Package weblogolib :: Module logomath |

# Source Code for Module weblogolib.logomath

      1  #!/usr/bin/env python
      2
      3  # -------------------------------- WebLogo --------------------------------
      4
      5  #  Copyright (c) 2003-2004 The Regents of the University of California.
      6  #  Copyright (c) 2005 Gavin E. Crooks
      7  #  Copyright (c) 2006-2011, The Regents of the University of California, through
      8  #  Lawrence Berkeley National Laboratory (subject to receipt of any required
      9  #  approvals from the U.S. Dept. of Energy).  All rights reserved.
     10
     11  #  This software is distributed under the new BSD Open Source License.
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     40
     41  from math import log, sqrt, exp
     42
     43  from numpy import array, asarray, float64, ones, zeros, int32,all,any, shape
     44  import numpy as na
     45  from corebio.moremath import *
     46
     47  import random
     48
     49 -class Dirichlet(object) :
     50      """The Dirichlet probability distribution. The Dirichlet is a continuous
     51      multivariate probability distribution across non-negative unit length
     52      vectors. In other words, the Dirichlet is a probability distribution of
     53      probability distributions. It is conjugate to the multinomial
     54      distribution and is widely used in Bayesian statistics.
     55
     56      The Dirichlet probability distribution of order K-1 is
     57
     58       p(theta_1,...,theta_K) d theta_1 ... d theta_K =
     59          (1/Z) prod_i=1,K theta_i^{alpha_i - 1} delta(1 -sum_i=1,K theta_i)
     60
     61      The normalization factor Z can be expressed in terms of gamma functions:
     62
     63        Z = {prod_i=1,K Gamma(alpha_i)} / {Gamma( sum_i=1,K alpha_i)}
     64
     65      The K constants, alpha_1,...,alpha_K, must be positive. The K parameters,
     66      theta_1,...,theta_K are nonnegative and sum to 1.
     67
     68      Status:
     69          Alpha
     70      """
     71      __slots__ = 'alpha', '_total', '_mean',
     72
     73
     74
     75
     76 -    def __init__(self, alpha) :
     77          """
     78          Args:
     79              - alpha  -- The parameters of the Dirichlet prior distribution.
     80                          A vector of non-negative real numbers.
     81          """
     82          # TODO: Check that alphas are positive
     83          #TODO : what if alpha's not one dimensional?
     84          self.alpha = asarray(alpha, float64)
     85
     86          self._total = sum(alpha)
     87          self._mean = None
     88
     89
     90 -    def sample(self) :
     91          """Return a randomly generated probability vector.
     92
     93          Random samples are generated by sampling K values from gamma
     94          distributions with parameters a=\alpha_i, b=1, and renormalizing.
     95
     96          Ref:
     97              A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).
     98          Authors:
     99              Gavin E. Crooks <gec@compbio.berkeley.edu> (2002)
    100          """
    101          alpha = self.alpha
    102          K = len(alpha)
    103          theta = zeros( (K,), float64)
    104
    105          for k in range(K):
    106              theta[k] = random.gammavariate(alpha[k], 1.0)
    107          theta /= sum(theta)
    108
    109          return theta
    110
    111 -    def mean(self) :
    112          if  self._mean ==None:
    113              self._mean = self.alpha / self._total
    114          return self._mean
    115
    116 -    def covariance(self) :
    117          alpha = self.alpha
    118          A = sum(alpha)
    119          #A2 = A * A
    120          K = len(alpha)
    121          cv = zeros( (K,K), float64)
    122
    123          for i in range(K) :
    124              cv[i,i] = alpha[i] * (1. - alpha[i]/A) / (A * (A+1.) )
    125
    126          for i in range(K) :
    127              for j in range(i+1,K) :
    128                  v = - alpha[i] * alpha[j] / (A * A * (A+1.) )
    129                  cv[i,j] = v
    130                  cv[j,i] = v
    131          return cv
    132
    133 -    def mean_x(self, x) :
    134          x = asarray(x, float64)
    135          if shape(x) != shape(self.alpha) :
    136              raise ValueError("Argument must be same dimension as Dirichlet")
    137          return sum( x * self.mean())
    138
    139 -    def variance_x(self, x) :
    140          x = asarray(x, float64)
    141          if shape(x) != shape(self.alpha) :
    142              raise ValueError("Argument must be same dimension as Dirichlet")
    143
    144          cv = self.covariance()
    145          var = na.dot(na.dot(na.transpose( x), cv), x)
    146          return var
    147
    148
    149 -    def mean_entropy(self) :
    150          """Calculate the average entropy of probabilities sampled
    151          from this Dirichlet distribution.
