 2 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) median (s2); (%o6) [10.06, 9.855, 10.73, 15.48, 14.105] |
The median is the 1/2-quantile
.
See also function quantile
.
The interquartilic range is the difference between the third and first quartiles, quantile(list,3/4) - quantile(list,1/4)
,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) qrange (s1); 21 (%o4) -- 4 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) qrange (s2); (%o6) [5.385, 5.572499999999998, 6.0225, 8.729999999999999, 6.650000000000002] |
See also function quantile
.
The mean deviation, defined as
n ==== 1 \ _ - > |x - x| n / i ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) mean_deviation (s1); 51 (%o4) -- 20 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) mean_deviation (s2); (%o6) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001, 4.028546000000002] |
See also function mean
.
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) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) median_deviation (s1); 5 (%o4) - 2 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) median_deviation (s2); (%o6) [2.75, 2.755, 3.08, 4.315, 3.31] |
See also function mean
.
The harmonic mean, defined as
n -------- n ==== \ 1 > -- / x ==== i i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$ (%i4) harmonic_mean (y), numer; (%o4) 3.901858027632205 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) harmonic_mean (s2); (%o6) [6.948015590052786, 7.391967752360356, 9.055658197151745, 13.44199028193692, 13.01439145898509] |
See also functions mean
and geometric_mean
.
The geometric mean, defined as
/ n \ 1/n | /===\ | | ! ! | | ! ! x | | ! ! i| | i = 1 | \ / |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$ (%i4) geometric_mean (y), numer; (%o4) 4.454845412337012 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) geometric_mean (s2); (%o6) [8.82476274347979, 9.22652604739361, 10.0442675714889, 14.61274126349021, 13.96184163444275] |
See also functions mean
and harmonic_mean
.
The kurtosis coefficient, defined as
n ==== 1 \ _ 4 ---- > (x - x) - 3 4 / i n s ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) kurtosis (s1), numer; (%o4) - 1.273247946514421 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) kurtosis (s2); (%o6) [- .2715445622195385, 0.119998784429451, - .4275233490482866, - .6405361979019522, - .4952382132352935] |
See also functions mean
, var
and skewness
.
The skewness coefficient, defined as
n ==== 1 \ _ 3 ---- > (x - x) 3 / i n s ==== i = 1 |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) skewness (s1), numer; (%o4) .009196180476450306 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) skewness (s2); (%o6) [.1580509020000979, .2926379232061854, .09242174416107717, .2059984348148687, .2142520248890832] |
See also functions mean
, var
and kurtosis
.
Pearson's skewness coefficient, defined as
_ 3 (x - med) ----------- s |
where med is the median of list.
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) pearson_skewness (s1), numer; (%o4) .2159484029093895 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) pearson_skewness (s2); (%o6) [- .08019976629211892, .2357036272952649, .1050904062491204, .1245042340592368, .4464181795804519] |
See also functions mean
, var
and median
.
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) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) quartile_skewness (s1), numer; (%o4) .04761904761904762 (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) quartile_skewness (s2); (%o6) [- 0.0408542246982353, .1467025572005382, 0.0336239103362392, .03780068728522298, 0.210526315789474] |
See also function quantile
.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) fpprintprec : 7$ /* change precision for pretty output */ (%i5) cov (s2); [ 17.22191 13.61811 14.37217 19.39624 15.42162 ] [ ] [ 13.61811 14.98774 13.30448 15.15834 14.9711 ] [ ] (%o5) [ 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
.
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) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) fpprintprec : 7$ /* change precision for pretty output */ (%i5) cov1 (s2); [ 17.39587 13.75567 14.51734 19.59216 15.5774 ] [ ] [ 13.75567 15.13913 13.43887 15.31145 15.12232 ] [ ] (%o5) [ 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
.
Function global_variances
returns a list of global variance measures:
trace(S_1)
,
trace(S_1)/p
,
determinant(S_1)
,
sqrt(determinant(S_1))
,
determinant(S_1)^(1/p)
, (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
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) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) global_variances (s2); (%o4) [105.338342060606, 21.06766841212119, 12874.34690469686, 113.4651792608502, 6.636590811800794, 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) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) s : cov1 (s2)$ (%i5) global_variances (s, false); (%o5) [105.338342060606, 21.06766841212119, 12874.34690469686, 113.4651792608502, 6.636590811800794, 2.576158149609762] |
See also cov
and cov1
.
The correlation matrix of the multivariate sample.
