def Spearman(poly, dist, sample=1e4, retall=False, **kws):
"""
Calculate Spearman's rank-order correlation coefficient
Parameters
----------
poly : Poly
Polynomial of interest.
dist : Dist
Defines the space where correlation is taken.
sample : int
Number of samples used in estimation.
retall : bool
If true, return p-value as well.
**kws : optional
Extra keywords passed to dist.sample.
Returns
-------
rho[, p-value]
rho : float or ndarray
Correlation output. Of type float if two-dimensional problem.
Correleation matrix if larger.
p-value : float or ndarray
The two-sided p-value for a hypothesis test whose null
hypothesis is that two sets of data are uncorrelated, has same
dimension as rho.
"""
samples = dist.sample(sample, **kws)
poly = po.flatten(poly)
Y = poly(*samples)
if retall:
return spearmanr(Y.T)
return spearmanr(Y.T)[0]
def E_cond(poly, freeze, dist, **kws):
assert not dist.dependent()
if poly.dim<len(dist):
poly = po.setdim(poly, len(dist))
freeze = po.Poly(freeze)
freeze = po.setdim(freeze, len(dist))
keys = freeze.A.keys()
if len(keys)==1 and keys[0]==(0,)*len(dist):
freeze = freeze.A.values()[0]
else:
freeze = np.array(keys)
freeze = freeze.reshape(freeze.size/len(dist), len(dist))
shape = poly.shape
poly = po.flatten(poly)
kmax = np.max(poly.keys, 0)+1
keys = [i for i in np.ndindex(*kmax)]
vals = dist.mom(np.array(keys).T, **kws).T
mom = dict(zip(keys, vals))
A = poly.A.copy()
keys = A.keys()
out = {}
zeros = [0]*poly.dim
for i in xrange(len(keys)):
key = list(keys[i])
a = A[tuple(key)]
for d in xrange(poly.dim):
for j in xrange(len(freeze)):
if freeze[j,d]:
key[d], zeros[d] = zeros[d], key[d]
break
tmp = a*mom[tuple(key)]
if tuple(zeros) in out:
out[tuple(zeros)] = out[tuple(zeros)] + tmp
else:
out[tuple(zeros)] = tmp
for d in xrange(poly.dim):
for j in xrange(len(freeze)):
if freeze[j,d]:
key[d], zeros[d] = zeros[d], key[d]
break
out = po.Poly(out, poly.dim, poly.shape, float)
out = po.reshape(out, shape)
return out
def Skew(poly, dist=None, **kws):
"""
Skewness, or element by element 3rd order statistics of a
distribution or polynomial.
Parameters
----------
poly : Poly, Dist
Input to take skewness on.
dist : Dist
Defines the space the skewness is taken on.
It is ignored if `poly` is a distribution.
**kws : optional
Extra keywords passed to dist.mom.
Returns
-------
skewness : ndarray
Element for element variance along `poly`, where
`skewness.shape==poly.shape`.
See Also
--------
Corr Correlation matrix
Cov Covariance matrix
E Expected value
Kurt Kurtosis operator
Var Variance operator
Examples
--------
>>> x = cp.variable()
>>> Z = cp.Gamma()
>>> print cp.Skew(Z)
2.0
"""
if isinstance(poly, di.Dist):
x = po.variable(len(poly))
poly, dist = x, poly
else:
poly = po.Poly(poly)
if poly.dim<len(dist):
po.setdim(poly, len(dist))
shape = poly.shape
poly = po.flatten(poly)
m1 = E(poly, dist)
m2 = E(poly**2, dist)
m3 = E(poly**3, dist)
out = (m3-3*m2*m1+2*m1**3)/(m2-m1**2)**1.5
out = np.reshape(out, shape)
return out
def orth_ttr(order, dist, normed=False, sort="GR", retall=False, **kws):
"""
Create orthogonal polynomial expansion from three terms recursion
formula.
Parameters
----------
order : int
Order of polynomial expansion
dist : Dist
Distribution space where polynomials are orthogonal
If dist.ttr exists, it will be used,
othervice Clenshaw-Curtis integration will be used.
Must be stochastically independent.
normed : bool
If True orthonormal polynomials will be used instead of monic.
sort : str
Polynomial sorting. Same as in basis
retall : bool
If true return norms as well.
kws : optional
Keyword argument passed to stieltjes method.
