''' A class for the distribution of a non-linear monotonic transformation of a continuous random variable simplest usage: example: create log-gamma distribution, i.e. y = log(x), where x is gamma distributed (also available in scipy.stats) loggammaexpg = Transf_gen(stats.gamma, np.log, np.exp) example: what is the distribution of the discount factor y=1/(1+x) where interest rate x is normally distributed with N(mux,stdx**2)')? (just to come up with a story that implies a nice transformation) invnormalg = Transf_gen(stats.norm, inversew, inversew_inv, decr=True, a=-np.inf) This class does not work well for distributions with difficult shapes, e.g. 1/x where x is standard normal, because of the singularity and jump at zero. Note: I'm working from my version of scipy.stats.distribution. But this script runs under scipy 0.6.0 (checked with numpy: 1.2.0rc2 and python 2.4) This is not yet thoroughly tested, polished or optimized ''' from scipy import integrate # for scipy 0.6.0 from scipy import stats, info from scipy.stats import distributions import numpy as np class Transf_gen(distributions.rv_continuous): #a class for non-linear monotonic transformation of a continuous random variable def __init__(self, kls, func, funcinv, *args, **kwargs): #print args #print kwargs self.kls = kls self.func = func self.funcinv = funcinv #explicit for self.__dict__.update(kwargs) if 'numargs' in kwargs: self.numargs = kwargs['numargs'] else: self.numargs = 1 if 'name' in kwargs: name = kwargs['name'] else: name = 'transfdist' if 'longname' in kwargs: longname = kwargs['longname'] else: longname = 'Non-linear transformed distribution' if 'extradoc' in kwargs: extradoc = kwargs['extradoc'] else: extradoc = None if 'a' in kwargs: a = kwargs['a'] else: a = 0 if 'decr' in kwargs: #defines whether it is a decreasing (True) # or increasing (False) monotone transformation self.decr = kwargs['decr'] else: self.decr = False super(Transf_gen,self).__init__(a=a, name = name, longname = longname, extradoc = extradoc) def _cdf(self,x,*args, **kwargs): #print args if not self.decr: return self.kls._cdf(self.funcinv(x),*args, **kwargs) else: return 1-self.kls.cdf(self.funcinv(x),*args, **kwargs) def _ppf(self, q, *args, **kwargs): if not self.decr: return self.func(self.kls._ppf(q,*args, **kwargs)) else: return self.func(self.kls._ppf(1-q,*args, **kwargs)) def inverse(x): return np.divide(1.0,x) mux, stdx = 0.05, 0.1 def inversew(x): return 1.0/(1+mux+x*stdx) def inversew_inv(x): return (1.0/x - 1.0 - mux)/stdx #.np.divide(1.0,x)-10 def identit(x): return x print '\nResults for invnormal with regularization y=1/(1+x), x is N(0, 0.1*0.1)' print '-----------------------------------------------------------------------' invnormalg = Transf_gen(stats.norm, inversew, inversew_inv, decr=True, a=-np.inf, name = 'discf', longname = 'normal-based discount factor', extradoc = '\ndistribution of discount factor y=1/(1+x)) with x N(0,0.1**2)') l,s = 0,1 print print invnormalg.__doc__ print print 'cdf for [0.95,1.0,1.1]:', invnormalg.cdf([0.95,1.0,1.1],loc=l, scale=s) print 'pdf for [0.95,1.0,1.1]:', invnormalg.pdf([0.95,1.0,1.1],loc=l, scale=s) print 'rvs:', invnormalg.rvs(loc=l, scale=s,size=5) print 'stats: ', invnormalg.stats(loc=l, scale=s) print 'stats kurtosis, skew: ', invnormalg.stats(moments='ks') print 'median:', invnormalg.ppf(0.5) rvs = invnormalg.rvs(loc=l, scale=s,size=10000) print 'sample stats: ', rvs.mean(), rvs.var() rvs = inversew(stats.norm.rvs(loc=l, scale=s,size=10000)) print 'std norm sample stats: ', rvs.mean(), rvs.var() rvs = inversew(stats.norm.rvs(size=100000))*s + l print 'transf. norm sample stats: ', rvs.mean(), rvs.var() print print 'Results for lognormal' print '---------------------' lognormalg = Transf_gen(stats.norm, np.exp, np.log, a=0, name = 'lnnorm', longname = 'Exp transformed