NAME

PDL::Primitive - primitive operations for pdl

DESCRIPTION

This module provides some primitive and useful functions defined using PDL::PP and able to use the new indexing tricks.

See PDL::Indexing for how to use indices creatively. For explanation of the signature format, see PDL::PP.

SYNOPSIS

# Pulls in PDL::Primitive, among other modules.
use PDL;

# Only pull in PDL::Primitive:
use PDL::Primitive;

FUNCTIONS

inner

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

Inner product over one dimension

c = sum_i a_i * b_i

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

outer

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

outer product over one dimension

Naturally, it is possible to achieve the effects of outer product simply by threading over the "*" operator but this function is provided for convenience.

outer 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.

x

Signature: (a(i,z), b(x,i),[o]c(x,z))

Matrix multiplication

PDL overloads the x operator (normally the repeat operator) for matrix multiplication. The number of columns (size of the 0 dimension) in the left-hand argument must normally equal the number of rows (size of the 1 dimension) in the right-hand argument.

Row vectors are represented as (N x 1) two-dimensional PDLs, or you may be sloppy and use a one-dimensional PDL. Column vectors are represented as (1 x N) two-dimensional PDLs.

Threading occurs in the usual way, but as both the 0 and 1 dimension (if present) are included in the operation, you must be sure that you don't try to thread over either of those dims.

EXAMPLES

Here are some simple ways to define vectors and matrices:

pdl> $r = pdl(1,2);                # A row vector
pdl> $c = pdl([[3],[4]]);          # A column vector
pdl> $c = pdl(3,4)->(*1);          # A column vector, using NiceSlice
pdl> $m = pdl([[1,2],[3,4]]);      # A 2x2 matrix

Now that we have a few objects prepared, here is how to matrix-multiply them:

pdl> print $r x $m                 # row x matrix = row
[
 [ 7 10]
]

pdl> print $m x $r                 # matrix x row = ERROR
PDL: Dim mismatch in matmult of [2x2] x [2x1]: 2 != 1

pdl> print $m x $c                 # matrix x column = column
[
 [ 5]
 [11]
]

pdl> print $m x 2                  # Trivial case: scalar mult.
[
 [2 4]
 [6 8]
]

pdl> print $r x $c                 # row x column = scalar
[
 [11]
]

pdl> print $c x $r                 # column x row = matrix
[
 [3 6]
 [4 8]
]

INTERNALS

The mechanics of the multiplication are carried out by the "matmult" method.

matmult

Signature: (a(t,h); b(w,t); [o]c(w,h))

Matrix multiplication

Notionally, matrix multiplication $x x $y is equivalent to the threading expression

$x->dummy(1)->inner($y->xchg(0,1)->dummy(2),$c);

but for large matrices that breaks CPU cache and is slow. Instead, matmult calculates its result in 32x32x32 tiles, to keep the memory footprint within cache as long as possible on most modern CPUs.

For usage, see "x", a description of the overloaded 'x' operator

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

innerwt

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

Weighted (i.e. triple) inner product

d = sum_i a(i) b(i) c(i)

innerwt 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.

inner2

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

Inner product of two vectors and a matrix

d = sum_ij a(i) b(i,j) c(j)

Note that you should probably not thread over a and c since that would be very wasteful. Instead, you should use a temporary for b*c.

inner2 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.

inner2d

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

Inner product over 2 dimensions.

Equivalent to

$c = inner($x->clump(2), $y->clump(2))

inner2d 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.

inner2t

Signature: (a(j,n); b(n,m); c(m,k); [t]tmp(n,k); [o]d(j,k)))

Efficient Triple matrix product a*b*c

Efficiency comes from by using the temporary tmp. This operation only scales as N**3 whereas threading using "inner2" would scale as N**4.

The reason for having this routine is that you do not need to have the same thread-dimensions for tmp as for the other arguments, which in case of large numbers of matrices makes this much more memory-efficient.

It is hoped that things like this could be taken care of as a kind of closures at some point.

inner2t 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.

crossp

Signature: (a(tri=3); b(tri); [o] c(tri))

Cross product of two 3D vectors

After

$c = crossp $x, $y

the inner product $c*$x and $c*$y will be zero, i.e. $c is orthogonal to $x and $y

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

norm

Signature: (vec(n); [o] norm(n))

Normalises a vector to unit Euclidean length

norm 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.

indadd

Signature: (a(); indx ind(); [o] sum(m))

Threaded Index Add: Add a to the ind element of sum, i.e:

sum(ind) += a

Simple Example:

$x = 2;
$ind = 3;
$sum = zeroes(10);
indadd($x,$ind, $sum);
print $sum
#Result: ( 2 added to element 3 of $sum)
# [0 0 0 2 0 0 0 0 0 0]

Threaded Example:

$x = pdl( 1,2,3);
$ind = pdl( 1,4,6);
$sum = zeroes(10);
indadd($x,$ind, $sum);
print $sum."\n";
#Result: ( 1, 2, and 3 added to elements 1,4,6 $sum)
# [0 1 0 0 2 0 3 0 0 0]

The routine barfs if any of the indices are bad.

conv1d

Signature: (a(m); kern(p); [o]b(m); int reflect)

1D convolution along first dimension

The m-th element of the discrete convolution of an input ndarray $a of size $M, and a kernel ndarray $kern of size $P, is calculated as

                            n = ($P-1)/2
                            ====
                            \
($a conv1d $kern)[m]   =     >      $a_ext[m - n] * $kern[n]
                            /
                            ====
                            n = -($P-1)/2

where $a_ext is either the periodic (or reflected) extension of $a so it is equal to $a on 0..$M-1 and equal to the corresponding periodic/reflected image of $a outside that range.

