# NAME

PDL::Stats::Basic -- basic statistics and related utilities such as standard deviation, Pearson correlation, and t-tests.

# DESCRIPTION

The terms FUNCTIONS and METHODS are arbitrarily used to refer to methods that are threadable and methods that are NOT threadable, respectively.

Does not have mean or median function here. see SEE ALSO.

# SYNOPSIS

```
use PDL::LiteF;
use PDL::NiceSlice;
use PDL::Stats::Basic;
my $stdv = $data->stdv;
```

or

` my $stdv = stdv( $data ); `

# FUNCTIONS

## stdv

` Signature: (a(n); float+ [o]b())`

Sample standard deviation.

stdv processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## stdv_unbiased

` Signature: (a(n); float+ [o]b())`

Unbiased estimate of population standard deviation.

stdv_unbiased processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## var

` Signature: (a(n); float+ [o]b())`

Sample variance.

var processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## var_unbiased

` Signature: (a(n); float+ [o]b())`

Unbiased estimate of population variance.

var_unbiased processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## se

` Signature: (a(n); float+ [o]b())`

Standard error of the mean. Useful for calculating confidence intervals.

```
# 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;
```

se processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## ss

` Signature: (a(n); float+ [o]b())`

Sum of squared deviations from the mean.

ss processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## skew

` Signature: (a(n); float+ [o]b())`

Sample skewness, measure of asymmetry in data. skewness == 0 for normal distribution.

skew processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## skew_unbiased

` Signature: (a(n); float+ [o]b())`

Unbiased estimate of population skewness. This is the number in GNumeric Descriptive Statistics.

skew_unbiased processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## kurt

` Signature: (a(n); float+ [o]b())`

Sample kurtosis, measure of "peakedness" of data. kurtosis == 0 for normal distribution.

kurt processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## kurt_unbiased

` Signature: (a(n); float+ [o]b())`

Unbiased estimate of population kurtosis. This is the number in GNumeric Descriptive Statistics.

kurt_unbiased processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## cov

` Signature: (a(n); b(n); float+ [o]c())`

Sample covariance. see **corr** for ways to call

cov processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## cov_table

` Signature: (a(n,m); float+ [o]c(m,m))`

Square covariance table. Gives the same result as threading using **cov** but it calculates only half the square, hence much faster. And it is easier to use with higher dimension pdls.

Usage:

```
# 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]
]
```

for the same cov matrix table using **cov**,

```
perldl> p $a->dummy(2)->cov($a->dummy(1))
```

cov_table processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## corr

` Signature: (a(n); b(n); float+ [o]c())`

Pearson correlation coefficient. r = cov(X,Y) / (stdv(X) * stdv(Y)).

Usage:

```
perldl> $a = random 5, 3
perldl> $b = sequence 5,3
perldl> p $a->corr($b)
[0.20934208 0.30949881 0.26713007]
```

for square corr table

```
perldl> p $a->corr($a->dummy(1))
[
[ 1 -0.41995259 -0.029301192]
[ -0.41995259 1 -0.61927619]
[-0.029301192 -0.61927619 1]
]
```

but it is easier and faster to use **corr_table**.

corr processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## corr_table

` Signature: (a(n,m); float+ [o]c(m,m))`

Square Pearson correlation table. Gives the same result as threading using **corr** but it calculates only half the square, hence much faster. And it is easier to use with higher dimension pdls.

Usage:

```
# 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]
]
]
```

for the same result using **corr**,

` perldl> p $a->dummy(2)->corr($a->dummy(1)) `

This is also how to use **t_corr** and **n_pair** with such a table.

corr_table processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## t_corr

` Signature: (r(); n(); [o]t())`

```
$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 ));
```

t significance test for Pearson correlations.

t_corr processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## n_pair

` Signature: (a(n); b(n); indx [o]c())`

Returns the number of good pairs between 2 lists. Useful with **corr** (esp. when bad values are involved)

n_pair processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## corr_dev

` Signature: (a(n); b(n); float+ [o]c())`

` $corr = $a->dev_m->corr_dev($b->dev_m);`

Calculates correlations from **dev_m** vals. Seems faster than doing **corr** from original vals when data pdl is big

corr_dev processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## t_test

` Signature: (a(n); b(m); float+ [o]t(); [o]d())`

```
my ($t, $df) = t_test( $pdl1, $pdl2 );
use PDL::GSL::CDF;
my $p_2tail = 2 * (1 - gsl_cdf_tdist_P( $t->abs, $df ));
```

Independent sample t-test, assuming equal var.

t_test processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## t_test_nev

` Signature: (a(n); b(m); float+ [o]t(); [o]d())`

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.

` my ($t, $df) = $pdl1->t_test( $pdl2 );`

t_test_nev processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## t_test_paired

` Signature: (a(n); b(n); float+ [o]t(); [o]d())`

Paired sample t-test.

t_test_paired processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## binomial_test

` Signature: (x(); n(); p_expected(); [o]p())`

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.

This function does NOT currently support bad value in the number of successes.

Usage:

```
# 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.
```

# METHODS

## 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 $opt{MISSN} then set to BAD*. Can specify num of data rows to read from top but not arbitrary range.

*If passed handle, it will not be closed here.

*PDL::Bad::setvaltobad only works consistently with the default TYPE double before PDL-2.4.4_04.

Default options (case insensitive):

```
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 => "\t", # can take regex qr//
MISSN => -999, # this value treated as missing and set to BAD
NROW => '', # set to read specified num of data rows
```

Usage:

Sample file diet.txt:

```
uid height weight diet
akw 72 320 1
bcm 68 268 1
clq 67 180 2
dwm 70 200 2
($data, $idv, $ido) = rtable 'diet.txt';
# By default prints out data info and @$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
```

Another way of using it,

` $data = rtable( \*STDIN, {TYPE=>long} );`

## 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.

Usage:

```
# 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 = [
[
'e_l',
'o_l'
],
[
'e_h',
'o_h'
]
];
```

## which_id

Lookup specified var (obs) ids in $idv ($ido) (see **rtable**) and return indices in $idv ($ido) as pdl if found. The indices are ordered by the specified subset. Useful for selecting data by var (obs) id.

```
my $ind = which_id $ido, ['smith', 'summers', 'tesla'];
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, );
```

# SEE ALSO

PDL::Basic (hist for frequency counts)

PDL::Ufunc (sum, avg, median, min, max, etc.)

PDL::GSL::CDF (various cumulative distribution functions)

# REFERENCES

Hays, W.L. (1994). Statistics (5th ed.). Fort Worth, TX: Harcourt Brace College Publishers.

# AUTHOR

Copyright (C) 2009 Maggie J. Xiong <maggiexyz users.sourceforge.net>

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.