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 # cython: profile=True # Time-stamp: <2012-08-01 18:06:19 Tao Liu> """Module Description Copyright (c) 2008,2009,2010,2011 Hyunjin Shin, Tao Liu This code is free software; you can redistribute it and/or modify it under the terms of the BSD License (see the file COPYING included with the distribution). @status: experimental @version: $Revision$ @author: Hyunjin Gene Shin, Tao Liu @contact: taoliu@jimmy.harvard.edu """ # ------------------------------------ # python modules # ------------------------------------ from libc.math cimport exp,log,log10, M_LN10 #,fabs,log1p from math import fabs from math import log1p #as py_log1p from math import sqrt import numpy as np cimport numpy as np from cpython cimport bool # ------------------------------------ # constants # ------------------------------------ cdef int LSTEP = 200 cdef double EXPTHRES = exp(LSTEP) cdef double EXPSTEP = exp(-1*LSTEP) # ------------------------------------ # Normal distribution functions # ------------------------------------ # x is the value, u is the mean, v is the variance cpdef pnorm(int x, int u, int v): """The probability of X=x when X=Norm(u,v) """ return 1.0/sqrt(2.0 * 3.141592653589793 * v) * exp(-(x-u)**2 / (2.0 * v)) # ------------------------------------ # Misc functions # ------------------------------------ cpdef factorial ( unsigned int n ): """Calculate N!. """ cdef double fact = 1 cdef unsigned long i if n < 0: return 0 for i in xrange( 2,n+1 ): fact = fact * i return fact cpdef double poisson_cdf ( unsigned int n, double lam, bool lower=False, bool log10=False ): """Poisson CDF evaluater. This is a more stable CDF function. It can tolerate large lambda value. While the lambda is larger than 700, the function will be a little slower. Parameters: n : your observation lam : lambda of poisson distribution lower : if lower is False, calculate the upper tail CDF, otherwise, to calculate lower tail; Default is False. log10 : if log10 is True, calculation will be in log space. Default is False. """ assert lam > 0.0, "Lambda must > 0, however we got %d" % lam if log10: if lower: # lower tail return log10_poisson_cdf_P_large_lambda(n, lam) else: # upper tail return log10_poisson_cdf_Q_large_lambda(n, lam) if lower: if lam > 700: return __poisson_cdf_large_lambda (n, lam) else: return __poisson_cdf(n,lam) else: # upper tail if lam > 700: return __poisson_cdf_Q_large_lambda (n, lam) else: return __poisson_cdf_Q(n,lam) cdef inline double __poisson_cdf ( unsigned int k, double a ): """Poisson CDF For small lambda. If a > 745, this will return incorrect result. Parameters: k : observation a : lambda """ if k < 0: return 0.0 # special cases cdef double nextcdf = exp( -1 * a ) cdef double cdf = nextcdf cdef unsigned int i cdef double lastcdf for i in xrange(1,k+1): lastcdf = nextcdf nextcdf = lastcdf * a / i cdf = cdf + nextcdf if cdf > 1.0: return 1.0 else: return cdf cdef inline double __poisson_cdf_large_lambda ( unsigned int k, double a ): """Slower poisson cdf for large lambda ( > 700 ) Parameters: k : observation a : lambda """ assert a > 700 if k < 0: return 0.0 # special cases cdef int num_parts = int(a/LSTEP) cdef double lastexp = exp(-1 * (a % LSTEP) ) cdef double nextcdf = EXPSTEP num_parts -= 1 cdef double