$con = conv1d sequence(10), pdl(-1,0,1);

$con = conv1d sequence(10), pdl(-1,0,1), {Boundary => 'reflect'};

By default, periodic boundary conditions are assumed (i.e. wrap around). Alternatively, you can request reflective boundary conditions using the Boundary option:

{Boundary => 'reflect'} # case in 'reflect' doesn't matter

The convolution is performed along the first dimension. To apply it across another dimension use the slicing routines, e.g.

$y = $x->mv(2,0)->conv1d($kernel)->mv(0,2); # along third dim

This function is useful for threaded filtering of 1D signals.

Compare also conv2d, convolve, fftconvolve, fftwconv, rfftwconv

WARNING: conv1d processes bad values in its inputs as the numeric value of $pdl->badvalue so it is not recommended for processing pdls with bad values in them unless special care is taken.

conv1d ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

in

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

test if a is in the set of values b

 $goodmsk = $labels->in($goodlabels);
 print pdl(3,1,4,6,2)->in(pdl(2,3,3));
[1 0 0 0 1]

in is akin to the is an element of of set theory. In principle, PDL threading could be used to achieve its functionality by using a construct like

$msk = ($labels->dummy(0) == $goodlabels)->orover;

However, in doesn't create a (potentially large) intermediate and is generally faster.

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

uniq

return all unique elements of an ndarray

The unique elements are returned in ascending order.

PDL> p pdl(2,2,2,4,0,-1,6,6)->uniq
[-1 0 2 4 6]     # 0 is returned 2nd (sorted order)

PDL> p pdl(2,2,2,4,nan,-1,6,6)->uniq
[-1 2 4 6 nan]   # NaN value is returned at end

Note: The returned pdl is 1D; any structure of the input ndarray is lost. NaN values are never compare equal to any other values, even themselves. As a result, they are always unique. uniq returns the NaN values at the end of the result ndarray. This follows the Matlab usage.

See "uniqind" if you need the indices of the unique elements rather than the values.

Bad values are not considered unique by uniq and are ignored.

$x=sequence(10);
$x=$x->setbadif($x%3);
print $x->uniq;
[0 3 6 9]

uniqind

Return the indices of all unique elements of an ndarray The order is in the order of the values to be consistent with uniq. NaN values never compare equal with any other value and so are always unique. This follows the Matlab usage.

PDL> p pdl(2,2,2,4,0,-1,6,6)->uniqind
[5 4 1 3 6]     # the 0 at index 4 is returned 2nd, but...

PDL> p pdl(2,2,2,4,nan,-1,6,6)->uniqind
[5 1 3 6 4]     # ...the NaN at index 4 is returned at end

Note: The returned pdl is 1D; any structure of the input ndarray is lost.

See "uniq" if you want the unique values instead of the indices.

Bad values are not considered unique by uniqind and are ignored.

uniqvec

Return all unique vectors out of a collection

NOTE: If any vectors in the input ndarray have NaN values
they are returned at the end of the non-NaN ones.  This is
because, by definition, NaN values never compare equal with
any other value.

NOTE: The current implementation does not sort the vectors
containing NaN values.

The unique vectors are returned in lexicographically sorted ascending order. The 0th dimension of the input PDL is treated as a dimensional index within each vector, and the 1st and any higher dimensions are taken to run across vectors. The return value is always 2D; any structure of the input PDL (beyond using the 0th dimension for vector index) is lost.

See also "uniq" for a unique list of scalars; and qsortvec for sorting a list of vectors lexicographcally.

If a vector contains all bad values, it is ignored as in "uniq". If some of the values are good, it is treated as a normal vector. For example, [1 2 BAD] and [BAD 2 3] could be returned, but [BAD BAD BAD] could not. Vectors containing BAD values will be returned after any non-NaN and non-BAD containing vectors, followed by the NaN vectors.

hclip

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

clip (threshold) $a by $b ($b is upper bound)

hclip 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.

lclip

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

clip (threshold) $a by $b ($b is lower bound)

lclip 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.

clip

Clip (threshold) an ndarray by (optional) upper or lower bounds.

$y = $x->clip(0,3);
$c = $x->clip(undef, $x);

clip handles bad values since it is just a wrapper around "hclip" and "lclip".

clip

Signature: (a(); l(); h(); [o] c())

info not available

clip 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.

wtstat

Signature: (a(n); wt(n); avg(); [o]b(); int deg)

Weighted statistical moment of given degree

This calculates a weighted statistic over the vector a. The formula is

b() = (sum_i wt_i * (a_i ** degree - avg)) / (sum_i wt_i)

Bad values are ignored in any calculation; $b will only have its bad flag set if the output contains any bad data.

statsover

Signature: (a(n); w(n); float+ [o]avg(); float+ [o]prms(); int+ [o]median(); int+ [o]min(); int+ [o]max(); float+ [o]adev(); float+ [o]rms())

Calculate useful statistics over a dimension of an ndarray

($mean,$prms,$median,$min,$max,$adev,$rms) = statsover($ndarray, $weights);

This utility function calculates various useful quantities of an ndarray. These are:

the mean:
```
MEAN = sum (x)/ N
```
with N being the number of elements in x
the population RMS deviation from the mean:
```
PRMS = sqrt( sum( (x-mean(x))^2 )/(N-1)
```
The population deviation is the best-estimate of the deviation of the population from which a sample is drawn.
the median

The median is the 50th percentile data value. Median is found by medover, so WEIGHTING IS IGNORED FOR THE MEDIAN CALCULATION.
the minimum
the maximum
the average absolute deviation:
```
AADEV = sum( abs(x-mean(x)) )/N
```
RMS deviation from the mean:
```
RMS = sqrt(sum( (x-mean(x))^2 )/N)
```
(also known as the root-mean-square deviation, or the square root of the variance)

This operator is a projection operator so the calculation will take place over the final dimension. Thus if the input is N-dimensional each returned value will be N-1 dimensional, to calculate the statistics for the entire ndarray either use clump(-1) directly on the ndarray or call stats.

