Primitive

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

 use PDL::Primitive;

FUNCTIONS

inner

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

Inner product over one dimension

 c = sum_i a_i * b_i

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.

matmult

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

Matrix multiplication

We peruse the inner product to define matrix multiplication via a threaded inner product

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)

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".

inner2d

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

Inner product over 2 dimensions.

Equivalent to

 $c = inner($a->clump(2), $b->clump(2))

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.

crossp

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

Cross product of two 3D vectors

After

 $c = crossp $a, $b

the inner product "$c*$a" and "$c*$b" will be zero, i.e. $c is orthogonal to $a and $b

norm

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

Normalises a vector to unit Euclidean length

indadd

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

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

 sum(ind) += a

Simple Example:

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

Threaded Example:

  $a = pdl( 1,2,3);
  $ind = pdl( 1,4,6);
  $sum = zeroes(10);
  indadd($a,$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]

conv1d

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

1d convolution along first dimension

  $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.

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

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

Compare also conv2d, convolve, fftconvolve, fftwconv, rfftwconv

in

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

test if a is in the set of values b

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

"in" is akin to the is an element of of set theory. In priciple, 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.

uniq

return all unique elements of a piddle

The unique elements are returned in ascending order.

  print pdl(2,2,2,4,0,-1,6,6)->uniq;
 [-1 0 2 4 6]

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

uniqvec

return all unique vectors out of a collection

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 uniqe list of scalars; and qsortvec for sorting a list of vectors lexicographcally.

hclip

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

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

lclip

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

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

clip

Clip a piddle by (optional) upper or lower bounds.

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

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)

statsover

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

Calculate useful statistics over a dimension of a piddle

  ($mean, $rms, $median, $min, $max, $adev) = statover($piddle, $weights);

This utility function calculates various useful quantities of a piddle. These are the mean:

  MEAN = sum (x)/ N

with "N" being the number of elements in x, the root mean square deviation from the mean, RMS, given as,

  RMS = sqrt(sum( (x-mean(x))^2 )/(N-1));

Note the use of "N-1" which for almost all cases should be the right normalisation factor. The routine also returns the median, minimum and maximum of the piddle as well as the mean absolute deviation, defined as:

  ADEV = sqrt(sum( abs(x-mean(x)) )/N)

note here that we use the mean and not the median. This could possibly be changed in future versions of the code.

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 piddle either use "clump(-1)" directly on the piddle or call "stats".

stats

Calculates useful statistics on a piddle

 ($mean,$rms,$median,$min,$max) = stats($piddle,[$weights]);

This utility calculates all the most useful quantities in one call.

Note: The RMS value that this function returns in the RMS deviation from the mean, also known as the population standard- deviation: $rms = sqrt(sum(($val-$mean)**2)) / ($n-1)

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 $numbins;  # Put histogram in existing piddle.
 histogram($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.

Use hist instead for a high-level interface.

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

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 $numbins;  # Put histogram in existing piddle.
 whistogram($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.

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

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, $nbiny;  # Put histogram in existing piddle.
 histogram2d($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.

 perldl> 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]
 ]

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, $nbiny;  # Put histogram in existing piddle.
 whistogram2d($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.

 perldl> 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]
 ]

fibonacci

  Signature: ([o]x(n))

Constructor - a vector with Fibonacci's sequence

append

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

append two or more piddles by concatenating along their first dimensions

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

"append" appends two piddles along their first dims. Rest of the dimensions must be compatible in the threading sense. Resulting size of first dim is the sum of the sizes of the first dims of the two argument piddles - ie "n + m".

glue

  $c = $a->glue(<dim>,$b,...)

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

Sticks $a, $b, and all following arguments together using a combination of xchg() and append(). All other dimensions must be compatible in the threading sense.

"glue" is implemented in pdl, and should probably be updated (one day) to a pure PP function.

axisvalues

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

Internal routine

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

random

Constructor which returns piddle of random numbers

 $a = random([type], $nx, $ny, $nz,...);
 $a = random $b;

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 perl function srand to seed the random generator. For further details consult Perl's srand documentation.

randsym

Constructor which returns piddle of random numbers

 $a = randsym([type], $nx, $ny, $nz,...);
 $a = randsym $b;

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 perl function srand to seed the random generator. For further details consult Perl's srand documentation.

grandom

Constructor which returns piddle of Gaussian random numbers

 $a = grandom([type], $nx, $ny, $nz,...);
 $a = grandom $b;

etc (see zeroes).

This is generated using the math library routine "ndtri".

Mean = 0, Stddev = 1

You can use the perl function srand to seed the random generator. For further details consult Perl's srand documentation.

vsearch

  Signature: (i(); x(n); int [o]ip())

routine for searching 1D values i.e. step-function interpolation.

 $inds = vsearch($vals, $xs);

Returns for each value of $vals the index of the least larger member of $xs (which need to be in increasing order). If the value is larger than any member of $xs, the index to the last element of $xs is returned.

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

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

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

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 piddle.

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 piddle, 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 piddle to a set of ND coordinates

 @coords=one2nd($a, $indices)

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

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

which

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

Returns piddle of indices of non-zero values.

 $i = which($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 want to index into the original mask or a similar piddle 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.

If you want to return both the indices of non-zero values and the complement, use the function which_both.

 perldl> $x = sequence(10); p $x
 [0 1 2 3 4 5 6 7 8 9]
 perldl> $indx = which($x>6); p $indx
 [7 8 9]

which_both

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

Returns piddle of indices of non-zero values and their complement

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

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

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

where

Return the values from a piddle for whose indices a condition piddle is nonzero.

 $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 piddles 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.

whichND

Returns the coordinates for non-zero values.

For historical reasons the return value is different in list and scalar context. In scalar context, you get back a PDL containing coordinates suitable for use in indexND or range; in list context, the coordinates are broken out into separate PDLs.

 $coords = whichND($mask);

returns a PDL containing the coordinates of the elements that are non-zero in $mask, suitable for use in indexND. 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.

 @coords=whichND($mask);

returns a perl list of piddles containing the coordinates of the elements that are non-zero in $mask. Each element corresponds to a particular index dimension. For example, if $mask has rank 4 and 100 matching elements, then @coords has 4 elements, each of which is a pdl of size 100.

 perldl> $a=sequence(10,10,3,4)
 perldl> ($x, $y, $z, $w)=whichND($a == 203); p $x, $y, $z, $w
 [3] [0] [2] [0]
 perldl> print $a->at(list(cat($x,$y,$z,$w)))
 203

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), and Craig DeForest (deforest@boulder.swri.edu). 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.

Index

NAME
DESCRIPTION
SYNOPSIS
FUNCTIONS
inner
outer
matmult
innerwt
inner2
inner2d
inner2t
crossp
norm
indadd
conv1d
in
uniq
uniqvec
hclip
lclip
clip
wtstat
statsover
stats
histogram
whistogram
histogram2d
whistogram2d
fibonacci
append
glue
axisvalues
random
randsym
grandom
vsearch
interpolate
interpol
interpND
one2nd
which
which_both
where
whichND
AUTHOR