CharFunc.cf2DistFFT

SecTheory.cf2DistFFT.py

Translated from MATLAB code by Viktor Witkovsky at witkovsky/CharFunTool

Copyright (c) 2022, SAXS Team, KEK-PF

%cf2DistFFT(cf,x,prob,options) Evaluates the CDF/PDF/QF (quantiles) from % the characteristic function CF of a (continuous) distribution F by using % the Fast Fourier Transform (FFT) algorithm. % % The algorithm cf2DistFFT evaluates the approximate values CDF(x), PDF(x), % and/or the quantiles QF(prob) for given x and prob, by interpolation from % the PDF-estimate computed by the numerical inversion of the given % characteristic function CF by using the FFT algorithm. % % SYNTAX: % result = cf2DistFFT(cf,x,prob,options) % [result,cdf,pdf,qf] = cf2DistFFT(cf,x,prob,options) % % INPUT: % cf - function handle for the characteristic function CF, % x - vector of values from domain of the distribution F, if x = [], % cf2DistFFT automatically selects vector x from the domain. % prob - vector of values from [0,1] for which the quantiles will be % estimated, if prob = [], cf2DistFFT automatically selects % vector prob = [0.9,0.95,0.975,0.99,0.995,0.999]. % options - structure with the following default parameters: % options.isCompound = false % treat the compound distributions % % of the RV Y = X_1 + … + X_N, % % where N is discrete RV and X>=0 % % are iid RVs from nonnegative % % continuous distribution. % options.isInterp = false % create and use the interpolant % functions for PDF/CDF/QF/RND % options.N = 2^10 % N points used by FFT % options.xMin = -Inf % set the lower limit of X % options.xMax = Inf % set the lower limit of X % options.xMean = [] % set the MEAN value of X % options.xStd = [] % set the STD value of X % options.dt = [] % set grid step dt = 2*pi/xRange % options.T = [] % set upper limit of (0,T), T = N*dt % options.SixSigmaRule = 6 % set the rule for computing domain % options.tolDiff = 1e-4 % tol for numerical differentiation % options.isPlot = true % plot the graphs of PDF/CDF % options.DIST - structure with information for future evaluations. % options.DIST is created automatically after first call: % options.DIST.xMin = xMin % the lower limit of X % options.DIST.xMax = xMax % the upper limit of X % options.DIST.xMean = xMean % the MEAN value of X, % options.DIST.cft = cft % CF evaluated at t_j : cf(t_j). % % OUTPUTS: % result - structure with the following results values: % result.x = x; % result.cdf = cdf; % result.pdf = pdf; % result.prob = prob; % result.qf = qf; % result.xFTT = xFFT; % result.pdfFFT = pdfFFT; % result.cdfFFT = cdfFFT; % result.SixSigmaRule = options.SixSigmaRule; % result.N = N; % result.dt = dt; % result.T = t(end); % result.PrecisionCrit = PrecisionCrit; % result.myPrecisionCrit = options.crit; % result.isPrecisionOK = isPrecisionOK; % result.xMean = xMean; % result.xStd = xStd; % result.xMin = xMin; % result.xMax = xMax; % result.cf = cf; % result.options = options; % result.tictoc = toc; % % EXAMPLE 1: % % DISTRIBUTION OF A LINEAR COMBINATION OF THE INDEPENDENT RVs % % (Normal, Student’s t, Rectangular, Triangular & Arcsine distribution) % % Y = X_{N} + X_{t} + 5*X_{R} + X_{T} + 10*X_{U} % % CFs: Normal, Student’s t, Rectangular, Triangular, and Arcsine % cf_N = @(t) exp(-t.^2/2); % cf_t = @(t,nu) min(1,besselk(nu/2, abs(t).*sqrt(nu),1) .* … % exp(-abs(t).*sqrt(nu)) .* (sqrt(nu).*abs(t)).^(nu/2) / … % 2^(nu/2-1)/gamma(nu/2)); % cf_R = @(t) min(1,sin(t)./t); % cf_T = @(t) min(1,(2-2*cos(t))./t.