Returns
-------
orth[, norms]
orth : Poly
Orthogonal polynomial expansion
norms : np.ndarray
Norms of the orthogonal expansion on the form
E(orth**2, dist)
Calculated using recurrence coefficients for stability.
Examples
--------
>>> Z = cp.Normal()
>>> print cp.orth_ttr(4, Z)
[1.0, q0, q0^2-1.0, q0^3-3.0q0, -6.0q0^2+3.0+q0^4]
"""
P, norms, A, B = qu.stieltjes(dist, order, retall=True, **kws)
if normed:
for i in xrange(len(P)):
P[i] = P[i]/np.sqrt(norms[:,i])
norms = norms**0
dim = len(dist)
if dim>1:
Q, G = [], []
indices = ber.bindex(0,order,dim,sort)
for I in indices:
q = [P[I[i]][i] for i in xrange(dim)]
q = reduce(lambda x,y: x*y, q)
Q.append(q)
if retall:
for I in indices:
g = [norms[i,I[i]] for i in xrange(dim)]
G.append(np.prod(g))
P = Q
else:
G = norms[0]
P = po.flatten(po.Poly(P))
if retall:
return P, np.array(G)
return P
def orth_ttr(order, dist, normed=False, sort="GR", retall=False, **kws):
"""
Create orthogonal polynomial expansion from three terms recursion
formula.
Parameters
----------
order : int
Order of polynomial expansion
dist : Dist
Distribution space where polynomials are orthogonal
If dist.ttr exists, it will be used,
othervice Clenshaw-Curtis integration will be used.
Must be stochastically independent.
normed : bool
If True orthonormal polynomials will be used instead of monic.
sort : str
Polynomial sorting. Same as in basis
retall : bool
If true return norms as well.
kws : optional
Keyword argument passed to stieltjes method.
Returns
-------
orth[, norms]
orth : Poly
Orthogonal polynomial expansion
norms : np.ndarray
Norms of the orthogonal expansion on the form
E(orth**2, dist)
Calculated using recurrence coefficients for stability.
Examples
--------
>>> Z = cp.Normal()
>>> print cp.orth_ttr(4, Z)
[1.0, q0, q0^2-1.0, q0^3-3.0q0, -6.0q0^2+3.0+q0^4]
"""
P, norms, A, B = qu.stieltjes(dist, order, retall=True, **kws)
if normed:
for i in xrange(len(P)):
P[i] = P[i] / np.sqrt(norms[:, i])
norms = norms**0
dim = len(dist)
if dim > 1:
Q, G = [], []
indices = ber.bindex(0, order, dim, sort)
for I in indices:
q = [P[I[i]][i] for i in xrange(dim)]
q = reduce(lambda x, y: x * y, q)
Q.append(q)
if retall:
for I in indices:
g = [norms[i, I[i]] for i in xrange(dim)]
G.append(np.prod(g))
P = Q
else:
G = norms[0]
P = po.flatten(po.Poly(P))
if retall:
return P, np.array(G)
return P
def Perc(poly, q, dist, sample=1e4, **kws):
"""
Percentile function
Parameters
----------
poly : Poly
Polynomial of interest.
q : array_like
positions where percentiles are taken. Must be a number or an
array, where all values are on the interval `[0,100]`.
dist : Dist
Defines the space where percentile is taken.
sample : int
Number of samples used in estimation.
**kws : optional
Extra keywords passed to dist.sample.
Returns
-------
Q : ndarray
Percentiles of `poly` with `Q.shape=poly.shape+q.shape`.
Examples
--------
>>> cp.seed(1000)
>>> x,y = cp.variable(2)
>>> poly = cp.Poly([x, x*y])
>>> Z = cp.J(cp.Uniform(3,6), cp.Normal())
>>> print cp.Perc(poly, [0, 50, 100], Z)
[[ 3. -45. ]
[ 4.5080777 -0.05862173]
[ 6. 45. ]]
"""
shape = poly.shape
poly = po.flatten(poly)
q = np.array(q)/100.