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) fpprintprec:7$ (%i4) s2 : read_matrix (file_search ("wind.data"))$ (%i5) cor (s2); [ 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 ] |
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) load (numericalio)$ (%i3) fpprintprec:7$ (%i4) s2 : read_matrix (file_search ("wind.data"))$ (%i5) s : cov1 (s2)$ (%i6) cor (s, false); /* this is faster */ [ 1.0 .8476339 .8803515 .8239624 .7519506 ] [ ] [ .8476339 1.0 .8735834 .6902622 0.782502 ] [ ] (%o6) [ .8803515 .8735834 1.0 .7764065 .8323358 ] [ ] [ .8239624 .6902622 .7764065 1.0 .7293848 ] [ ] [ .7519506 0.782502 .8323358 .7293848 1.0 ] |
See also cov
and cov1
.
Function list_correlations
returns a list of correlation measures:
-1 ij S = (s ) 1 i,j = 1,2,...,p |
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.
ij s r = - ------------ ij.rest / ii jj\ 1/2 |s s | \ / |
Example:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) z : list_correlations (s2)$ (%i5) fpprintprec : 5$ /* for pretty output */ (%i6) z[1]; /* precision matrix */ [ .38486 - .13856 - .15626 - .10239 .031179 ] [ ] [ - .13856 .34107 - .15233 .038447 - .052842 ] [ ] (%o6) [ - .15626 - .15233 .47296 - .024816 - .10054 ] [ ] [ - .10239 .038447 - .024816 .10937 - .034033 ] [ ] [ .031179 - .052842 - .10054 - .034033 .14834 ] (%i7) z[2]; /* multiple correlation vector */ (%o7) [.85063, .80634, .86474, .71867, .72675] (%i8) z[3]; /* partial correlation matrix */ [ - 1.0 .38244 .36627 .49908 - .13049 ] [ ] [ .38244 - 1.0 .37927 - .19907 .23492 ] [ ] (%o8) [ .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
.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Funtion dataplot
permits direct visualization of sample data, both univariate (list) and multivariate (matrix). Giving values to the following options some aspects of the plot can be controlled:
'outputdev
, default "x"
, indicates the output device; correct values are "x"
, "eps"
and "png"
, for the screen, postscript and png format files, respectively.
'maintitle
, default ""
, is the main title between double quotes.
'axisnames
, default ["x","y","z"]
, is a list with the names of axis x
, y
and z
.
'joined
, default false
, a logical value to select points in 2D to be joined or isolated.
'picturescales
, default [1.0, 1.0]
, scaling factors for the size of the plot.
'threedim
, default true
, tells Maxima whether to plot a three column matrix with a 3D diagram or a multivariate scatterplot. See examples bellow.
'axisrot
, default [60, 30]
, changes the point of view when 'threedim
is set to true
and data are stored in a three column matrix. The first number is the rotation angle of the x-axis, and the second number is the rotation angle of the z-axis, both measured in degrees.
'nclasses
, default 10
, is the number of classes for the histograms in the diagonal of multivariate scatterplots.
'pointstyle
, default 1
, is an integer to indicate how to display sample points.
For example, with the following input a simple plot of the first twenty digits of %pi
is requested and the output stored in an eps file.
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) dataplot (makelist (s1[k], k, 1, 20), 'pointstyle = 3)$ |
Note that one dimensional data are plotted as a time series. In the next case, same more data with different settings,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) dataplot (makelist (s1[k], k, 1, 50), 'maintitle = "First pi digits", 'axisnames = ["digit order", "digit value"], 'pointstyle = 2, 'joined = true)$ |
Function dataplot
can be used to plot points in the plane. The next example is a scatterplot of the pairs of wind speeds corresponding to the first and fifth meteorological stations,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) dataplot (submatrix (s2, 2, 3, 4), 'pointstyle = 2, 'maintitle = "Pairs of wind speeds measured in knots", 'axisnames = ["Wind speed in A", "Wind speed in E"])$ |
If points are stored in a two column matrix, dataplot
can plot them directly, but if they are formatted as a list of pairs, their must be transformed to a matrix as in the following example.
(%i1) load (descriptive)$ (%i2) x : [[-1, 2], [5, 7], [5, -3], [-6, -9], [-4, 6]]$ (%i3) dataplot (apply ('matrix, x), 'maintitle = "Points", 'joined = true, 'axisnames = ["", ""], 'picturescales = [0.5, 1.0])$ |
Points in three dimensional space can be seen as a projection on the plane. In this example, plots of wind speeds corresponding to three meteorological stations are requested, first in a 3D plot and then in a multivariate scatterplot.
