'''estimate distribution parameters by various methods method of moments or matching quantiles, and Maximum Likelihood estimation based on binned data and Maximum Product-of-Spacings Warning: I'm still finding cut-and-paste and refactoring errors, e.g. hardcoded variables from outer scope in functions some results do not seem to make sense for Pareto case, looks better now after correcting some name errors initially loosely based on a paper and blog for quantile matching by John D. Cook formula for gamma quantile (ppf) matching by him (from paper) http://www.codeproject.com/KB/recipes/ParameterPercentile.aspx http://www.johndcook.com/blog/2010/01/31/parameters-from-percentiles/ this is what I actually used (in parts): http://www.bepress.com/mdandersonbiostat/paper55/ quantile based estimator ^^^^^^^^^^^^^^^^^^^^^^^^ only special cases for number or parameters so far Is there a literature for GMM estimation of distribution parameters? check found one: Wu/Perloff 2007 binned estimator ^^^^^^^^^^^^^^^^ * I added this also * use it for chisquare tests with estimation distribution parameters * move this to distribution_extras (next to gof tests powerdiscrepancy and continuous) or add to distribution_patch example: t-distribution * works with quantiles if they contain tail quantiles * results with momentcondquant do not look as good as mle estimate TODOs * rearange and make sure I do not use module globals (as I did initially) DONE make two version exactly identified method of moments with fsolve and GMM (?) version with fmin and maybe the special cases of JD Cook update: maybe exact (MM) version is not so interesting compared to GMM * add semifrozen version of moment and quantile based estimators, e.g. for beta (both loc and scale fixed), or gamma (loc fixed) * add beta example to the semifrozen MLE, fitfr, code -> added method of moment estimator to _fitstart for beta * start a list of how well different estimators, especially current mle work for the different distributions * need general GMM code (with optimal weights ?), looks like a good example for it * get example for binned data estimation, mailing list a while ago * any idea when these are better than mle ? * check language: I use quantile to mean the value of the random variable, not quantile between 0 and 1. * for GMM: move moment conditions to separate function, so that they can be used for further analysis, e.g. covariance matrix of parameter estimates * question: Are GMM properties different for matching quantiles with cdf or ppf? Estimate should be the same, but derivatives of moment conditions differ. * add maximum spacings estimator, Wikipedia, Per Brodtkorb -> basic version Done * add parameter estimation based on empirical characteristic function (Carrasco/Florens), especially for stable distribution * provide a model class based on estimating all distributions, and collect all distribution specific information References ---------- Ximing Wu, Jeffrey M. Perloff, GMM estimation of a maximum entropy distribution with interval data, Journal of Econometrics, Volume 138, Issue 2, 'Information and Entropy Econometrics' - A Volume in Honor of Arnold Zellner, June 2007, Pages 532-546, ISSN 0304-4076, DOI: 10.1016/j.jeconom.2006.05.008. http://www.sciencedirect.com/science/article/B6VC0-4K606TK-4/2/78bc07c6245546374490f777a6bdbbcc http://escholarship.org/uc/item/7jf5w1ht (working paper) Johnson, Kotz, Balakrishnan: Volume 2 Author : josef-pktd License : BSD