dim = len(dist)
# Interior
sample = kws.pop("sample", 1e4)
Z = dist.sample(sample, **kws)
if dim==1:
Z = (Z,)
q = np.array([q])
poly1 = poly(*Z)
# Min/max
mi, ma = dist.range().reshape(2,dim)
ext = np.mgrid[(slice(0,2,1),)*dim].reshape(dim, 2**dim).T
ext = np.where(ext, mi, ma).T
poly2 = poly(*ext)
poly2 = np.array([_ for _ in poly2.T if not np.any(np.isnan(_))]).T
# Finish
if poly2.shape:
poly1 = np.concatenate([poly1,poly2], -1)
samples = poly1.shape[-1]
poly1.sort()
out = poly1.T[np.asarray(q*(samples-1), dtype=int)]
out = out.reshape(q.shape + shape)
return out
def Cov(poly, dist=None, **kws):
"""
Covariance matrix, or 2rd order statistics of a distribution or
polynomial.
Parameters
----------
poly : Poly, Dist
Input to take covariance on. Must have `len(poly)>=2`.
dist : Dist
Defines the space the covariance is taken on.
It is ignored if `poly` is a distribution.
**kws : optional
Extra keywords passed to dist.mom.
Returns
-------
covariance : ndarray
Covariance matrix with
`covariance.shape==poly.shape+poly.shape`.
See Also
--------
Corr Correlation matrix
E Expected value
Kurt Kurtosis operator
Skew Skewness operator
Var Variance operator
Examples
--------
>>> Z = cp.MvNormal([0,0], [[2,.5],[.5,1]])
>>> print cp.Cov(Z)
[[ 2. 0.5]
[ 0.5 1. ]]
>>> x = cp.variable()
>>> Z = cp.Normal()
>>> print cp.Cov([x, x**2], Z)
[[ 1. 0.]
[ 0. 2.]]
"""
if not isinstance(poly, (di.Dist, po.Poly)):
poly = po.Poly(poly)
if isinstance(poly, di.Dist):
x = po.variable(len(poly))
poly, dist = x, poly
else:
poly = po.Poly(poly)
dim = len(dist)
shape = poly.shape
poly = po.flatten(poly)
keys = poly.keys
N = len(keys)
A = poly.A
keys1 = np.array(keys).T
if dim==1:
keys1 = keys1[0]
keys2 = sum(np.meshgrid(keys, keys))
else:
keys2 = np.empty((dim, N, N))
for i in xrange(N):
for j in xrange(N):
keys2[:, i,j] = keys1[:,i]+keys1[:,j]
m1 = dist.mom(keys1, **kws)
m2 = dist.mom(keys2, **kws)
mom = m2-np.outer(m1, m1)
out = np.zeros((len(poly), len(poly)))
for i in xrange(len(keys)):
a = A[keys[i]]
out += np.outer(a,a)*mom[i,i]
for j in xrange(i+1, len(keys)):
b = A[keys[j]]
ab = np.outer(a,b)
out += (ab+ab.T)*mom[i,j]
out = np.reshape(out, shape+shape)
return out
def Kurt(poly, dist=None, fisher=True, **kws):
"""
Kurtosis, or element by element 4rd order statistics of a
distribution or polynomial.
Parameters
----------
P : Poly, Dist
Input to take skewness on.
dist : Dist
Defines the space the skewness is taken on.
It is ignored if `poly` is a distribution.
fisher : bool
If True, Fisher's definition is used (Normal -> 0.0). If False,
Pearson's definition is used (normal -> 3.0)
**kws : optional
Extra keywords passed to dist.mom.
Returns
-------
skewness : ndarray
Element for element variance along `poly`, where
`skewness.shape==poly.shape`.
See Also
--------
Corr Correlation matrix
Cov Covariance matrix
E Expected value
Skew Skewness operator
Var Variance operator
Examples
--------
>>> x = cp.variable()
>>> Z = cp.Uniform()
>>> print cp.Kurt(Z)
-1.2
>>> Z = cp.Normal()
>>> print cp.Kurt(x, Z)
4.4408920985e-16
"""
if isinstance(poly, di.Dist):
x = po.variable(len(poly))
poly, dist = x, poly
else:
poly = po.Poly(poly)
if fisher: adjust = 3
else: adjust = 0
shape = poly.shape
poly = po.flatten(poly)
m1 = E(poly, dist)
m2 = E(poly**2, dist)
m3 = E(poly**3, dist)
m4 = E(poly**4, dist)
out = (m4-4*m3*m1 + 6*m2*m1**2 - 3*m1**4) /\
(m2**2-2*m2*m1**2+m1**4) - adjust
out = np.reshape(out, shape)
return out
def E(poly, dist=None, **kws):
"""
Expected value, or 1st order statistics of a probability
distribution or polynomial on a given probability space.