    152
    153          Returns:
    154              The average entropy.
    155
    156          Ref:
    157              Wolpert & Wolf, PRE 53:6841-6854 (1996) Theorem 7
    158              (Warning: this paper contains typos.)
    159          Status:
    160              Alpha
    161          Authors:
    162              GEC 2005
    163
    164          """
    165          # TODO: Optimize
    166          alpha = self.alpha
    167          A = float(sum(alpha))
    168          ent = 0.0
    169          for a in alpha:
    170              if a>0 : ent += - 1.0 * a * digamma( 1.0+a) # FIXME: Check
    171          ent /= A
    172          ent += digamma(A+1.0)
    173          return ent
    174
    175
    176
    177 -    def variance_entropy(self):
    178          """Calculate the variance of the Dirichlet entropy.
    179
    180          Ref:
    181              Wolpert & Wolf, PRE 53:6841-6854 (1996) Theorem 8
    182              (Warning: this paper contains typos.)
    183          """
    184          alpha = self.alpha
    185          A = float(sum(alpha))
    186          A2 = A * (A+1)
    187          L = len(alpha)
    188
    189          dg1 = zeros( (L) , float64)
    190          dg2 = zeros( (L) , float64)
    191          tg2 = zeros( (L) , float64)
    192
    193          for i in range(L) :
    194              dg1[i] = digamma(alpha[i] + 1.0)
    195              dg2[i] = digamma(alpha[i] + 2.0)
    196              tg2[i] = trigamma(alpha[i] + 2.0)
    197
    198          dg_Ap2 = digamma( A+2. )
    199          tg_Ap2 = trigamma( A+2. )
    200
    201          mean = self.mean_entropy()
    202          var = 0.0
    203
    204          for i in range(L) :
    205              for j in range(L) :
    206                  if i != j :
    207                      var += (
    208                          ( dg1[i] - dg_Ap2 ) * (dg1[j] - dg_Ap2 ) - tg_Ap2
    209                          ) * (alpha[i] * alpha[j] ) / A2
    210                  else :
    211                      var += (
    212                          ( dg2[i] - dg_Ap2 ) **2 + ( tg2[i] - tg_Ap2 )
    213                          ) * ( alpha[i] * (alpha[i]+1.) ) / A2
    214
    215          var -= mean**2
    216          return var
    217
    218
    219
    220 -    def mean_relative_entropy(self, pvec) :
    221          ln_p = na.log(pvec)
    222          return - self.mean_x(ln_p) - self.mean_entropy()
    223
    224
    225 -    def variance_relative_entropy(self, pvec) :
    226          ln_p = na.log(pvec)
    227          return self.variance_x(ln_p) + self.variance_entropy()
    228
    229
    230 -    def interval_relative_entropy(self, pvec, frac) :
    231          mean = self.mean_relative_entropy(pvec)
    232          variance = self.variance_relative_entropy(pvec)
    233          sd = sqrt(variance)
    234
    235          # If the variance is small, use the standard 95%
    236          # confidence interval: mean +/- 1.96 * sd
    237          if mean/sd >3.0 :
    238              return max(0.0, mean - sd*1.96), mean + sd*1.96
    239
    240          g = Gamma.from_mean_variance(mean, variance)
    241          low_limit = g.inverse_cdf( (1.-frac)/2.)
    242          high_limit = g.inverse_cdf( 1. - (1.-frac)/2. )
    243
    244          return low_limit, high_limit
    245
    246
    247 -class Gamma(object) :
    248      """The gamma probability distribution. (Not to be confused with the
    249      gamma function.)