created : 2010-04-20 changes: added Maximum Product-of-Spacings 2010-05-12 ''' import numpy as np from scipy import stats, optimize, special cache = {} #module global storage for temp results, not used # the next two use distfn from module scope - not anymore def gammamomentcond(distfn, params, mom2, quantile=None): '''estimate distribution parameters based method of moments (mean, variance) for distributions with 1 shape parameter and fixed loc=0. Returns ------- cond : function Notes ----- first test version, quantile argument not used ''' def cond(params): alpha, scale = params mom2s = distfn.stats(alpha, 0.,scale) #quantil return np.array(mom2)-mom2s return cond def gammamomentcond2(distfn, params, mom2, quantile=None): '''estimate distribution parameters based method of moments (mean, variance) for distributions with 1 shape parameter and fixed loc=0. Returns ------- difference : ndarray difference between theoretical and empirical moments Notes ----- first test version, quantile argument not used The only difference to previous function is return type. ''' alpha, scale = params mom2s = distfn.stats(alpha, 0.,scale) return np.array(mom2)-mom2s ######### fsolve does not move in small samples, fmin not very accurate def momentcondunbound(distfn, params, mom2, quantile=None): '''moment conditions for estimating distribution parameters using method of moments, uses mean, variance and one quantile for distributions with 1 shape parameter. Returns ------- difference : ndarray difference between theoretical and empirical moments and quantiles ''' shape, loc, scale = params mom2diff = np.array(distfn.stats(shape, loc,scale)) - mom2 if quantile is not None: pq, xq = quantile #ppfdiff = distfn.ppf(pq, alpha) cdfdiff = distfn.cdf(xq, shape, loc, scale) - pq return np.concatenate([mom2diff, cdfdiff[:1]]) return mom2diff ###### loc scale only def momentcondunboundls(distfn, params, mom2, quantile=None, shape=None): '''moment conditions for estimating loc and scale of a distribution with method of moments using either 2 quantiles or 2 moments (not both). Returns ------- difference : ndarray difference between theoretical and empirical moments or quantiles ''' loc, scale = params mom2diff = np.array(distfn.stats(shape, loc, scale)) - mom2 if quantile is not None: pq, xq = quantile #ppfdiff = distfn.ppf(pq, alpha) cdfdiff = distfn.cdf(xq, shape, loc, scale) - pq #return np.concatenate([mom2diff, cdfdiff[:1]]) return cdfdiff return mom2diff ######### try quantile GMM with identity weight matrix #(just a guess that's what it is def momentcondquant(distfn, params, mom2, quantile=None, shape=None): '''moment conditions for estimating distribution parameters by matching quantiles, defines as many moment conditions as quantiles. Returns ------- difference : ndarray difference between theoretical and empirical quantiles Notes ----- This can be used for method of moments or for generalized method of moments. ''' #this check looks redundant/unused know if len(params) == 2: loc, scale = params elif len(params) == 3: shape, loc, scale = params else: #raise NotImplementedError pass #see whether this might work, seems to work for beta with 2 shape args #mom2diff = np.array(distfn.stats(*params)) - mom2 #if not quantile is None: pq, xq = quantile #ppfdiff = distfn.ppf(pq, alpha) cdfdiff = distfn.cdf(xq, *params) - pq #return np.concatenate([mom2diff, cdfdiff[:1]]) return cdfdiff #return mom2diff def fitquantilesgmm(distfn, x, start=None, pquant=None, frozen=None): if