cdf = nextcdf cdef unsigned int i cdef double lastcdf for i in xrange(1,k+1): lastcdf = nextcdf nextcdf = lastcdf * a / i cdf = cdf + nextcdf if nextcdf > EXPTHRES or cdf > EXPTHRES: if num_parts>=1: cdf *= EXPSTEP nextcdf *= EXPSTEP num_parts -= 1 else: cdf *= lastexp lastexp = 1 for i in xrange(num_parts): cdf *= EXPSTEP cdf *= lastexp return cdf cdef inline double __poisson_cdf_Q ( unsigned int k, double a ): """internal Poisson CDF evaluater for upper tail with small lambda. Parameters: k : observation a : lambda """ cdef unsigned int i if k < 0: return 1.0 # special cases cdef double nextcdf nextcdf = exp( -1 * a ) cdef double lastcdf for i in xrange(1,k+1): lastcdf = nextcdf nextcdf = lastcdf * a / i cdef double cdf = 0.0 i = k+1 while nextcdf >0.0: lastcdf = nextcdf nextcdf = lastcdf * a / i cdf += nextcdf i+=1 return cdf cdef inline double __poisson_cdf_Q_large_lambda ( unsigned int k, double a ): """Slower internal Poisson CDF evaluater for upper tail with large lambda. Parameters: k : observation a : lambda """ assert a > 700 if k < 0: return 1.0 # special cases cdef unsigned int num_parts = int(a/LSTEP) cdef double lastexp = exp(-1 * (a % LSTEP) ) cdef double nextcdf = EXPSTEP cdef unsigned int i cdef double lastcdf num_parts -= 1 for i in xrange(1,k+1): lastcdf = nextcdf nextcdf = lastcdf * a / i if nextcdf > EXPTHRES: if num_parts>=1: nextcdf *= EXPSTEP num_parts -= 1 else: # simply raise an error raise Exception("Unexpected error") #cdf *= lastexp #lastexp = 1 cdef double cdf = 0.0 i = k+1 while nextcdf >0.0: lastcdf = nextcdf nextcdf = lastcdf * a / i cdf += nextcdf i+=1 if nextcdf > EXPTHRES or cdf > EXPTHRES: if num_parts>=1: cdf *= EXPSTEP nextcdf *= EXPSTEP num_parts -= 1 else: cdf *= lastexp lastexp = 1 for i in xrange(num_parts): cdf *= EXPSTEP cdf *= lastexp return cdf cdef inline double log10_poisson_cdf_P_large_lambda ( unsigned int k, double lbd ): """Slower Poisson CDF evaluater for lower tail which allow calculation in log space. Better for the pvalue < 10^-310. Parameters: k : observation lbd : lambda ret = -lambda + \ln( \sum_{i=k+1}^{\inf} {lambda^i/i!} = -lambda + \ln( sum{ exp{ln(F)} } ), where F=lambda^m/m! \ln{F(m)} = m*ln{lambda} - \sum_{x=1}^{m}\ln(x) Calculate \ln( sum{exp{N} ) by logspace_add function Return the log10(pvalue) """ cdef double residue = 0 cdef double logx = 0 cdef double ln_lbd = log(lbd) # first residue cdef int m = k cdef double sum_ln_m = 0 cdef int i = 0 for i in range(1,m+1): sum_ln_m += log(i) logx = m*ln_lbd - sum_ln_m residue = logx while m > 1: m -= 1 logy = logx-ln_lbd+log(m) pre_residue = residue residue = logspace_add(pre_residue,logy) if fabs(pre_residue-residue) < 1e-10: break logx = logy return round((residue-lbd)/M_LN10,5) cdef inline double log10_poisson_cdf_Q_large_lambda ( unsigned int k, double lbd ): """Slower Poisson CDF evaluater for upper tail which allow calculation in log space. Better for the pvalue < 10^-310. Parameters: k : observation lbd : lambda ret = -lambda + \ln( \sum_{i=k+1}^{\inf} {lambda^i/i!