National Radio
Astronomy Observatory

image.fitprofile - Function

1.1.1 Fit gaussians and/or polynomials to a 1-dimensional profile.


Description

This application simultaneously fits any number of gaussian singlets, any number of lorentzian singlets, and any number of gaussian multiplets, and/or a polynomial to one dimensional profiles using the non-linear, least squares Levenberg-Marquardt algorithm. A description of the fitting algorithm may be found in AIPS++ Note 224 (http://www.astron.nl/casacore/trunk/casacore/doc/notes/224.html) and in Numerical Recipes by W.H. Press et al., Cambridge University Press. A gaussian/lorentzian singlet is a gaussian/lorentzian whose parameters (amplitude, center position, and width) are all independent from any other feature that may be simultaneously fit. A gaussian multiplet is a set of two or more gaussian lines in which at least one (and possibly two or three) parameter of each line is dependent on the parameter of another, single (reference) profile in the multiplet. For example, one can specify a doublet in which the amplitude of the first line is 0.6 times the amplitude of the zeroth line and/or the center of the first line is 20 pixels from the center of the zeroth line, and/or the fwhm of the first line is identical (in pixels) to that of the zeroth line. There is no limit to the number of components one can specify in a multiplet (except of course that the number of parameters to be fit should be significantly less than the number of data points), but there can be only a single reference profile in a multiplet to which to tie constraints of parameters of the other profiles in the set.

Additionally, a power logarithmic polynomial (plp) or a logarithmic tranformed polynomial (ltp) can be fit. In this case, each of these functions cannot be fit simultaneously with any other supported function. These functions are most often used for fitting the spectral index and higher order terms of a spectrum. A power logarithmic polynomial has the form

y = c0*x/div**(c1 + c2*ln(x/div) + c3*ln(x/div)**2 + ... + cn*ln(x/div)**(n - 1))

and a logarithmic transformed polynomial is simply the result of this equation after taking the natural log of both sides so that it has the form

ln(y) = c0 + c1*ln(x/div) + c2*ln(x/div)**2 + ... + cn*ln(x/div)**n

The coefficients of the two forms correspond with each other except that c0 in the second equation is equal to ln(c0) of the first. In the case of fitting a spectral index, the spectral index, traditionally represented as alpha, is equal to c1.

In both cases, div is a numerical value used to scale abscissa values so they are closer to unity when they are sent to the fitter. This generally improves the probability that the fit will converge. This parameter may be specified via the div parameter. A value of 0 (the default) indicates that the application should determine a reasonable value for div, which is determined via

div = 10**int(log10(sqrt(min(x)*max(x))))

where min(x) and max(x) are the minimum and maximum abscissa values, respectively.

So, for example, if S(nu) is proportional to nu**alpha and you expect alpha to be near -0.8 and the value of S(nu) is 1.5 at 1e9 Hz and your image(s) have spectral units of Hz, you would specify spxest=[1.5, -0.8] and div=1e9 when fitting a plp function, or spxest=[0.405, -0.8] and div=1e9 if fitting an ltp function.

More details of fitting all of these functions are described in following sections.

A CAUTIONARY NOTE Note that the likelihood of getting a reliable solution increases with the number of good data points as well as the goodness of the initial estimate. It is possible that the first solution found might not be the best one, and so, if a solution is found, it is recommended that the fit be repeated using the solution of the previous fit as the initial estimatE for the new fit. This process should be repeated until the solutions from one fit to the next differ only insignificantly. The convergent solution is very likely the best solution.

AXIS The axis parameter indicates on which axis profiles should be fit; a value ¡0 indicates the spectral axis should be used, or if one does not exist, that the zeroth axis should be used.

MINIMUM NUMBER OF PIXELS The minpts parameter indicates the minimum number of unmasked pixels that must be present in order for a fit to be attempted. When multifit=True, positions with too few good points will be masked in any output images.

ONE FIT OF REGION AVERAGE OR PIXEL BY PIXEL FIT The multifit parameter indicates if profiles should be fit at each pixel in the selected region (true), or if the profiles in that region should be averaged and the fit done to that average profile (false).

