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.\" ========================================================================
.\"
.IX Title "Basic 3pm"
.TH Basic 3pm 2024-01-10 "perl v5.38.2" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
.if n .ad l
.nh
.SH NAME
PDL::Stats::Basic \-\- basic statistics and related utilities such as standard deviation, Pearson correlation, and t\-tests.
.SH DESCRIPTION
.IX Header "DESCRIPTION"
The terms FUNCTIONS and METHODS are arbitrarily used to refer to methods that are threadable and methods that are NOT threadable, respectively.
.PP
Does not have mean or median function here. see SEE ALSO.
.SH SYNOPSIS
.IX Header "SYNOPSIS"
.Vb 3
\&    use PDL::LiteF;
\&    use PDL::NiceSlice;
\&    use PDL::Stats::Basic;
\&
\&    my $stdv = $data\->stdv;
.Ve
.PP
or
.PP
.Vb 1
\&    my $stdv = stdv( $data );
.Ve
.SH FUNCTIONS
.IX Header "FUNCTIONS"
.SS stdv
.IX Subsection "stdv"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Sample standard deviation.
.PP
stdv processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS stdv_unbiased
.IX Subsection "stdv_unbiased"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Unbiased estimate of population standard deviation.
.PP
stdv_unbiased processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS var
.IX Subsection "var"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Sample variance.
.PP
var processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS var_unbiased
.IX Subsection "var_unbiased"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Unbiased estimate of population variance.
.PP
var_unbiased processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS se
.IX Subsection "se"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Standard error of the mean. Useful for calculating confidence intervals.
.PP
.Vb 1
\&    # 95% confidence interval for samples with large N
\&
\&    $ci_95_upper = $data\->average + 1.96 * $data\->se;
\&    $ci_95_lower = $data\->average \- 1.96 * $data\->se;
.Ve
.PP
se processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS ss
.IX Subsection "ss"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Sum of squared deviations from the mean.
.PP
ss processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS skew
.IX Subsection "skew"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Sample skewness, measure of asymmetry in data. skewness == 0 for normal distribution.
.PP
skew processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS skew_unbiased
.IX Subsection "skew_unbiased"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Unbiased estimate of population skewness. This is the number in GNumeric Descriptive Statistics.
.PP
skew_unbiased processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS kurt
.IX Subsection "kurt"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Sample kurtosis, measure of "peakedness" of data. kurtosis == 0 for normal distribution.
.PP
kurt processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS kurt_unbiased
.IX Subsection "kurt_unbiased"
.Vb 1
\&  Signature: (a(n); float+ [o]b())
.Ve
.PP
Unbiased estimate of population kurtosis. This is the number in GNumeric Descriptive Statistics.
.PP
kurt_unbiased processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS cov
.IX Subsection "cov"
.Vb 1
\&  Signature: (a(n); b(n); float+ [o]c())
.Ve
.PP
Sample covariance. see \fBcorr\fR for ways to call
.PP
cov processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS cov_table
.IX Subsection "cov_table"
.Vb 1
\&  Signature: (a(n,m); float+ [o]c(m,m))
.Ve
.PP
Square covariance table. Gives the same result as threading using \fBcov\fR but it calculates only half the square, hence much faster. And it is easier to use with higher dimension pdls.
.PP
Usage:
.PP
.Vb 1
\&    # 5 obs x 3 var, 2 such data tables
\&
\&    perldl> $a = random 5, 3, 2
\&
\&    perldl> p $cov = $a\->cov_table
\&    [
\&     [
\&      [ 8.9636438 \-1.8624472 \-1.2416588]
\&      [\-1.8624472  14.341514 \-1.4245366]
\&      [\-1.2416588 \-1.4245366  9.8690655]
\&     ]
\&     [
\&      [   10.32644 \-0.31311789 \-0.95643674]
\&      [\-0.31311789   15.051779  \-7.2759577]
\&      [\-0.95643674  \-7.2759577   5.4465141]
\&     ]
\&    ]
\&    # diagonal elements of the cov table are the variances
\&    perldl> p $a\->var
\&    [
\&     [ 8.9636438  14.341514  9.8690655]
\&     [  10.32644  15.051779  5.4465141]
\&    ]
.Ve
.PP
for the same cov matrix table using \fBcov\fR,
.PP
.Vb 1
\&    perldl> p $a\->dummy(2)\->cov($a\->dummy(1))
.Ve
.PP
cov_table processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS corr
.IX Subsection "corr"
.Vb 1
\&  Signature: (a(n); b(n); float+ [o]c())
.Ve
.PP
Pearson correlation coefficient. r = cov(X,Y) / (stdv(X) * stdv(Y)).
