MATERNFIT Parametric spectral fit to the Matern form. [with A. Sykulski] MATERNFIT performs a parametric fit of the spectrum of a time series to that expected for a Matern process plus optional spin. The time series may either be real-valued or complex-valued. The Matern process plus spin has a spectrum given by, see MATERNSPEC, SPP(F) = SIGMA^2 / [(F-NU)^2/LAMBDA^2 + 1]^ALPHA / (LAMBDA * MATERNC(ALPHA)) + 2*pi * EPSILON^2 where SIGMA is the standard deviation, NU is a frequency shift, LAMBDA is a damping coefficient, ALPHA is 1/2 of the spectral slope, and EPSILON^2 is the variance of an optional additive noise component. The coefficient LAMBDA * MATERNC(ALPHA) in the above form lets us parameterize the spectrum in terms of the process variance SIGMA^2. The optimal parameters are found using a frequency-domain maximum likelihood method, accounting for both aliasing and spectral blurring. For further details on the parameter inference method method, see Sykulski, Olhede, Lilly, and Danioux (2016). Lagrangian time series models for ocean surface drifter trajectories. Journal of the Royal Statisical Society, Series C. 65 (1): 29--50. Sykulski, Olhede, Guillaumin, Lilly, and Early (2019). The de-biased Whittle likelihood. Biometrika, 106 (2): 251--266. For details on the Matern process and its spectrum, see: Lilly, Sykulski, Early, and Olhede, (2017). Fractional Brownian motion, the Matern process, and stochastic modeling of turbulent dispersion. Nonlinear Processes in Geophysics, 24: 481--514. __________________________________________________________________ Usage FIT=MATERNFIT(DT,Z,FO) where Z is a times series oriented as column vector, returns the result of a fitting the periodogram of Z to a spectrum having a Matern form, fit over all frequencies. The range of frequencies involved in the fit can be modified, as described below. DT is the sample interval, which has units of days, while FO is a signed reference frequency in units of radians per day. FO is used in determining search ranges and initial guesses. In oceanographic applications, one would normally choose FO as the Coriolis frequency. The output argument FIT is a structure which will be described later. The default search ranges and initial guess values are as follows: Low Guess High SIGMA = [0 1 100] * STD(Z) ALPHA = [1/2 1 10] LAMBDA = [1e-3 2e-3 10] * ABS(FO) EPSILON = [0 0 0] * STD(Z) NU = [0 0 0] * ABS(FO) Note that by default, neither the EPSILON nor NU parameters are used. The search ranges and guesses can all be modified, as described below. A common special case is the inclusion of an additive noise component. Calling flag MATERNFIT(...'noisy'), which sets EPSILON = [0 0.01 2] * STD(Z) for the range of the EPSILON parameter. According to the spectral normalizations used here, a white noice time series with standard deviation EPSILON will have a spectral __________________________________________________________________ Output format The following parameters are output as fields of the structure FIT. SIGMA Standard deviation of currents in cm/s ALPHA Slope parameter LAMBDA Damping parameter in rad / day NU Oscillation frequency in rad /day RANGE Ranges of fit search for each parameter PARAMS Sub-structure with fields described below The associated spectra can be then created from SIGMA, ALPHA, LAMBDA, and NU using MATERNSPEC. RANGE is substructure with fields SIGMA, ALPHA, LAMBDA, and NU, such that RANGE.SIGMA specifies the *dimensional* values of the associated range and initial guess with format [MIN GUESS MAX], and so forth for all the other parameters. PARAMS is a substructure containing various parameter values characterizing the fit itself: PARAMS.DT Sample rate, as input to MATERNFIT PARAMS.FO Reference frequency, as input to MATERNFIT PARAMS.A Index into first frequency F(A) used in the fit PARAMS.B Index into last frequency F(B) used in the fit PARAMS.P Number of free parameters used in the fit LIKE Negative of the log-likelihood AICC Akaike Information Criterion, corrected version ERR Normalized error of fit to log spectra EXITFLAG The exit flag from the optimization routine ITER The number of iterations in the optimization routine PARAMS.SIDE String specifying frequency side options PARAMS.ALG String specifying algorithm options PARAMS.VER String specifiying raw or difference option PARAMS.CORES String specifying series or parallel computation These parameters are described in more detail