Second initial import
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@ -201,16 +201,16 @@ def eigen(A,k=6,M=None,ncv=None,which='LM',
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dr=sb.zeros(k+1,typ)
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di=sb.zeros(k+1,typ)
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zr=sb.zeros((n,k+1),typ)
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dr,di,z,info=\
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dr,di,zr,info=\
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eigextract(rvec,howmny,sselect,sigmar,sigmai,workev,
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bmat,which,k,tol,resid,v,iparam,ipntr,
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workd,workl,info)
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# make eigenvalues complex
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d=dr+1.0j*di
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d=(dr+1.0j*di)[:k]
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# futz with the eigenvectors:
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# complex are stored as real,imaginary in consecutive columns
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z=zr.astype(typ.upper())
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z=zr.astype(typ.upper())[:,:k]
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for i in range(k): # fix c.c. pairs
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if di[i] > 0 :
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z[:,i]=zr[:,i]+1.0j*zr[:,i+1]
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@ -1,5 +1,6 @@
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from crossvalidation import lpls_val
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from statistics import lpls_qvals
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from engines import pca, pcr, pls
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from engines import nipals_lpls as lpls
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#from tests import *
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@ -2,11 +2,12 @@
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The primary use is crossvalidation.
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"""
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__all__ = ['lpls_val', 'lpls_jk']
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__all__ = ['lpls_val', 'pls_jk', 'lpls_jk']
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__docformat__ = "restructuredtext en"
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from numpy import dot,empty,zeros,sqrt,atleast_2d,argmax,asarray,median,\
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array_split
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array_split,arange
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from numpy.random import shuffle
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from engines import nipals_lpls as lpls
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@ -93,8 +94,92 @@ def lpls_val(X, Y, Z, a_max=2, nsets=None,alpha=.5, mean_ctr=[2,0,2],verbose=Tru
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return rmsep, Yhat, aopt
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def pca_jk(a, aopt, n_blocks=None):
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"""Returns jack-knife segements from PCA.
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Parameters:
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def lpls_jk(X, Y, Z, a_max, nsets=None, xz_alpha=.5, mean_ctr=[2,0,2], verbose=False):
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a : {array}
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data matrix (n x m)
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aopt : {integer}
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number of components in model.
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nsets : {integer}
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number of segments
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Returns:
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Pcv : {array}
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Loadings collected in a three way matrix (n_segments, m, aopt)
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Notes:
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- The loadings are scaled with the (1/samples)*eigenvalues.
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- Crossvalidation method is currently set to random blocks of samples.
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- todo: add support for T
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- fixme: more efficient to add this in validation loop?
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"""
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if nsets == None:
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nsets = a.shape[0]
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Pcv = empty((nsets, a.shape[1], aopt), dtype=a.dtype)
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mn_a = .5*(a.mean(0) + a.mean(1)[:,newaxis])
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for i, (cal, val) in enumerate(cv_diag(a.shape, nsets)):
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old_values = a.take(ind)
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new_values = mn_a.take(ind)
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a.put(ind, new_values)
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dat = pca(a, aopt, mode='fast', scale='loads', center=center)
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PP[nn,:,:] = dat['P']
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a.put(ind, old_values)
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return PP
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def pls_jk(X, Y, a_opt, nsets=None, center=True, verbose=False):
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""" Returns jack-knife segements of W.
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*Parameters*:
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X : {array}
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Main data matrix (m, n)
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Y : {array}
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External row data (m, l)
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a_opt : {integer}
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The number of components to calculate (0, min(m,n))
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nsets : (integer), optional
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Number of jack-knife segments
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center : {boolean}, optional
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- -1 : nothing
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- 0 : row center
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- 1 : column center
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- 2 : double center
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verbose : {boolean}, optional
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Verbosity of console output. For use in debugging.
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*Returns*:
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Wcv : {array}
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Loading-weights jack-knife segements
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"""
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m, n = X.shape
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k, l = Y.shape
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assert(m==k)
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if nsets == None:
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nsets = X.shape[0]
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Wcv = empty((nsets, X.shape[1], a_opt), dtype=X.dtype)
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for i, (cal, val) in enumerate(cv(k, nsets)):
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if verbose:
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print "Segement number: %d" %i
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dat = pls(X[cal,:], Y[cal,:], a_opt, scale='loads', mode='fast')
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Wcv[nn,:,:] = dat['W']
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return Wcv
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def lpls_jk(X, Y, Z, a_opt, nsets=None, xz_alpha=.5, mean_ctr=[2,0,2], verbose=False):
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"""Returns jack-knifed segments of lpls model.