.PP
Usage:
.PP
.Vb 3
\&    perldl> $a = random 5, 3
\&    perldl> $b = sequence 5,3
\&    perldl> p $a\->corr($b)
\&
\&    [0.20934208 0.30949881 0.26713007]
.Ve
.PP
for square corr table
.PP
.Vb 1
\&    perldl> p $a\->corr($a\->dummy(1))
\&
\&    [
\&     [           1  \-0.41995259 \-0.029301192]
\&     [ \-0.41995259            1  \-0.61927619]
\&     [\-0.029301192  \-0.61927619            1]
\&    ]
.Ve
.PP
but it is easier and faster to use \fBcorr_table\fR.
.PP
corr processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS corr_table
.IX Subsection "corr_table"
.Vb 1
\&  Signature: (a(n,m); float+ [o]c(m,m))
.Ve
.PP
Square Pearson correlation table. Gives the same result as threading using \fBcorr\fR but it calculates only half the square, hence much faster. And it is easier to use with higher dimension pdls.
.PP
Usage:
.PP
.Vb 1
\&    # 5 obs x 3 var, 2 such data tables
\& 
\&    perldl> $a = random 5, 3, 2
\&    
\&    perldl> p $a\->corr_table
\&    [
\&     [
\&     [          1 \-0.69835951 \-0.18549048]
\&     [\-0.69835951           1  0.72481605]
\&     [\-0.18549048  0.72481605           1]
\&    ]
\&    [
\&     [          1  0.82722569 \-0.71779883]
\&     [ 0.82722569           1 \-0.63938828]
\&     [\-0.71779883 \-0.63938828           1]
\&     ]
\&    ]
.Ve
.PP
for the same result using \fBcorr\fR,
.PP
.Vb 1
\&    perldl> p $a\->dummy(2)\->corr($a\->dummy(1))
.Ve
.PP
This is also how to use \fBt_corr\fR and \fBn_pair\fR with such a table.
.PP
corr_table processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS t_corr
.IX Subsection "t_corr"
.Vb 1
\&  Signature: (r(); n(); [o]t())
.Ve
.PP
.Vb 3
\&    $corr   = $data\->corr( $data\->dummy(1) );
\&    $n      = $data\->n_pair( $data\->dummy(1) );
\&    $t_corr = $corr\->t_corr( $n );
\&
\&    use PDL::GSL::CDF;
\&
\&    $p_2tail = 2 * (1 \- gsl_cdf_tdist_P( $t_corr\->abs, $n\-2 ));
.Ve
.PP
t significance test for Pearson correlations.
.PP
t_corr processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS n_pair
.IX Subsection "n_pair"
.Vb 1
\&  Signature: (a(n); b(n); indx [o]c())
.Ve
.PP
Returns the number of good pairs between 2 lists. Useful with \fBcorr\fR (esp. when bad values are involved)
.PP
n_pair processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS corr_dev
.IX Subsection "corr_dev"
.Vb 1
\&  Signature: (a(n); b(n); float+ [o]c())
.Ve
.PP
.Vb 1
\&    $corr = $a\->dev_m\->corr_dev($b\->dev_m);
.Ve
.PP
Calculates correlations from \fBdev_m\fR vals. Seems faster than doing \fBcorr\fR from original vals when data pdl is big
.PP
corr_dev processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS t_test
.IX Subsection "t_test"
.Vb 1
\&  Signature: (a(n); b(m); float+ [o]t(); [o]d())
.Ve
.PP
.Vb 1
\&    my ($t, $df) = t_test( $pdl1, $pdl2 );
\&
\&    use PDL::GSL::CDF;
\&
\&    my $p_2tail = 2 * (1 \- gsl_cdf_tdist_P( $t\->abs, $df ));
.Ve
.PP
Independent sample t\-test, assuming equal var.