below. __________________________________________________________________ Options for multiple input time series FIT=MATERNFIT(DT,Z,FO) may have Z being a matrix with N columns, or a cell array of N different time series. In both of these cases, DT and FO may be scalars or length N arrays. In these cases, the fields SIGMA, ALPHA, LAMBDA, and NU of FIT will also be arrays with N elements, as all non-string fields of PARAMS. The fields of RANGE will then all be N x 3 arrays. Alternatively, MATERNFIT(...,'average') with Z a matrix or a cell array of matrices causes the columns of each matrix to be interpreted as members of an ensemble, averaging over columns to create an average spectrum. One fit per matrix is returned, rather than one per column. __________________________________________________________________ Specifiying frequencies MATERNFIT(DT,Z,FO,RA,RB), where RA and RB are both real-valued scalars, applies the fit to only frequencies in the range ABS(FO)*RA < F < ABS(FO)*RB with the default behavior corresponding to MATERNFIT(DT,Z,FO,0,INF). Thus RA is the smallest permissible ratio of F to the reference frequency, while RB is similarly the largest permissible ratio. MATERNFIT(DT,Z,FO,RA,[RB,RN]) also works, where the fifth argument is an array of length two. In this case the fit is applied to the range ABS(FO)*RA < F < MIN( ABS(FO)*RB, PI/DT*RN ) RB is the largest permissible ratio of F to the reference frequency, while RN is the largest permissible ratio of F to the Nyquist PI/DT. The fit extends to the smaller of these two frequencies. If RA and RB are imaginary numbers, rather than real numbers, then the fit is only applied to frequencies in the range IMAG(RA) < F < IMAG(RB) that is, the range is found without scaling by the reference frequency. MATERNFIT can create the fit by utilizing both positive and negative frequency sides of the spectrum, the default behavior, or to only one side (plus the zero frequency). This is modified as follows: MATERNFIT(...,'both',...), the default, uses both sides. MATERNFIT(...,'positive',...), uses the side where F/FO is positive. MATERNFIT(...,'negative',...), uses the side where F/FO is negative. Thus, changing the sign of the reference frequency also changes the side of the spectrum to be fit using the 'postive' or 'negative' flags. __________________________________________________________________ Parameter range specification The default search ranges and guess values can be modified. As an example, to modify the default range and guess for the SIGMA parameter corresponding to the background flow, use MATERNFIT(...,'range.sigma',[MIN GUESS MAX],...) and so forth for other parameters. SIGMA ranges input to MATERNFIT represent *fractions* of the total signal standard deviation. Thus MATERNFIT(...,'range.sigma',[0 1 100],...) corresponds to the default setting. The upper limit is set very high because occasionally an optimum spectral fit is found that has much larger total variance than the signal. The ranges of LAMBDA and NU are nondimensional values representing fractions of the magnitude of the reference frequency ABS(FO). Thus MATERNFIT(...,'range.lambda',[1/1000 2/1000 1],...) corresponds to the default range for LAMBDA. The slope parameter ALPHA can take on a value of no less than 1/2, so the lower range for ALPHA cannot be less than 1/2. This approach can be used to omit parameters from the fit, by setting the MIN, GUESS, and MAX values to be identical. For example, MATERNFIT(...,'range.alpha',[2 2 2]) sets the ALPHA parameter to a value of 2. Fixed parameters are then not included in the optimization, thus speeding up the fit. __________________________________________________________________ Parameter value specification Parameters can also be set to particular values for each time series. This is done by setting the 'value' field as follows MATERNFIT(...,'value.sigma',SIGMA,...) and so forth for the other parameters. Here SIGMA is an array of the same length as the number of time series in Z. Thus SIGMA is a scalar if Z is a single array, an array of LENGTH(Z) is Z is a cell array, or of SIZE(Z,2) if Z is a matrix. Note that the values specified in this way are actual dimensional values, not nondimensional values as with setting the range. This approach works by internally setting the dimension MIN, GUESS, and MAX to the value specified for each time series, overriding the default choices. This will be reflected