Primitive(e) User Contributed Perl Documentation Primitive(e) 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(a); b(b); [o]c()) Inner product over one dimension c = sum_i a_i * b_i outer Signature: (a(a); b(b); [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 multiplica- tion via a threaded inner product innerwt Signature: (a(a); b(b); c(c); [o]d()) Weighted (i.e. triple) inner product d = sum_i a(a) b(b) c(c) inner2 Signature: (a(a); b(n,m); c(c); [o]d()) Inner product of two vectors and a matrix d = sum_ij a(a) b(i,j) c(c) 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(p), $b->clump(p)) 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 matri- ces 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(b); [o] c(c)) 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(c); [o] norm(m)) 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(m) += a Simple Example: $a = 2; $ind = 3; $sum = zeroes(s); indadd($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); $sum = zeroes(s); indadd($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(a); kern(n); [o]b(b); int reflect) 1d convolution along first dimension $con = conv1d sequence(e), 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 rou- tines, e.g. $b = $a->mv(2,0)->conv1d($kernel)->mv(0,2); # along third dim This function is useful for threaded filtering of 1D sig- nals. Compare also conv2d, convolve, fftconvolve, fftwconv, rfftwconv in Signature: (a(); b(b); [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 func- tionality by using a construct like $msk = ($labels->dummy(y) == $goodlabels)->orover; However, "in" doesn't create a (potentially large) inter- mediate 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. 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, $x); wtstat Signature: (a(a); wt(t); avg(); [o]b(); int deg) Weighted statistical moment of given degree This calculates a weighted statistic over the vector "a". The formula is b() = (sum_i wt_i * (a_i ** degree - avg)) / (sum_i wt_i) statsover Signature: (a(a); w(w); int+ [o]avg(); int+ [o]rms(); int+ [o]min(); int+ [o]max(); int+ [o]adev()) Calculate useful statistics over a dimension of a piddle ($mean, $rms, $median, $min, $max, $adev) = statover($piddle, $weights); This utility function calculates various useful quantities of a piddle. These are the mean: MEAN = sum (x)/ N with "N" being the number of elements in x, the root mean square deviation from the mean, RMS, given as, RMS = sqrt(sum( (x-mean(n))^2 )/(N-1)); Note the use of "N-1" which for almost all cases should be the right normalisation factor. The routine also returns the median, minimum and maximum of the piddle as well as the mean absolute deviation, defined as: ADEV = sqrt(sum( abs(x-mean(n)) )/N) note here that we use the mean and not the median. This could possibly be changed in future versions of the code. This operator is a projection operator so the calculation will take place over the final dimension. Thus if the input is N-dimensional each returned value will be N-1 dimensional, to calculate the statistics for the entire piddle either use "clump(-1)" directly on the piddle or call "stats". stats Calculates useful statistics on a piddle ($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. histogram Signature: (in(n); int+[o] hist(t); 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(b)" 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(t);float+[o] hist(t); double step; double min; int msize => m) Calculates a histogram from weighted data for given step- size 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(b)" 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(a); inb(b); 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(a); inb(b); float+ wt(t);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 lim- its. 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(x)) Constructor - a vector with Fibonacci's sequence append Signature: (a(a); b(b); [o] c(c)) append two piddles by concantening along their respective 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". axisvalues Signature: ([o,nc]a(a)) 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 docu- mentation. 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 (exclud- ing 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 docu- mentation. grandom Constructor which returns piddle of Gaussian random num- bers $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 docu- mentation. vsearch Signature: (i(); x(x); int [o]ip()) routine for searching 1D values i.e. step-function inter- polation. $inds = vsearch($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 prob- abilities 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(x); y(y); [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 decreas- ing). For interpolation at lots of closely spaced abscis- sas 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(x); y(y); [o] yi()) routine for 1D linear interpolation $yi = interpol($xi, $x, $y) "interpol" uses the same search method as interpolate, hence $x must be strictly ordered (either increasing or decreasing). The difference occurs in the handling of out-of-bounds values; here an error message is printed. one2nd Converts a one dimensional index piddle to a set of ND coordinates @coords=one2nd($a, $indices) returns an array of piddles containing the ND indexes cor- responding 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(k); int [o] inds(s)) 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(e); p $x [0 1 2 3 4 5 6 7 8 9] perldl> $indx = which($x>6); p $indx [7 8 9] which_both Signature: (mask(k); int [o] inds(s); int [o]notinds(s)) Returns piddle of indices of non-zero values and their complement ($i, $c_i) = which_both($mask); This works just as which, but the complement of $i will be in $c_i. perldl> $x = sequence(e); 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 Returns indices to non-zero values or those values from another piddle. $i = $x->where($x+5 > 0); # $i contains elements of $x # where mask ($x+5 > 0) is 1 Note: $i is always 1-D, even if $x is >1-D. The first argument (the values) and the second argument (the mask) currently have to have the same initial dimensions (or horrible things happen). It is also possible to use the same mask for several pid- dles with the same call: ($i,$j,$k) = where($x,$y,$z, $x+5>0); whichND Returns the coordinates for non-zero values @coords=whichND($mask); returns an array of piddles containing the coordinates of the elements that are non-zero in $mask. 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.har- vard.edu). Contributions by Christian Soeller (c.soeller@auckland.ac.nz) and Karl Glazebrook (kgb@aaoepp.aao.gov.au). 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. perl v5.6.1 2002-04-08 Primitive(e)