.PP
t_test processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS t_test_nev
.IX Subsection "t_test_nev"
.Vb 1
\&  Signature: (a(n); b(m); float+ [o]t(); [o]d())
.Ve
.PP
Independent sample t\-test, NOT assuming equal var. ie Welch two sample t test. Df follows Welch-Satterthwaite equation instead of Satterthwaite (1946, as cited by Hays, 1994, 5th ed.). It matches GNumeric, which matches R.
.PP
.Vb 1
\&    my ($t, $df) = $pdl1\->t_test( $pdl2 );
.Ve
.PP
t_test_nev processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS t_test_paired
.IX Subsection "t_test_paired"
.Vb 1
\&  Signature: (a(n); b(n); float+ [o]t(); [o]d())
.Ve
.PP
Paired sample t\-test.
.PP
t_test_paired processes bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
.SS binomial_test
.IX Subsection "binomial_test"
.Vb 1
\&  Signature: (x(); n(); p_expected(); [o]p())
.Ve
.PP
Binomial test. One-tailed significance test for two-outcome distribution. Given the number of successes, the number of trials, and the expected probability of success, returns the probability of getting this many or more successes.
.PP
This function does NOT currently support bad value in the number of successes.
.PP
Usage:
.PP
.Vb 2
\&  # assume a fair coin, ie. 0.5 probablity of getting heads
\&  # test whether getting 8 heads out of 10 coin flips is unusual
\&
\&  my $p = binomial_test( 8, 10, 0.5 );  # 0.0107421875. Yes it is unusual.
.Ve
.SH METHODS
.IX Header "METHODS"
.SS rtable
.IX Subsection "rtable"
Reads either file or file handle*. Returns observation x variable pdl and var and obs ids if specified. Ids in perl @ ref to allow for non-numeric ids. Other non-numeric entries are treated as missing, which are filled with \f(CW$opt\fR{MISSN} then set to BAD*. Can specify num of data rows to read from top but not arbitrary range.
.PP
*If passed handle, it will not be closed here.
.PP
Default options (case insensitive):
.PP
.Vb 8
\&    V       => 1,        # verbose. prints simple status
\&    TYPE    => double,
\&    C_ID    => 1,        # boolean. file has col id.
\&    R_ID    => 1,        # boolean. file has row id.
\&    R_VAR   => 0,        # boolean. set to 1 if var in rows
\&    SEP     => "\et",     # can take regex qr//
\&    MISSN   => \-999,     # this value treated as missing and set to BAD
\&    NROW    => \*(Aq\*(Aq,       # set to read specified num of data rows
.Ve
.PP
Usage:
.PP
Sample file diet.txt:
.PP
.Vb 5
\&    uid height  weight  diet
\&    akw 72      320     1
\&    bcm 68      268     1
\&    clq 67      180     2
\&    dwm 70      200     2
\&  
\&    ($data, $idv, $ido) = rtable \*(Aqdiet.txt\*(Aq;
\&
\&    # By default prints out data info and @$idv index and element
\&
\&    reading diet.txt for data and id... OK.
\&    data table as PDL dim o x v: PDL: Double D [4,3]
\&    0   height
\&    1   weight
\&    2   diet
.Ve
.PP
Another way of using it,
.PP
.Vb 1
\&    $data = rtable( \e*STDIN, {TYPE=>long} );
.Ve
.SS group_by
.IX Subsection "group_by"
Returns pdl reshaped according to the specified factor variable. Most useful when used in conjunction with other threading calculations such as average, stdv, etc. When the factor variable contains unequal number of cases in each level, the returned pdl is padded with bad values to fit the level with the most number of cases. This allows the subsequent calculation (average, stdv, etc) to return the correct results for each level.