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Jack-knifing is a method to perturb the model paramters, hopefully
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@ -113,8 +198,8 @@ def lpls_jk(X, Y, Z, a_max, nsets=None, xz_alpha=.5, mean_ctr=[2,0,2], verbose=F
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External row data (m, l)
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Z : {array}
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External column data (n, o)
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a_max : {integer}, optional
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Maximum number of components to calculate (0, min(m,n))
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a_opt : {integer}, optional
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The number of components to calculate (0, min(m,n))
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nsets : (integer), optional
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Number of jack-knife segments
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xz_alpha : {float}, optional
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@ -145,11 +230,11 @@ def lpls_jk(X, Y, Z, a_max, nsets=None, xz_alpha=.5, mean_ctr=[2,0,2], verbose=F
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assert(n==p)
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if nsets==None:
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nsets = m
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WWx = empty((nsets, n, a_max), 'd')
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WWz = empty((nsets, o, a_max), 'd')
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#WWy = empty((nsets, l, a_max), 'd')
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WWx = empty((nsets, n, a_opt), 'd')
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WWz = empty((nsets, o, a_opt), 'd')
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#WWy = empty((nsets, l, a_opt), 'd')
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for i, (cal, val) in enumerate(cv(k, nsets)):
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dat = lpls(X[cal],Y[cal],Z,a_max=a_max,alpha=xz_alpha,mean_ctr=mean_ctr,
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dat = lpls(X[cal], Y[cal], Z, a_max=a_opt,alpha=xz_alpha, mean_ctr=mean_ctr,
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scale='loads', verbose=verbose)
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WWx[i,:,:] = dat['W']
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WWz[i,:,:] = dat['L']
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@ -219,6 +304,7 @@ def cv(N, K, randomise=True, sequential=False):
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*Notes*:
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If randomise is true, a copy of index is shuffled before partitioning,
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otherwise its order is preserved in training and validation.
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Randomise overrides the sequential argument. If randomise is true,
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@ -245,13 +331,40 @@ def cv(N, K, randomise=True, sequential=False):
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validation = [i for i in index if i % K == k]
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yield training, validation
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def diag_cv(shape, nsets=9):
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"""Generates K (training, validation) index pairs.
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Parameters:
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N : {integer}
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alpha -- scalar, approx. portion of data perturbed
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"""
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try:
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m, n = shape
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except:
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raise ValueError("shape needs to be a two-tuple")
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if nsets>m or nsets>n:
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msg = "You may not use more subsets than max(n_rows, n_cols)"
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raise ValueError, msg
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nm = n*m
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index = arange(nm)
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n_ind = arange(n)
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shuffle(n_ind) # random start diag
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start_inds = array_split(n_ind, nsets)
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for v in range(nsets):
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validation = []
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for start in start_inds[v]:
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ind = arange(start+v, nm, n+1)
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[validation.append(i) for i in ind]
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training = [j for j in index if j not in validation]
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yield training, validation
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def class_error(Yhat, Y, method='vanilla'):
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""" Not used.
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"""
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a_max, k, l = Yhat.shape
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a_opt, k, l = Yhat.shape
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Yhat_c = zeros((k, l), dtype='d')
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for a in range(a_max):
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for a in range(a_opt):
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for i in range(k):
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Yhat_c[a,val,argmax(Yhat[a,val,:])] = 1.0
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err = 100*((Yhat_c + Y) == 2).sum(1)/Y.sum(0).astype('d')
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475
pyblm/engines.py
475
pyblm/engines.py
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@ -2,14 +2,414 @@
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"""
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__all__ = ['nipals_lpls']
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__all__ = ['pca', 'pcr', 'pls', 'nipals_lpls']
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__docformat__ = "restructuredtext en"
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from math import sqrt as msqrt
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from numpy import dot,empty,zeros,apply_along_axis,newaxis,finfo
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from numpy.linalg import inv
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from numpy import dot,empty,zeros,apply_along_axis,newaxis,finfo,sqrt,r_,expand_dims,\
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minimum
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from numpy.linalg import inv, svd
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from scipy.sandbox import arpack
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def pca(X, aopt, scale='scores', mode='normal', center_axis=0):
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""" Principal Component Analysis.
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PCA is a low rank bilinear aprroximation to a data matrix that sequentially
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extracts orthogonal components of maximum variance.