POLYNOMIAL FITTING The order of the polynomial to fit is specified only via the poly parameter. If poly¡0, no polynomial will be fit. No initial estimates of coefficients can be specified; these are determined automatically.

GAUSSIAN SINGLET FITTING In the absence of an estimates file and no estimates being specified by the p*est parameters, and gmncomps=0 or is empty, the ngauss parameter indicates the maximum number of gaussian singlets that should be fit. The initial estimates of the parameters for these gaussians will be attempted automatically in this case. If it deems appropriate, the fitter will fit fewer than this number. In the case where an estimates file is supplied, ngauss is ignored (see below). ngauss is also ignored if the p*est parameters are specified or if gmncomps is not an empty array or, if an integer, is greater than zero. If estimates is not specified or the p*est parameters are not specified and ngauss=0, gmncomps is empty or 0, and poly¡0, an error will occur as this indicates there is nothing to fit.

One can specify initial estimates of gaussian singlet parameters via an estimates file or the pampest, pcenterest, pfwhmest, and optionally, the pfix parameters. The latter is the recommended way to specify these estimates as support for estimates files may be deprecated in the future. No matter which option is used, an amplitude initial estimate must always be nonzero. A negative fwhm estimate will be silently changed to positve.

SPECIFYING INITIAL ESTIMATES FOR GAUSSIAN AND LORENTZIAN SINGLETS (RECOMMENDED METHOD) One may specify initial estimates via the pampest, pcenterest, and pfwhmest parameters. In the case of a single gaussian or lorentzian singlet, these parameters can be numbers. pampest must be specified in image brightness units, pcenterest must be given in the number of pixels from the zeroth pixel, and pfwhmest must be given in pixels. Optionally pfix can be specified and in the case of a single gaussian or lorentzian singlet can be a string. In it is coded which parameters should be held constant during the fix. Any combination of ”p” (amplitude), ”c” (center), or ”f” (fwhm) is allowed; eg pfix=”pc” means fix both the amplitude and center during the fit. In the case of more than one gaussian and/or lorentzian singlets, these parameters must be specified as arrays of numbers. The length of the arrays indicates the number of singlets to fit and must be the same for all the p*est parameters.

If no parameters are to be fixed for any of the singlets, pfix can be set to the empty string. However, if at least one parameter of one singlet is to be fixed, pfix must be an array of strings and have a length equal to the p*est arrays. Singlets which are not to have any parameters fixed should be represented as an empty string in the pfix array. So, for example, if one desires to fit three singlets and fix the fwhm of the middle one, one must specify pfix=[””, ”f”, ””], the empty strings indicating no parameters of the zeroth and second singlet should be held constant.

In the case of multifit=True, the initial estimates, whether from the p*est parameters or from a file (see below), will be applied to the location of the first fit. This is normally the bottom left corner of the region selected. If masked, not enough good points to perform a fit, or the attempted fit fails, the fitting proceeds to the next pixel with the pixel value of the lowest numbered axis changing the fastest. Once a successful fit has been performed, subsequent fits will use the results of a fit for a nearest pixel for which a previous fit was successful as the initial estimate for the parameters at the current location. The fixed parameter string will be honored for every fit performed when multifit=True.

One specifies what type of PCF profile to fit via the pfunc parameter. A PCF function is one that can be parameterized by a peak, center, and FWHM, as both gaussian and lorentzian singlets can. If all singlets to be fit are gaussians, one can set pfunc equal to the empty string and all snglets will be assumed to be gaussians. If at least one lorentzian is to be fit, pfunc must be specified as a string (in the case of a single singlet) or an array of strings (in the case of multiple singlets). The position of each string corresponds to the positions of the initial estimates in the p*est and pfix arrays. Minimal match (”g”, ”G”, ”l”, or ”L”) is supported. So, if one wanted to simultaneously fit two gaussian and two lorentzian singlets, the zeroth and last of which were lorentzians, one would specify pfunc=[”L”, ”G”, ”G”, ”L”].