.PP
Usage:
.PP
.Vb 1
\&    # simple case with 1d pdl and equal number of n in each level of the factor
\&
\&        pdl> p $a = sequence 10
\&        [0 1 2 3 4 5 6 7 8 9]
\&
\&        pdl> p $factor = $a > 4
\&        [0 0 0 0 0 1 1 1 1 1]
\&
\&        pdl> p $a\->group_by( $factor )\->average
\&        [2 7]
\&
\&    # more complex case with threading and unequal number of n across levels in the factor
\&
\&        pdl> p $a = sequence 10,2
\&        [
\&         [ 0  1  2  3  4  5  6  7  8  9]
\&         [10 11 12 13 14 15 16 17 18 19]
\&        ]
\&
\&        pdl> p $factor = qsort $a( ,0) % 3
\&        [
\&         [0 0 0 0 1 1 1 2 2 2]
\&        ]
\&
\&        pdl> p $a\->group_by( $factor )
\&        [
\&         [
\&          [ 0  1  2  3]
\&          [10 11 12 13]
\&         ]
\&         [
\&          [  4   5   6 BAD]
\&          [ 14  15  16 BAD]
\&         ]
\&         [
\&          [  7   8   9 BAD]
\&          [ 17  18  19 BAD]
\&         ]
\&        ]
\&     ARRAY(0xa2a4e40)
\&
\&    # group_by supports perl factors, multiple factors
\&    # returns factor labels in addition to pdl in array context
\&
\&    pdl> p $a = sequence 12
\&    [0 1 2 3 4 5 6 7 8 9 10 11]
\&
\&    pdl> $odd_even = [qw( e o e o e o e o e o e o )]
\&
\&    pdl> $magnitude = [qw( l l l l l l h h h h h h )]
\&
\&    pdl> ($a_grouped, $label) = $a\->group_by( $odd_even, $magnitude )
\&
\&    pdl> p $a_grouped
\&    [
\&     [
\&      [0 2 4]
\&      [1 3 5]
\&     ]
\&     [
\&      [ 6  8 10]
\&      [ 7  9 11]
\&     ]
\&    ]
\&
\&    pdl> p Dumper $label
\&    $VAR1 = [
\&              [
\&                \*(Aqe_l\*(Aq,
\&                \*(Aqo_l\*(Aq
\&              ],
\&              [
\&                \*(Aqe_h\*(Aq,
\&                \*(Aqo_h\*(Aq
\&              ]
\&            ];
.Ve
.SS which_id
.IX Subsection "which_id"
Lookup specified var (obs) ids in \f(CW$idv\fR ($ido) (see \fBrtable\fR) and return indices in \f(CW$idv\fR ($ido) as pdl if found. The indices are ordered by the specified subset. Useful for selecting data by var (obs) id.
.PP
.Vb 1
\&    my $ind = which_id $ido, [\*(Aqsmith\*(Aq, \*(Aqsummers\*(Aq, \*(Aqtesla\*(Aq];
\&
\&    my $data_subset = $data( $ind, );
\&
\&    # take advantage of perl pattern matching
\&    # e.g. use data from people whose last name starts with s
\&
\&    my $i = which_id $ido, [ grep { /^s/ } @$ido ];
\&
\&    my $data_s = $data($i, );
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
PDL::Basic (hist for frequency counts)
.PP
PDL::Ufunc (sum, avg, median, min, max, etc.)
.PP
PDL::GSL::CDF (various cumulative distribution functions)
.SH REFERENCES
.IX Header "REFERENCES"
Hays, W.L. (1994). Statistics (5th ed.). Fort Worth, TX: Harcourt Brace College Publishers.
.SH AUTHOR
.IX Header "AUTHOR"
Copyright (C) 2009 Maggie J. Xiong <maggiexyz users.sourceforge.net>
.PP
All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution.