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smoothn.py
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smoothn.py
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from numpy import *
from pylab import *
import scipy.optimize.lbfgsb as lbfgsb
import numpy.linalg
from scipy.fftpack.realtransforms import dct,idct
import numpy as np
import numpy.ma as ma
def smoothn(y,nS0=10,axis=None,smoothOrder=2.0,sd=None,verbose=False,\
s0=None,z0=None,isrobust=False,W=None,s=None,MaxIter=100,TolZ=1e-3,weightstr='bisquare'):
'''
function [z,s,exitflag,Wtot] = smoothn(varargin)
SMOOTHN Robust spline smoothing for 1-D to N-D data.
SMOOTHN provides a fast, automatized and robust discretized smoothing
spline for data of any dimension.
Z = SMOOTHN(Y) automatically smoothes the uniformly-sampled array Y. Y
can be any N-D noisy array (time series, images, 3D data,...). Non
finite data (NaN or Inf) are treated as missing values.
Z = SMOOTHN(Y,S) smoothes the array Y using the smoothing parameter S.
S must be a real positive scalar. The larger S is, the smoother the
output will be. If the smoothing parameter S is omitted (see previous
option) or empty (i.e. S = []), it is automatically determined using
the generalized cross-validation (GCV) method.
Z = SMOOTHN(Y,W) or Z = SMOOTHN(Y,W,S) specifies a weighting array W of
real positive values, that must have the same size as Y. Note that a
nil weight corresponds to a missing value.
Robust smoothing
----------------
Z = SMOOTHN(...,'robust') carries out a robust smoothing that minimizes
the influence of outlying data.
[Z,S] = SMOOTHN(...) also returns the calculated value for S so that
you can fine-tune the smoothing subsequently if needed.
An iteration process is used in the presence of weighted and/or missing
values. Z = SMOOTHN(...,OPTION_NAME,OPTION_VALUE) smoothes with the
termination parameters specified by OPTION_NAME and OPTION_VALUE. They
can contain the following criteria:
-----------------
TolZ: Termination tolerance on Z (default = 1e-3)
TolZ must be in ]0,1[
MaxIter: Maximum number of iterations allowed (default = 100)
Initial: Initial value for the iterative process (default =
original data)
-----------------
Syntax: [Z,...] = SMOOTHN(...,'MaxIter',500,'TolZ',1e-4,'Initial',Z0);
[Z,S,EXITFLAG] = SMOOTHN(...) returns a boolean value EXITFLAG that
describes the exit condition of SMOOTHN:
1 SMOOTHN converged.
0 Maximum number of iterations was reached.
Class Support
-------------
Input array can be numeric or logical. The returned array is of class
double.
Notes
-----
The N-D (inverse) discrete cosine transform functions <a
href="matlab:web('http://www.biomecardio.com/matlab/dctn.html')"
>DCTN</a> and <a
href="matlab:web('http://www.biomecardio.com/matlab/idctn.html')"
>IDCTN</a> are required.
To be made
----------
Estimate the confidence bands (see Wahba 1983, Nychka 1988).
Reference
---------
Garcia D, Robust smoothing of gridded data in one and higher dimensions
with missing values. Computational Statistics & Data Analysis, 2010.
