def inverse(a, d0=None):
if shape(a) == (2, 2):
_a = array(shape=shape(a), type=Float64)
ai = array(shape=shape(a), type=Float64)
_a[:] = a[:]
if d0 == None: d0 = determinant(_a)
ai[0, 0] = _a[1, 1]
ai[0, 1] = -_a[0, 1]
ai[1, 0] = -_a[1, 0]
ai[1, 1] = _a[0, 0]
return ai / d0
elif shape(a) == (3, 3):
_a = array(shape=shape(a), type=Float64)
ai = array(shape=shape(a), type=Float64)
_a[:] = a[:]
if d0 == None: d0 = determinant(_a)
ai[0, 0] = _a[1, 1] * _a[2, 2] - _a[2, 1] * _a[1, 2]
ai[0, 1] = _a[2, 1] * _a[0, 2] - _a[0, 1] * _a[2, 2]
ai[0, 2] = _a[0, 1] * _a[1, 2] - _a[1, 1] * _a[0, 2]
ai[1, 0] = _a[2, 0] * _a[1, 2] - _a[1, 0] * _a[2, 2]
ai[1, 1] = _a[0, 0] * _a[2, 2] - _a[2, 0] * _a[0, 2]
ai[1, 2] = _a[1, 0] * _a[0, 2] - _a[0, 0] * _a[1, 2]
ai[2, 0] = _a[1, 0] * _a[2, 1] - _a[2, 0] * _a[1, 1]
ai[2, 1] = _a[2, 0] * _a[0, 1] - _a[0, 0] * _a[2, 1]
ai[2, 2] = _a[0, 0] * _a[1, 1] - _a[1, 0] * _a[0, 1]
return ai / d0
return linear_algebra.inverse(a)
class _Matrix(NumArray):
def _rc(self, a):
if len(shape(a)) == 0:
return a
else:
return Matrix(a)
def __mul__(self, other):
aother = asarray(other)
if len(aother.shape) == 0:
return self._rc(self * aother)
else:
return self._rc(_dot(self, aother))
def __rmul__(self, other):
aother = asarray(other)
if len(aother.shape) == 0:
return self._rc(aother * self)
else:
return self._rc(_dot(aother, self))
def __imul__(self, other):
aother = asarray(other)
self[:] = _dot(self, aother)
return self
def __pow__(self, other):
shape = self.shape
if len(shape) != 2 or shape[0] != shape[1]:
raise TypeError, "matrix is not square"
if type(other) not in (type(1), type(1L)):
raise TypeError, "exponent must be an integer"
if other == 0:
return Matrix(identity(shape[0]))
if other < 0:
result = Matrix(LinearAlgebra.inverse(self))
x = Matrix(result)
other = -other
else:
result = self
x = result
if other <= 3:
while (other > 1):
result = result * x
other = other - 1
return result
# binary decomposition to reduce the number of Matrix
# Multiplies for other > 3.
beta = _binary(other)
t = len(beta)
Z, q = x.copy(), 0
while beta[t - q - 1] == '0':
Z *= Z
q += 1
result = Z.copy()
for k in range(q + 1, t):
Z *= Z
if beta[t - k - 1] == '1':
result *= Z
return result
def _simple_logistic_regression(x,y,beta_start=None,verbose=False,
CONV_THRESH=1.e-3,MAXIT=500):
"""
Faster than logistic_regression when there is only one predictor.
"""
if len(x) != len(y):
raise ValueError, "x and y should be the same length!"