ESTIMATES FILE FOR GAUSSIAN SINGLETS (NONRECOMMENDED METHOD) Initial estimates for gaussian singlets can be specified in an estimates file. Estimates files may be deprecated in the future in favor of the p*est parameters, so it is recommended users use those parameters instead. If an estimates file is desired to be used, the p*est parameters must be 0 or empty and mgncomps must be 0 or empty. Only gaussian singlets can be specified in an estimates file. If one desires to fit one or more gaussian multiplets and/or one or more lorentzian singlets simultaneously, the p*est parameters must be used to specify the initial parameters of all gaussian singlets to fit; one cannot use an estimates file in this case. If an estimates file is specified, a polynomial can be fit simultaneously by specifying the poly parameter. The estimates file must contain initial estimates of parameters for all gaussian singlets to be fit. The number of gaussian singlets to fit is gotten from the number of estimates in the file. The file can contain comments which are indicated by a ”#” at the beginning of a line. All non-comment lines will be interpreted as initial estimates. The format of such a line is

[peak intensity], [center], [fwhm], [optional fixed parameter string]

The first three values are required and must be numerical values. The peak intensity must be expressed in image brightness units, while the center must be specified in pixels offset from the zeroth pixel, and fwhm must be specified in pixels. The fourth value is optional and if present, represents the parameter(s) that should be held constant during the fit. Any combination of the characters ’p’ (peak), ’c’ (center), and ’f’ (fwhm) are permitted, eg ”fc” means hold the fwhm and the center constant during the fit. Fixed parameters will have no error associated with them. Here is an example file:

GAUSSIAN MULTIPLET FITTING Any number of gaussian multiplets, each containing any number of two or more components, can be simultaneously fit, optionally with a polynomial and/or any number of gaussian and/or lorentzian singlets, the only caveat being that the number of parameters to be fit should be significantly less than the number of data points. The gmncomps parameter indicates the number of multiplets to fit and the number of components in each multiplet. In the case of a single multiplet, an integer (¿1) can be specified. For example, mgncomps=4 means fit a single quadruplet of gaussians. In the case of 2 or more multiplets, and array of integers (all ¿1) must be specified. For example, gmncomps=[2, 4, 3] means 3 seperate multiples are to be fit, the zeroth being a doublet, the first being a quadruplet, and the second being a triplet.

Initial estimates of all gaussians in all multiplets are specified via the gm*est parameters which must be arrays of numbers. The order starts with the zeroth component of the zeroth multiplet to the last component of the zeroth multiplet, then the zeroth component of the first multiplet to the last compoenent of the first multiplet, etc to the zeroth component of the last multiplet to the last element of the last multiplet. The zeroth element of a multiplet is defined as the reference component of that multiplet and has the special significance that it is the profile to which all constraints of all other profiles in that multiplet are referenced (see below). So, in our example of gmncomps=[2, 4, 3], gmampest, gmcenterest, and gmfwhmest must each be nine (the total number of individual gaussian profiles summed over all multiplets) element arrays. The zeroth, second, and sixth elements represent parameters of the reference profiles in the zeroth, first, and second multiplet, respectively.

The fixed relationships between the non-reference profile(s) and the reference profile of a multiplet are specified via the gmampcon, gmcentercon, and gmfwhmcon parameters. At least one, and any combination, of constraints can be specified for any non-reference component of a multiplet. The amplitude ratio of a non-reference line to that of the reference line is set in gmampcon. The ratio of the fwhm of a non-reference line to that of the reference line is set in gmfwhmcon. The offset in pixels of the center position of a non-reference line to that of the reference line is set in gmcentercon. In the case where a parameter is not constrained for any non-reference line of any multiplet, the value of the associated parameter must be 0. In the case of a single doublet, a constraint may be specified as a number or an array of a single number. For example, mgncomps=2 and gmampcon=0.65 and gmcentercon=[32.4] means there is a single doublet to fit where the amplitude ratio of the first to the zeroth line is constained to be 0.65 and the center of the first line is constrained to be offset by 32.4 pixels from the center of the zeroth line. In cases of a total of three or more gaussians, the constraints parameters must be specified as arrays with lengths equal to the total number of gaussians summed over all multiplets minus the number of reference lines (one per multiplet, or just number of multiplets, since reference lines cannot be constrained by themselves). In the cases where an array must be specified but a component in that array does not have that constraint, 0 should be specified. Here’s an example

