## DESCRIPTION

This module provides some primitive and useful functions defined using PDL::PP based on functionality of what are sometimes called*ufuncs*(for example NumPY and Mathematica talk about these). It collects all the functions generally used to

`"reduce"`or

`"accumulate"`along a dimension. These all do their job across the first dimension but by using the slicing functions you can do it on any dimension.

The PDL::Reduce module provides an alternative interface to many of the functions in this module.

## SYNOPSIS

use PDL::Ufunc;

## FUNCTIONS

## prodover

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

Project via product to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the product along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = prodover($a);

$spectrum = prodover $image->xchg(0,1)

prodover 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.

## dprodover

Signature: (a(n); double [o]b())

Project via product to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the product along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = dprodover($a);

$spectrum = dprodover $image->xchg(0,1)

Unlike prodover, the calculations are performed in double precision.

dprodover 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.

## cumuprodover

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

Cumulative product

This function calculates the cumulative product along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

The sum is started so that the first element in the cumulative product is the first element of the parameter.

$b = cumuprodover($a);

$spectrum = cumuprodover $image->xchg(0,1)

cumuprodover 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.

## dcumuprodover

Signature: (a(n); double [o]b(n))

Cumulative product

This function calculates the cumulative product along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

The sum is started so that the first element in the cumulative product is the first element of the parameter.

$b = cumuprodover($a);

$spectrum = cumuprodover $image->xchg(0,1)

Unlike cumuprodover, the calculations are performed in double precision.

dcumuprodover 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.

## sumover

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

Project via sum to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the sum along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = sumover($a);

$spectrum = sumover $image->xchg(0,1)

sumover 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.

## dsumover

Signature: (a(n); double [o]b())

Project via sum to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the sum along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = dsumover($a);

$spectrum = dsumover $image->xchg(0,1)

Unlike sumover, the calculations are performed in double precision.

dsumover 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.

## cumusumover

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

Cumulative sum

This function calculates the cumulative sum along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

The sum is started so that the first element in the cumulative sum is the first element of the parameter.

$b = cumusumover($a);

$spectrum = cumusumover $image->xchg(0,1)

cumusumover 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.

## dcumusumover

Signature: (a(n); double [o]b(n))

Cumulative sum

This function calculates the cumulative sum along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

The sum is started so that the first element in the cumulative sum is the first element of the parameter.

$b = cumusumover($a);

$spectrum = cumusumover $image->xchg(0,1)

Unlike cumusumover, the calculations are performed in double precision.

dcumusumover 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.

## andover

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

Project via and to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the and along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = andover($a);

$spectrum = andover $image->xchg(0,1)

If `"a()"` contains only bad data (and its bad flag is set),
`"b()"` is set bad. Otherwise `"b()"` will have its bad flag cleared,
as it will not contain any bad values.

## bandover

Signature: (a(n); [o]b())

Project via bitwise and to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the bitwise and along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = bandover($a);

$spectrum = bandover $image->xchg(0,1)

If `"a()"` contains only bad data (and its bad flag is set),
`"b()"` is set bad. Otherwise `"b()"` will have its bad flag cleared,
as it will not contain any bad values.

## borover

Signature: (a(n); [o]b())

Project via bitwise or to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the bitwise or along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = borover($a);

$spectrum = borover $image->xchg(0,1)

If `"a()"` contains only bad data (and its bad flag is set),
`"b()"` is set bad. Otherwise `"b()"` will have its bad flag cleared,
as it will not contain any bad values.

## orover

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

Project via or to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the or along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = orover($a);

$spectrum = orover $image->xchg(0,1)

`"a()"` contains only bad data (and its bad flag is set),
`"b()"` is set bad. Otherwise `"b()"` will have its bad flag cleared,
as it will not contain any bad values.