Bad values are simply ignored in the calculation, effectively reducing the sample size. If all data are bad then the output data are marked bad.

stats

Calculates useful statistics on an ndarray

($mean,$prms,$median,$min,$max,$adev,$rms) = stats($ndarray,[$weights]);

This utility calculates all the most useful quantities in one call. It works the same way as "statsover", except that the quantities are calculated considering the entire input PDL as a single sample, rather than as a collection of rows. See "statsover" for definitions of the returned quantities.

Bad values are handled; if all input values are bad, then all of the output values are flagged bad.

histogram

Signature: (in(n); int+[o] hist(m); double step; double min; int msize => m)

Calculates a histogram for given stepsize and minimum.

$h = histogram($data, $step, $min, $numbins);
$hist = zeroes $numbins;  # Put histogram in existing ndarray.
histogram($data, $hist, $step, $min, $numbins);

The histogram will contain $numbins bins starting from $min, each $step wide. The value in each bin is the number of values in $data that lie within the bin limits.

Data below the lower limit is put in the first bin, and data above the upper limit is put in the last bin.

The output is reset in a different threadloop so that you can take a histogram of $a(10,12) into $b(15) and get the result you want.

For a higher-level interface, see hist.

pdl> p histogram(pdl(1,1,2),1,0,3)
[0 2 1]

histogram 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.

whistogram

Signature: (in(n); float+ wt(n);float+[o] hist(m); double step; double min; int msize => m)

Calculates a histogram from weighted data for given stepsize and minimum.

$h = whistogram($data, $weights, $step, $min, $numbins);
$hist = zeroes $numbins;  # Put histogram in existing ndarray.
whistogram($data, $weights, $hist, $step, $min, $numbins);

The histogram will contain $numbins bins starting from $min, each $step wide. The value in each bin is the sum of the values in $weights that correspond to values in $data that lie within the bin limits.

Data below the lower limit is put in the first bin, and data above the upper limit is put in the last bin.

The output is reset in a different threadloop so that you can take a histogram of $a(10,12) into $b(15) and get the result you want.

pdl> p whistogram(pdl(1,1,2), pdl(0.1,0.1,0.5), 1, 0, 4)
[0 0.2 0.5 0]

whistogram 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.

histogram2d

  Signature: (ina(n); inb(n); int+[o] hist(ma,mb); double stepa; double mina; int masize => ma;
	             double stepb; double minb; int mbsize => mb;)

Calculates a 2d histogram.

$h = histogram2d($datax, $datay, $stepx, $minx,
      $nbinx, $stepy, $miny, $nbiny);
$hist = zeroes $nbinx, $nbiny;  # Put histogram in existing ndarray.
histogram2d($datax, $datay, $hist, $stepx, $minx,
      $nbinx, $stepy, $miny, $nbiny);

The histogram will contain $nbinx x $nbiny bins, with the lower limits of the first one at ($minx, $miny), and with bin size ($stepx, $stepy). The value in each bin is the number of values in $datax and $datay that lie within the bin limits.

Data below the lower limit is put in the first bin, and data above the upper limit is put in the last bin.

pdl> p histogram2d(pdl(1,1,1,2,2),pdl(2,1,1,1,1),1,0,3,1,0,3)
[
 [0 0 0]
 [0 2 2]
 [0 1 0]
]

histogram2d 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.

whistogram2d

  Signature: (ina(n); inb(n); float+ wt(n);float+[o] hist(ma,mb); double stepa; double mina; int masize => ma;
	             double stepb; double minb; int mbsize => mb;)

Calculates a 2d histogram from weighted data.

$h = whistogram2d($datax, $datay, $weights,
      $stepx, $minx, $nbinx, $stepy, $miny, $nbiny);
$hist = zeroes $nbinx, $nbiny;  # Put histogram in existing ndarray.
whistogram2d($datax, $datay, $weights, $hist,
      $stepx, $minx, $nbinx, $stepy, $miny, $nbiny);

The histogram will contain $nbinx x $nbiny bins, with the lower limits of the first one at ($minx, $miny), and with bin size ($stepx, $stepy). The value in each bin is the sum of the values in $weights that correspond to values in $datax and $datay that lie within the bin limits.

Data below the lower limit is put in the first bin, and data above the upper limit is put in the last bin.

pdl> p whistogram2d(pdl(1,1,1,2,2),pdl(2,1,1,1,1),pdl(0.1,0.2,0.3,0.4,0.5),1,0,3,1,0,3)
[
 [  0   0   0]
 [  0 0.5 0.9]
 [  0 0.1   0]
]

whistogram2d 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.

fibonacci

Signature: ([o]x(n))

Constructor - a vector with Fibonacci's sequence

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

append

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

append two ndarrays by concatenating along their first dimensions

$x = ones(2,4,7);
$y = sequence 5;
$c = $x->append($y);  # size of $c is now (7,4,7) (a jumbo-ndarray ;)

append appends two ndarrays along their first dimensions. The rest of the dimensions must be compatible in the threading sense. The resulting size of the first dimension is the sum of the sizes of the first dimensions of the two argument ndarrays - i.e. n + m.