if beta_start is None:
beta_start = NA.zeros(2,x.typecode())
iter = 0; diff = 1.; beta = beta_start # initial values
if verbose:
print 'iteration beta log-likliehood |beta-beta_old|'
while iter < MAXIT:
beta_old = beta
p = NA.exp(beta[0]+beta[1]*x)/(1.+NA.exp(beta[0]+beta[1]*x))
l = NA.sum(y*NA.log(p) + (1.-y)*NA.log(1.-p)) # log-likliehood
s = NA.array([NA.sum(y-p), NA.sum((y-p)*x)]) # scoring function
# information matrix
J_bar = NA.array([[NA.sum(p*(1-p)),NA.sum(p*(1-p)*x)],
[NA.sum(p*(1-p)*x),NA.sum(p*(1-p)*x*x)]])
beta = beta_old + NA.dot(LA.inverse(J_bar),s) # new value of beta
diff = NA.sum(NA.fabs(beta-beta_old)) # sum of absolute differences
if verbose:
print iter+1, beta, l, diff
if diff <= CONV_THRESH: break
iter = iter + 1
return beta, J_bar, l
def __pow__(self, other):
shape = self.shape
if len(shape)!=2 or shape[0]!=shape[1]:
raise TypeError, "matrix is not square"
if type(other) not in (type(1), type(1L)):
raise TypeError, "exponent must be an integer"
if other==0:
return Matrix(identity(shape[0]))
if other<0:
result=Matrix(LinearAlgebra.inverse(self))
x=Matrix(result)
other=-other
else:
result=self
x=result
if other <= 3:
while(other>1):
result=result*x
other=other-1
return result
# binary decomposition to reduce the number of Matrix
# Multiplies for other > 3.
beta = _binary(other)
t = len(beta)
Z,q = x.copy(),0
while beta[t-q-1] == '0':
Z *= Z
q += 1
result = Z.copy()
for k in range(q+1,t):
Z *= Z
if beta[t-k-1] == '1':
result *= Z
return result
def _simple_logistic_regression(x,y,beta_start=None,verbose=False,
CONV_THRESH=1.e-3,MAXIT=500):
"""
Faster than logistic_regression when there is only one predictor.
"""
if len(x) != len(y):
raise ValueError, "x and y should be the same length!"
if beta_start is None:
beta_start = NA.zeros(2,x.dtype.char)
iter = 0; diff = 1.; beta = beta_start # initial values
if verbose:
print 'iteration beta log-likliehood |beta-beta_old|'
while iter < MAXIT:
beta_old = beta
p = NA.exp(beta[0]+beta[1]*x)/(1.+NA.exp(beta[0]+beta[1]*x))
l = NA.sum(y*NA.log(p) + (1.-y)*NA.log(1.-p)) # log-likliehood
s = NA.array([NA.sum(y-p), NA.sum((y-p)*x)]) # scoring function
# information matrix
J_bar = NA.array([[NA.sum(p*(1-p)),NA.sum(p*(1-p)*x)],
[NA.sum(p*(1-p)*x),NA.sum(p*(1-p)*x*x)]])
beta = beta_old + NA.dot(LA.inverse(J_bar),s) # new value of beta
diff = NA.sum(NA.fabs(beta-beta_old)) # sum of absolute differences
if verbose:
print iter+1, beta, l, diff
if diff <= CONV_THRESH: break
iter = iter + 1
return beta, J_bar, l
def inverse(a, d0 = None): if shape(a) == (2,2): _a = array(shape=shape(a), type=Float64) ai = array(shape=shape(a), type=Float64) _a[:] = a[:] if d0 == None: d0 = determinant(_a) ai[0,0] = _a[1,1] ai[0,1] = -_a[0,1] ai[1,0] = -_a[1,0] ai[1,1] = _a[0,0] return ai/d0 elif shape(a) == (3,3): _a = array(shape=shape(a), type=Float64) ai = array(shape=shape(a), type=Float64) _a[:] = a[:] if d0 == None: d0 = determinant(_a) ai[0,0] = _a[1,1] * _a[2,2] - _a[2,1] * _a[1,2] ai[0,1] = _a[2,1] * _a[0,2] - _a[0,1] * _a[2,2] ai[0,2] = _a[0,1] * _a[1,2] - _a[1,1] * _a[0,2] ai[1,0] = _a[2,0] * _a[1,2] - _a[1,0] * _a[2,2] ai[1,1] = _a[0,0] * _a[2,2] - _a[2,0] * _a[0,2] ai[1,2] = _a[1,0] * _a[0,2] - _a[0,0] * _a[1,2] ai[2,0] = _a[1,0] * _a[2,1] - _a[2,0] * _a[1,1] ai[2,1] = _a[2,0] * _a[0,1] - _a[0,0] * _a[2,1] ai[2,2] = _a[0,0] * _a[1,1] - _a[1,0] * _a[0,1] return ai/d0 return linear_algebra.inverse(a)