## zcover

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

Project via == 0 to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the == 0 along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = zcover($a);

$spectrum = zcover $image->xchg(0,1)

`"a()"` contains only bad data (and its bad flag is set),
`"b()"` is set bad. Otherwise `"b()"` will have its bad flag cleared,
as it will not contain any bad values.

## intover

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

Project via integral to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the integral along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = intover($a);

$spectrum = intover $image->xchg(0,1)

Notes:

`"intover"` uses a point spacing of one (i.e., delta-h==1). You will
need to scale the result to correct for the true point delta).

For `"n > 3"`, these are all `"O(h^4)"` (like Simpson's rule), but are
integrals between the end points assuming the pdl gives values just at
these centres: for such `functions', sumover is correct to `O(h)`, but
is the natural (and correct) choice for binned data, of course.

intover ignores the bad-value flag of the input piddles. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## average

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

Project via average to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the average along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = average($a);

$spectrum = average $image->xchg(0,1)

average 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.

## avgover

Synonym for average.

## daverage

Signature: (a(n); double [o]b())

Project via average to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the average along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = daverage($a);

$spectrum = daverage $image->xchg(0,1)

Unlike average, the calculation is performed in double precision.

daverage 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.

## davgover

Synonym for daverage.

## medover

Signature: (a(n); [o]b(); [t]tmp(n))

Project via median to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the median along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = medover($a);

$spectrum = medover $image->xchg(0,1)

medover 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.

## oddmedover

Signature: (a(n); [o]b(); [t]tmp(n))

Project via oddmedian to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the oddmedian along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = oddmedover($a);

$spectrum = oddmedover $image->xchg(0,1)

The median is sometimes not a good choice as if the array has an even number of elements it lies half-way between the two middle values - thus it does not always correspond to a data value. The lower-odd median is just the lower of these two values and so it ALWAYS sits on an actual data value which is useful in some circumstances.

oddmedover 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.

## modeover

Signature: (data(n); [o]out(); [t]sorted(n))

Project via mode to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the mode along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = modeover($a);

$spectrum = modeover $image->xchg(0,1)

The mode is the single element most frequently found in a discrete data set.

It *only* makes sense for integer data types, since
floating-point types are demoted to integer before the
mode is calculated.

`"modeover"` treats BAD the same as any other value: if
BAD is the most common element, the returned value is also BAD.

modeover does not process bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## pctover

Signature: (a(n); p(); [o]b(); [t]tmp(n))

Project via percentile to N-1 dimensions

This function reduces the dimensionality of a piddle by one by finding the specified percentile (p) along the 1st dimension. The specified percentile must be between 0.0 and 1.0. When the specified percentile falls between data points, the result is interpolated. Values outside the allowed range are clipped to 0.0 or 1.0 respectively. The algorithm implemented here is based on the interpolation variant described at <http://en.wikipedia.org/wiki/Percentile> as used by Microsoft Excel and recommended by NIST.

By using xchg etc. it is possible to use
*any* dimension.

$b = pctover($a, $p);

$spectrum = pctover $image->xchg(0,1), $p

pctover 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.

## oddpctover

Signature: (a(n); p(); [o]b(); [t]tmp(n))

Project via percentile to N-1 dimensions

This function reduces the dimensionality of a piddle by one by finding the specified percentile along the 1st dimension. The specified percentile must be between 0.0 and 1.0. When the specified percentile falls between two values, the nearest data value is the result. The algorithm implemented is from the textbook version described first at <http://en.wikipedia.org/wiki/Percentile>.

By using xchg etc. it is possible to use
*any* dimension.

$b = oddpctover($a, $p);

$spectrum = oddpctover $image->xchg(0,1), $p

oddpctover 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.

## pct

Return the specified percentile of all elements in a piddle. The specified percentile (p) must be between 0.0 and 1.0. When the specified percentile falls between data points, the result is interpolated.

$x = pct($data, $pct);

## oddpct

Return the specified percentile of all elements in a piddle. The specified percentile must be between 0.0 and 1.0. When the specified percentile falls between two values, the nearest data value is the result.