Similar functions include "glue" (below), which can append more than two ndarrays along an arbitrary dimension, and cat, which can append more than two ndarrays that all have the same sized dimensions.

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

glue

$c = $x->glue(<dim>,$y,...)

Glue two or more PDLs together along an arbitrary dimension (N-D "append").

Sticks $x, $y, and all following arguments together along the specified dimension. All other dimensions must be compatible in the threading sense.

Glue is permissive, in the sense that every PDL is treated as having an infinite number of trivial dimensions of order 1 -- so $x->glue(3,$y) works, even if $x and $y are only one dimensional.

If one of the PDLs has no elements, it is ignored. Likewise, if one of them is actually the undefined value, it is treated as if it had no elements.

If the first parameter is a defined perl scalar rather than a pdl, then it is taken as a dimension along which to glue everything else, so you can say $cube = PDL::glue(3,@image_list); if you like.

glue is implemented in pdl, using a combination of xchg and "append". It should probably be updated (one day) to a pure PP function.

Similar functions include "append" (above), which appends only two ndarrays along their first dimension, and cat, which can append more than two ndarrays that all have the same sized dimensions.

axisvalues

Signature: ([o,nc]a(n))

Internal routine

axisvalues is the internal primitive that implements axisvals and alters its argument.

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

srand

Signature: (a())

Seed random-number generator with a 64-bit int. Will generate seed data for a number of threads equal to the return-value of "online_cpus" in PDL::Core.

srand(); # uses current time
srand(5); # fixed number e.g. for testing

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

random

Signature: (a())

Constructor which returns ndarray of random numbers

$x = random([type], $nx, $ny, $nz,...);
$x = random $y;

etc (see zeroes).

This is the uniform distribution between 0 and 1 (assumedly excluding 1 itself). The arguments are the same as zeroes (q.v.) - i.e. one can specify dimensions, types or give a template.

You can use the PDL function "srand" to seed the random generator. If it has not been called yet, it will be with the current time.

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

randsym

Signature: (a())

Constructor which returns ndarray of random numbers

$x = randsym([type], $nx, $ny, $nz,...);
$x = randsym $y;

etc (see zeroes).

This is the uniform distribution between 0 and 1 (excluding both 0 and 1, cf "random"). The arguments are the same as zeroes (q.v.) - i.e. one can specify dimensions, types or give a template.

You can use the PDL function "srand" to seed the random generator. If it has not been called yet, it will be with the current time.

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

grandom

Constructor which returns ndarray of Gaussian random numbers

$x = grandom([type], $nx, $ny, $nz,...);
$x = grandom $y;

etc (see zeroes).

This is generated using the math library routine ndtri.

Mean = 0, Stddev = 1

You can use the PDL function "srand" to seed the random generator. If it has not been called yet, it will be with the current time.

vsearch

Signature: ( vals(); xs(n); [o] indx(); [\%options] )

Efficiently search for values in a sorted ndarray, returning indices.

$idx = vsearch( $vals, $x, [\%options] );
vsearch( $vals, $x, $idx, [\%options ] );

vsearch performs a binary search in the ordered ndarray $x, for the values from $vals ndarray, returning indices into $x. What is a "match", and the meaning of the returned indices, are determined by the options.

The mode option indicates which method of searching to use, and may be one of:

sample: invoke vsearch_sample, returning indices appropriate for sampling within a distribution.
insert_leftmost: invoke vsearch_insert_leftmost, returning the left-most possible insertion point which still leaves the ndarray sorted.
insert_rightmost: invoke vsearch_insert_rightmost, returning the right-most possible insertion point which still leaves the ndarray sorted.
match: invoke vsearch_match, returning the index of a matching element, else -(insertion point + 1)
bin_inclusive: invoke vsearch_bin_inclusive, returning an index appropriate for binning on a grid where the left bin edges are inclusive of the bin. See below for further explanation of the bin.
bin_exclusive: invoke vsearch_bin_exclusive, returning an index appropriate for binning on a grid where the left bin edges are exclusive of the bin. See below for further explanation of the bin.

The default value of mode is sample.

use PDL;

my @modes = qw( sample insert_leftmost insert_rightmost match
                bin_inclusive bin_exclusive );

# Generate a sequence of 3 zeros, 3 ones, ..., 3 fours.
my $x = zeroes(3,5)->yvals->flat;

for my $mode ( @modes ) {
  # if the value is in $x
  my $contained = 2;
  my $idx_contained = vsearch( $contained, $x, { mode => $mode } );
  my $x_contained = $x->copy;
  $x_contained->slice( $idx_contained ) .= 9;
  
  # if the value is not in $x
  my $not_contained = 1.5;
  my $idx_not_contained = vsearch( $not_contained, $x, { mode => $mode } );
  my $x_not_contained = $x->copy;
  $x_not_contained->slice( $idx_not_contained ) .= 9;
  
  print sprintf("%-23s%30s\n", '$x', $x);
  print sprintf("%-23s%30s\n",   "$mode ($contained)", $x_contained);
  print sprintf("%-23s%30s\n\n", "$mode ($not_contained)", $x_not_contained);
}

