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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 ]
Categories: Descriptive statistics · Share packages · Package descriptive
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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
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
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
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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
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
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) 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
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
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) 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
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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:
draw
package. See also bars
and barsplot
.
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
Plots scatter diagrams both for univariate (list) and multivariate (matrix) samples.
Available options are:
draw
package.
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
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:
draw
package.
3/4
by default). This
value must be in the range [0,1]
.
[]
. When there are more samples than specified
colors, the extra necesary colors are chosen at random. See color
to learn more about them.
false
, absolute frequencies are
used; if true
, tics on the y-axis are relative frequencies.
Default value is false
.
orderlessp
or ordergreatp
,
indicating how statistical outcomes should be ordered on the x-axis.
Default value is orderlessp
.
[]
. 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
Similar to barsplot
, but plots sectors instead of rectangles.
Available options are:
draw
package.
[0,0]
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
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:
draw
package.
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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This document was generated by Robert Dodier on April, 24 2010 using texi2html 1.76.