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) /* 3D plot */ dataplot (submatrix (s2, 4, 5), 'pointstyle = 2, 'maintitle = "Pairs of wind speeds measured in knots", 'axisnames = ["Station A", "Station B", "Station C"])$ (%i5) /* Multivariate scatterplot */ dataplot (submatrix (s2, 4, 5), 'nclasses = 6, 'threedim = false)$ |
Note that in the last example, the number of classes in the histograms of the diagonal is set to 6, and that option 'threedim
is set to false
.
For more than three dimensions only multivariate scatterplots are possible, as in
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) dataplot (s2)$ |
This function plots an histogram. Sample data must be stored in a list of numbers or a one column matrix. Giving values to the following options some aspects of the plot can be controlled:
'outputdev
, default "x"
, indicates the output device; correct values are "x"
, "eps"
and "png"
, for the screen, postscript and png format files, respectively.
'maintitle
, default ""
, is the main title between double quotes.
'axisnames
, default ["x", "Fr."]
, is a list with the names of axis x
and y
.
'picturescales
, default [1.0, 1.0]
, scaling factors for the size of the plot.
'nclasses
, default 10
, is the number of classes or bars.
'relbarwidth
, default 0.9
, a decimal number between 0 and 1 to control bars width.
'barcolor
, default 1
, an integer to indicate bars color.
'colorintensity
, default 1
, a decimal number between 0 and 1 to fix color intensity.
In the next two examples, histograms are requested for the first 100 digits of number %pi
and for the wind speeds in the third meteorological station.
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s1 : read_list (file_search ("pidigits.data"))$ (%i4) histogram (s1, 'maintitle = "pi digits", 'axisnames = ["", "Absolute frequency"], 'relbarwidth = 0.2, 'barcolor = 3, 'colorintensity = 0.6)$ (%i5) s2 : read_matrix (file_search ("wind.data"))$ (%i6) histogram (col (s2, 3), 'colorintensity = 0.3)$ |
Note that in the first case, s1
is a list and in the second example, col(s2,3)
is a matrix.
See also function barsplot
.
Similar to histogram
but for discrete, numeric or categorical, statistical variables. These are the options,
'outputdev
, default "x"
, indicates the output device; correct values are "x"
, "eps"
and "png"
, for the screen, postscript and png format files, respectively.
'maintitle
, default ""
, is the main title between double quotes.
'axisnames
, default ["x", "Fr."]
, is a list with the names of axis x
and y
.
'picturescales
, default [1.0, 1.0]
, scaling factors for the size of the plot.
'relbarwidth
, default 0.9
, a decimal number between 0 and 1 to control bars width.
'barcolor
, default 1
, an integer to indicate bars color.
'colorintensity
, default 1
, a decimal number between 0 and 1 to fix color intensity.
This example plots the barchart for groups A
and B
of patients in sample s3
,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s3 : read_matrix (file_search ("biomed.data"))$ (%i4) barsplot (col (s3, 1), 'maintitle = "Groups of patients", 'axisnames = ["Group", "# of individuals"], 'colorintensity = 0.2)$ |
The first column in sample s3
stores the categorical values A
and B
, also known sometimes as factors. On the other hand, the positive integer numbers in the second column are ages, in years, which is a discrete variable, so we can plot the absolute frequencies for these values,
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s3 : read_matrix (file_search ("biomed.data"))$ (%i4) barsplot (col (s3, 2), 'maintitle = "Ages", 'axisnames = ["Years", "# of individuals"], 'colorintensity = 0.2, 'relbarwidth = 0.6)$ |
See also function histogram
.
This function plots box 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. See example bellow. These are the options,
'outputdev
, default "x"
, indicates the output device; correct values are "x"
, "eps"
and "png"
, for the screen, postscript and png format files, respectively.
'maintitle
, default ""
, is the main title between double quotes.
'axisnames
, default ["sample", "y"]
, is a list with the names of axis x
and y
.
'picturescales
, default [1.0, 1.0]
, scaling factors for the size of the plot.
Examples:
(%i1) load (descriptive)$ (%i2) load (numericalio)$ (%i3) s2 : read_matrix (file_search ("wind.data"))$ (%i4) boxplot (s2, 'maintitle = "Windspeed in knots", 'axisnames = ["Seasons", ""])$ (%i5) 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]]$ (%i6) boxplot (A)$ |
[ << ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This document was generated by Robert Dodier on May, 2 2007 using texi2html 1.76.