gmncomps=[2, 4, 3] gmampcon= [ 0 , 0.2, 0 , 0.1, 4.5, 0 ] gcentercon=[24.2, 45.6, 92.7, 0 , -22.8, -33.5] gfwhmcon=””

In this case we have our previous example of one doublet, one quadruplet, and one triplet. The first component of the doublet has the constraint that its center is offset by 24.2 pixels from the zeroth (reference) component. The first component of the quadruplet is constrained to have an amplitude of 0.2 times that of the quadruplet’s zeroth component and its center is constrained to be offset by 45.6 pixels from the reference component. The second component of the quadruplet is constained to have its center offset by 92.7 pixels from the associated reference component and the third component is constrained to have an amplitude of 0.1 times that of the associated reference component. The first component of the triplet is constrained to have an amplitude of 4.5 times that of its associated reference component and its center is constrained to be offset by -22.8 pixels from the reference component’s center. The second component of the triplet is constrained to have its center offset by -33.5 pixels from the center of the reference component. No lines have FWHM constraints, so the empty string can be given for that parameter. Note that using 0 to indicate no constraint for line center means that one cannot specify a line centered at the same position as the reference component but having a different FWHM from the reference component. If you must specify this very unusual case, try using a very small positive (or even negative) value for the center constraint.

Note that when a parameter for a line is constrained, the corresponding value for that component in the corresponding gm*est array is ignored and the value of the constrained parameter is automatically used instead. So let’s say, for our example above, we had specified the following estimates:

gmampest = [ 1, .2, 2, .1, .1, .5, 3, 2, 5] gmcenterest = [20, 10 , 30, 45.2, 609 , -233, 30, -859, 1]

Before any fitting is done, the constraints would be taken into account and these arrays would be implicitly rewritten as:

gmampest = [ 1, .2, 2, .4, .1, .2, 3, 13.5, 5 ] gmcenterest = [20, 44.2, 30, 75.6, 127.7, -233, 30, 7.2, -3.5]

The value of gmfwhmest would be unchanged since there are no FWHM constraints in this example.

In addition to be constrained by values of the reference component, parameters of individual components can be fixed. Fixed parameters are specified via the gmfix parameter. If no parameters are to be fixed, gmfix can be specified as the empty string or a zero element array. In the case where any parameter is to be fixed, gmfix must be specified as an array of strings with length equal to the total number of components summed over all multiplets. These strings encode which parameters to be fixed for the corresponding components. If a component is to have no parameters fixed, an empty string is used. In other cases one or more of any combination of parameters can be fixed using ”p”, ”c”, and/or ”f” described above for fixing singlet parameters. There are a couople of special cases to be aware of. In the case where a non-reference component parameter is constrained and the corresponding reference component parameter is set as fixed, that parameter in the non-reference parameter will automatically be fixed even if it was specified not to be fixed in the gmfix array. This is the only way the constraint can be honored afterall. In the converse case of when a constrained parameter of a non-reference component is specified as fixed, but the corresponding parameter in the reference component is not specified to be fixed, an error will occur. Fixing an unconstrained parameter in a non-reference component is always legal as is fixing any combination of parameters in a reference component (with the above caveat that corresponding constrained parameters in non-reference components will be silently held fixed as well).

The same rules that apply to singlets when multifit=True apply to multiplets.

LIMITING RANGES FOR SOLUTION PARAMETERS In cases of low (or no) signal to noise spectra, it is still possible for the fit to converge, but often to a nonsensical solution. The astronomer can use her knowledge of the source to filter out obviously bogus solutions. Any solution which contains a NaN value as a value or error in any one of its parameters is automatically marked as invalid.