<a
href="matlab:web('http://www.biomecardio.com/pageshtm/publi/csda10.pdf')">PDF download</a>
Examples:
--------
# 1-D example
x = linspace(0,100,2**8);
y = cos(x/10)+(x/50)**2 + randn(size(x))/10;
y[[70, 75, 80]] = [5.5, 5, 6];
z = smoothn(y); # Regular smoothing
zr = smoothn(y,'robust'); # Robust smoothing
subplot(121), plot(x,y,'r.',x,z,'k','LineWidth',2)
axis square, title('Regular smoothing')
subplot(122), plot(x,y,'r.',x,zr,'k','LineWidth',2)
axis square, title('Robust smoothing')
# 2-D example
xp = 0:.02:1;
[x,y] = meshgrid(xp);
f = exp(x+y) + sin((x-2*y)*3);
fn = f + randn(size(f))*0.5;
fs = smoothn(fn);
subplot(121), surf(xp,xp,fn), zlim([0 8]), axis square
subplot(122), surf(xp,xp,fs), zlim([0 8]), axis square
# 2-D example with missing data
n = 256;
y0 = peaks(n);
y = y0 + rand(size(y0))*2;
I = randperm(n^2);
y(I(1:n^2*0.5)) = NaN; # lose 1/2 of data
y(40:90,140:190) = NaN; # create a hole
z = smoothn(y); # smooth data
subplot(2,2,1:2), imagesc(y), axis equal off
title('Noisy corrupt data')
subplot(223), imagesc(z), axis equal off
title('Recovered data ...')
subplot(224), imagesc(y0), axis equal off
title('... compared with original data')
# 3-D example
[x,y,z] = meshgrid(-2:.2:2);
xslice = [-0.8,1]; yslice = 2; zslice = [-2,0];
vn = x.*exp(-x.^2-y.^2-z.^2) + randn(size(x))*0.06;
subplot(121), slice(x,y,z,vn,xslice,yslice,zslice,'cubic')
title('Noisy data')
v = smoothn(vn);
subplot(122), slice(x,y,z,v,xslice,yslice,zslice,'cubic')
title('Smoothed data')
# Cardioid
t = linspace(0,2*pi,1000);
x = 2*cos(t).*(1-cos(t)) + randn(size(t))*0.1;
y = 2*sin(t).*(1-cos(t)) + randn(size(t))*0.1;
z = smoothn(complex(x,y));
plot(x,y,'r.',real(z),imag(z),'k','linewidth',2)
axis equal tight
# Cellular vortical flow
[x,y] = meshgrid(linspace(0,1,24));
Vx = cos(2*pi*x+pi/2).*cos(2*pi*y);
Vy = sin(2*pi*x+pi/2).*sin(2*pi*y);
Vx = Vx + sqrt(0.05)*randn(24,24); # adding Gaussian noise
Vy = Vy + sqrt(0.05)*randn(24,24); # adding Gaussian noise
I = randperm(numel(Vx));
Vx(I(1:30)) = (rand(30,1)-0.5)*5; # adding outliers
Vy(I(1:30)) = (rand(30,1)-0.5)*5; # adding outliers
Vx(I(31:60)) = NaN; # missing values
Vy(I(31:60)) = NaN; # missing values
Vs = smoothn(complex(Vx,Vy),'robust'); # automatic smoothing
subplot(121), quiver(x,y,Vx,Vy,2.5), axis square
title('Noisy velocity field')
subplot(122), quiver(x,y,real(Vs),imag(Vs)), axis square
title('Smoothed velocity field')
See also SMOOTH, SMOOTH3, DCTN, IDCTN.
-- Damien Garcia -- 2009/03, revised 2010/11
Visit my <a
href="matlab:web('http://www.biomecardio.com/matlab/smoothn.html')">website</a> for more details about SMOOTHN
# Check input arguments
error(nargchk(1,12,nargin));
z0=None,W=None,s=None,MaxIter=100,TolZ=1e-3
'''
if type(y) == ma.core.MaskedArray: # masked array
is_masked = True
mask = y.mask
y = np.array(y)
y[mask] = 0.
if W != None:
W = np.array(W)
W[mask] = 0.
if sd != None:
W = np.array(1./sd**2)
W[mask] = 0.
sd = None
y[mask] = np.nan
if sd != None:
sd_ = np.array(sd)
mask = (sd > 0.)
