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#!/usr/bin/env python
# 4 implementations of P = exp(Q*t)
# APIs along the lines of:
# exponentiator = WhateverExponenentiator(Q or Q derivative(s))
# P = exponentiator(t)
#
# Class(Q) instance(t) Limitations
# Eigen slow fast not too asymm
# SemiSym slow fast mprobs > 0
# Pade instant slow
# Taylor instant very slow
from cogent.util.modules import importVersionedModule, ExpectedImportError
import numpy
Float = numpy.core.numerictypes.sctype2char(float)
from numpy.linalg import inv as _inv, eig as _eig,\
solve as solve_linear_equations, LinAlgError
import logging
LOG = logging.getLogger('cogent.maths.exponentiation')
__author__ = "Peter Maxwell"
__copyright__ = "Copyright 2007-2009, The Cogent Project"
__credits__ = ["Peter Maxwell", "Gavin Huttley", "Zongzhi Liu"]
__license__ = "GPL"
__version__ = "1.4.1"
__maintainer__ = "Gavin Huttley"
__email__ = "gavin.huttley@anu.edu.au"
__status__ = "Production"
def eig(*a, **kw):
"""make eig to return the same eigvectors as numpy ones"""
vals, vecs = _eig(*a, **kw)
return vals, vecs.T
def inv(a):
"""make inv return the same contiguous matrix as numpy one"""
return numpy.ascontiguousarray(_inv(a))
def inv(a):
"""make inv return the same contiguous matrix as numpy one"""
return numpy.ascontiguousarray(_inv(a))
try:
pyrex = importVersionedModule('_matrix_exponentiation', globals(),
(1, 2), LOG, "pure Python/NumPy exponentiation")
except ExpectedImportError:
pyrex = None
else:
pyrex.setNumPy(numpy)
if pyrex.version_info == (1, 2):
def _pyrex_eigenvectors(q, orig=pyrex.eigenvectors):
try:
return orig(q)
except RuntimeError, detail:
raise ArithmeticError, detail
pyrex.eigenvectors = _pyrex_eigenvectors
class _Exponentiator(object):
def __init__(self, Q):
self.Q = Q
def __repr__(self):
return "%s(%s)" % (self.__class__.__name__, repr(self.Q))
class EigenExponentiator(_Exponentiator):
"""A matrix ready for fast exponentiation. P=exp(Q*t)"""
__slots__ = ['Q', 'ev', 'roots', 'evI', 'evT']
def __init__(self, Q, roots, ev, evI=None):
self.Q = Q
if evI is None:
evI = inv(ev)
self.evI = evI
self.evT = numpy.transpose(ev)
self.ev = ev
self.roots = roots
def __call__(self, t):
exp_roots = numpy.exp(t*self.roots)
result = numpy.inner(self.evT * exp_roots, self.evI)
if result.dtype.kind == "c":
result = numpy.asarray(result.real)
result = numpy.maximum(result, 0.0)
return result
def SemiSymmetricExponentiator(motif_probs, Q, ex):
"""Like EigenExponentiator, but more numerically stable and
30% faster when the rate matrix (Q/motif_probs) is symmetrical.
Only usable when all motif probs > 0. Unlike the others
it needs to know the motif probabilities."""
H = numpy.sqrt(motif_probs)
H2 = numpy.divide.outer(H, H)
#A = Q * H2
#assert numpy.allclose(A, numpy.transpose(A)), A
(roots, R) = ex.eigenvectors(Q*H2)
ev = R / H2
evI = numpy.transpose(R*H2)
#self.evT = numpy.transpose(self.ev)
return ex.exponentiator(Q, roots, ev, evI)
# These next two are slow exponentiators, they don't get any speed up
# from reusing Q with a new t, but for compatability with the diagonalising
# approach they look like they do. They are derived from code in SciPy.
class TaylorExponentiator(_Exponentiator):
def __init__(self, Q):
self.Q = Q
self.q = 21
def __call__(self, t=1.0):
"""Compute the matrix exponential using a Taylor series of order q."""