Parameters
----------
poly : Poly, Dist
Input to take expected value on.
dist : Dist
Defines the space the expected value is taken on.
It is ignored if `poly` is a distribution.
**kws : optional
Extra keywords passed to dist.mom.
Returns
-------
expected : ndarray
The expected value of the polynomial or distribution, where
`expected.shape==poly.shape`.
See Also
--------
Corr Correlation matrix
Cov Covariance matrix
Kurt Kurtosis operator
Skew Skewness operator
Var Variance operator
Examples
--------
For distributions:
>>> x = cp.variable()
>>> Z = cp.Uniform()
>>> print cp.E(Z)
0.5
>>> print cp.E(x**3, Z)
0.25
"""
if not isinstance(poly, (di.Dist, po.Poly)):
print type(poly)
print "Approximating expected value..."
out = qu.quad(poly, dist, veceval=True, **kws)
print "done"
return out
if isinstance(poly, di.Dist):
dist = poly
poly = po.variable(len(poly))
if not poly.keys:
return np.zeros(poly.shape, dtype=int)
if isinstance(poly, (list, tuple, np.ndarray)):
return [E(_, dist, **kws) for _ in poly]
if poly.dim<len(dist):
poly = po.setdim(poly, len(dist))
shape = poly.shape
poly = po.flatten(poly)
keys = poly.keys
mom = dist.mom(np.array(keys).T, **kws)
A = poly.A
if len(dist)==1:
mom = mom[0]
out = np.zeros(poly.shape)
for i in xrange(len(keys)):
out += A[keys[i]]*mom[i]
out = np.reshape(out, shape)
return out
def Std(poly, dist=None, **kws):
"""
Standard deviation, or element by element 2nd order statistics of a
distribution or polynomial.
Parameters
----------
poly : Poly, Dist
Input to take variance on.
dist : Dist
Defines the space the variance is taken on.
It is ignored if `poly` is a distribution.
**kws : optional
Extra keywords passed to dist.mom.
Returns
-------
variation : ndarray
Element for element variance along `poly`, where
`variation.shape==poly.shape`.
See Also
--------
Corr Correlation matrix
Cov Covariance matrix
E Expected value
Kurt Kurtosis operator
Skew Skewness operator
Examples
--------
>>> x = cp.variable()
>>> Z = cp.Uniform()
>>> print cp.Var(Z)
0.0833333333333
>>> print cp.Var(x**3, Z)
0.0803571428571
"""
if isinstance(poly, di.Dist):
x = po.variable(len(poly))
poly, dist = x, poly
else:
poly = po.Poly(poly)
dim = len(dist)
if poly.dim<dim:
po.setdim(poly, dim)
shape = poly.shape
poly = po.flatten(poly)
keys = poly.keys
N = len(keys)
A = poly.A
keys1 = np.array(keys).T
if dim==1:
keys1 = keys1[0]
keys2 = sum(np.meshgrid(keys, keys))
else:
keys2 = np.empty((dim, N, N))
for i in xrange(N):
for j in xrange(N):
keys2[:,i,j] = keys1[:,i]+keys1[:,j]
m1 = np.outer(*[dist.mom(keys1, **kws)]*2)
m2 = dist.mom(keys2, **kws)
mom = m2-m1
out = np.zeros(poly.shape)
for i in xrange(N):
a = A[keys[i]]
out += a*a*mom[i,i]
for j in xrange(i+1, N):
b = A[keys[j]]
out += 2*a*b*mom[i,j]
out = out.reshape(shape)
return np.sqrt(out)
def weightgen(nodes, dist):
poly = stieltjes(dist, len(nodes) - 1, retall=True)[0]
poly = po.flatten(po.Poly(poly))
V = poly(nodes)
Vi = np.linalg.inv(V)
return Vi[:, 0]
def weightgen(nodes, dist):
poly = stieltjes(dist, len(nodes)-1, retall=True)[0]
poly = po.flatten(po.Poly(poly))
V = poly(nodes)
Vi = np.linalg.inv(V)
return Vi[:,0]