def __init__(self, p, webmatrix):
assert p>=0 and p <= 1
self.p = float(p)
if type(webmatrix)in [type([]), type(())]:
webmatrix = array(webmatrix, Float)
assert webmatrix.shape[0]==webmatrix.shape[1]
self.webmatrix = webmatrix
# create the deltamatrix
imatrix = identity(webmatrix.shape[0], Float)
for i in range(webmatrix.shape[0]):
imatrix[i] = imatrix[i]*sum(webmatrix[i,:])
deltamatrix = la.inverse(imatrix)
self.deltamatrix = deltamatrix
# create the fmatrix
self.fmatrix = ones(webmatrix.shape, Float)
self.sigma = webmatrix.shape[0]
# calculate the Stochastic matrix
_f_normalized = (self.sigma**-1)*self.fmatrix
_randmatrix = (1-p)*_f_normalized
_linkedmatrix = p * matrixmultiply(deltamatrix, webmatrix)
M = _randmatrix + _linkedmatrix
self.stochasticmatrix = M
self.invariantmeasure = ones((1, webmatrix.shape[0]), Float)
def findmec(th0, th1):
delta = 1e-3
if th0 < 0 and th0 - th1 < .1:
sm = 5.
sp = 6.
else:
sm = 3.
sp = 5.
params = [sm, sp]
lasterr = 1e6
for i in range(25):
sm, sp = params
ath0, ath1 = trymec(sm, sp)
c1c, c2c = th0 - ath0, th1 - ath1
err = c1c * c1c + c2c * c2c
if 0:
print '%findmec', sm, sp, ath0, ath1, err
sys.stdout.flush()
if err < 1e-9:
return params
if err > lasterr:
return None
lasterr = err
dc1s = []
dc2s = []
for j in range(len(params)):
params1 = N.array(params)
params1[j] += delta
sm, sp = params1
ath0, ath1 = trymec(sm, sp)
c1p, c2p = th0 - ath0, th1 - ath1
params1 = N.array(params)
params1[j] -= delta
sm, sp = params1
ath0, ath1 = trymec(sm, sp)
c1m, c2m = th0 - ath0, th1 - ath1
dc1s.append((c1p - c1m) / (2 * delta))
dc2s.append((c2p - c2m) / (2 * delta))
jm = N.array([dc1s, dc2s])
ji = la.inverse(jm)
dp = N.dot(ji, [c1c, c2c])
if i < 4:
scale = .5
else:
scale = 1
params -= scale * dp
if params[0] < 0: params[0] = 0.
return params
def findmec(th0, th1):
delta = 1e-3
if th0 < 0 and th0 - th1 < .1:
sm = 5.
sp = 6.
else:
sm = 3.
sp = 5.
params = [sm, sp]
lasterr = 1e6
for i in range(25):
sm, sp = params
ath0, ath1 = trymec(sm, sp)
c1c, c2c = th0 - ath0, th1 - ath1
err = c1c * c1c + c2c * c2c
if 0:
print '%findmec', sm, sp, ath0, ath1, err
sys.stdout.flush()
if err < 1e-9:
return params
if err > lasterr:
return None
lasterr = err
dc1s = []
dc2s = []
for j in range(len(params)):
params1 = N.array(params)
params1[j] += delta
sm, sp = params1
ath0, ath1 = trymec(sm, sp)
c1p, c2p = th0 - ath0, th1 - ath1
params1 = N.array(params)
params1[j] -= delta
sm, sp = params1
ath0, ath1 = trymec(sm, sp)
c1m, c2m = th0 - ath0, th1 - ath1
dc1s.append((c1p - c1m) / (2 * delta))
dc2s.append((c2p - c2m) / (2 * delta))
jm = N.array([dc1s, dc2s])
ji = la.inverse(jm)
dp = N.dot(ji, [c1c, c2c])
if i < 4:
scale = .5
else:
scale = 1
params -= scale * dp
if params[0] < 0: params[0] = 0.
return params
def toWorld(self,v,trl=False):
"""Transform a vertex from camera to world coordinates.