normal', extradoc = '\ndistribution of y = exp(x), with x standard normal') print 'cdf for [2.0,2.5,3.0,3.5]: ',lognormalg.cdf([2.0,2.5,3.0,3.5]) print 'scipy cdf for [2.0,2.5,3.0,3.5]:', stats.lognorm.cdf([2.0,2.5,3.0,3.5],1) print 'pdf for [2.0,2.5,3.0,3.5]: ', lognormalg.pdf([2.0,2.5,3.0,3.5]) print 'scipy pdf for [2.0,2.5,3.0,3.5]:', stats.lognorm.pdf([2.0,2.5,3.0,3.5],1) print 'stats: ', lognormalg.stats() print 'scipy stats:', stats.lognorm.stats(1) print 'rvs:', lognormalg.rvs(size=5) print info(lognormalg) print '\nResults for idnormal' print '--------------------' idnormalg = Transf_gen(stats.norm, identit, identit, a=-np.inf, name = 'normal') print idnormalg.cdf(1,loc=100,scale=10) print stats.norm.cdf(1,loc=100,scale=10) print idnormalg.stats(loc=100,scale=10) print stats.norm.stats(loc=100,scale=10) print idnormalg.rvs(loc=100,scale=10,size=5) rvs = idnormalg.rvs(loc=100,scale=10,size=10000) print rvs.mean(), rvs.var() print '\nResults for expgamma' print '--------------------' loggammaexpg = Transf_gen(stats.gamma, np.log, np.exp) print loggammaexpg._cdf(1,10) print stats.loggamma.cdf(1,10) print loggammaexpg._cdf(2,15) print stats.loggamma.cdf(2,15) print 'cdf for [2.0,2.5,3.0,3.5]: ', loggammaexpg._cdf([2.0,2.5,3.0,3.5],10) print 'scipy cdf for [2.0,2.5,3.0,3.5]:', stats.loggamma.cdf([2.0,2.5,3.0,3.5],10) #print 'pdf for [2.0,2.5,3.0,3.5]: ', loggammaexpg.pdf([2.0,2.5,3.0,3.5],10) # not in scipy 0.6.0 print loggammaexpg._pdf(2.0,10) # ok in scipy 0.6.0 print 'scipy pdf for [2.0,2.5,3.0,3.5]:', stats.loggamma.pdf([2.0,2.5,3.0,3.5],10) print '\n\n the rest is not so good\n\n' print '\nResults for invnormal' print '--------------------' invnormalg2 = Transf_gen(stats.norm, inverse, inverse, decr=True, a=-np.inf, longname = 'inverse normal') print invnormalg2.cdf(1,loc=10) print stats.invnorm.cdf(1,1,loc=10) print invnormalg2.stats(loc=10) print stats.invnorm.stats(1,loc=10) print invnormalg2.rvs(loc=10,size=5) print '\nResults for invexpon' print '--------------------' invexpong = Transf_gen(stats.expon, inverse, inverse, a=0.0, name = 'Log transformed normal general') print invexpong.cdf(1) #print stats.invnorm.cdf(1,1) print invexpong.stats() #print stats.invnorm.stats(1) print invexpong.rvs(size=5) print '\nResults for inv gamma' print '---------------------' invgammag = Transf_gen(stats.gamma, inverse, inverse, a=0.0, name = 'Log transformed normal general') print invgammag._cdf(1,5) # .cdf #not in scipy 0.6.0 print stats.invgamma.cdf(1,1) #print invgammag.stats(1) #not in scipy 0.6.0 print stats.invgamma.stats(1) #print invgammag.rvs(1,size=5) #not in scipy 0.6.0 print invgammag._rvs(1) print '\nResults for loglaplace' print '----------------------' #no idea what the relationship is loglaplaceg = Transf_gen(stats.laplace, np.log, np.exp) print loglaplaceg._cdf(2) print stats.loglaplace.cdf(2,10) loglaplaceexpg = Transf_gen(stats.laplace, np.exp, np.log) print loglaplaceexpg._cdf(2) #this are the results, that I get: results = \ ''' Results for invnormal with regularization y=1/(1+x), x is N(0, 0.1*0.1) ----------------------------------------------------------------------- normal-based discount factor continuous random variable. Continuous random variables are defined from a standard form chosen for simplicity of representation. The standard form may require some shape parameters to complete its specification. The distributions also take optional location and scale parameters using loc= and scale= keywords (defaults: loc=0, scale=1) These shape, scale, and location parameters can be passed to any of the methods of the RV object such as the following: discf.rvs(loc=0,scale=1) - random variates discf.pdf(x,loc=0,scale=1) - probability density function discf.cdf(x,loc=0,scale=1) - cumulative density function discf.sf(x,loc=0,scale=1) - survival function (1-cdf --- sometimes more accurate) discf.ppf(q,loc=0,scale=1) - percent point function (inverse of cdf --- percentiles) discf.isf(q,loc=0,scale=1) - inverse survival function (inverse of sf) discf.stats(loc=0,scale=1,moments='mv') - mean('m',axis=0), variance('v'), skew('s'), and/or kurtosis('k') discf.entropy(loc=0,scale=1) - (differential) entropy of the RV. Alternatively, the object may be called (as a function) to fix the shape, location, and scale parameters returning a "frozen" continuous RV object: myrv = discf(loc=0,scale=1) - frozen RV object with the same methods but holding the given shape, location, and scale fixed distribution of discount factor y=1/(1+x)) with x N(0,0.1**2) cdf for [0.95,1.0,1.1]: [ 0.48950273 0.69146246 0.92059586] pdf for [0.95,1.0,1.1]: [ 4.41888273 3.52065327 1.22171744] rvs: [ 0.83915166 1.05916973 0.92895054 0.97653308 0.88997135] stats: (array(0.9612657841613822), array(0.0088754849141799985)) stats kurtosis, skew: (array(0.61482268025485831), array(0.78380821248995103)) median: 0.952380952381 sample stats: 0.962079869848 0.00884562090518 std norm sample stats: 0.961089776974 0.00874620747261 transf. norm sample stats: 0.961224369048 0.00883332533467 Results for lognormal --------------------- cdf for [2.0,2.5,3.0,3.5]: [ 0.7558914 0.82024279 0.86403139 0.89485401] scipy cdf for [2.0,2.5,3.0,3.5]: [ 0.7558914 0.82024279 0.86403139 0.89485401] pdf for [2.0,2.5,3.0,3.5]: [ 0.15687402 0.10487107 0.07272826 0.05200533] scipy pdf for [2.0,2.5,3.0,3.5]: [ 0.15687402 0.10487107 0.07272826 0.05200533] stats: (array(1.6487212707089971), array(4.6707742108633177)) scipy stats: (array(1.6487212707001282), array(4.670774270471604)) rvs: [ 0.45054343 0.17302502 0.15131355 0.74181241 0.30077315] Exp transformed normal continuous random variable. Continuous random variables are defined from a standard form chosen for simplicity of representation. The standard form may require some shape parameters to complete its specification. The distributions also take optional location and scale parameters using loc= and scale= keywords (defaults: loc=0, scale=1) These shape, scale, and location parameters can be passed to any of the methods of the RV object such as the following: lnnorm.rvs(loc=0,scale=1) - random variates lnnorm.pdf(x,loc=0,scale=1) - probability density function lnnorm.cdf(x,loc=0,scale=1) - cumulative density function lnnorm.sf(x,loc=0,scale=1) - survival function (1-cdf --- sometimes more accurate) lnnorm.ppf(q,loc=0,scale=1) - percent point function (inverse of cdf --- percentiles) lnnorm.isf(q,loc=0,scale=1) - inverse survival function (inverse of sf) lnnorm.stats(loc=0,scale=1,moments='mv') - mean('m',axis=0), variance('v'), skew('s'), and/or kurtosis('k') lnnorm.entropy(loc=0,scale=1) - (differential) entropy of the RV. Alternatively, the object may be called (as a function) to fix the shape, location, and scale parameters returning a "frozen" continuous RV object: myrv = lnnorm(loc=0,scale=1) - frozen RV object with the same methods but holding the given shape, location, and scale fixed distribution of y = exp(x), with x standard normal None Results for idnormal -------------------- 2.08137521949e-023 2.08137521949e-023 (array(100.0), array(99.999999999476046)) (array(100.0), array(100.0)) [ 108.95135206 105.97192526 96.56950463 103.55316046 95.69133148] 99.9529496989 102.460238876 Results for expgamma -------------------- 0.00052773949978 0.00052773949978 0.00906529986325 0.00906529986325 cdf for [2.0,2.5,3.0,3.5]: [ 0.21103986 0.77318778 0.99524757 0.99999926] scipy cdf for [2.0,2.5,3.0,3.5]: [ 0.21103986 0.77318778 0.99524757 0.99999926] pdf for [2.0,2.5,3.0,3.5]: [ 8.26228773e-01 1.01580211e+00 5.57228823e-02 1.81420198e-05] scipy pdf for [2.0,2.5,3.0,3.5]: [ 8.26228773e-01 1.01580211e+00 5.57228823e-02 1.81420259e-05] the rest is not so good '''