# $x                     [0 0 0 1 1 1 2 2 2 3 3 3 4 4 4]
# sample (2)             [0 0 0 1 1 1 9 2 2 3 3 3 4 4 4]
# sample (1.5)           [0 0 0 1 1 1 9 2 2 3 3 3 4 4 4]
# 
# $x                     [0 0 0 1 1 1 2 2 2 3 3 3 4 4 4]
# insert_leftmost (2)    [0 0 0 1 1 1 9 2 2 3 3 3 4 4 4]
# insert_leftmost (1.5)  [0 0 0 1 1 1 9 2 2 3 3 3 4 4 4]
# 
# $x                     [0 0 0 1 1 1 2 2 2 3 3 3 4 4 4]
# insert_rightmost (2)   [0 0 0 1 1 1 2 2 2 9 3 3 4 4 4]
# insert_rightmost (1.5) [0 0 0 1 1 1 9 2 2 3 3 3 4 4 4]
# 
# $x                     [0 0 0 1 1 1 2 2 2 3 3 3 4 4 4]
# match (2)              [0 0 0 1 1 1 2 9 2 3 3 3 4 4 4]
# match (1.5)            [0 0 0 1 1 1 2 2 9 3 3 3 4 4 4]
# 
# $x                     [0 0 0 1 1 1 2 2 2 3 3 3 4 4 4]
# bin_inclusive (2)      [0 0 0 1 1 1 2 2 9 3 3 3 4 4 4]
# bin_inclusive (1.5)    [0 0 0 1 1 9 2 2 2 3 3 3 4 4 4]
# 
# $x                     [0 0 0 1 1 1 2 2 2 3 3 3 4 4 4]
# bin_exclusive (2)      [0 0 0 1 1 9 2 2 2 3 3 3 4 4 4]
# bin_exclusive (1.5)    [0 0 0 1 1 9 2 2 2 3 3 3 4 4 4]

Also see vsearch_sample, vsearch_insert_leftmost, vsearch_insert_rightmost, vsearch_match, vsearch_bin_inclusive, and vsearch_bin_exclusive

vsearch_sample

Signature: (vals(); x(n); indx [o]idx())

Search for values in a sorted array, return index appropriate for sampling from a distribution

$idx = vsearch_sample($vals, $x);

$x must be sorted, but may be in decreasing or increasing order.

vsearch_sample returns an index I for each value V of $vals appropriate for sampling $vals

I has the following properties:

if $x is sorted in increasing order

        V <= x[0]  : I = 0
x[0]  < V <= x[-1] : I s.t. x[I-1] < V <= x[I]
x[-1] < V          : I = $x->nelem -1

if $x is sorted in decreasing order

         V > x[0]  : I = 0
x[0]  >= V > x[-1] : I s.t. x[I] >= V > x[I+1]
x[-1] >= V         : I = $x->nelem - 1

If all elements of $x are equal, I = $x->nelem - 1.

If $x contains duplicated elements, I is the index of the leftmost (by position in array) duplicate if V matches.

This function is useful e.g. when you have a list of probabilities for events and want to generate indices to events:

$x = pdl(.01,.86,.93,1); # Barnsley IFS probabilities cumulatively
$y = random 20;
$c = vsearch_sample($y, $x); # Now, $c will have the appropriate distr.

It is possible to use the cumusumover function to obtain cumulative probabilities from absolute probabilities.

needs major (?) work to handles bad values

vsearch_insert_leftmost

Signature: (vals(); x(n); indx [o]idx())

Determine the insertion point for values in a sorted array, inserting before duplicates.

$idx = vsearch_insert_leftmost($vals, $x);

$x must be sorted, but may be in decreasing or increasing order.

vsearch_insert_leftmost returns an index I for each value V of $vals equal to the leftmost position (by index in array) within $x that V may be inserted and still maintain the order in $x.

Insertion at index I involves shifting elements I and higher of $x to the right by one and setting the now empty element at index I to V.

I has the following properties:

if $x is sorted in increasing order

        V <= x[0]  : I = 0
x[0]  < V <= x[-1] : I s.t. x[I-1] < V <= x[I]
x[-1] < V          : I = $x->nelem

if $x is sorted in decreasing order

         V >  x[0]  : I = -1
x[0]  >= V >= x[-1] : I s.t. x[I] >= V > x[I+1]
x[-1] >= V          : I = $x->nelem -1

If all elements of $x are equal,

i = 0

If $x contains duplicated elements, I is the index of the leftmost (by index in array) duplicate if V matches.

needs major (?) work to handles bad values

vsearch_insert_rightmost

Signature: (vals(); x(n); indx [o]idx())

Determine the insertion point for values in a sorted array, inserting after duplicates.

$idx = vsearch_insert_rightmost($vals, $x);

$x must be sorted, but may be in decreasing or increasing order.

vsearch_insert_rightmost returns an index I for each value V of $vals equal to the rightmost position (by index in array) within $x that V may be inserted and still maintain the order in $x.

Insertion at index I involves shifting elements I and higher of $x to the right by one and setting the now empty element at index I to V.

I has the following properties:

if $x is sorted in increasing order

         V < x[0]  : I = 0
x[0]  <= V < x[-1] : I s.t. x[I-1] <= V < x[I]
x[-1] <= V         : I = $x->nelem

if $x is sorted in decreasing order

        V >= x[0]  : I = -1
x[0]  > V >= x[-1] : I s.t. x[I] >= V > x[I+1]
x[-1] > V          : I = $x->nelem -1

If all elements of $x are equal,

i = $x->nelem - 1

If $x contains duplicated elements, I is the index of the leftmost (by index in array) duplicate if V matches.

needs major (?) work to handles bad values

vsearch_match

Signature: (vals(); x(n); indx [o]idx())

Match values against a sorted array.