W = np.zeros_like(sd_)
W[mask] = 1./sd_[mask]**2
sd = None
if W != None:
W = W/W.max()
sizy = y.shape;
# sort axis
if axis == None:
axis = tuple(np.arange(y.ndim))
noe = y.size # number of elements
if noe<2:
z = y
exitflag = 0;Wtot=0
return z,s,exitflag,Wtot
#---
# Smoothness parameter and weights
#if s != None:
# s = []
if W == None:
W = ones(sizy);
#if z0 == None:
# z0 = y.copy()
#---
# "Weighting function" criterion
weightstr = weightstr.lower()
#---
# Weights. Zero weights are assigned to not finite values (Inf or NaN),
# (Inf/NaN values = missing data).
IsFinite = np.array(isfinite(y)).astype(bool);
nof = IsFinite.sum() # number of finite elements
W = W*IsFinite;
if any(W<0):
error('smoothn:NegativeWeights',\
'Weights must all be >=0')
else:
#W = W/np.max(W)
pass
#---
# Weighted or missing data?
isweighted = any(W != 1);
#---
# Robust smoothing?
#isrobust
#---
# Automatic smoothing?
isauto = not s;
#---
# DCTN and IDCTN are required
try:
from scipy.fftpack.realtransforms import dct,idct
except:
z = y
exitflag = -1;Wtot=0
return z,s,exitflag,Wtot
## Creation of the Lambda tensor
#---
# Lambda contains the eingenvalues of the difference matrix used in this
# penalized least squares process.
axis = tuple(np.array(axis).flatten())
d = y.ndim;
Lambda = zeros(sizy);
for i in axis:
# create a 1 x d array (so e.g. [1,1] for a 2D case
siz0 = ones((1,y.ndim))[0];
siz0[i] = sizy[i];
# cos(pi*(reshape(1:sizy(i),siz0)-1)/sizy(i)))
# (arange(1,sizy[i]+1).reshape(siz0) - 1.)/sizy[i]
Lambda = Lambda + (cos(pi*(arange(1,sizy[i]+1) - 1.)/sizy[i]).reshape(siz0))
#else:
# Lambda = Lambda + siz0
Lambda = -2.*(len(axis)-Lambda);
if not isauto:
Gamma = 1./(1+(s*abs(Lambda))**smoothOrder);
## Upper and lower bound for the smoothness parameter
# The average leverage (h) is by definition in [0 1]. Weak smoothing occurs
# if h is close to 1, while over-smoothing appears when h is near 0. Upper
# and lower bounds for h are given to avoid under- or over-smoothing. See
# equation relating h to the smoothness parameter (Equation #12 in the
# referenced CSDA paper).
N = sum(array(sizy) != 1); # tensor rank of the y-array
hMin = 1e-6; hMax = 0.99;
# (h/n)**2 = (1 + a)/( 2 a)
# a = 1/(2 (h/n)**2 -1)
# where a = sqrt(1 + 16 s)
# (a**2 -1)/16
try:
sMinBnd = np.sqrt((((1+sqrt(1+8*hMax**(2./N)))/4./hMax**(2./N))**2-1)/16.);
sMaxBnd = np.sqrt((((1+sqrt(1+8*hMin**(2./N)))/4./hMin**(2./N))**2-1)/16.);
except:
sMinBnd = None
sMaxBnd = None
## Initialize before iterating
#---
Wtot = W;
#--- Initial conditions for z
if isweighted:
#--- With weighted/missing data
# An initial guess is provided to ensure faster convergence. For that
# purpose, a nearest neighbor interpolation followed by a coarse
# smoothing are performed.
#---
if z0 != None: # an initial guess (z0) has been provided
z = z0;
else:
z = y #InitialGuess(y,IsFinite);
z[~IsFinite] = 0.
else:
z = zeros(sizy);
#---
z0 = z;
y[~IsFinite] = 0; # arbitrary values for missing y-data
#---
tol = 1.;
RobustIterativeProcess = True;
RobustStep = 1;
nit = 0;
#--- Error on p. Smoothness parameter s = 10^p
errp = 0.1;
#opt = optimset('TolX',errp);
#--- Relaxation factor RF: to speedup convergence
RF = 1 + 0.75*isweighted;
# ??
## Main iterative process
#---
if isauto:
try:
xpost = array([(0.9*log10(sMinBnd) + log10(sMaxBnd)*0.1)])
except:
array([100.])
