def lowess2(x, y, xest, f=2./3., iter=3):
"""Returns estimated values of y in data points xest (or None if estimation fails).
Lowess smoother: Robust locally weighted regression.
The lowess function fits a nonparametric regression curve to a scatterplot.
The arrays x and y contain an equal number of elements; each pair
(x[i], y[i]) defines a data point in the scatterplot. The function returns
the estimated (smooth) values of y.
The smoothing span is given by f. A larger value for f will result in a
smoother curve. The number of robustifying iterations is given by iter. The
function will run faster with a smaller number of iterations."""
x = Numeric.asarray(x, 'd')
y = Numeric.asarray(y, 'd')
xest = Numeric.asarray(xest, 'd')
n = len(x)
nest = len(xest)
r = min(int(Numeric.ceil(f*n)),n-1) # radius: num. of points to take into LR
h = [Numeric.sort(abs(x-x[i]))[r] for i in range(n)] # distance of the r-th point from x[i]
w = Numeric.clip(abs(([x]-Numeric.transpose([x]))/h),0.0,1.0)
w = 1-w*w*w
w = w*w*w
hest = [Numeric.sort(abs(x-xest[i]))[r] for i in range(nest)] # r-th min. distance from xest[i] to x
west = Numeric.clip(abs(([xest]-Numeric.transpose([x]))/hest),0.0,1.0) # shape: (len(x), len(xest)
west = 1-west*west*west
west = west*west*west
yest = Numeric.zeros(n,'d')
yest2 = Numeric.zeros(nest,'d')
delta = Numeric.ones(n,'d')
try:
for iteration in range(iter):
# fit xest
for i in range(nest):
weights = delta * west[:,i]
b = Numeric.array([sum(weights*y), sum(weights*y*x)])
A = Numeric.array([[sum(weights), sum(weights*x)], [sum(weights*x), sum(weights*x*x)]])
beta = LinearAlgebra.solve_linear_equations(A,b)
yest2[i] = beta[0] + beta[1]*xest[i]
# fit x (to calculate residuals and delta)
for i in range(n):
weights = delta * w[:,i]
b = Numeric.array([sum(weights*y), sum(weights*y*x)])
A = Numeric.array([[sum(weights), sum(weights*x)], [sum(weights*x), sum(weights*x*x)]])
beta = LinearAlgebra.solve_linear_equations(A,b)
yest[i] = beta[0] + beta[1]*x[i]
residuals = y-yest
s = MLab.median(abs(residuals))
delta = Numeric.clip(residuals/(6*s),-1,1)
delta = 1-delta*delta
delta = delta*delta
except LinearAlgebra.LinAlgError:
print "Warning: NumExtn.lowess2: LinearAlgebra.solve_linear_equations: Singular matrix"
yest2 = None
return yest2
def newgeo(self,geo,forces):
err = -ravel(array(forces))
maxerr = max(abs(err))
print "Maxerror = ",maxerr
if maxerr < self.errcut and not self.started:
self.started = 1
print "GDIIS started"
newgeo = []
nat = len(geo)
nit = len(self.errors)
if self.started:
self.errors.append(err)
self.geos.append(geo)
if self.started and nit > 1:
#print "Nit = ",nit
a = zeros((nit+1,nit+1),Float)
b = zeros(nit+1,Float)
for i in range(nit):
for j in range(nit):
a[i,j] = dot(self.errors[i],
self.errors[j])
for i in range(nit):
a[nit,i] = a[i,nit] = -1.
b[nit] = -1.
c = solve_linear_equations(a,b)
newf = zeros(len(self.errors[0]),Float)
for i in range(nit): newf += c[i]*self.errors[i]
newf = reshape(newf,(nat,3))
for i in range(nat):
atno = 0
x = y = z = 0
fx = fy = fz = 0
for j in range(nit):
atno,(xj,yj,zj) = self.geos[j][i]
x += c[j]*xj
y += c[j]*yj
z += c[j]*zj
newgeo.append((atno,(x,y,z)))
for i in range(nat):
atno,(xi,yi,zi) = newgeo[i]
fxi,fyi,fzi = newf[i]
newgeo[i] = atno,(xi-fxi,yi-fyi,zi-fzi)
else:
for i in range(nat):
atno,(x,y,z) = geo[i]
fx,fy,fz = forces[i]
x += self.sdstep*fx
y += self.sdstep*fy
z += self.sdstep*fz
newgeo.append((atno,(x,y,z)))
return newgeo
def _getBasisCoef(self, x, T):
"""returns coefficients of basis polynomials given T, i.e. their values at x: (poly0, xi)
where polynomials in rows, coefficients [a0,a1,...ak] where k=len(x)-1
similar goes for T
"""
numPoints = T.shape[1] # number of points that polynomials are calculated == len(x)
assert len(x) == numPoints, "len(x)=" + str(x) + ", T.shape[1]=" + str(numPoints) + " do not match"
numLinSys = T.shape[0] # number of polynomials
lin_sys_b = Numeric.transpose(T)
lin_sys_a = Numeric.ones((numPoints, numPoints), Numeric.Float)
lin_sys_a[:,1] = x
for idx in range(2,numPoints):
lin_sys_a[:,idx] = lin_sys_a[:,idx-1] * x
return Numeric.transpose(LinearAlgebra.solve_linear_equations(lin_sys_a, lin_sys_b))
def quadFit(X, Y):
"""quadFit(X, Y) --> [a1, a2, ... , b1, b2, ... , c]
Fits (X,Y) data to a generalized quadratic:
X, Y -- arrays of the (x,y) data, where x is either a scalar
or a tuple (x1, x2, x3, ...)