The specified vector can have 3 or 4 (homogoneous) components.
This uses the currently saved rotation matrix.
"""
a = la.inverse(array(self.rot))
if len(v) == 3:
v = v + [ 1. ]
v = matrixmultiply(array(v),a)
return v[0:3] / v[3]
def setNewX(self):
"""Calculate new X values"""
Jacobian = linear_algebra.inverse(self.Jacobian)
self.DeltaX = matrixmultiply(Jacobian, self.OldY)
self.NewX = self.OldX-self.DeltaX
for h in range(self.HarmonicSize)
if self.NewX > self.MAX_X:
NewX[i] = complex(1.0)
def regress(X):
"""
Apply a linear regression on X.
X is an numarray array of form [Y|X].
We split this up, do the regression and return the
regression coefficients, beta_i.
"""
X = N.array(X)
Y = X[:,0].copy()
X[:,0] = 1.0
a = N.dot(NL.inverse(N.dot(N.transpose(X), X)), N.transpose(X))
return N.dot(a, Y)
def makeSPlot(categories, samples):
"""Construct the sPlot weights function.
'categories' -- A sequence of categories. Each is a pair '(pdf,
num)', where 'pdf' is the normalized PDF of the *reduced* variables
for the category, and 'num' is the number of samples in that
category.
'samples' -- A iterable of samples.
returns -- A function that takes a sample as its argument and
returns an array of sPlot weights for the given categories."""
num_categories = len(categories)
# Split the 'categories' argument into a sequence of PDFs and an
# array of numbers.
pdfs = [ f for (f, n) in categories ]
nums = numarray.array([ n for (f, n) in categories ], type="Float64")
# Accumulate the inverse of the covariance matrix.
V_inv = numarray.zeros((num_categories, num_categories), "Float64")
for sample in samples:
# Evaluate PDFs for this sample.
pds = numarray.array([ f(sample) for f in pdfs ])
# Compute the contribution for this sample.
denominator = numarray.dot(nums, pds) ** 2
if denominator != 0:
V_inv += outer_product(pds, pds) / denominator
# Invert to obtain the covariance matrix.
V = inverse(V_inv)
def getWeights(sample):
# Evaluate PDFs for this sample.
pds = numarray.array([ f(sample) for f in pdfs ])
# Compute an array of the weights.
denominator = numarray.dot(nums, pds)
if denominator != 0:
return numarray.matrixmultiply(V, pds) / denominator
else:
return numarray.zeros((num_categories, ), "Float32")
getWeights.covariance_matrix = V
return getWeights
def solve_mec_3constr(constraint_fnl, n = 30, initparams = None):
delta = 1e-3
if initparams:
params = N.array(initparams)
else:
params = [3.14, 0, 0]
for i in range(n):
k, lam1, lam2 = params
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1c, c2c, c3c = constraint_fnl(cost, x, y, th)
print '% constraint_fnl =', c1c, c2c, c3c
dc1s = []
dc2s = []
dc3s = []
for j in range(len(params)):
params1 = N.array(params)
params1[j] += delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1p, c2p, c3p = constraint_fnl(cost, x, y, th)
params1 = N.array(params)
params1[j] -= delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1m, c2m, c3m = constraint_fnl(cost, x, y, th)
dc1s.append((c1p - c1m) / (2 * delta))
dc2s.append((c2p - c2m) / (2 * delta))
dc3s.append((c3p - c3m) / (2 * delta))