else:
xpost = array([log10(s)])
while RobustIterativeProcess:
#--- "amount" of weights (see the function GCVscore)
aow = sum(Wtot)/noe; # 0 < aow <= 1
#---
while tol>TolZ and nit<MaxIter:
if verbose:
print 'tol',tol,'nit',nit
nit = nit+1;
DCTy = dctND(Wtot*(y-z)+z,f=dct,axis=axis);
if isauto and not remainder(log2(nit),1):
#---
# The generalized cross-validation (GCV) method is used.
# We seek the smoothing parameter s that minimizes the GCV
# score i.e. s = Argmin(GCVscore).
# Because this process is time-consuming, it is performed from
# time to time (when nit is a power of 2)
#---
# errp in here somewhere
#xpost,f,d = lbfgsb.fmin_l_bfgs_b(gcv,xpost,fprime=None,factr=10.,\
# approx_grad=True,bounds=[(log10(sMinBnd),log10(sMaxBnd))],\
# args=(Lambda,aow,DCTy,IsFinite,Wtot,y,nof,noe))
# if we have no clue what value of s to use, better span the
# possible range to get a reasonable starting point ...
# only need to do it once though. nS0 is teh number of samples used
if not s0:
ss = np.arange(nS0)*(1./(nS0-1.))*(log10(sMaxBnd)-log10(sMinBnd))+ log10(sMinBnd)
g = np.zeros_like(ss)
for i,p in enumerate(ss):
g[i] = gcv(p,Lambda,aow,DCTy,IsFinite,Wtot,y,nof,noe,smoothOrder,axis)
#print 10**p,g[i]
xpost = [np.median(ss[g==g.min()])]
#print '==============='
#print nit,tol,g.min(),xpost[0],s
#print '==============='
else:
xpost = [s0]
xpost,f,d = lbfgsb.fmin_l_bfgs_b(gcv,xpost,fprime=None,factr=10.,\
approx_grad=True,bounds=[(log10(sMinBnd),log10(sMaxBnd))],\
args=(Lambda,aow,DCTy,IsFinite,Wtot,y,nof,noe,smoothOrder,axis))
s = 10**xpost[0];
# update the value we use for the initial s estimate
s0 = xpost[0]
Gamma = 1./(1+(s*abs(Lambda))**smoothOrder);
z = RF*dctND(Gamma*DCTy,f=idct,axis=axis) + (1-RF)*z;
# if no weighted/missing data => tol=0 (no iteration)
tol = isweighted*norm(z0-z)/norm(z);
z0 = z; # re-initialization
exitflag = nit<MaxIter;
if isrobust: #-- Robust Smoothing: iteratively re-weighted process
#--- average leverage
h = sqrt(1+16.*s);
h = sqrt(1+h)/sqrt(2)/h;
h = h**N;
#--- take robust weights into account
Wtot = W*RobustWeights(y-z,IsFinite,h,weightstr);
#--- re-initialize for another iterative weighted process
isweighted = True; tol = 1; nit = 0;
#---
RobustStep = RobustStep+1;
RobustIterativeProcess = RobustStep<3; # 3 robust steps are enough.
else:
RobustIterativeProcess = False; # stop the whole process
## Warning messages
#---
if isauto:
if abs(log10(s)-log10(sMinBnd))<errp:
warning('MATLAB:smoothn:SLowerBound',\
['s = %.3f '%(s) + ': the lower bound for s '\
+ 'has been reached. Put s as an input variable if required.'])
elif abs(log10(s)-log10(sMaxBnd))<errp:
warning('MATLAB:smoothn:SUpperBound',\
['s = %.3f '%(s) + ': the upper bound for s '\
+ 'has been reached. Put s as an input variable if required.'])