} = -lambda + \ln( sum{ exp{ln(F)} } ), where F=lambda^m/m! \ln{F(m)} = m*ln{lambda} - \sum_{x=1}^{m}\ln(x) Calculate \ln( sum{exp{N} ) by logspace_add function Return the log10(pvalue) """ cdef double residue = 0 cdef double logx = 0 cdef double ln_lbd = log(lbd) # first residue cdef int m = k+1 cdef double sum_ln_m = 0 cdef int i = 0 for i in range(1,m+1): sum_ln_m += log(i) logx = m*ln_lbd - sum_ln_m residue = logx while True: m += 1 logy = logx+ln_lbd-log(m) pre_residue = residue residue = logspace_add(pre_residue,logy) if fabs(pre_residue-residue) < 1e-5: break logx = logy return round((residue-lbd)/log(10),5) cdef inline double logspace_add ( double logx, double logy ): return max (logx, logy) + log1p (exp (-fabs (logx - logy))); cpdef poisson_cdf_inv ( double cdf, double lam, int maximum=1000 ): """inverse poisson distribution. cdf : the CDF lam : the lambda of poisson distribution note: maxmimum return value is 1000 and lambda must be smaller than 740. """ assert lam < 740 if cdf < 0 or cdf > 1: raise Exception ("CDF must >= 0 and <= 1") elif cdf == 0: return 0 cdef double sum2 = 0 cdef double newval = exp( -1*lam ) sum2 = newval cdef int i cdef double sumold cdef double lastval for i in xrange(1,maximum+1): sumold = sum2 lastval = newval newval = lastval * lam / i sum2 = sum2 + newval if sumold <= cdf and cdf <= sum2: return i return maximum cpdef poisson_cdf_Q_inv ( double cdf, double lam, int maximum=1000 ): """inverse poisson distribution. cdf : the CDF lam : the lambda of poisson distribution note: maxmimum return value is 1000 and lambda must be smaller than 740. """ assert lam < 740 if cdf < 0 or cdf > 1: raise Exception ("CDF must >= 0 and <= 1") elif cdf == 0: return 0 cdef double sum2 = 0 cdef double newval = exp( -1 * lam ) sum2 = newval cdef int i cdef double lastval cdef double sumold for i in xrange(1,maximum+1): sumold = sum2 lastval = newval newval = lastval * lam / i sum2 = sum2 + newval if sumold <= cdf and cdf <= sum2: return i return maximum cpdef poisson_pdf ( unsigned int k, double a ): """Poisson PDF. PDF(K,A) is the probability that the number of events observed in a unit time period will be K, given the expected number of events in a unit time as A. """ if a <= 0: return 0 return exp(-a) * pow (a, k, None) / factorial (k) cdef binomial_coef ( long n, long k ): """BINOMIAL_COEF computes the Binomial coefficient C(N,K) n,k are integers. """ cdef long mn = min (k, n-k) cdef long mx cdef double cnk cdef long i if mn < 0: return 0 elif mn == 0: return 1 else: mx = max(k,n-k) cnk = float(mx+1) for i in xrange(2,mn+1): cnk = cnk * (mx+i) / i return cnk cpdef binomial_cdf ( long x, long a, double b, bool lower=True ): """ BINOMIAL_CDF compute the binomial CDF. CDF(x)(A,B) is the probability of at most X successes in A trials, given that the probability of success on a single trial is B. """ if lower: return _binomial_cdf_f (x,a,b) else: return _binomial_cdf_r (x,a,b) cpdef binomial_sf ( long x, long a, double b, bool lower=True ): """ BINOMIAL_SF compute the binomial survival function (1-CDF) SF(x)(A,B) is the probability of more than X successes in A trials, given that the probability of success on a single trial is B. """ if lower: return 1.0 - _binomial_cdf_f (x,a,b) else: return 1.0 - _binomial_cdf_r (x,a,b) cpdef pduplication (np.ndarray[np.float64_t] pmf, int N_obs): """return the probability of a duplicate fragment given a pmf and a number of observed fragments N_obs """ cdef: n = pmf.shape[0] float p, sf = 0.0 for p in pmf: sf += binomial_sf(2, N_obs, p) return