$idx = vsearch_match($vals, $x);

$x must be sorted, but may be in decreasing or increasing order.

vsearch_match returns an index I for each value V of $vals. If V matches an element in $x, I is the index of that element, otherwise it is -( insertion_point + 1 ), where insertion_point is an index in $x where V may be inserted while maintaining the order in $x. If $x has duplicated values, I may refer to any of them.

needs major (?) work to handles bad values

vsearch_bin_inclusive

Signature: (vals(); x(n); indx [o]idx())

Determine the index for values in a sorted array of bins, lower bound inclusive.

$idx = vsearch_bin_inclusive($vals, $x);

$x must be sorted, but may be in decreasing or increasing order.

$x represents the edges of contiguous bins, with the first and last elements representing the outer edges of the outer bins, and the inner elements the shared bin edges.

The lower bound of a bin is inclusive to the bin, its outer bound is exclusive to it. vsearch_bin_inclusive returns an index I for each value V of $vals

I has the following properties:

if $x is sorted in increasing order

         V < x[0]  : I = -1
x[0]  <= V < x[-1] : I s.t. x[I] <= V < x[I+1]
x[-1] <= V         : I = $x->nelem - 1

if $x is sorted in decreasing order

         V >= x[0]  : I = 0
x[0]  >  V >= x[-1] : I s.t. x[I+1] > V >= x[I]
x[-1] >  V          : I = $x->nelem

If all elements of $x are equal,

i = $x->nelem - 1

If $x contains duplicated elements, I is the index of the righmost (by index in array) duplicate if V matches.

needs major (?) work to handles bad values

vsearch_bin_exclusive

Signature: (vals(); x(n); indx [o]idx())

Determine the index for values in a sorted array of bins, lower bound exclusive.

$idx = vsearch_bin_exclusive($vals, $x);

$x must be sorted, but may be in decreasing or increasing order.

$x represents the edges of contiguous bins, with the first and last elements representing the outer edges of the outer bins, and the inner elements the shared bin edges.

The lower bound of a bin is exclusive to the bin, its upper bound is inclusive to it. vsearch_bin_exclusive returns an index I for each value V of $vals.

I has the following properties:

if $x is sorted in increasing order

         V <= x[0]  : I = -1
x[0]  <  V <= x[-1] : I s.t. x[I] < V <= x[I+1]
x[-1] <  V          : I = $x->nelem - 1

if $x is sorted in decreasing order

         V >  x[0]  : I = 0
x[0]  >= V >  x[-1] : I s.t. x[I-1] >= V > x[I]
x[-1] >= V          : I = $x->nelem

If all elements of $x are equal,

i = $x->nelem - 1

If $x contains duplicated elements, I is the index of the righmost (by index in array) duplicate if V matches.

needs major (?) work to handles bad values

interpolate

Signature: (xi(); x(n); y(n); [o] yi(); int [o] err())

routine for 1D linear interpolation

( $yi, $err ) = interpolate($xi, $x, $y)

Given a set of points ($x,$y), use linear interpolation to find the values $yi at a set of points $xi.

interpolate uses a binary search to find the suspects, er..., interpolation indices and therefore abscissas (ie $x) have to be strictly ordered (increasing or decreasing). For interpolation at lots of closely spaced abscissas an approach that uses the last index found as a start for the next search can be faster (compare Numerical Recipes hunt routine). Feel free to implement that on top of the binary search if you like. For out of bounds values it just does a linear extrapolation and sets the corresponding element of $err to 1, which is otherwise 0.

See also "interpol", which uses the same routine, differing only in the handling of extrapolation - an error message is printed rather than returning an error ndarray.

needs major (?) work to handles bad values

interpol

Signature: (xi(); x(n); y(n); [o] yi())

routine for 1D linear interpolation

$yi = interpol($xi, $x, $y)

interpol uses the same search method as "interpolate", hence $x must be strictly ordered (either increasing or decreasing). The difference occurs in the handling of out-of-bounds values; here an error message is printed.

interpND

Interpolate values from an N-D ndarray, with switchable method

$source = 10*xvals(10,10) + yvals(10,10);
$index = pdl([[2.2,3.5],[4.1,5.0]],[[6.0,7.4],[8,9]]);
print $source->interpND( $index );

InterpND acts like indexND, collapsing $index by lookup into $source; but it does interpolation rather than direct sampling. The interpolation method and boundary condition are switchable via an options hash.

By default, linear or sample interpolation is used, with constant value outside the boundaries of the source pdl. No dataflow occurs, because in general the output is computed rather than indexed.

All the interpolation methods treat the pixels as value-centered, so the sample method will return $a->(0) for coordinate values on the set [-0.5,0.5), and all methods will return $a->(1) for a coordinate value of exactly 1.

Recognized options:

method

Values can be:

0, s, sample, Sample (default for integer source types)

The nearest value is taken. Pixels are regarded as centered on their respective integer coordinates (no offset from the linear case).
1, l, linear, Linear (default for floating point source types)

The values are N-linearly interpolated from an N-dimensional cube of size 2.
3, c, cube, cubic, Cubic

The values are interpolated using a local cubic fit to the data. The fit is constrained to match the original data and its derivative at the data points. The second derivative of the fit is not continuous at the data points. Multidimensional datasets are interpolated by the successive-collapse method.

(Note that the constraint on the first derivative causes a small amount of ringing around sudden features such as step functions).
f, fft, fourier, Fourier

The source is Fourier transformed, and the interpolated values are explicitly calculated from the coefficients. The boundary condition option is ignored -- periodic boundaries are imposed.