^2); % cf_U = @(t) besselj(0,t); % % Characteristic function of the linear combination Y % c = [1 1 5 1 10]; nu = 1; % cf_Y = @(t) cf_N(c(1)*t) .* cf_t(c(2)*t,nu) .* cf_R(c(3)*t) .* … % cf_T(c(4)*t) .* cf_U(c(5)*t); % clear options % options.N = 2^10; % options.xMin = -50; % options.xMax = 50; % result = cf2DistFFT(cf_Y,[],[],options); % title(‘CDF of Y = X_{N} + X_{t} + 5*X_{R} + X_{T} + 10*X_{U}’) % % EXAMPLE 2: % % DISTRIBUTION OF A LINEAR COMBINATION OF THE INDEPENDENT CHI2 RVs % % (Chi-squared RVs with 1 and 10 degrees of freedom) % % Y = 10*X_{Chi2_1} + X_{Chi2_10} % % Characteristic functions of X_{Chi2_1} and X_{Chi2_10} % % % df1 = 1; % df2 = 10; % cfChi2_1 = @(t) (1-2i*t).^(-df1/2); % cfChi2_10 = @(t) (1-2i*t).^(-df2/2); % cf_Y = @(t) cfChi2_1(10*t) .* cfChi2_10(t); % clear options % options.xMin = 0; % result = cf2DistFFT(cf_Y,[],[],options); % title(‘CDF of Y = 10*X_{chi^2_{1}} + X_{chi^2_{10}}’) % % EXAMPLE3 (PDF/CDF of the compound Poisson-Exponential distribution) % lambda1 = 10; % lambda2 = 5; % cfX = @(t) cfX_Exponential(t,lambda2); % cf = @(t) cfN_Poisson(t,lambda1,cfX); % x = linspace(0,8,101); % prob = [0.9 0.95 0.99]; % clear options % options.isCompound = 1; % result = cf2DistFFT(cf,x,prob,options) % % REMARKS: % The outputs of the algorithm cf2DistFFT are approximate values! The % precission of the presented results depends on several different % factors: % - application of the FFT algorithm for numerical inversion of the CF % - selected number of points used by the FFT algorithm (by default % options.N = 2^10), % - estimated/calculated domain [A,B] of the distribution F, used with the % FFT algorithm. Optimally, [A,B] covers large part of the % distribution domain, say more than 99%. However, the default % automatic procedure for selection of the domain [A,B] may fail. It % is based on the ‘SixSigmaRule’: A = MEAN - SixSigmaRule * STD, and % B = MEAN + SixSigmaRule * STD. Alternatively, change the % options.SixSigmaRule to different value, say 12, or use the % options.xMin and options.xMax to set manually the values of A and B. % % REFERENCES: % [1] WITKOVSKY, V.: On the exact computation of the density and of % the quantiles of linear combinations of t and F random % variables. Journal of Statistical Planning and Inference 94 % (2001), 1-13. % [2] WITKOVSKY, V.: Matlab algorithm TDIST: The distribution of a % linear combination of Student’s t random variables. In COMPSTAT % 2004 Symposium (2004), J. Antoch, Ed., Physica-Verlag/Springer % 2004, Heidelberg, Germany, pp. 1995-2002. % [3] WITKOVSKY, V.: WIMMER,G., DUBY, T. Logarithmic Lambert W x F % random variables for the family of chi-squared distributions % and their applications. Statistics & Probability Letters 96 % (2015), 223-231. % [4] WITKOVSKY V. (2016). Numerical inversion of a characteristic % function: An alternative tool to form the probability distribution of % output quantity in linear measurement models. Acta IMEKO, 5(3), 32-44. % % SEE ALSO: cf2Dist, cf2DistGP, cf2DistGPT, cf2DistGPA, cf2DistFFT, % cf2DistBV, cf2CDF, cf2PDF, cf2QF

% (c) Viktor Witkovsky (witkovsky@gmail.com) % Ver.: 01-Sep-2020 13:25:21 % % Revision history: % Ver.: 15-Nov-2016 13:36:26

class DistInfo

Bases: object

class Options

Bases: object

cf2DistFFT(cf, x=None, prob=None, options=<molass_legacy.CharFunc.cf2DistFFT.Options object>)