sf / n cdef _binomial_cdf_r ( long x, long a, double b ): """ Binomial CDF for upper tail. """ if x < 0: return 1 elif a < x: return 0 elif b == 0: return 0 elif b == 1: return 1 cdef long argmax=int(a*b) cdef double seedpdf cdef double cdf cdef double pdf cdef long i if xargmax: seedpdf=binomial_pdf(argmax,a,b) pdf=seedpdf cdf = pdf for i in xrange(argmax-1,-1,-1): pdf/=(a-i)*b/(1-b)/(i+1) if pdf==0.0: break cdf += pdf pdf = seedpdf for i in xrange(argmax,x): pdf*=(a-i)*b/(1-b)/(i+1) if pdf==0.0: break cdf+=pdf cdf=min(1,cdf) cdf = float("%.10e" %cdf) return cdf else: pdf=binomial_pdf(x,a,b) cdf = pdf for i in xrange(x-1,-1,-1): pdf/=(a-i)*b/(1-b)/(i+1) if pdf==0.0: break cdf += pdf cdf=min(1,cdf) cdf = float("%.10e" %cdf) return cdf cpdef binomial_cdf_inv ( double cdf, long a, double b ): """BINOMIAL_CDF_INV inverts the binomial CDF. For lower tail only! """ if cdf < 0 or cdf >1: raise Exception("CDF must >= 0 or <= 1") cdef double cdf2 = 0 cdef long x for x in xrange(0,a+1): pdf = binomial_pdf (x,a,b) cdf2 = cdf2 + pdf if cdf < cdf2: return x return a cpdef binomial_pdf( long x, long a, double b ): """binomial PDF by H. Gene Shin """ if a<1: return 0 elif x<0 or aa-x: p=1-b mn=a-x mx=x else: p=b mn=x mx=a-x pdf=1 t = 0 for q in xrange(1,mn+1): pdf*=(a-q+1)*p/(mn-q+1) if pdf < 1e-100: while pdf < 1e-3: pdf /= 1-p t-=1 if pdf > 1e+100: while pdf > 1e+3 and t 1: # raise Exception("Illegal argument %f for qnorm(p)." % p) # split = 0.42 # a0 = 2.50662823884 # a1 = -18.61500062529 # a2 = 41.39119773534 # a3 = -25.44106049637 # b1 = -8.47351093090 # b2 = 23.08336743743 # b3 = -21.06224101826 # b4 = 3.13082909833 # c0 = -2.78718931138 # c1 = -2.29796479134 # c2 = 4.85014127135 # c3 = 2.32121276858 # d1 = 3.54388924762 # d2 = 1.63706781897 # q = p - 0.5 # r = 0.0 # ppnd = 0.0 # if abs(q) <= split: # r = q * q # ppnd = q * (((a3 * r + a2) * r + a1) * r + a0) / ((((b4 * r + b3) * r + b2) * r + b1) * r + 1) # else: # r = p # if q > 0: # r = 1 - p # if r > 0: # r = math.sqrt(- math.log(r)) # ppnd = (((c3 * r + c2) * r + c1) * r + c0) / ((d2 * r + d1) * r + 1) # if q < 0: # ppnd = - ppnd # else: # ppnd = 0 # if upper: # ppnd = - ppnd # if mu != None and sigma2 != None: # return ppnd * math.sqrt(sigma2) + mu # else: # return ppnd # def normal_cdf (z, mu = 0.0, sigma2 = 1.0, lower=True): # upper = not lower # z = (z - mu) / math.sqrt(sigma2) # ltone = 7.0 # utzero = 18.66 # con = 1.28 # a1 = 0.398942280444 # a2 = 0.399903438504 # a3 = 5.75885480458 # a4 = 29.8213557808 # a5 = 2.62433121679 # a6 = 48.6959930692 # a7 = 5.92885724438 # b1 = 0.398942280385 # b2 = 3.8052e-8 # b3 = 1.00000615302 # b4 = 3.98064794e-4 # b5 = 1.986153813664 # b6 = 0.151679116635 # b7 = 5.29330324926 # b8 = 4.8385912808 # b9 = 15.1508972451 # b10 = 0.742380924027 # b11 = 30.789933034 # b12 = 3.99019417011 # y = 0.0 # alnorm = 0.0 # if z < 0: # upper = not upper # z = - z # if z <= ltone or upper and z <= utzero: # y = 0.5 * z * z # if z > con: # alnorm = b1 * math.exp(- y) / (z - b2 + b3 / (z + b4 + b5 / (z - b6 + b7 / (z + b8 - b9 / (z + b10 + b11 / (z + b12)))))) # else: # alnorm = 0.5 - z * (a1 - a2 * y / (y + a3 - a4 / (y + a5 + a6 / (y + a7)))) # else: # alnorm = 0.0 # if not upper: # alnorm = 1.0 - alnorm # return alnorm
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