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Parameters:
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X : {array}
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Data measurement matrix, (samples x variables)
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aopt : {integer}
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Number of components to use, aopt<=min(samples, variables)
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center_axis : {integer}
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Center along given axis. If neg.: no centering (-inf,..., matrix modes)
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Returns:
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T : {array}
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Scores, (samples, components)
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P : {array}
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Loadings, (variables, components)
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E : {array}
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Residuals, (samples, variables)
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evx : {array}
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X-explained variance, (components,)
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mnx : {array}
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X location, (variables,)
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aopt : {integer}
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The number of components used
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ssqx : {list}
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Sum of squared residuals in X along each dimesion
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[(samples, ), (variables,)]
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leverage : {array}
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Leverages, (samples,)
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OtherParameters:
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scale : {string}, optional
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Where to put the weights [['scores'], 'loadings']
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mode : {string}, optional
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Amount of info retained, [['normal'], 'fast', 'detailed']
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:SeeAlso:
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`center` : Data centering
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*Notes*
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Uses kernel speed-up if m>>n or m<<n.
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If residuals turn rank deficient, a lower number of component than given
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in input will be used.
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*Examples*:
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>>> import scipy,engines
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>>> a=scipy.asarray([[1,2,3],[2,4,5]])
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>>> dat=engines.pca(a, 2)
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>>> dat['evx']
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array([0.,99.8561562, 100.])
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"""
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m, n = X.shape
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assert(aopt<=min(m,n))
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if center_axis>=0:
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X = X - expand_dims(X.mean(center_axis), center_axis)
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if m>(n+100) or n>(m+100):
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u, s, v = esvd(X, aopt)
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else:
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u, s, vt = svd(X, 0)
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v = vt.T
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u = u[:,:aopt]
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s = s[:aopt]
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v = v[:,:aopt]
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# ranktest
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tol = 1e-10
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eff_rank = sum(s>s[0]*tol)
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aopt = minimum(aopt, eff_rank)
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T = u*s
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s = s[:aopt]
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T = T[:,:aopt]
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P = v[:,:aopt]
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e = s**2
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if scale=='loads':
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T = T/s
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P = P*s
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if mode == 'fast':
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return {'T':T, 'P':P, 'aopt':aopt}
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if mode=='detailed':
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E = empty((aopt, m, n))
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ssq = []
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lev = []
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for ai in range(aopt):
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E[ai,:,:] = X - dot(T[:,:ai+1], P[:,:ai+1].T)
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ssq.append([(E[ai,:,:]**2).mean(0), (E[ai,:,:]**2).mean(1)])
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if scale=='loads':
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lev.append([((s*T)**2).sum(1), (P**2).sum(1)])
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else:
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lev.append([(T**2).sum(1), ((s*P)**2).sum(1)])
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else:
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# residuals
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E = X - dot(T, P.T)
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sep = E**2
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ssq = [sep.sum(0), sep.sum(1)]
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# leverages
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if scale=='loads':
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lev = [(1./m)+(T**2).sum(1), (1./n)+((P/s)**2).sum(1)]
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else:
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lev = [(1./m)+((T/s)**2).sum(1), (1./n)+(P**2).sum(1)]
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# variances
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expvarx = r_[0, 100*e.cumsum()/(X*X).sum()]
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return {'T': T, 'P': P, 'E': E, 'evx': expvarx, 'leverage': lev, 'ssqx': ssq,
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'aopt': aopt, 'eigvals': e}
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def pcr(a, b, aopt, scale='scores',mode='normal',center_axis=0):
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""" Principal Component Regression.
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Performs PCR on given matrix and returns results in a dictionary.
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Parameters:
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a : array
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Data measurement matrix, (samples x variables)
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b : array
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Data response matrix, (samples x responses)
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aopt : int
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Number of components to use, aopt<=min(samples, variables)
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Returns:
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results : dict
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keys -- values, T -- scores, P -- loadings, E -- residuals,
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levx -- leverages, ssqx -- sum of squares, expvarx -- cumulative
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explained variance, aopt -- number of components used
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OtherParameters:
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mode : str
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Amount of info retained, ('fast', 'normal', 'detailed')
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center_axis : int
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Center along given axis. If neg.: no centering (-inf,..., matrix modes)
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SeeAlso:
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- pca : other blm
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- pls : other blm
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- lpls : other blm
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*Notes*
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-----
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Uses kernel speed-up if m>>n or m<<n.
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If residuals turn rank deficient, a lower number of component than given
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in input will be used. The number of components used is given in results-dict.
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Examples
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--------
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>>> import scipy,engines
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>>> a=scipy.asarray([[1,2,3],[2,4,5]])
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>>> b=scipy.asarray([[1,1],[2,3]])
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>>> dat=engines.pcr(a, 2)
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>>> dat['evx']
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array([0.,99.8561562, 100.])
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"""
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try:
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k, l = b.shape
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except:
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b = atleast_2d(b).T
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k, l = b.shape
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if center_axis>=0:
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b = b - expand_dims(b.mean(center_axis), center_axis)
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dat = pca(a, aopt=aopt, scale=scale, mode=mode, center_axis=center_axis)
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T = dat['T']
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weights = apply_along_axis(vnorm, 0, T)**2
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if scale=='loads':
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Q = dot(b.T, T*weights)
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else:
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Q = dot(b.T, T/weights)
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if mode=='fast':
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dat.update({'Q':Q})
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return dat
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if mode=='detailed':
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F = empty((aopt, k, l))
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ssqy = []
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for i in range(aopt):
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F[i,:,:] = b - dot(T[:,:i+1], Q[:,:i+1].T)
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ssqy.append([(F[i,:,:]**2).mean(0), (F[i,:,:]**2).mean(1)])
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else:
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F = b - dot(T, Q.T)
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sepy = F**2
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ssqy = [sepy.sum(0), sepy.sum(1)]
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expvary = r_[0, 100*((T**2).sum(0)*(Q**2).sum(0)/(b**2).sum()).cumsum()[:aopt]]
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dat.update({'Q': Q, 'F': F, 'evy': expvary, 'ssqy': ssqy})
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return dat
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def pls(X, Y, aopt=2, scale='scores', mode='normal', center_axis=-1):
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"""Partial Least Squares Regression.
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Performs PLS on given matrix and returns results in a dictionary.
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*Parameters*:
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X : {array}
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Data measurement matrix, (samples x variables)
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Y : {array}
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Data response matrix, (samples x responses)
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aopt : {integer}, optional
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Number of components to use, aopt<=min(samples, variables)
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scale : ['scores', 'loadings'], optional
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Which component should get the scale
|
||||
center_axis : {-1, integer}
|
||||
Perform centering across given axis, (-1 is no centering)
|
||||
|
||||
*Returns*:
|
||||
|
||||
T : {array}
|
||||
X-scores
|
||||
W : {array}
|
||||
X-loading-weights
|
||||
P : {array}
|
||||
X-loadings
|
||||
R : {array}
|
||||
X-loadings-basis
|
||||
Q : {array}
|
||||
Y-loadings
|
||||
B : {array}
|
||||
Regression coefficients
|
||||
E : {array}
|
||||
X-block residuals
|
||||
F : {array}
|
||||
Y-block residuals
|
||||
evx : {array}
|
||||
X-explained variance
|
||||
evy : {array}
|
||||
Y-explained variance
|
||||
mnx : {array}
|
||||
X location
|
||||
mny : {array}
|
||||
Y location
|
||||
aopt : {array}
|
||||
The number of components used
|
||||
ssqx : {list}, optional
|
||||
Sum of squared residuals in X along each dimesion
|
||||
ssqy : {list}
|
||||
Sum of squared residuals in Y along each dimesion
|
||||
leverage : {array}
|
||||
Sample leverages
|
||||
|
||||
*OtherParameters*:
|
||||
|
||||
mode : ['normal', 'fast', 'detailed'], optional
|
||||
How much details to compute
|
||||
|
||||
*SeeAlso*:
|
||||
|
||||
`center` : data centering
|
||||
|
||||
*Notes*
|
||||
|
||||
- The output with mode='fast' will only return T and W
|
||||
|
||||
- If residuals turn rank deficient, a lower number of component than given in input will be used. The number of components used is given in results.
|
||||
|
||||
|
||||
*Examples*
|
||||
|
||||
>>> import numpy, engines
|
||||
>>> a = numpy.asarray([[1,2,3],[2,4,5]])
|
||||
>>> b = numpy.asarray([[1,1],[2,3]])
|
||||
>>> dat =engines.pls(a, b, 2)
|
||||
>>> dat['evx']
|
||||
array([0.,99.8561562, 100.])