# Make Jacobian matrix to invert
jm = N.array([dc1s, dc2s, dc3s])
#print jm
ji = la.inverse(jm)
dp = N.dot(ji, [c1c, c2c, c3c])
if i < n/2: scale = .25
else: scale = 1
params -= scale * dp
print '%', params
return params
def solve_mec_3constr(constraint_fnl, n=30, initparams=None):
delta = 1e-3
if initparams:
params = N.array(initparams)
else:
params = [3.14, 0, 0]
for i in range(n):
k, lam1, lam2 = params
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1c, c2c, c3c = constraint_fnl(cost, x, y, th)
print '% constraint_fnl =', c1c, c2c, c3c
dc1s = []
dc2s = []
dc3s = []
for j in range(len(params)):
params1 = N.array(params)
params1[j] += delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1p, c2p, c3p = constraint_fnl(cost, x, y, th)
params1 = N.array(params)
params1[j] -= delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1m, c2m, c3m = constraint_fnl(cost, x, y, th)
dc1s.append((c1p - c1m) / (2 * delta))
dc2s.append((c2p - c2m) / (2 * delta))
dc3s.append((c3p - c3m) / (2 * delta))
# Make Jacobian matrix to invert
jm = N.array([dc1s, dc2s, dc3s])
#print jm
ji = la.inverse(jm)
dp = N.dot(ji, [c1c, c2c, c3c])
if i < n / 2: scale = .25
else: scale = 1
params -= scale * dp
print '%', params
return params
#境界条件
T[0][0] = 300.0 #温度固定
T[M][0] = T[M - 1][0] #断熱
#行列の作成
A = numarray.array([[0. for i in range(M + 1)] for j in range(M + 1)])
for i in range(1, M):
A[i][i - 1] = -a
A[i][i] = 1 + 2 * a
A[i][i + 1] = -a
#境界条件
A[0][0] = 1. #一定温度
A[M][M] = 1. #断熱
#Aの逆行列
A_inv = la.inverse(A)
f = open('output', 'w')
for j in range(1, N):
preT = T
T = numarray.dot(A_inv, preT)
T[0][0] = 300.0
T[M][0] = T[M - 1][0]
#計算結果をファイルへ出力
f.write('# t=%ss\n' % (j * dt))
for i in range(M + 1):
f.write('%s, %s\n' % (i * dx, T[i][0]))
f.write('\n\n')
def logistic_regression(x,y,beta_start=None,verbose=False,CONV_THRESH=1.e-3,
MAXIT=500):
"""
Uses the Newton-Raphson algorithm to calculate a maximum
likelihood estimate logistic regression.
The algorithm is known as 'iteratively re-weighted least squares', or IRLS.
x - rank-1 or rank-2 array of predictors. If x is rank-2,
the number of predictors = x.shape[0] = N. If x is rank-1,
it is assumed N=1.
y - binary outcomes (if N>1 len(y) = x.shape[1], if N=1 len(y) = len(x))
beta_start - initial beta vector (default zeros(N+1,x.dtype.char))
if verbose=True, diagnostics printed for each iteration (default False).
MAXIT - max number of iterations (default 500)
CONV_THRESH - convergence threshold (sum of absolute differences
of beta-beta_old, default 0.001)
returns beta (the logistic regression coefficients, an N+1 element vector),
J_bar (the (N+1)x(N+1) information matrix), and l (the log-likeliehood).
J_bar can be used to estimate the covariance matrix and the standard
error for beta.
l can be used for a chi-squared significance test.
covmat = inverse(J_bar) --> covariance matrix of coefficents (beta)
stderr = sqrt(diag(covmat)) --> standard errors for beta
deviance = -2l --> scaled deviance statistic
chi-squared value for -2l is the model chi-squared test.