    250
    251
    252      """
    253      __slots__ = 'alpha', 'beta'
    254
    255 -    def __init__(self, alpha, beta) :
    256          if alpha <=0.0 :
    257              raise ValueError("alpha must be positive")
    258          if beta <=0.0 :
    259              raise ValueError("beta must be positive")
    260          self.alpha = alpha
    261          self.beta = beta
    262
    263
    264 -    def from_shape_scale(cls, shape, scale) :
    265          return cls( shape, 1./scale)
    266      from_shape_scale = classmethod(from_shape_scale)
    267
    268
    269 -    def from_mean_variance(cls, mean, variance) :
    270          alpha = mean **2 / variance
    271          beta = alpha/mean
    272          return cls(alpha, beta)
    273      from_mean_variance = classmethod(from_mean_variance)
    274
    275 -    def mean(self) :
    276          return self.alpha / self.beta
    277
    278 -    def variance(self) :
    279          return self.alpha / (self.beta**2)
    280
    281
    282 -    def sample(self) :
    283          return random.gammavariate(self.alpha, 1./self.beta)
    284
    285 -    def pdf(self, x) :
    286          if x==0.0 : return 0.0
    287          a = self.alpha
    288          b = self.beta
    289          return (x**(a-1.)) * exp(- b*x )* (b**a) / gamma(a)
    290
    291 -    def cdf(self, x) :
    292          return 1.0-normalized_incomplete_gamma(self.alpha, self.beta*x)
    293
    294 -    def inverse_cdf(self, p) :
    295          def rootof(x) :
    296              return self.cdf(exp(x)) - p
    297
    298          return exp(find_root(  rootof, log(self.mean()) ) )
    299
    300  #
    301
    302 -def find_root(f, x, y=None, fprime=None, tolerance=1.48e-8, max_iterations=50):
    303      """Return argument 'x' of the function f(x), such that f(x)=0 to
    304      within the given tolerance.
    305
    306      f : The function to optimize, f(x)
    307      x : The initial guess
    308      y : An optional second guess that shoudl bracket the root.
    309      fprime : The derivate of f'(x) (Optional)
    310      tolerance : The error bounds
    311      max_iterations : Maximum number of iterations
    312
    313      Raises:
    314          ArithmeticError :
    315              Failure to converge to the given tolerence
    316
    317      Notes:
    318          Uses Newton-Raphson algorihtm if f'(x) is given, else uses bisect if
    319          y is given and brackets the root, else uses secant.
    320
    321      Status : Beta (Not fully tested)
    322      """
    323
    324
    325      def secant (f, x, tolerance, max_iterations):
    326          x0 = x
    327          x1 = x0+ 1e-4
    328          v0 = f(x0)
    329          v1 = f(x1)
    330          x2 = 0
    331
    332          for i in range(max_iterations):
    333              # print x0, x1, v0, v1, x2-x0
    334              x2 = x1 - v1*(x1-x0)/(v1-v0)
    335              if abs(x2-x1) < tolerance : return x2
    336              x0 = x1
    337              v0 = v1
    338              x1 = x2
    339              v1 = f(x1)
    340
    341          raise ArithmeticError(
    342              "Failed to converge after %d iterations, value is %f" \
    343                  % (max_iterations,x1) )
    344
    345
    346      def bisect(f, a, b, tolerance, max_iterations) :
    347          fa = f(a)
    348          fb = f(b)
    349
    350          if fa==0: return a
    351          if fb==0: return b
    352
    353          if fa*fb >0 :
    354              raise ArithmeticError("Start points do not bracket root.")
    355
    356          for i in range(max_iterations):
    357              #print a,b, fa, fb
    358              delta = b-a
    359              xm = a + 0.5*delta # Minimize roundoff in computing the midpoint
    360              fm = f(xm)
    361              if delta < tolerance : return xm
    362
    363              if fm * fa >0 : #   Root lies in interval [xm,b], replace a
    364                  a = xm
    365                  fa = fm
    366              else : #   Root lies in interval [a,xm], replace b
    367                  b = xm
    368                  fb = fm
    369
    370          raise ArithmeticError(
    371              "Failed to converge after %d iterations, value is %f" \
    372                  % (max_iterations, x) )
    373
    374
    375      def newton(f, x, fprime, tolerance, max_iterations) :
    376          x0 = x
    377          for i in range(max_iterations) :
    378              x1 = x0 - f(x0)/fprime(x0)
    379              if abs(x1-x0) < tolerance: return x1
    380              x0 = x1
    381
    382          raise ArithmeticError(
    383              "Failed to converge after %d iterations, value is %f" \
    384                  % (max_iterations,x1) )
    385
    386      if fprime is not None :
    387          return newton(f, x, fprime, tolerance, max_iterations)
    388      elif y is not None :
    389          return bisect(f, x, y, tolerance, max_iterations)
    390      else :
    391          return secant(f, x, tolerance, max_iterations)
    392