in the output RANGE fields. If both ranges and values are specified for the same parameter, the value settings take precedence. __________________________________________________________________ Numerical options MATERNFIT has options for specifying numerical details of the fit and the optimization algorithm. MATERNFIT can use one of two different optimization algorithms. MATERNFIT(...,'bnd',...), the default, uses FMINSEARCHBND by J. D'Errico included with JLAB in accordance with its license terms. This in turn calls Matlab's FMINSEARCH using Nelder-Mead. MATERNFIT(...,'con',...) alternately uses FMINCON with the default interior-point algorithm. This requires Matlab's Optimization Toolbox to be installed. This is mainly for testing. MATERNFIT(...,'nlo',...) uses the Nelder-Mead algorithm from the NLopt toolbox at https://nlopt.readthedocs.io. This requires NLopt to be installed. Again, this is mainly for testing at the moment. In tests, FMINCON is generally faster, and most of the fits agree closely with those using FMINSEARCHBND. However, occasionally FMINCON produces fits that are significantly worse than those obtained from FMINSEARCHBND, which is why the latter is preferred by default. MATERNFIT can employ two different versions of the fit. MATERNFIT(...,'difference',...) estimates the Matern parameters by fitting the first difference of the time series to the first difference of a Matern process. This amounts to a form of pre- whitening, and is the default behavior when a taper is not input. MATERNFIT(...,'raw',...) fits the time series directly to a Matern. This is the default behavior when a taper *is* input. These choices are reflected in the output fields PARAMS.ALG and PARAMS.VER, respectively. __________________________________________________________________ Tapering The default behavior of fitting the first difference of the spectrum is usually sufficient to account for spectral blurring. For very steep spectra or those with a very large dynamic range, this is no longer the case, because leakage from high-energy portions of the spectra will obscure the structure of low-energy portions. To addess this, MATERNFIT can optionally perform the fit by fitting to the tapered and aliased spectrum, correctly accounting for the influence of tapering. This is accomplished with MATERNFIT(...,'taper',PSI,...) where PSI is a data taper of the same length as Z. If Z is a cell array, then PSI is a cell array of data tapers having the same length as the components of Z. PSI is typically computed by SLEPTAP. When a taper is input, the default behavior is *not* to difference the time series, corresponding to the 'raw' option. If both a taper and the 'difference' flag are both input, then the taper length must be one less than the data length. __________________________________________________________________ Error While the maximum likelikehood method is not about finding the best fit in a least squares sense, a measure of the mean squared error provides a useful measure of evaluating the degree of misfit. The error ERR returned by MATERNFIT is the squared difference between the *natural log* of the periodogram and that of the fit spectrum, summed over all frequencies used in the fit, and divided by the sum of the squared natural log of the periodogram over the same frequencies. The error is computed over the inertial side, anti-inertial side, or both sides, depending on which frequency range is used for the fit. When the 'difference' behavior is employed, as is the default, ERR will reflect the error between the periodogram of the first difference of the time series and the spectrum of the first difference of the fit. __________________________________________________________________ Parallelization With Matlab's Parallel Computing toolbox installed, when Z is a cell array or a matrix, MATERNFIT(...,'parallel') will loop over the elements of Z using a parfor loop to speed things up. This choice is reflected in the output field PARAMS.CORES, which will take on the values 'series' (the default) or 'parallel'. __________________________________________________________________ 'maternfit --t' runs a test. Usage: fit=maternfit(dt,z,fo); fit=maternfit(dt,z,fo,a,b); __________________________________________________________________ This is part of JLAB --- type 'help jlab' for more information (C) 2014--2021 J.M. Lilly and A.M. Sykulski --- type 'help jlab_license' for details