"""
if x.shape[-1] != len(y):
raise ValueError, "x.shape[-1] and y should be the same length!"
try:
N, npreds = x.shape[1], x.shape[0]
except: # single predictor, use simple logistic regression routine.
return _simple_logistic_regression(x,y,beta_start=beta_start,
CONV_THRESH=CONV_THRESH,MAXIT=MAXIT,verbose=verbose)
if beta_start is None:
beta_start = NA.zeros(npreds+1,x.dtype.char)
X = NA.ones((npreds+1,N), x.dtype.char)
X[1:, :] = x
Xt = NA.transpose(X)
iter = 0; diff = 1.; beta = beta_start # initial values
if verbose:
print 'iteration beta log-likliehood |beta-beta_old|'
while iter < MAXIT:
beta_old = beta
ebx = NA.exp(NA.dot(beta, X))
p = ebx/(1.+ebx)
l = NA.sum(y*NA.log(p) + (1.-y)*NA.log(1.-p)) # log-likeliehood
s = NA.dot(X, y-p) # scoring function
J_bar = NA.dot(X*p,Xt) # information matrix
beta = beta_old + NA.dot(LA.inverse(J_bar),s) # new value of beta
diff = NA.sum(NA.fabs(beta-beta_old)) # sum of absolute differences
if verbose:
print iter+1, beta, l, diff
if diff <= CONV_THRESH: break
iter = iter + 1
if iter == MAXIT and diff > CONV_THRESH:
print 'warning: convergence not achieved with threshold of %s in %s iterations' % (CONV_THRESH,MAXIT)
return beta, J_bar, l
def solve_mec(constraint_fnl):
delta = 1e-3
params = [pi, 0, 0]
for i in range(20):
k, lam1, lam2 = params
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
#print i * .05, 'setgray'
#plot(xys)
c1c, c2c, costc = constraint_fnl(cost, x, y, th)
print '% constraint_fnl =', c1c, c2c, 'cost =', costc
dc1s = []
dc2s = []
for j in range(len(params)):
params1 = N.array(params)
params1[j] += delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1p, c2p, costp = constraint_fnl(cost, x, y, th)
params1 = N.array(params)
params1[j] -= delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1m, c2m, costm = constraint_fnl(cost, x, y, th)
dc1s.append((c1p - c1m) / (2 * delta))
dc2s.append((c2p - c2m) / (2 * delta))
xp = cross_prod(dc1s, dc2s)
xp = N.divide(xp, sqrt(N.dot(xp, xp))) # Normalize to unit length
print '% dc1s =', dc1s
print '% dc2s =', dc2s
print '% xp =', xp
# Compute second derivative wrt orthogonal vec
params1 = N.array(params)
for j in range(len(params)):
params1[j] += delta * xp[j]
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1p, c2p, costp = constraint_fnl(cost, x, y, th)
print '% constraint_fnl+ =', c1p, c2p, 'cost =', costp
params1 = N.array(params)
for j in range(len(params)):
params1[j] -= delta * xp[j]
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1m, c2m, costm = constraint_fnl(cost, x, y, th)
print '% constraint_fnl- =', c1m, c2m, 'cost =', costm
d2cost = (costp + costm - 2 * costc) / (delta * delta)
dcost = (costp - costm) / (2 * delta)
print '% dcost =', dcost, 'd2cost =', d2cost
if d2cost < 0: d2cost = .1
# Make Jacobian matrix to invert
jm = N.array([dc1s, dc2s, [x * d2cost for x in xp]])
#print jm
ji = la.inverse(jm)
#print ji
dp = N.dot(ji, [c1c, c2c, dcost])
print '% dp =', dp
print '% [right]=', [c1c, c2c, dcost]
params -= dp * .1
print '%', params
sys.stdout.flush()
return params
def solve_mec(constraint_fnl):
delta = 1e-3
params = [pi, 0, 0]
for i in range(20):
k, lam1, lam2 = params
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
#print i * .05, 'setgray'
#plot(xys)
c1c, c2c, costc = constraint_fnl(cost, x, y, th)
print '% constraint_fnl =', c1c, c2c, 'cost =', costc
dc1s = []
dc2s = []
for j in range(len(params)):
params1 = N.array(params)