|
||||
|
||||
"""
|
||||
|
||||
m, n = X.shape
|
||||
try:
|
||||
k, l = Y.shape
|
||||
except:
|
||||
Y = atleast_2d(Y).T
|
||||
k, l = Y.shape
|
||||
assert(m==k)
|
||||
assert(aopt<min(m, n))
|
||||
mnx, mny = 0,0
|
||||
if center_axis>=0:
|
||||
mnx = expand_dims(X.mean(center_axis), center_axis)
|
||||
X = X - mnx
|
||||
mny = expand_dims(Y.mean(center_axis), center_axis)
|
||||
Y = Y - mny
|
||||
|
||||
W = empty((n, aopt))
|
||||
P = empty((n, aopt))
|
||||
R = empty((n, aopt))
|
||||
Q = empty((l, aopt))
|
||||
T = empty((m, aopt))
|
||||
B = empty((aopt, n, l))
|
||||
tt = empty((aopt,))
|
||||
|
||||
XY = dot(X.T, Y)
|
||||
for i in range(aopt):
|
||||
if XY.shape[1]==1: #pls 1
|
||||
w = XY.reshape(n, l)
|
||||
w = w/vnorm(w)
|
||||
elif n<l: # more yvars than xvars
|
||||
s, w = arpack.eigen_symmetric(dot(XY, XY.T),k=1, tol=1e-10, maxiter=100)
|
||||
#w, s, vh = svd(dot(XY, XY.T))
|
||||
#w = w[:,:1]
|
||||
else: # more xvars than yvars
|
||||
s, q = arpack.eigen_symmetric(dot(XY.T, XY), k=1, tol=1e-10, maxiter=100)
|
||||
#q, s, vh = svd(dot(XY.T, XY))
|
||||
#q = q[:,:1]
|
||||
|
||||
w = dot(XY, q)
|
||||
w = w/vnorm(w)
|
||||
r = w.copy()
|
||||
if i>0:
|
||||
for j in range(0, i, 1):
|
||||
r = r - dot(P[:,j].T, w)*R[:,j][:,newaxis]
|
||||
|
||||
t = dot(X, r)
|
||||
tt[i] = tti = dot(t.T, t).ravel()
|
||||
p = dot(X.T, t)/tti
|
||||
q = dot(r.T, XY).T/tti
|
||||
XY = XY - dot(p, q.T)*tti
|
||||
T[:,i] = t.ravel()
|
||||
W[:,i] = w.ravel()
|
||||
|
||||
if mode=='fast' and i==aopt-1:
|
||||
if scale=='loads':
|
||||
tnorm = sqrt(tt)
|
||||
T = T/tnorm
|
||||
W = W*tnorm
|
||||
return {'T':T, 'W':W}
|
||||
|
||||
P[:,i] = p.ravel()
|
||||
R[:,i] = r.ravel()
|
||||
Q[:,i] = q.ravel()
|
||||
B[i] = dot(R[:,:i+1], Q[:,:i+1].T)
|
||||
|
||||
qnorm = apply_along_axis(vnorm, 0, Q)
|
||||
tnorm = sqrt(tt)
|
||||
pp = (P**2).sum(0)
|
||||
if mode=='detailed':
|
||||
E = empty((aopt, m, n))
|
||||
F = empty((aopt, k, l))
|
||||
ssqx, ssqy = [], []
|
||||
leverage = empty((aopt, m))
|
||||
#h2x = [] #hotellings T^2
|
||||
#h2y = []
|
||||
for ai in range(aopt):
|
||||
E[ai,:,:] = X - dot(T[:,:ai+1], P[:,:ai+1].T)
|
||||
F[ai,:,:] = Y - dot(T[:,:i], Q[:,:i].T)
|
||||
ssqx.append([(E[ai,:,:]**2).mean(0), (E[ai,:,:]**2).mean(1)])
|
||||
ssqy.append([(F[ai,:,:]**2).mean(0), (F[ai,:,:]**2).mean(1)])
|
||||
leverage[ai,:] = 1./m + ((T[:,:ai+1]/tnorm[:ai+1])**2).sum(1)
|
||||
#h2y.append(1./k + ((Q[:,:ai+1]/qnorm[:ai+1])**2).sum(1))
|
||||
else:
|
||||
# residuals
|
||||
E = X - dot(T, P.T)
|
||||
F = Y - dot(T, Q.T)
|
||||
sepx = E**2
|
||||
ssqx = [sepx.sum(0), sepx.sum(1)]
|
||||
sepy = F**2
|
||||
ssqy = [sepy.sum(0), sepy.sum(1)]
|
||||
leverage = 1./m + ((T/tnorm)**2).sum(1)
|
||||
|
||||
# variances
|
||||
tp= tt*pp
|
||||
tq = tt*qnorm*qnorm
|
||||
expvarx = r_[0, 100*tp/(X*X).sum()]
|
||||
expvary = r_[0, 100*tq/(Y*Y).sum()]
|
||||
|
||||
if scale=='loads':
|
||||
T = T/tnorm
|
||||
W = W*tnorm
|
||||
Q = Q*tnorm
|
||||
P = P*tnorm
|
||||
|
||||
return {'Q': Q, 'P': P, 'T': T, 'W': W, 'R': R, 'E': E, 'F': F, 'B': B,
|
||||
'evx': expvarx, 'evy': expvary, 'ssqx': ssqx, 'ssqy': ssqy,
|
||||
'leverage': leverage, 'mnx': mnx, 'mny': mny}
|
||||
|
||||
def nipals_lpls(X, Y, Z, a_max, alpha=.7, mean_ctr=[2, 0, 1], scale='scores', verbose=False):
|
||||
""" L-shaped Partial Least Sqaures Regression by the nipals algorithm.