#warning('MATLAB:smoothn:MaxIter',\
# ['Maximum number of iterations (%d'%(MaxIter) + ') has '\
# + 'been exceeded. Increase MaxIter option or decrease TolZ value.'])
return z,s,exitflag,Wtot
def warning(s1,s2):
print s1
print s2[0]
## GCV score
#---
#function GCVscore = gcv(p)
def gcv(p,Lambda,aow,DCTy,IsFinite,Wtot,y,nof,noe,smoothOrder,axis):
# Search the smoothing parameter s that minimizes the GCV score
#---
s = 10**p;
Gamma = 1./(1+(s*abs(Lambda))**smoothOrder);
#--- RSS = Residual sum-of-squares
if aow>0.9: # aow = 1 means that all of the data are equally weighted
# very much faster: does not require any inverse DCT
RSS = norm(DCTy*(Gamma-1.))**2;
else:
# take account of the weights to calculate RSS:
yhat = dctND(Gamma*DCTy,f=idct,axis=axis);
RSS = norm(sqrt(Wtot[IsFinite])*(y[IsFinite]-yhat[IsFinite]))**2;
#---
TrH = sum(Gamma);
GCVscore = RSS/float(nof)/(1.-TrH/float(noe))**2;
return GCVscore
## Robust weights
#function W = RobustWeights(r,I,h,wstr)
def RobustWeights(r,I,h,wstr):
# weights for robust smoothing.
MAD = median(abs(r[I]-median(r[I]))); # median absolute deviation
u = abs(r/(1.4826*MAD)/sqrt(1-h)); # studentized residuals
if wstr == 'cauchy':
c = 2.385; W = 1./(1+(u/c)**2); # Cauchy weights
elif wstr == 'talworth':
c = 2.795; W = u<c; # Talworth weights
else:
c = 4.685; W = (1-(u/c)**2)**2.*((u/c)<1); # bisquare weights
W[isnan(W)] = 0;
return W
## Initial Guess with weighted/missing data
#function z = InitialGuess(y,I)
def InitialGuess(y,I):
#-- nearest neighbor interpolation (in case of missing values)
if any(~I):
try:
from scipy.ndimage.morphology import distance_transform_edt
#if license('test','image_toolbox')
#[z,L] = bwdist(I);
L = distance_transform_edt(1-I)
z = y;
z[~I] = y[L[~I]];
except:
# If BWDIST does not exist, NaN values are all replaced with the
# same scalar. The initial guess is not optimal and a warning
# message thus appears.
z = y;
z[~I] = mean(y[I]);
else:
z = y;
# coarse fast smoothing
z = dctND(z,f=dct,axis=axis)
k = array(z.shape)
m = ceil(k/10)+1
d = []
for i in xrange(len(k)):
d.append(arange(m[i],k[i]))
d = np.array(d).astype(int)
z[d] = 0.
z = dctND(z,f=idct,axis=axis)
return z
#-- coarse fast smoothing using one-tenth of the DCT coefficients
#siz = z.shape;
#z = dct(z,norm='ortho',type=2);
#for k in np.arange(len(z.shape)):
# z[ceil(siz[k]/10)+1:-1] = 0;
# ss = tuple(roll(array(siz),1-k))
# z = z.reshape(ss)
# z = np.roll(z.T,1)
#z = idct(z,norm='ortho',type=2);
# NB: filter is 2*I - (np.roll(I,-1) + np.roll(I,1))
def dctND(data,f=dct,axis=None):
nd = len(data.shape)
axes = np.array(axis)
ff = f(data,norm='ortho',type=2,axis=axes[-1])
for i in xrange(1,len(axes)):
ff = f(ff,norm='ortho',type=2,axis=axes[-1-i])
return ff
def peaks(n):
'''
Mimic basic of matlab peaks fn
'''
xp = arange(n)
[x,y] = meshgrid(xp,xp)
z = np.zeros_like(x).astype(float)
for i in xrange(n/5):
x0 = random()*n
y0 = random()*n
sdx = random()*n/4.
sdy = sdx
c = random()*2 - 1.