params1[j] += delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1p, c2p, costp = constraint_fnl(cost, x, y, th)
params1 = N.array(params)
params1[j] -= delta
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1m, c2m, costm = constraint_fnl(cost, x, y, th)
dc1s.append((c1p - c1m) / (2 * delta))
dc2s.append((c2p - c2m) / (2 * delta))
xp = cross_prod(dc1s, dc2s)
xp = N.divide(xp, sqrt(N.dot(xp, xp))) # Normalize to unit length
print '% dc1s =', dc1s
print '% dc2s =', dc2s
print '% xp =', xp
# Compute second derivative wrt orthogonal vec
params1 = N.array(params)
for j in range(len(params)):
params1[j] += delta * xp[j]
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1p, c2p, costp = constraint_fnl(cost, x, y, th)
print '% constraint_fnl+ =', c1p, c2p, 'cost =', costp
params1 = N.array(params)
for j in range(len(params)):
params1[j] -= delta * xp[j]
k, lam1, lam2 = params1
xys, cost, x, y, th = run_elastica(-.5, .5, k, lam1, lam2)
c1m, c2m, costm = constraint_fnl(cost, x, y, th)
print '% constraint_fnl- =', c1m, c2m, 'cost =', costm
d2cost = (costp + costm - 2 * costc) / (delta * delta)
dcost = (costp - costm) / (2 * delta)
print '% dcost =', dcost, 'd2cost =', d2cost
if d2cost < 0: d2cost = .1
# Make Jacobian matrix to invert
jm = N.array([dc1s, dc2s, [x * d2cost for x in xp]])
#print jm
ji = la.inverse(jm)
#print ji
dp = N.dot(ji, [c1c, c2c, dcost])
print '% dp =', dp
print '% [right]=', [c1c, c2c, dcost]
params -= dp * .1
print '%', params
sys.stdout.flush()
return params
# random draws from trivariate normal distribution
x = multivariate_normal(NA.array([0,0,0]),NA.array([[1,r12,r13],[r12,1,r23],[r13,r23,1]]), nsamps)
x2 = multivariate_normal(NA.array([0,0,0]),NA.array([[1,r12,r13],[r12,1,r23],[r13,r23,1]]), nsamps)
print 'correlations (r12,r13,r23) = ',r12,r13,r23
print 'number of realizations = ',nsamps
# training data.
obs = x[:,0]
climprob = NA.sum((obs > 0).astype('f'))/nsamps
fcst = NA.transpose(x[:,1:]) # 2 predictors.
obs_binary = obs > 0.
# independent data for verification.
obs2 = x2[:,0]
fcst2 = NA.transpose(x2[:,1:])
# compute logistic regression.
beta,Jbar,llik = logistic_regression(fcst,obs_binary,verbose=True)
covmat = LA.inverse(Jbar)
stderr = NA.sqrt(mlab.diag(covmat))
print 'beta =' ,beta
print 'standard error =',stderr
# forecasts from independent data.
prob = calcprob(beta, fcst2)
# compute Brier Skill Score
verif = (obs2 > 0.).astype('f')
bs = mlab.mean((0.01*prob - verif)**2)
bsclim = mlab.mean((climprob - verif)**2)
bss = 1.-(bs/bsclim)
print 'Brier Skill Score (should be within +/- 0.1 of 0.18) = ',bss
# calculate reliability.
# see http://www.bom.gov.au/bmrc/wefor/staff/eee/verif/verif_web_page.html
# for information on the Brier Skill Score and reliability diagrams.
totfreq = NA.zeros(10,'f')
def logistic_regression(x,
y,
beta_start=None,
verbose=False,
CONV_THRESH=1.e-3,
MAXIT=500):
"""
Uses the Newton-Raphson algorithm to calculate a maximum
likelihood estimate logistic regression.
The algorithm is known as 'iteratively re-weighted least squares', or IRLS.
x - rank-1 or rank-2 array of predictors. If x is rank-2,
the number of predictors = x.shape[0] = N. If x is rank-1,
it is assumed N=1.
y - binary outcomes (if N>1 len(y) = x.shape[1], if N=1 len(y) = len(x))
beta_start - initial beta vector (default zeros(N+1,x.dtype.char))
if verbose=True, diagnostics printed for each iteration (default False).