|
||||
|
|
@ -38,11 +438,7 @@ def nipals_lpls(X, Y, Z, a_max, alpha=.7, mean_ctr=[2, 0, 1], scale='scores', ve
|
|||
-1 : nothing
|
||||
0 : row center
|
||||
1 : column center
|
||||
2 : double center
|
||||
scale : {'scores', 'loads'}, optional
|
||||
Option to decide on where the scale goes.
|
||||
verbose : {boolean}, optional
|
||||
Verbosity of console output. For use in debugging.
|
||||
2 : double center
|
||||
|
||||
*Returns*:
|
||||
|
||||
|
|
@ -75,6 +471,13 @@ def nipals_lpls(X, Y, Z, a_max, alpha=.7, mean_ctr=[2, 0, 1], scale='scores', ve
|
|||
mnz : {array}
|
||||
Z location
|
||||
|
||||
*OtherParameters*:
|
||||
|
||||
scale : {'scores', 'loads'}, optional
|
||||
Option to decide on where the scale goes.
|
||||
verbose : {boolean}, optional
|
||||
Verbosity of console output. For use in debugging.
|
||||
|
||||
*References*
|
||||
Saeboe et al., LPLS-regression: a method for improved prediction and
|
||||
classification through inclusion of background information on
|
||||
|
|
@ -296,3 +699,59 @@ def _scale(a, axis):
|
|||
raise IOError("input error: axis must be in [-1,0,1]")
|
||||
|
||||
return a - sc, sc
|
||||
|
||||
def esvd(data, a_max=None):
|
||||
""" SVD with kernel calculation
|
||||
|
||||
Calculate subspaces of X'X or XX' depending on the shape
|
||||
of the matrix.
|
||||
|
||||
Parameters:
|
||||
|
||||
data : {array}
|
||||
Data matrix
|
||||
a_max : {integer}
|
||||
Number of components to extract
|
||||
|
||||
Returns:
|
||||
|
||||
u : {array}
|
||||
Right hand eigenvectors
|
||||
s : {array}
|
||||
Singular values
|
||||
v : {array}
|
||||
Left hand eigenvectors
|
||||
|
||||
notes:
|
||||
|
||||
Uses Anoldi iterations (ARPACK)
|
||||
|
||||
"""
|
||||
|
||||
m, n = data.shape
|
||||
if m>=n:
|
||||
kernel = dot(data.T, data)
|
||||
|
||||
if a_max==None:
|
||||
a_max = n - 1
|
||||
s, v = arpack.eigen_symmetric(kernel,k=a_max, which='LM',
|
||||
maxiter=200,tol=1e-5)
|
||||
s = s[::-1]
|
||||
v = v[:,::-1]
|
||||
#u, s, vt = svd(kernel)
|
||||
#v = vt.T
|
||||
s = sqrt(s)
|
||||
u = dot(data, v)/s
|
||||
else:
|
||||
kernel = dot(data, data.T)
|
||||
if a_max==None:
|
||||
a_max = m -1
|
||||
s, u = arpack.eigen_symmetric(kernel,k=a_max, which='LM',
|
||||
maxiter=200,tol=1e-5)
|
||||
s = s[::-1]
|
||||
u = u[:,::-1]
|
||||
#u, s, vt = svd(kernel)
|
||||
s = sqrt(s)
|
||||
v = dot(data.T, u)/s
|
||||
|
||||
return u, s, v
|
||||
|
|
|
|||
Loading…
Reference in New Issue