f = exp(-((x-x0)/sdx)**2-((y-y0)/sdy)**2 - (((x-x0)/sdx))*((y-y0)/sdy)*c)
#f /= f.sum()
f *= random()
z += f
return z
def test1():
plt.figure(1)
plt.clf()
# 1-D example
x = linspace(0,100,2**8);
y = cos(x/10)+(x/50)**2 + randn(size(x))/10;
y[[70, 75, 80]] = [5.5, 5, 6];
z = smoothn(y)[0]; # Regular smoothing
zr = smoothn(y,isrobust=True)[0]; # Robust smoothing
subplot(121)
plot(x,y,'r.')
plot(x,z,'k')
title('Regular smoothing')
subplot(122)
plot(x,y,'r.')
plot(x,zr,'k')
title('Robust smoothing')
def test2(axis=None):
# 2-D example
plt.figure(2)
plt.clf()
xp = arange(0,1,.02)
[x,y] = meshgrid(xp,xp);
f = exp(x+y) + sin((x-2*y)*3);
fn = f + (randn(f.size)*0.5).reshape(f.shape);
fs = smoothn(fn,axis=axis)[0];
subplot(121); plt.imshow(fn,interpolation='Nearest');# axis square
subplot(122); plt.imshow(fs,interpolation='Nearest'); # axis square
def test3(axis=None):
# 2-D example with missing data
plt.figure(3)
plt.clf()
n = 256;
y0 = peaks(n);
y = (y0 + random(shape(y0))*2 - 1.0).flatten();
I = np.random.permutation(range(n**2));
y[I[1:n**2*0.5]] = nan; # lose 50% of data
y = y.reshape(y0.shape)
y[40:90,140:190] = nan; # create a hole
yData = y.copy()
z0,s,exitflag,Wtot = smoothn(yData,axis=axis); # smooth data
yData = y.copy()
z,s,exitflag,Wtot = smoothn(yData,isrobust=True,axis=axis); # smooth data
y = yData
vmin = np.min([np.min(z),np.min(z0),np.min(y),np.min(y0)])
vmax = np.max([np.max(z),np.max(z0),np.max(y),np.max(y0)])
subplot(221); plt.imshow(y,interpolation='Nearest',vmin=vmin,vmax=vmax);
title('Noisy corrupt data')
subplot(222); plt.imshow(z0,interpolation='Nearest',vmin=vmin,vmax=vmax);
title('Recovered data #1')
subplot(223); plt.imshow(z,interpolation='Nearest',vmin=vmin,vmax=vmax);
title('Recovered data #2')
subplot(224); plt.imshow(y0,interpolation='Nearest',vmin=vmin,vmax=vmax);
title('... compared with original data')
def test4(i=10,step=0.2,axis=None):
[x,y,z] = mgrid[-2:2:step,-2:2:step,-2:2:step]
x = array(x);y=array(y);z=array(z)
xslice = [-0.8,1]; yslice = 2; zslice = [-2,0];
v0 = x*exp(-x**2-y**2-z**2)
vn = v0 + randn(x.size).reshape(x.shape)*0.06
v = smoothn(vn)[0];
plt.figure(4)
plt.clf()
vmin = np.min([np.min(v[:,:,i]),np.min(v0[:,:,i]),np.min(vn[:,:,i])])
vmax = np.max([np.max(v[:,:,i]),np.max(v0[:,:,i]),np.max(vn[:,:,i])])
subplot(221); plt.imshow(v0[:,:,i],interpolation='Nearest',vmin=vmin,vmax=vmax);
title('clean z=%d'%i)
subplot(223); plt.imshow(vn[:,:,i],interpolation='Nearest',vmin=vmin,vmax=vmax);
title('noisy')
subplot(224); plt.imshow(v[:,:,i],interpolation='Nearest',vmin=vmin,vmax=vmax);
title('cleaned')
def test5():
t = linspace(0,2*pi,1000);
x = 2*cos(t)*(1-cos(t)) + randn(size(t))*0.1;
y = 2*sin(t)*(1-cos(t)) + randn(size(t))*0.1;
zx = smoothn(x)[0];
zy = smoothn(y)[0];
plt.figure(5)
plt.clf()
plt.title('Cardioid')
plot(x,y,'r.')