MAXIT - max number of iterations (default 500)
CONV_THRESH - convergence threshold (sum of absolute differences
of beta-beta_old, default 0.001)
returns beta (the logistic regression coefficients, an N+1 element vector),
J_bar (the (N+1)x(N+1) information matrix), and l (the log-likeliehood).
J_bar can be used to estimate the covariance matrix and the standard
error for beta.
l can be used for a chi-squared significance test.
covmat = inverse(J_bar) --> covariance matrix of coefficents (beta)
stderr = sqrt(diag(covmat)) --> standard errors for beta
deviance = -2l --> scaled deviance statistic
chi-squared value for -2l is the model chi-squared test.
"""
if x.shape[-1] != len(y):
raise ValueError, "x.shape[-1] and y should be the same length!"
try:
N, npreds = x.shape[1], x.shape[0]
except: # single predictor, use simple logistic regression routine.
return _simple_logistic_regression(x,
y,
beta_start=beta_start,
CONV_THRESH=CONV_THRESH,
MAXIT=MAXIT,
verbose=verbose)
if beta_start is None:
beta_start = NA.zeros(npreds + 1, x.dtype.char)
X = NA.ones((npreds + 1, N), x.dtype.char)
X[1:, :] = x
Xt = NA.transpose(X)
iter = 0
diff = 1.
beta = beta_start # initial values
if verbose:
print 'iteration beta log-likliehood |beta-beta_old|'
while iter < MAXIT:
beta_old = beta
ebx = NA.exp(NA.dot(beta, X))
p = ebx / (1. + ebx)
l = NA.sum(y * NA.log(p) +
(1. - y) * NA.log(1. - p)) # log-likeliehood
s = NA.dot(X, y - p) # scoring function
J_bar = NA.dot(X * p, Xt) # information matrix
beta = beta_old + NA.dot(LA.inverse(J_bar), s) # new value of beta
diff = NA.sum(NA.fabs(beta - beta_old)) # sum of absolute differences
if verbose:
print iter + 1, beta, l, diff
if diff <= CONV_THRESH: break
iter = iter + 1
if iter == MAXIT and diff > CONV_THRESH:
print 'warning: convergence not achieved with threshold of %s in %s iterations' % (
CONV_THRESH, MAXIT)
return beta, J_bar, l
x2 = multivariate_normal(
NA.array([0, 0, 0]),
NA.array([[1, r12, r13], [r12, 1, r23], [r13, r23, 1]]), nsamps)
print 'correlations (r12,r13,r23) = ', r12, r13, r23
print 'number of realizations = ', nsamps
# training data.
obs = x[:, 0]
climprob = NA.sum((obs > 0).astype('f')) / nsamps
fcst = NA.transpose(x[:, 1:]) # 2 predictors.
obs_binary = obs > 0.
# independent data for verification.
obs2 = x2[:, 0]
fcst2 = NA.transpose(x2[:, 1:])
# compute logistic regression.
beta, Jbar, llik = logistic_regression(fcst, obs_binary, verbose=True)
covmat = LA.inverse(Jbar)
stderr = NA.sqrt(mlab.diag(covmat))
print 'beta =', beta
print 'standard error =', stderr
# forecasts from independent data.
prob = calcprob(beta, fcst2)
# compute Brier Skill Score
verif = (obs2 > 0.).astype('f')
bs = mlab.mean((0.01 * prob - verif)**2)
bsclim = mlab.mean((climprob - verif)**2)
bss = 1. - (bs / bsclim)
print 'Brier Skill Score (should be within +/- 0.1 of 0.18) = ', bss
# calculate reliability.
# see http://www.bom.gov.au/bmrc/wefor/staff/eee/verif/verif_web_page.html
# for information on the Brier Skill Score and reliability diagrams.
totfreq = NA.zeros(10, 'f')
