Perl module Math::Collocate - Collocation with prediction and filtering for scattered data

# Include module
use Math::Collocate;

# Create new object using available scattered data
my $this = Math::Collocate->new(sample => \@data);

# Adjust signal and noise variances
$this->sigs($sigs);
$this->sign($sign);

# Apply collocation to create regular grid data
my @result = $this->interpolate({ range => \@range, grid => \@grid });

• List::Util
• Math::MatrixReal
• Moose
• Scalar::Util
• Spreadsheet::Read::Simple

To install this module, run the following commands:

$ perl Makefile.PL
$ make
$ make test
$ make install

In order to consider an alternate installation location for scripts, manpages and libraries, use the PREFIX value as a parameter on the command line:

$ perl Makefile.PL PREFIX=~/testing

This module implements a least-squares collocation method based on Moritz [1972]. Collocation is a data processing method which simultaneously perfoms regression, filtering and spatial prediction based on given data. Collocation methods generally assume that measurements consist of a systematic component (trend surface) and a random component (signal and noise), i.e.

w = A x + s + n

where:

• A Design matrix from constituent parts of mathematical model
• x Vector of unknown parameters (if any)
• n Vector of noise components in measurements
• s Vector of signal components in measurements
• w Vector of measurements

The purpose of least-squares collocation is the prediction and the separation of the random components s and n and furthermore, the estimation of x if required.

When the design matrix is known and removed from the measurements, the task reduces to separate the signal and the noise components:

z = w - A x = s + n.

The constitutents of the vector z can be calculated from the statistical quadratic form which is the expression which weighted least-squares adjustment would minimise:

sqf(v) = v^T Cov(v,v)^-1 v.

The vectors v contain the distances between the known samples and the distances to the required point. The covariance matrix Cov(v,v) is the sum of the signal and noise covariance matrices. The element ij of the signal covariance matrix can be computed using the Gaussian function:

σ_s ( s_ij ) = σ_s² * exp( - s_ij² / s_h² )

where s_ij is the separator factor representing the distance between the points of interest, and s_h represents some constant control parameter. The signal variance σ_s is set by the user.

If the observed measurements are error-free, then the noise covariance matrix is zero. Otherwise, a small error is assigned to each available point i:

σ_n ( s_ii ) = σ_n²

The noise variance σ_n is set by the user. By adjusting the variances σ_s and σ_n, the user can control the separation of the corresponding constituents.

Nothing.

• new( key => value )

  The constructor new expects a list of key-value pairs and returns the reference to the created instance. The following keys are accepted.

  • dim => int

    The key dim is required if you want to specify the dimension of the problem only. An instance is created without any samples. Samples can be added later using the method add described below.

  • sample => arrayref

    The key sample can be used to submit existing data points of arbitrary dimension. If the submitted array reference contains numeric values only, the problem is assumed to be 1D and the key value is mandatory to submit the corresponding values using a reference to an array of the same size. If the submitted array reference contains arrays, then the problem is assumed to be N-dimensional according to the size of the largest array. If the key value is omitted, the values are taken from the last entry of the available arrays and the dimension of the problem is reduced to N - 1.

  • value => arrayref

    The key value receives a reference to an array containing numeric values that describe the properties to the corresponding samples given by the key sample. The user has to take care that the sizes of the arrays referenced by value and sample are the same.

Here is a list of object methods available. Object methods are applied to the object in question, in contrast with class methods which are applied to a class.

• size

  The method size returns the number of available samples.

• sigs( value )

  The method sigs reads or sets the signal variance σ_s for the corresponding object. The default value is 5.

• sign( value )

  The method sign reads or sets the noise variance σ_n for the corresponding object. Set the noise variance to zero, if the values at the given samples are free of any measurement errors. The default value is 1.

• bucket( value )

  The method bucket reads or sets the number of samples to consider for collocation. If undefined, all samples are used for collocation. Use this option for large datasets.

• add({ sample => arrayref, value => arrayref })

  The method add adds a single sample to the corresponding object. The method accepts a single argument, which can either be a hash reference or an array reference.

• prediction

  The method prediction returns the normalized prediction fraction in the range [0..1].

• filter

  The method filter returns the normalized filtering fraction in the range [0..1].

• interpolate({ range => arrayref, grid => arrayref })

  The method interpolate applies the collocation algorithm and serves as primary method of this class. The method receives an optional argument which can be of any type. If the method is called in scalar context, the method will return a reference to the array containing the results of the collocation. If the method is called in array context, the method will return the resulting array directly.

  • hashref (see example above)

    If the submitted argument is a hash reference, then the keys range and grid receive the boundaries and the coarseness of the domain of interest.

    • range => arrayref

      The optional key range receives an array reference either to scalars indicating the lower and upper boundary of the requested range separated by a colon (:) or to arrays with 2 values, i.e. the lower and the upper boudnary itselves. If several array elements are undefined or the array is not defined at all, the range is determined automatically based on the minimum and the maximum value given by the samples.

    • grid => int | arrayref

      The optional key grid may receive a positive integer which is applied to all dimensions available or it may receive a array of integers which indicate the requested grid size for each dimension. If key is omitted, the default grid of 20 is applied.

  • arrayref

    If the submitted value is an array reference, the value is assumed to be a single sample with size corresponding to the dimension of the object. The method will return the resulting value in scalar context.

  • scalar

    A scalar value is allowed for objects with dimension 1 only, otherwise an error is raised. The method will return the resulting value as a scalar.

• newline( index )

  The method newline returns true or false whether the sample referred to by the given index closes a block, i.e. is a multiple of the grid points along the first dimension. This method is particularly useful to insert a newline when the output data shall be forwarded to gnuplot.

The following routines are provided.

• isequal( value1, value2 [, tol ] )

  This subroutine compares two floating-point numbers value1 and value2 and checks whether they are equal within a reasonable tolerance tol.

• Höpcke, W. (1980):

  Fehlerlehre und Ausgleichsrechnung. De Gruyter, Pages 211 f.

• Moritz, H. (1972):

  Advanced least-squares methods. Report No. 175, Department of Geodetic Science, Ohio State University. 132 Pages.

• Ruffhead, A. (1987):

  An introduction to least-squares collocation. Survey Review, Volume 29, Number 224, Pages 85-94.

My father Joachim Boljen for guiding me the way through the mists of statistical geodesy for filtering, prediction and modeling.

MIT License

Copyright (c) 2020, 2021 Matthias Boljen

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