plot(zx,zy,'k')
def test6(noise=0.05,nout=30):
plt.figure(6)
plt.clf()
[x,y] = meshgrid(linspace(0,1,24),linspace(0,1,24))
Vx0 = cos(2*pi*x+pi/2)*cos(2*pi*y);
Vy0 = sin(2*pi*x+pi/2)*sin(2*pi*y);
Vx = Vx0 + noise*randn(24,24); # adding Gaussian noise
Vy = Vy0 + noise*randn(24,24); # adding Gaussian noise
I = np.random.permutation(range(Vx.size))
Vx = Vx.flatten()
Vx[I[0:nout]] = (rand(nout,1)-0.5)*5; # adding outliers
Vx = Vx.reshape(Vy.shape)
Vy = Vy.flatten()
Vy[I[0:nout]] = (rand(nout,1)-0.5)*5; # adding outliers
Vy = Vy.reshape(Vx.shape)
Vsx = smoothn(Vx,isrobust=True)[0];
Vsy = smoothn(Vy,isrobust=True)[0];
subplot(131);quiver(x,y,Vx,Vy,2.5)
title('Noisy')
subplot(132); quiver(x,y,Vsx,Vsy)
title('Recovered')
subplot(133); quiver(x,y,Vx0,Vy0)
title('Original')
def sparseSVD(D):
import scipy.sparse
try:
import sparsesvd
except:
print 'bummer ... better get sparsesvd'
exit(0)
Ds = scipy.sparse.csc_matrix(D)
a = sparsesvd.sparsesvd(Ds,Ds.shape[0])
return a
def sparseTest(n=1000):
I = np.identity(n)
# define a 'traditional' D1 matrix
# which is a right-side difference
# and which is *not* symmetric :-(
D1 = np.matrix(I - np.roll(I,1))
# so define a symemtric version
D1a = D1.T - D1
U, s, Vh = scipy.linalg.svd(D1a)
# now, get eigenvectors for D1a
Ut,eigenvalues,Vt = sparseSVD(D1a)
Ut = np.matrix(Ut)
# then, an equivalent 2nd O term would be
D2a = D1a**2
# show we can recover D1a
D1a_est = Ut.T * np.diag(eigenvalues) * Ut
# Now, because D2a (& the target D1a) are symmetric:
D1a_est = Ut.T * np.diag(eigenvalues**0.5) * Ut
D = 2*I - (np.roll(I,-1) + np.roll(I,1))
a = sparseSVD(-D)
eigenvalues = np.matrix(a[1])
Ut = np.matrix(a[0])
Vt = np.matrix(a[2])
orig = (Ut.T * np.diag(np.array(eigenvalues).flatten()) * Vt)
Feigenvalues = np.diag(np.array(np.c_[eigenvalues,0]).flatten())
FUt = np.c_[Ut.T,np.zeros(Ut.shape[1])]
# confirm: FUt * Feigenvalues * FUt.T ~= D
# m is a 1st O difference matrix
# with careful edge conditions
# such that m.T * m = D2
# D2 being a 2nd O difference matrix
m = np.matrix(np.identity(100) - np.roll(np.identity(100),1))
m[-1,-1] = 0
m[0,0] = 1
a = sparseSVD(m)
eigenvalues = np.matrix(a[1])
Ut = np.matrix(a[0])
Vt = np.matrix(a[2])
orig = (Ut.T * np.diag(np.array(eigenvalues).flatten()) * Vt)
# Vt* Vt.T = I
# Ut.T * Ut = I
# ((Vt.T * (np.diag(np.array(eigenvalues).flatten())**2)) * Vt)
# we see you get the same as m.T * m by squaring the eigenvalues