If you pass in the option "fft", and it is a list (ARRAY) ref, then it is a stash for the magnitude and phase of the source FFT. If the list has two elements then they are taken as already computed; otherwise they are calculated and put in the stash.

b, bound, boundary, Boundary

This option is passed unmodified into indexND, which is used as the indexing engine for the interpolation. Some current allowed values are 'extend', 'periodic', 'truncate', and 'mirror' (default is 'truncate').

bad

contains the fill value used for 'truncate' boundary. (default 0)

fft

An array ref whose associated list is used to stash the FFT of the source data, for the FFT method.

one2nd

Converts a one dimensional index ndarray to a set of ND coordinates

@coords=one2nd($x, $indices)

returns an array of ndarrays containing the ND indexes corresponding to the one dimensional list indices. The indices are assumed to correspond to array $x clumped using clump(-1). This routine is used in the old vector form of "whichND", but is useful on its own occasionally.

Returned ndarrays have the indx datatype. $indices can have values larger than $x->nelem but negative values in $indices will not give the answer you expect.

pdl> $x=pdl [[[1,2],[-1,1]], [[0,-3],[3,2]]]; $c=$x->clump(-1)
pdl> $maxind=maximum_ind($c); p $maxind;
6
pdl> print one2nd($x, maximum_ind($c))
0 1 1
pdl> p $x->at(0,1,1)
3

which

Signature: (mask(n); indx [o] inds(m))

Returns indices of non-zero values from a 1-D PDL

$i = which($mask);

returns a pdl with indices for all those elements that are nonzero in the mask. Note that the returned indices will be 1D. If you feed in a multidimensional mask, it will be flattened before the indices are calculated. See also "whichND" for multidimensional masks.

If you want to index into the original mask or a similar ndarray with output from which, remember to flatten it before calling index:

$data = random 5, 5;
$idx = which $data > 0.5; # $idx is now 1D
$bigsum = $data->flat->index($idx)->sum;  # flatten before indexing

Compare also "where" for similar functionality.

which_both

Signature: (mask(n); indx [o] inds(m); indx [o]notinds(q))

Returns indices of zero and nonzero values in a mask PDL

($i, $c_i) = which_both($mask);

This works just as "which", but the complement of $i will be in $c_i.

pdl> $x = sequence(10); p $x
[0 1 2 3 4 5 6 7 8 9]
pdl> ($small, $big) = which_both ($x >= 5); p "$small\n $big"
[5 6 7 8 9]
[0 1 2 3 4]

which_both 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.

where

Use a mask to select values from one or more data PDLs

where accepts one or more data ndarrays and a mask ndarray. It returns a list of output ndarrays, corresponding to the input data ndarrays. Each output ndarray is a 1-dimensional list of values in its corresponding data ndarray. The values are drawn from locations where the mask is nonzero.

The output PDLs are still connected to the original data PDLs, for the purpose of dataflow.

where combines the functionality of "which" and index into a single operation.

BUGS:

While where works OK for most N-dimensional cases, it does not thread properly over (for example) the (N+1)th dimension in data that is compared to an N-dimensional mask. Use whereND for that.

$i = $x->where($x+5 > 0); # $i contains those elements of $x
                          # where mask ($x+5 > 0) is 1
$i .= -5;  # Set those elements (of $x) to -5. Together, these
           # commands clamp $x to a maximum of -5.

It is also possible to use the same mask for several ndarrays with the same call:

($i,$j,$k) = where($x,$y,$z, $x+5>0);

Note: $i is always 1-D, even if $x is >1-D.

WARNING: The first argument (the values) and the second argument (the mask) currently have to have the exact same dimensions (or horrible things happen). You *cannot* thread over a smaller mask, for example.

whereND

where with support for ND masks and threading

whereND accepts one or more data ndarrays and a mask ndarray. It returns a list of output ndarrays, corresponding to the input data ndarrays. The values are drawn from locations where the mask is nonzero.

whereND differs from where in that the mask dimensionality is preserved which allows for proper threading of the selection operation over higher dimensions.

As with where the output PDLs are still connected to the original data PDLs, for the purpose of dataflow.

$sdata = whereND $data, $mask
($s1, $s2, ..., $sn) = whereND $d1, $d2, ..., $dn, $mask

where

  $data is M dimensional
  $mask is N < M dimensional
  dims($data) 1..N == dims($mask) 1..N
  with threading over N+1 to M dimensions

$data   = sequence(4,3,2);   # example data array
$mask4  = (random(4)>0.5);   # example 1-D mask array, has $n4 true values
$mask43 = (random(4,3)>0.5); # example 2-D mask array, has $n43 true values
$sdat4  = whereND $data, $mask4;   # $sdat4 is a [$n4,3,2] pdl
$sdat43 = whereND $data, $mask43;  # $sdat43 is a [$n43,2] pdl

Just as with where, you can use the returned value in an assignment. That means that both of these examples are valid:

# Used to create a new slice stored in $sdat4:
$sdat4 = $data->whereND($mask4);
$sdat4 .= 0;
# Used in lvalue context:
$data->whereND($mask4) .= 0;

whichND

Return the coordinates of non-zero values in a mask.

WhichND returns the N-dimensional coordinates of each nonzero value in a mask PDL with any number of dimensions. The returned values arrive as an array-of-vectors suitable for use in indexND or range.