One can also limit the ranges of solution parameters to known ”good” values via the goodamprange, goodcenterrange, and goodfwhmrange parameters. Any combination can be specified and the limit constraints will be ANDed together. The ranges apply to all PCF components that might be fit; choosing ranges on a component by component basis is not supported. If specified, an array of exactly two numerical values must be given to indicate the range of acceptable solution values for that parameter. goodamprange is expressed in terms of image brightness units. goodcenterrange is expressed in terms of pixels from the zeroth pixel in the specified region. goodfwhmrange is expressed in terms of pixels (only non-negative values should be given for FWHM range endpoints). In the case of a multiple-PCF fit, if any of the corresponding solutions are outside the specified ranges, the entire solution is considered to be invalid.

In addition, solutions for which the absolute value of the ratio of the amplitude error to the amplitude exceeds 100 or the ratio of the FWHM error to the FWHM exceeds 100 are automatically marked as invalid.

POWER LOGARITHMIC POLYNOMIAL AND LOGARITHMIC TRANSFORMED POLYNOMIAL FITTING Fitting of a sngle power logarithmic polynomial or a single logarithmic transformed polynomial function is supported. No other functions may be fit simultaneously with either of these; if parameters relating to other functions are supplied simultaneously with parameters relating to these functions, an exception will occur. For details of the functional forms, see the introduction of this document.

The set of c0 ... cn coefficients (as defined previously) can be solved for. Initial estimates for the c values should be supplied via the plpest or ltpest parameters, depending on which form is being fit. The number of values given in this array will be the number of coeffecients that are solved for. One may specify which coefficients should be held fixed during the fit in the plpfix or ltpfix array. If supplied, this array should have the same number of elements as its respective initial estimates array. A value of True means the corresponding coefficient will be held fixed during the fit. An empty array indicates that no parameters will be held fixed. This is the default.

Because the logarithm of the ordinate values must be taken before fitting a logarithmic transformed polynomial, all non-positive pixel values are effectively masked for the purposes of fitting.

INCLUDING STANDARD DEVIATIONS OF PIXEL VALUES If the standard deviations of the pixel values in the input image are known and they vary in the image (eg they are higher for pixels near the edge of the band), they can be included in the sigma parameter. This parameter takes either an array or an image name. The array or image must have one of three shapes: 1. the shape of the input image, 2. the same dimensions as the input image with the lengths of all axes being one except for the fit axis which must have length corresponding to its length in the input image, or 3. be one dimensional with lenght equal the the length of the fit axis in the input image. In cases 2 and 3, the array or pixels in sigma will be replicated such that the image that is ultimately used is the same shape as the input image. The values of sigma must be non-negative. It is only the relative values that are important. A value of 0 means that pixel should not be used in the fit. Other than that, if pixel A has a higher standard deviation than pixel B, then pixel A is noisier than pixel B and will receive a lower weight when the fit is done. The weight of a pixel is the usual

weight = 1/(sigma*sigma)

In the case of multifit=False, the sigma values at each pixel along the fit axis in the hyperplane perpendicular to the fit axis which includes that pixel are averaged and the resultant averaged standard deviation spectrum is the one used in the fit. Internally, sigma values are normalized such that the maximum value is 1. This mitigates a known overflow issue.

One can write the normalized standard deviation image used in the fit but specifying its name in outsigma. This image can then be used as sigma for subsequent runs.

RETURNED DICTIONARY STRUCTURE The returned dictionary has a (necessarily) complex structure. First, there are keys ”xUnit” and ”yUnit” whose values are the abscissa unit and the ordinate unit described by simple strings. Next there are arrays giving a broad overview of the fit quality. These arrays have the shape of the specified region collapsed along the fit axis with the axis corresponding to the fit axis having length of 1:

attempted: a boolean array indicating which fits were attempted (eg if too few unmasked points, a fit will not be attempted). converged: a boolean array indicating which fits converged. False if the fit was not attempted. valid: a boolean array indicating which solutions fall within the specified valid ranges of parameter space (see section LIMITING RANGES FOR SOLUTION PARAMETERS for details). niter: an int array indicating the number of iterations for each profile, ¡0 if the fit did not converge ncomps: the number of components (gaussian singlets + lorentzian singlets + gaussian multiplets + polynomial) fit for the profile, ¡0 if the fit did not converge direction: a string array containing the world direction coordinate for each profile