A = self.Q * t
M = A.shape[0]
eA = numpy.identity(M, Float)
trm = eA
for k in range(1, self.q):
trm = numpy.dot(trm, A/float(k))
eA += trm
while not numpy.allclose(eA, eA-trm):
k += 1
trm = numpy.dot(trm, A/float(k))
eA += trm
if k >= self.q:
LOG.warning("Taylor series lengthened from %s to %s" % (self.q, k+1))
self.q = k + 1
return eA
class PadeExponentiator(_Exponentiator):
def __init__(self, Q):
self.Q = Q
def __call__(self, t=1.0):
"""Compute the matrix exponential using Pade approximation of order q.
"""
A = self.Q * t
M = A.shape[0]
# Scale A so that norm is < 1/2
norm = numpy.maximum.reduce(numpy.sum(numpy.absolute(A), axis=1))
j = int(numpy.floor(numpy.log(max(norm, 0.5))/numpy.log(2.0))) + 1
A = A / 2.0**j
# How many iterations required
e = 1.0
q = 0
qf = 1.0
while e > 1e-12:
q += 1
q2 = 2.0 * q
qf *= q**2 / (q2 * (q2-1) * q2 * (q2+1))
e = 8 * (norm/(2**j))**(2*q) * qf
# Pade Approximation for exp(A)
X = A
c = 1.0/2
N = numpy.identity(M) + c*A
D = numpy.identity(M) - c*A
for k in range(2,q+1):
c = c * (q-k+1) / (k*(2*q-k+1))
X = numpy.dot(A,X)
cX = c*X
N = N + cX
if not k % 2:
D = D + cX;
else:
D = D - cX;
F = solve_linear_equations(D,N)
for k in range(1,j+1):
F = numpy.dot(F,F)
return F
import time
def _fastest(fs, *args):
if len(fs) == 1:
i = 0
else:
es = [[] for f in fs]
samples = 0
while samples < 10:
for (f, e) in zip(fs, es):
t0 = time.time()
f(*args)
t1 = time.time()
e.append(t1-t0)
samples += 1
m = []
for e in es:
e.sort()
m.append(e[samples/2])
i = numpy.argmin(m, -1)
return (fs[i], fs[i](*args))
def _chooseFastExponentiators(Q):
if pyrex is not None:
eigen_candidates = [eig, pyrex.eigenvectors]
(eigenvectors, (roots,ev)) = _fastest(eigen_candidates, Q)
inverse_candidates = [inv, pyrex.inverse]
(inverse, evI) = _fastest(inverse_candidates, ev)
exponentiator_candidates = [EigenExponentiator(Q, roots, ev, evI)]
if roots.dtype.kind == 'f':
assert ev.dtype.kind == 'f'
pyx = pyrex.EigenExponentiator(Q, roots, ev, evI)
exponentiator_candidates.append(pyx)
(exponentiator, P) = _fastest(exponentiator_candidates, 1.1)
FastestExponentiator = type(exponentiator)
else:
eigenvectors = eig
inverse = inv
FastestExponentiator = EigenExponentiator
# FastestExponentiator may be the pyrex Exponentiator which
# doesn't cope with complex inputs
e_class = {'c': EigenExponentiator, 'f':FastestExponentiator}
def Exp(Q, e_class=e_class):
(roots, ev) = eigenvectors(Q)
return e_class[roots.dtype.kind](Q, roots, ev, inverse(ev))
def Exp2(Q, e_class=e_class):
(roots, ev) = eigenvectors(Q)
evI = inverse(ev)
reQ = numpy.inner(ev.T * roots, evI).real
if not numpy.allclose(Q, reQ):
raise ArithmeticError, "eigen failed precision test"
return e_class[roots.dtype.kind](Q, roots, ev, evI)
# These function attributes are just for log / debug etc.
Exp.eigenvectors = eigenvectors
Exp.inverse = inverse
Exp.exponentiator = FastestExponentiator
return (Exp, Exp2)
def chooseFastExponentiators(Q):
ex = _chooseFastExponentiators(Q)
LOG.info('Strategy for Q Size %s: PyrexEig:%s PyrexInv:%s PyrexExp:%s'
% (
Q.shape[0],
(ex[0].eigenvectors is not eig),
(ex[0].inverse is not inv),
(ex[0].exponentiator is not EigenExponentiator)))
return ex
def FastExponentiator(Q):
size = Q.shape[0]
if pyrex is not None and size < 32:
(roots, ev) = pyrex.eigenvectors(Q)
else:
(roots, ev) = eig(Q)
ex = EigenExponentiator(Q, roots, ev)
return ex
def RobustExponentiator(Q):
return PadeExponentiator(Q)
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