$coords = whichND($mask);

returns a PDL containing the coordinates of the elements that are non-zero in $mask, suitable for use in "indexND" in PDL::Slices. The 0th dimension contains the full coordinate listing of each point; the 1st dimension lists all the points. For example, if $mask has rank 4 and 100 matching elements, then $coords has dimension 4x100.

If no such elements exist, then whichND returns a structured empty PDL: an Nx0 PDL that contains no values (but matches, threading-wise, with the vectors that would be produced if such elements existed).

DEPRECATED BEHAVIOR IN LIST CONTEXT:

whichND once delivered different values in list context than in scalar context, for historical reasons. In list context, it returned the coordinates transposed, as a collection of 1-PDLs (one per dimension) in a list. This usage is deprecated in PDL 2.4.10, and will cause a warning to be issued every time it is encountered. To avoid the warning, you can set the global variable "$PDL::whichND" to 's' to get scalar behavior in all contexts, or to 'l' to get list behavior in list context.

In later versions of PDL, the deprecated behavior will disappear. Deprecated list context whichND expressions can be replaced with:

@list = $x->whichND->mv(0,-1)->dog;

setops

Implements simple set operations like union and intersection

Usage: $set = setops($x, <OPERATOR>, $y);

The operator can be OR, XOR or AND. This is then applied to $x viewed as a set and $y viewed as a set. Set theory says that a set may not have two or more identical elements, but setops takes care of this for you, so $x=pdl(1,1,2) is OK. The functioning is as follows:

OR

The resulting vector will contain the elements that are either in $x or in $y or both. This is the union in set operation terms

XOR

The resulting vector will contain the elements that are either in $x or $y, but not in both. This is

Union($x, $y) - Intersection($x, $y)

in set operation terms.

AND

The resulting vector will contain the intersection of $x and $y, so the elements that are in both $x and $y. Note that for convenience this operation is also aliased to "intersect".

It should be emphasized that these routines are used when one or both of the sets $x, $y are hard to calculate or that you get from a separate subroutine.

Finally IDL users might be familiar with Craig Markwardt's cmset_op.pro routine which has inspired this routine although it was written independently However the present routine has a few less options (but see the examples)

You will very often use these functions on an index vector, so that is what we will show here. We will in fact something slightly silly. First we will find all squares that are also cubes below 10000.

Create a sequence vector:

pdl> $x = sequence(10000)

Find all odd and even elements:

pdl> ($even, $odd) = which_both( ($x % 2) == 0)

Find all squares

pdl> $squares= which(ceil(sqrt($x)) == floor(sqrt($x)))

Find all cubes (being careful with roundoff error!)

pdl> $cubes= which(ceil($x**(1.0/3.0)) == floor($x**(1.0/3.0)+1e-6))

Then find all squares that are cubes:

pdl> $both = setops($squares, 'AND', $cubes)

And print these (assumes that PDL::NiceSlice is loaded!)

pdl> p $x($both)
 [0 1 64 729 4096]

Then find all numbers that are either cubes or squares, but not both:

pdl> $cube_xor_square = setops($squares, 'XOR', $cubes)

pdl> p $cube_xor_square->nelem()
 112

So there are a total of 112 of these!

Finally find all odd squares:

pdl> $odd_squares = setops($squares, 'AND', $odd)

Another common occurrence is to want to get all objects that are in $x and in the complement of $y. But it is almost always best to create the complement explicitly since the universe that both are taken from is not known. Thus use "which_both" if possible to keep track of complements.

If this is impossible the best approach is to make a temporary:

This creates an index vector the size of the universe of the sets and set all elements in $y to 0

pdl> $tmp = ones($n_universe); $tmp($y) .= 0;

This then finds the complement of $y

pdl> $C_b = which($tmp == 1);

and this does the final selection:

pdl> $set = setops($x, 'AND', $C_b)

intersect

Calculate the intersection of two ndarrays

Usage: $set = intersect($x, $y);

This routine is merely a simple interface to "setops". See that for more information

Find all numbers less that 100 that are of the form 2*y and 3*x

pdl> $x=sequence(100)
pdl> $factor2 = which( ($x % 2) == 0)
pdl> $factor3 = which( ($x % 3) == 0)
pdl> $ii=intersect($factor2, $factor3)
pdl> p $x($ii)
[0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96]

AUTHOR

Copyright (C) Tuomas J. Lukka 1997 (lukka@husc.harvard.edu). Contributions by Christian Soeller (c.soeller@auckland.ac.nz), Karl Glazebrook (kgb@aaoepp.aao.gov.au), Craig DeForest (deforest@boulder.swri.edu) and Jarle Brinchmann (jarle@astro.up.pt) 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.

Updated for CPAN viewing compatibility by David Mertens.

To install PDL, copy and paste the appropriate command in to your terminal.

cpanm

cpanm PDL

CPAN shell

perl -MCPAN -e shell
install PDL

For more information on module installation, please visit the detailed CPAN module installation guide.

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NAME

DESCRIPTION

SYNOPSIS

FUNCTIONS

inner

outer

x

matmult

innerwt

inner2

inner2d

inner2t

crossp

norm

indadd

conv1d

in

uniq

uniqind

uniqvec

hclip

lclip

clip

clip

wtstat

statsover

stats

histogram

whistogram

histogram2d

whistogram2d

fibonacci

append

glue

axisvalues

srand

random

randsym

grandom

vsearch

vsearch_sample

vsearch_insert_leftmost

vsearch_insert_rightmost

vsearch_match

vsearch_bin_inclusive

vsearch_bin_exclusive

interpolate

interpol

interpND

one2nd

which

which_both

where

whereND

whichND

setops

intersect

AUTHOR

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