pquant is None: pquant = np.array([0.01, 0.05,0.1,0.4,0.6,0.9,0.95,0.99]) if start is None: if hasattr(distfn, '_fitstart'): start = distfn._fitstart(x) else: start = [1]*distfn.numargs + [0.,1.] #TODO: vectorize this: xqs = [stats.scoreatpercentile(x, p) for p in pquant*100] mom2s = None parest = optimize.fmin(lambda params:np.sum( momentcondquant(distfn, params, mom2s,(pquant,xqs), shape=None)**2), start) return parest def fitbinned(distfn, freq, binedges, start, fixed=None): '''estimate parameters of distribution function for binned data using MLE Parameters ---------- distfn : distribution instance needs to have cdf method, as in scipy.stats freq : ndarray, 1d frequency count, e.g. obtained by histogram binedges : ndarray, 1d binedges including lower and upper bound start : tuple or array_like ? starting values, needs to have correct length Returns ------- paramest : ndarray estimated parameters Notes ----- todo: add fixed parameter option added factorial ''' if fixed is not None: raise NotImplementedError nobs = np.sum(freq) lnnobsfact = special.gammaln(nobs+1) def nloglike(params): '''negative loglikelihood function of binned data corresponds to multinomial ''' prob = np.diff(distfn.cdf(binedges, *params)) return -(lnnobsfact + np.sum(freq*np.log(prob)- special.gammaln(freq+1))) return optimize.fmin(nloglike, start) def fitbinnedgmm(distfn, freq, binedges, start, fixed=None, weightsoptimal=True): '''estimate parameters of distribution function for binned data using GMM Parameters ---------- distfn : distribution instance needs to have cdf method, as in scipy.stats freq : ndarray, 1d frequency count, e.g. obtained by histogram binedges : ndarray, 1d binedges including lower and upper bound start : tuple or array_like ? starting values, needs to have correct length fixed : None not used yet weightsoptimal : bool If true, then the optimal weighting matrix for GMM is used. If false, then the identity matrix is used Returns ------- paramest : ndarray estimated parameters Notes ----- todo: add fixed parameter option added factorial ''' if fixed is not None: raise NotImplementedError nobs = np.sum(freq) if weightsoptimal: weights = freq/float(nobs) else: weights = np.ones(len(freq)) freqnormed = freq/float(nobs) # skip turning weights into matrix diag(freq/float(nobs)) def gmmobjective(params): '''negative loglikelihood function of binned data corresponds to multinomial ''' prob = np.diff(distfn.cdf(binedges, *params)) momcond = freqnormed - prob return np.dot(momcond*weights, momcond) return optimize.fmin(gmmobjective, start) #Addition from try_maxproductspacings: """Estimating Parameters of Log-Normal Distribution with Maximum Likelihood and Maximum Product-of-Spacings MPS definiton from JKB page 233 Created on Tue May 11 13:52:50 2010 Author: josef-pktd License: BSD """ def hess_ndt(fun, pars, args, options): import numdifftools as ndt if not ('stepMax' in options or 'stepFix' in options): options['stepMax'] = 1e-5 f = lambda params: fun(params, *args) h = ndt.Hessian(f, **options) return h(pars), h def logmps(params, xsorted, dist): '''calculate negative log of Product-of-Spacings Parameters ---------- params : array_like, tuple ? parameters of the distribution funciton xsorted : array_like data that is already sorted dist : instance of a distribution class only cdf method is used Returns ------- mps : float negative log of Product-of-Spacings Notes ----- MPS definiton from JKB page 233 ''' xcdf = np.r_[0., dist.cdf(xsorted, *params), 1.] D = np.diff(xcdf) return -np.log(D).mean() def getstartparams(dist, data): '''get starting values for estimation of distribution parameters Parameters ---------- dist : distribution instance the distribution instance needs to have either a method fitstart or an attribute numargs data : ndarray data for which preliminary estimator or starting value for parameter estimation is desired Returns ------- x0 : ndarray preliminary estimate or starting value for the parameters of the distribution given the data, including loc and scale ''' if hasattr(dist, 'fitstart'): #x0 = getattr(dist, 'fitstart')(data) x0 = dist.fitstart(data) else: if np.isfinite(dist.a): x0 = np.r_[[1.]*dist.numargs, (data.min()-1), 1.] else: x0 = np.r_[[1.]*dist.numargs, (data.mean()-1), 1.] return x0 def fit_mps(dist, data, x0=None): '''Estimate distribution parameters with Maximum Product-of-Spacings Parameters ---------- params : array_like, tuple ? parameters of the distribution funciton xsorted : array_like data that is already sorted dist : instance of a distribution class only cdf method is used Returns ------- x : ndarray estimates for the parameters of the distribution given the data, including loc and scale ''' xsorted = np.sort(data) if x0 is None: x0 = getstartparams(dist, xsorted) args = (xsorted, dist) print(x0) #print(args) return optimize.fmin(logmps, x0, args=args) if __name__ == '__main__': #Example: gamma - distribution #----------------------------- print('\n\nExample: gamma Distribution') print( '---------------------------') alpha = 2 xq = [0.5, 4] pq = [0.1, 0.9] print(stats.gamma.ppf(pq, alpha)) xq = stats.gamma.ppf(pq, alpha) print(np.diff(stats.gamma.ppf(pq, np.linspace(0.01,4,10)[:,None])*xq[::-1])) #optimize.bisect(lambda alpha: np.diff((stats.gamma.ppf(pq, alpha)*xq[::-1]))) print(optimize.fsolve(lambda alpha: np.diff(stats.gamma.ppf(pq, alpha)*xq[::-1]), 3.)) distfn = stats.gamma mcond = gammamomentcond(distfn, [5.,10], mom2=stats.gamma.stats(alpha, 0.,1.), quantile=None) print(optimize.fsolve(mcond, [1.,2.])) mom2 = stats.gamma.stats(alpha, 0.,1.) print(optimize.fsolve(lambda params:gammamomentcond2(distfn, params, mom2), [1.,2.])) grvs = stats.gamma.rvs(alpha, 0.,2., size=1000) mom2 = np.array([grvs.mean(), grvs.var()]) alphaestq = optimize.fsolve(lambda params:gammamomentcond2(distfn, params, mom2), [1.,3.]) print(alphaestq) print('scale = ', xq/stats.gamma.ppf(pq, alphaestq)) #Example beta - distribution #--------------------------- #Warning: this example had cut-and-paste errors print('\n\nExample: beta Distribution') print( '--------------------------') #monkey patching : ## if hasattr(stats.beta, '_fitstart'): ## del stats.beta._fitstart #bug in _fitstart #raises AttributeError: _fitstart #stats.distributions.beta_gen._fitstart = lambda self, data : np.array([1,1,0,1]) #_fitstart seems to require a tuple stats.distributions.beta_gen._fitstart = lambda self, data : (5,5,0,1) pq = np.array([0.01, 0.05,0.1,0.4,0.6,0.9,0.95,0.99]) #rvsb = stats.beta.rvs(0.5,0.15,size=200) rvsb = stats.beta.rvs(10,15,size=2000) print('true params', 10, 15, 0, 1) print(stats.beta.fit(rvsb)) xqsb = [stats.scoreatpercentile(rvsb, p) for p in pq*100] mom2s = np.array([rvsb.mean(), rvsb.var()]) betaparest_gmmquantile = optimize.fmin(lambda params:np.sum(momentcondquant(stats.beta, params, mom2s,(pq,xqsb), shape=None)**2), [10,10, 0., 1.], maxiter=2000) print('betaparest_gmmquantile', betaparest_gmmquantile) #result