There is a ”type” array having number of dimensions equal to the number of dimensions in the above arrays plus one. The shape of the first n-1 dimensions is the same as the shape of the above arrays. The length of the last dimension is equal to the number of components fit. The values of this array are strings describing the components that were fit at each possition (”POLYNOMIAL”, ”GAUSSIAN” in the case of gaussian singlets, ”LORENTZIAN” in the case of lorentzian singlets, and ””GAUSSIAN MULTPLET”).

If any gaussian singlets were fit, there will be a subdictionary accessible via the ”gs” key which will have subkeys ”amp”, ”ampErr”, ”center”, ”centerErr”, ”fwhm”, ”fwhmErr, ”integral”, and ”integralErr”. Each of these arrays will have one more dimension than the overview arrays described above. The shape of the first n-1 dimensions will be the same as the shape of the arrays described above, while the final dimension will have length equal to the maximum number of gaussian singlets that were fit. Along this axis will be the corresponding fit result or associated error (depending on the array’s associated key) of the fit for that singlet component number. In cases where the fit did not converge, or that particular component was excluded from the fit, a value of NAN will be present.

If any lorentzian singlets were fit, their solutions will be accessible via the ”ls” key. These arrays follow the same rules as the ”gs” arrays described above.

If any gaussian multiplets were fit, there will be subdictionaries accessible by keys ”gm0”, ”gm1”, ..., ”gmn-1” where n is the number of gaussian muliplets that were fit. Each of these dictionaries will have the same arrays described above for gaussian singlets. The last dimension will have length equal to the number of components in that particular multiplet. Each pixel along the last axis will be the parameter solution value or error for that component number in the multiplet, eg the zeroth pixel along that axis contains the parameter solution or error for the reference component of the multiplet.

The polynomial coefficient solutions and errors are not returned, although they are logged.

If a power logarithmic polynomial was fit, there will be a subdictionary accessible via the ”plp” key which will have subkeys ”soltuion” and ”error” which will each have an array value. Each of these arrays will have one more dimension than the overview arrays described above. The shape of the first n-1 dimensions will be the same as the shape of the overview arrays described above, while the final dimension will have length equal to the number of parameters that were fit. Along this axis will be the corresponding fit result or associated error (depending on the array’s associated key) of the fit. In cases where the fit was not attempted or did not converge, a value of NAN will be present.

OUTPUT IMAGES In addition to the returned dictionary, optionally one or more of any combination of output images can be written. The model and residual parameters indicate the names of the model and residual images to be written; blank values inidcate that these images should not be written.

One can also write none, any or all of the solution and error images for gaussian singlet, lorentzian singlet, and gaussian multiplet fits via the parameters amp, amperr, center, centererr, fwhm, fwhmerr, integral, integralerr when doing multi-pixel fits. For a power logarithmic polynomial or a logarithmic transformed polynomial fit, plpsol or ltpsol and plperr or ltpsol are the names of the solution and error images to write, respectively.

These images contain the arrays described for the associated parameter solutions or errors described in previous sections. Each component is written to a different image, and each image is distiguished by the component it represents by its name ending in an uderscore and the relevant component number (”_0”, ”_1”, etc). In the case of Gaussian multiplets, the image name ends with the number of the mulitplet group followed by the number of the component in that group (eg ”_3_4” represents component 4 in multiplet group 3). In the case of lorentzian singlets, ”_ls” is appended to the image names (but before the identifying component number), in the case of gaussian multiplets. Similarly ”_gm” is included in the name of Gaussian multiplet images. Pixels for which fits were not attempted, did not converge, or converged but have values of NaN (not a number) or INF (infinity) will be masked as bad.

Writing analogous images for polynomial coefficients is not supported.

Arguments

Example

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