$x = oddpct($data, $pct);

## avg

Return the average of all elements in a piddle.See the documentation for average for more information.

$x = avg($data);

This routine handles bad values.

## sum

Return the sum of all elements in a piddle.See the documentation for sumover for more information.

$x = sum($data);

This routine handles bad values.

## prod

Return the product of all elements in a piddle.See the documentation for prodover for more information.

$x = prod($data);

This routine handles bad values.

## davg

Return the average (in double precision) of all elements in a piddle.See the documentation for daverage for more information.

$x = davg($data);

This routine handles bad values.

## dsum

Return the sum (in double precision) of all elements in a piddle.See the documentation for dsumover for more information.

$x = dsum($data);

This routine handles bad values.

## dprod

Return the product (in double precision) of all elements in a piddle.See the documentation for dprodover for more information.

$x = dprod($data);

This routine handles bad values.

## zcheck

Return the check for zero of all elements in a piddle.See the documentation for zcover for more information.

$x = zcheck($data);

This routine handles bad values.

## and

Return the logical and of all elements in a piddle.See the documentation for andover for more information.

$x = and($data);

This routine handles bad values.

## band

Return the bitwise and of all elements in a piddle.See the documentation for bandover for more information.

$x = band($data);

This routine handles bad values.

## or

Return the logical or of all elements in a piddle.See the documentation for orover for more information.

$x = or($data);

This routine handles bad values.

## bor

Return the bitwise or of all elements in a piddle.See the documentation for borover for more information.

$x = bor($data);

This routine handles bad values.

## min

Return the minimum of all elements in a piddle.See the documentation for minimum for more information.

$x = min($data);

This routine handles bad values.

## max

Return the maximum of all elements in a piddle.See the documentation for maximum for more information.

$x = max($data);

This routine handles bad values.

## median

Return the median of all elements in a piddle.See the documentation for medover for more information.

$x = median($data);

This routine handles bad values.

## mode

Return the mode of all elements in a piddle.See the documentation for modeover for more information.

$x = mode($data);

This routine handles bad values.

## oddmedian

Return the oddmedian of all elements in a piddle.See the documentation for oddmedover for more information.

$x = oddmedian($data);

This routine handles bad values.

## any

Return true if any element in piddle setUseful in conditional expressions:

if (any $a>15) { print "some values are greater than 15\n" }

See or for comments on what happens when all elements in the check are bad.

## all

Return true if all elements in piddle setUseful in conditional expressions:

if (all $a>15) { print "all values are greater than 15\n" }

See and for comments on what happens when all elements in the check are bad.

## minmax

Returns an array with minimum and maximum values of a piddle.

($mn, $mx) = minmax($pdl);

This routine does *not* thread over the dimensions of `$pdl`;
it returns the minimum and maximum values of the whole array.
See minmaximum if this is not what is required.
The two values are returned as Perl scalars similar to min/max.

pdl> $x = pdl [1,-2,3,5,0] pdl> ($min, $max) = minmax($x); pdl> p "$min $max\n"; -2 5

## qsort

Signature: (a(n); [o]b(n))

Quicksort a vector into ascending order.

print qsort random(10);

Bad values are moved to the end of the array:

pdl> p $b [42 47 98 BAD 22 96 74 41 79 76 96 BAD 32 76 25 59 BAD 96 32 BAD] pdl> p qsort($b) [22 25 32 32 41 42 47 59 74 76 76 79 96 96 96 98 BAD BAD BAD BAD]

## qsorti

Signature: (a(n); indx [o]indx(n))

Quicksort a vector and return index of elements in ascending order.