sensitive to initial condition #Example t - distribution #------------------------ print('\n\nExample: t Distribution') print( '-----------------------') nobs = 1000 distfn = stats.t pq = np.array([0.1,0.9]) paramsdgp = (5, 0, 1) trvs = distfn.rvs(5, 0, 1, size=nobs) xqs = [stats.scoreatpercentile(trvs, p) for p in pq*100] mom2th = distfn.stats(*paramsdgp) mom2s = np.array([trvs.mean(), trvs.var()]) tparest_gmm3quantilefsolve = optimize.fsolve(lambda params:momentcondunbound(distfn,params, mom2s,(pq,xqs)), [10,1.,2.]) print('tparest_gmm3quantilefsolve', tparest_gmm3quantilefsolve) tparest_gmm3quantile = optimize.fmin(lambda params:np.sum(momentcondunbound(distfn,params, mom2s,(pq,xqs))**2), [10,1.,2.]) print('tparest_gmm3quantile', tparest_gmm3quantile) print(distfn.fit(trvs)) ## ##distfn = stats.t ##pq = np.array([0.1,0.9]) ##paramsdgp = (5, 0, 1) ##trvs = distfn.rvs(5, 0, 1, size=nobs) ##xqs = [stats.scoreatpercentile(trvs, p) for p in pq*100] ##mom2th = distfn.stats(*paramsdgp) ##mom2s = np.array([trvs.mean(), trvs.var()]) print(optimize.fsolve(lambda params:momentcondunboundls(distfn, params, mom2s,shape=5), [1.,2.])) print(optimize.fmin(lambda params:np.sum(momentcondunboundls(distfn, params, mom2s,shape=5)**2), [1.,2.])) print(distfn.fit(trvs)) #loc, scale, based on quantiles print(optimize.fsolve(lambda params:momentcondunboundls(distfn, params, mom2s,(pq,xqs),shape=5), [1.,2.])) ## pq = np.array([0.01, 0.05,0.1,0.4,0.6,0.9,0.95,0.99]) #paramsdgp = (5, 0, 1) xqs = [stats.scoreatpercentile(trvs, p) for p in pq*100] tparest_gmmquantile = optimize.fmin(lambda params:np.sum(momentcondquant(distfn, params, mom2s,(pq,xqs), shape=None)**2), [10, 1.,2.]) print('tparest_gmmquantile', tparest_gmmquantile) tparest_gmmquantile2 = fitquantilesgmm(distfn, trvs, start=[10, 1.,2.], pquant=None, frozen=None) print('tparest_gmmquantile2', tparest_gmmquantile2) ## #use trvs from before bt = stats.t.ppf(np.linspace(0,1,21),5) ft,bt = np.histogram(trvs,bins=bt) print('fitbinned t-distribution') tparest_mlebinew = fitbinned(stats.t, ft, bt, [10, 0, 1]) tparest_gmmbinewidentity = fitbinnedgmm(stats.t, ft, bt, [10, 0, 1]) tparest_gmmbinewoptimal = fitbinnedgmm(stats.t, ft, bt, [10, 0, 1], weightsoptimal=False) print(paramsdgp) #Note: this can be used for chisquare test and then has correct asymptotic # distribution for a distribution with estimated parameters, find ref again #TODO combine into test with binning included, check rule for number of bins #bt2 = stats.t.ppf(np.linspace(trvs.,1,21),5) ft2,bt2 = np.histogram(trvs,bins=50) 'fitbinned t-distribution' tparest_mlebinel = fitbinned(stats.t, ft2, bt2, [10, 0, 1]) tparest_gmmbinelidentity = fitbinnedgmm(stats.t, ft2, bt2, [10, 0, 1]) tparest_gmmbineloptimal = fitbinnedgmm(stats.t, ft2, bt2, [10, 0, 1], weightsoptimal=False) tparest_mle = stats.t.fit(trvs) np.set_printoptions(precision=6) print('sample size', nobs) print('true (df, loc, scale) ', paramsdgp) print('parest_mle ', tparest_mle) print print('tparest_mlebinel ', tparest_mlebinel) print('tparest_gmmbinelidentity ', tparest_gmmbinelidentity) print('tparest_gmmbineloptimal ', tparest_gmmbineloptimal) print print('tparest_mlebinew ', tparest_mlebinew) print('tparest_gmmbinewidentity ', tparest_gmmbinewidentity) print('tparest_gmmbinewoptimal ', tparest_gmmbinewoptimal) print print('tparest_gmmquantileidentity', tparest_gmmquantile) print('tparest_gmm3quantilefsolve ', tparest_gmm3quantilefsolve) print('tparest_gmm3quantile ', tparest_gmm3quantile) ''' example results: standard error for df estimate looks large note: iI do not impose that df is an integer, (b/c not necessary) need Monte Carlo to check variance of estimators sample size 1000 true (df, loc, scale) (5, 0, 1) parest_mle [ 4.571405 -0.021493 1.028584] tparest_mlebinel [ 4.534069 -0.022605 1.02962 ] tparest_gmmbinelidentity [ 2.653056 0.012807 0.896958] tparest_gmmbineloptimal [ 2.437261 -0.020491 0.923308] tparest_mlebinew [ 2.999124 -0.0199 0.948811] tparest_gmmbinewidentity [ 2.900939 -0.020159 0.93481 ] tparest_gmmbinewoptimal [ 2.977764 -0.024925 0.946487] tparest_gmmquantileidentity [ 3.940797 -0.046469 1.002001] tparest_gmm3quantilefsolve [ 10. 1. 2.] tparest_gmm3quantile [ 6.376101 -0.029322 1.112403] ''' #Example with Maximum Product of Spacings Estimation #=================================================== #Example: Lognormal Distribution #------------------------------- #tough problem for MLE according to JKB #but not sure for which parameters print('\n\nExample: Lognormal Distribution') print( '-------------------------------') sh = np.exp(10) sh = 0.01 print(sh) x = stats.lognorm.rvs(sh,loc=100, scale=10,size=200) print(x.min()) print(stats.lognorm.fit(x, 1.,loc=x.min()-1,scale=1)) xsorted = np.sort(x) x0 = [1., x.min()-1, 1] args = (xsorted, stats.lognorm) print(optimize.fmin(logmps,x0,args=args)) #Example: Lomax, Pareto, Generalized Pareto Distributions #-------------------------------------------------------- #partially a follow-up to the discussion about numpy.random.pareto #Reference: JKB #example Maximum Product of Spacings Estimation # current results: # does not look very good yet sensitivity to starting values # Pareto and Generalized Pareto look like a tough estimation problemprint('\n\nExample: Lognormal Distribution' print('\n\nExample: Lomax, Pareto, Generalized Pareto Distributions') print( '--------------------------------------------------------') p2rvs = stats.genpareto.rvs(2, size=500) #Note: is Lomax without +1; and classical Pareto with +1 p2rvssorted = np.sort(p2rvs) argsp = (p2rvssorted, stats.pareto) x0p = [1., p2rvs.min()-5, 1] print(optimize.fmin(logmps,x0p,args=argsp)) print(stats.pareto.fit(p2rvs, 0.5, loc=-20, scale=0.5)) print('gpdparest_ mle', stats.genpareto.fit(p2rvs)) parsgpd = fit_mps(stats.genpareto, p2rvs) print('gpdparest_ mps', parsgpd) argsgpd = (p2rvssorted, stats.genpareto) options = dict(stepFix=1e-7) #hess_ndt(fun, pars, argsgdp, options) #the results for the following look strange, maybe refactoring error he, h = hess_ndt(logmps, parsgpd, argsgpd, options) print(np.linalg.eigh(he)[0]) f = lambda params: logmps(params, *argsgpd) print(f(parsgpd)) #add binned fp2, bp2 = np.histogram(p2rvs, bins=50) 'fitbinned t-distribution' gpdparest_mlebinel = fitbinned(stats.genpareto, fp2, bp2, x0p) gpdparest_gmmbinelidentity = fitbinnedgmm(stats.genpareto, fp2, bp2, x0p) print('gpdparest_mlebinel', gpdparest_mlebinel) print('gpdparest_gmmbinelidentity', gpdparest_gmmbinelidentity) gpdparest_gmmquantile2 = fitquantilesgmm( stats.genpareto, p2rvs, start=x0p, pquant=None, frozen=None) print('gpdparest_gmmquantile2', gpdparest_gmmquantile2) print(fitquantilesgmm(stats.genpareto, p2rvs, start=x0p, pquant=np.linspace(0.01,0.99,10), frozen=None)) fp2, bp2 = np.histogram( p2rvs, bins=stats.genpareto(2).ppf(np.linspace(0,0.99,10))) print('fitbinnedgmm equal weight bins') print(fitbinnedgmm(stats.genpareto, fp2, bp2, x0p))