Returns [a1, a2, ..., b1, b2, ... , c]
such that y = a1*x1^2 + a2*x2^2 + .... b1*x1 + b2*x2 + ... + c"""
from numpy import asarray, reshape, ones, concatenate
X = asarray(X)
if len(X.shape) == 1:
X = reshape(X, (X.shape[0], 1))
A = concatenate((X ** 2, X, ones((X.shape[0], 1))), 1)
alpha = dot(transpose(A), A)
beta = dot(transpose(A), Y)
return solve_linear_equations(alpha, beta)
def _getBasisCoef(self, x, T):
"""returns coefficients of basis polynomials given T, i.e. their values at x: (poly0, xi)
where polynomials in rows, coefficients [a0,a1,...ak] where k=len(x)-1
similar goes for T
"""
numPoints = T.shape[
1] # number of points that polynomials are calculated == len(x)
assert len(x) == numPoints, "len(x)=" + str(x) + ", T.shape[1]=" + str(
numPoints) + " do not match"
numLinSys = T.shape[0] # number of polynomials
lin_sys_b = Numeric.transpose(T)
lin_sys_a = Numeric.ones((numPoints, numPoints), Numeric.Float)
lin_sys_a[:, 1] = x
for idx in range(2, numPoints):
lin_sys_a[:, idx] = lin_sys_a[:, idx - 1] * x
return Numeric.transpose(
LinearAlgebra.solve_linear_equations(lin_sys_a, lin_sys_b))
def lowessW(x, y, xest, f=2./3., iter=3, dWeights=None, callback=None):
"""Returns estimated values of y in data points xest (or None if estimation fails).
Lowess smoother: Robust locally weighted regression.
The lowess function fits a nonparametric regression curve to a scatterplot.
The arrays x and y contain an equal number of elements; each pair
(x[i], y[i]) defines a data point in the scatterplot. The function returns
the estimated (smooth) values of y.
The smoothing span is given by f. A larger value for f will result in a
smoother curve. The number of robustifying iterations is given by iter. The
function will run faster with a smaller number of iterations.
Data points may be assigned weights; if None, all weights equal 1.
"""
x = Numeric.asarray(x, 'd')
y = Numeric.asarray(y, 'd')
xest = Numeric.asarray(xest, 'd')
n = len(x)
if n <> len(y):
raise AttributeError, "Error: lowessW(x,y,xest,f,iter,dWeights): len(x)=%i not equal to len(y)=%i" % (len(x), len(y))
nest = len(xest)
# weights of data points (optional)
if dWeights <> None:
dWeights = Numeric.asarray(dWeights, 'd')
if len(dWeights) <> n:
raise AttributeError, "Error: lowessW(x,y,xest,f,iter,dWeights): len(dWeights)=%i not equal to len(x)=%i" % (len(dWeights), len(x))
## dWeights = dWeights.reshape((n,1))
else:
## dWeights = Numeric.ones((n,1))
dWeights = Numeric.ones((n,))
r = min(int(Numeric.ceil(f*n)),n-1) # radius: num. of points to take into LR
h = [Numeric.sort(abs(x-x[i]))[r] for i in range(n)] # distance of the r-th point from x[i]
w = Numeric.clip(abs(([x]-Numeric.transpose([x]))/h),0.0,1.0)
w = 1-w*w*w
w = w*w*w
hest = [Numeric.sort(abs(x-xest[i]))[r] for i in range(nest)] # r-th min. distance from xest[i] to x
west = Numeric.clip(abs(([xest]-Numeric.transpose([x]))/hest),0.0,1.0) # shape: (len(x), len(xest))
west = 1-west*west*west
west = west*west*west
yest = Numeric.zeros(n,'d')
yest2 = Numeric.zeros(nest,'d')
delta = Numeric.ones(n,'d')
try:
for iteration in range(int(iter)):
# fit xest
for i in range(nest):
## print delta.shape, west[:,i].shape, dWeights.shape
weights = delta * west[:,i] * dWeights
b = Numeric.array([sum(weights*y), sum(weights*y*x)])
A = Numeric.array([[sum(weights), sum(weights*x)], [sum(weights*x), sum(weights*x*x)]])
beta = LinearAlgebra.solve_linear_equations(A,b)
yest2[i] = beta[0] + beta[1]*xest[i]
# fit x (to calculate residuals and delta)
for i in range(n):
weights = delta * w[:,i] * dWeights
b = Numeric.array([sum(weights*y), sum(weights*y*x)])
A = Numeric.array([[sum(weights), sum(weights*x)], [sum(weights*x), sum(weights*x*x)]])
beta = LinearAlgebra.solve_linear_equations(A,b)
yest[i] = beta[0] + beta[1]*x[i]
residuals = y-yest
s = MLab.median(abs(residuals))
delta = Numeric.clip(residuals/(6*s),-1,1)
delta = 1-delta*delta
delta = delta*delta
if callback: callback()
except LinearAlgebra.LinAlgError:
print "Warning: NumExtn.lowessW: LinearAlgebra.solve_linear_equations: Singular matrix"
yest2 = None
return yest2