$ix = qsorti $a; print $a->index($ix); # Sorted list

Bad elements are moved to the end of the array:

pdl> p $b [42 47 98 BAD 22 96 74 41 79 76 96 BAD 32 76 25 59 BAD 96 32 BAD] pdl> p $b->index( qsorti($b) ) [22 25 32 32 41 42 47 59 74 76 76 79 96 96 96 98 BAD BAD BAD BAD]

## qsortvec

Signature: (a(n,m); [o]b(n,m))

Sort a list of vectors lexicographically.

The 0th dimension of the source piddle is dimension in the vector; the 1st dimension is list order. Higher dimensions are threaded over.

print qsortvec pdl([[1,2],[0,500],[2,3],[4,2],[3,4],[3,5]]); [ [ 0 500] [ 1 2] [ 2 3] [ 3 4] [ 3 5] [ 4 2] ]

Vectors with bad components should be moved to the end of the array:

## qsortveci

Signature: (a(n,m); indx [o]indx(m))

Sort a list of vectors lexicographically, returning the indices of the sorted vectors rather than the sorted list itself.

As with `"qsortvec"`, the input PDL should be an NxM array containing M
separate N-dimensional vectors. The return value is an integer M-PDL
containing the M-indices of original array rows, in sorted order.

As with `"qsortvec"`, the zeroth element of the vectors runs slowest in the
sorted list.

Additional dimensions are threaded over: each plane is sorted separately, so qsortveci may be thought of as a collapse operator of sorts (groan).

Vectors with bad components should be moved to the end of the array:

## minimum

Signature: (a(n); [o]c())

Project via minimum to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the minimum along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = minimum($a);

$spectrum = minimum $image->xchg(0,1)

Output is set bad if all elements of the input are bad, otherwise the bad flag is cleared for the output piddle.

Note that `"NaNs"` are considered to be valid values;
see isfinite and badmask
for ways of masking NaNs.

## minimum_ind

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

Like minimum but returns the index rather than the value

Output is set bad if all elements of the input are bad, otherwise the bad flag is cleared for the output piddle.

## minimum_n_ind

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

Returns the index of `"m"` minimum elements

Not yet been converted to ignore bad values

## maximum

Signature: (a(n); [o]c())

Project via maximum to N-1 dimensions

This function reduces the dimensionality of a piddle by one by taking the maximum along the 1st dimension.

By using xchg etc. it is possible to use
*any* dimension.

$b = maximum($a);

$spectrum = maximum $image->xchg(0,1)

Output is set bad if all elements of the input are bad, otherwise the bad flag is cleared for the output piddle.

Note that `"NaNs"` are considered to be valid values;
see isfinite and badmask
for ways of masking NaNs.

## maximum_ind

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

Like maximum but returns the index rather than the value

## maximum_n_ind

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

Returns the index of `"m"` maximum elements

Not yet been converted to ignore bad values

## maxover

Synonym for maximum.

## maxover_ind

Synonym for maximum_ind.

## maxover_n_ind

Synonym for maximum_n_ind.

## minover

Synonym for minimum.

## minover_ind

Synonym for minimum_ind.

## minover_n_ind

Synonym for minimum_n_ind

## minmaximum

Signature: (a(n); [o]cmin(); [o] cmax(); indx [o]cmin_ind(); indx [o]cmax_ind())

Find minimum and maximum and their indices for a given piddle;

pdl> $a=pdl [[-2,3,4],[1,0,3]] pdl> ($min, $max, $min_ind, $max_ind)=minmaximum($a) pdl> p $min, $max, $min_ind, $max_ind [-2 0] [4 3] [0 1] [2 2]

See also minmax, which clumps the piddle together.

If `"a()"` contains only bad data, then the output piddles will
be set bad, along with their bad flag.
Otherwise they will have their bad flags cleared,
since they will not contain any bad values.

## minmaxover

Synonym for minmaximum.

## AUTHOR

Copyright (C) Tuomas J. Lukka 1997 ([email protected]). Contributions by Christian Soeller ([email protected]) and Karl Glazebrook ([email protected]). All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For details, see the file COPYING in the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be included in the file.