417 lines
11 KiB
Python
417 lines
11 KiB
Python
"""Module contain algorithms for low-rank models.
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There is almost no typechecking of any kind here, just focus on speed
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"""
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import math
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from scipy.linalg import svd,inv
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from scipy import dot,empty,eye,newaxis,zeros,sqrt,diag,\
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apply_along_axis,mean,ones,randn,empty_like,outer,c_,\
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rand,sum,cumsum,matrix
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has_sym=True
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try:
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import symmeig
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except:
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has_sym = False
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def pca(a, aopt, scale='scores', mode='normal'):
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""" Principal Component Analysis model
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mode:
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-- fast : returns smallest dim scaled (T for n<=m, P for n>m )
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-- normal : returns all model params and residuals after aopt comp
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-- detailed : returns all model params and all residuals
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"""
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m, n = a.shape
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#print "rows: %s cols: %s" %(m,n)
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if m>(n+100) or n>(m+100):
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u, s, v = esvd(a)
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else:
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u, s, vt = svd(a, 0)
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v = vt.T
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eigvals = (1./m)*s
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T = u*s
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T = T[:,:aopt]
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P = v[:,:aopt]
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if scale=='loads':
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tnorm = apply_along_axis(vnorm, 0, T)
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T = T/tnorm
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P = P*tnorm
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if mode == 'fast':
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return {'T':T, 'P':P}
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if mode=='detailed':
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"""Detailed mode returns residual matrix for all comp.
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That is E, is a three-mode matrix: (amax, m, n) """
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E = empty((aopt, m, n))
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for ai in range(aopt):
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e = a - dot(T[:,:ai+1], P[:,:ai+1].T)
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E[ai,:,:] = e.copy()
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else:
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E = a - dot(T,P.T)
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return {'T':T, 'P':P, 'E':E}
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def pcr(a, b, aopt=2, scale='scores', mode='normal'):
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"""Principal Component Regression.
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Returns
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"""
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m, n = m_shape(a)
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B = empty((aopt, n, l))
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dat = pca(a, aopt=aopt, scale=scale, mode='normal', center_axis=0)
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T = dat['T']
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weigths = apply_along_axis(vnorm, 0, T)
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if scale=='loads':
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# fixme: check weights
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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**2)
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if mode=='fast':
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return {'T', T:, 'P':P, 'Q':Q}
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if mode=='detailed':
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for i in range(1, aopt+1, 1):
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F[i,:,:] = b - dot(T[:,i],Q[:,:i].T)
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else:
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F = b - dot(T, Q.T)
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#fixme: explained variance in Y + Y-var leverages
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dat.update({'Q',Q, 'F':F})
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return dat
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def pls(a, b, aopt=2, scale='scores', mode='normal', ab=None):
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"""Partial Least Squares Regression.
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Applies plsr to given matrices and returns results in a dictionary.
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Fast pls for calibration. Only inefficient for many Y-vars.
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"""
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m, n = a.shape
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if ab!=None:
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mm, ll = m_shape(ab)
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else:
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k, l = m_shape(b)
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assert(m==mm)
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assert(l==ll)
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W = empty((n, aopt))
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P = empty((n, aopt))
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R = empty((n, aopt))
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Q = empty((l, aopt))
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T = empty((m, aopt))
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B = empty((aopt, n, l))
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if ab==None:
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ab = dot(a.T, b)
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for i in range(aopt):
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if ab.shape[1]==1:
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w = ab.reshape(n, l)
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else:
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u, s, vh = svd(dot(ab.T, ab))
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w = dot(ab, u[:,:1])
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w = w/vnorm(w)
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r = w.copy()
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if i>0: # recursive estimate to
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for j in range(0, i, 1):
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r = r - dot(P[:,j].T, w)*R[:,j][:,newaxis]
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t = dot(a, r)
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tt = vnorm(t)**2
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p = dot(a.T, t)/tt
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q = dot(r.T, ab).T/tt
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ab = ab - dot(p, q.T)*tt
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T[:,i] = t.ravel()
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W[:,i] = w.ravel()
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P[:,i] = p.ravel()
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R[:,i] = r.ravel()
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if mode=='fast' and i==aopt-1:
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if scale=='loads':
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tnorm = apply_along_axis(vnorm, 0, T)
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T = T/tnorm
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W = W*tnorm
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return {'T':T, 'W':W}
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Q[:,i] = q.ravel()
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B[i] = dot(R[:,:i+1], Q[:,:i+1].T)
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if mode=='detailed':
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E = empty((aopt, m, n))
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F = empty((aopt, k, l))
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for i in range(1, aopt+1, 1):
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E[i-1] = a - dot(T[:,:i], P[:,:i].T)
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F[i-1] = b - dot(T[:,:i], Q[:,:i].T)
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else:
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E = a - dot(T[:,:aopt], P[:,:aopt].T)
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F = b - dot(T[:,:aopt], Q[:,:aopt].T)
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if scale=='loads':
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tnorm = apply_along_axis(vnorm, 0, T)
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T = T/tnorm
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W = W*tnorm
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Q = Q*tnorm
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P = P*tnorm
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return {'B':B, 'Q':Q, 'P':P, 'T':T, 'W':W, 'R':R, 'E':E, 'F':F}
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def w_simpls(aat, b, aopt):
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""" Simpls for wide matrices.
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Fast pls for crossval, used in calc rmsep for wide X
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There is no P or W. T is normalised
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"""
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bb = b.copy()
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m, m = aat.shape
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U = empty((m, aopt))
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T = empty((m, aopt))
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H = empty((m, aopt)) #just like W in simpls
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PROJ = empty((m, aopt)) #just like R in simpls
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for i in range(aopt):
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u, s, vh = svd(dot(dot(b.T, aat), b), full_matrices=0)
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u = dot(b, u[:,:1]) #y-factor scores
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U[:,i] = u.ravel()
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t = dot(aat, u)
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t = t/vnorm(t)
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T[:,i] = t.ravel()
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h = dot(aat, t) #score-weights
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H[:,i] = h.ravel()
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PROJ[:,:i+1] = dot(T[:,:i+1], inv(dot(T[:,:i+1].T, H[:,:i+1])) )
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if i<aopt:
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b = b - dot(PROJ[:,:i+1], dot(H[:,:i+1].T,b) )
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C = dot(bb.T, T)
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return {'T':T, 'U':U, 'Q':C, 'H':H}
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def bridge(a, b, aopt, scale='scores', mode='normal', r=0):
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"""Undeflated Ridged svd(X'Y)
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"""
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m, n = m_shape(a)
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k, l = m_shape(b)
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u, s, vt = svd(b, full_matrices=0)
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g0 = dot(u*s, u.T)
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g = (1 - r)*g0 + r*eye(m)
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ag = dot(a.T, g)
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u, s, vt = svd(ag, full_matrices=0)
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W = u[:,:aopt]
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K = vt[:aopt,:].T
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T = dot(a, W)
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tnorm = apply_along_axis(vnorm, 0, T) # norm of T-columns
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if mode == 'fast':
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if scale=='loads':
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T = T/tnorm
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W = W*tnorm
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return {'T':T, 'W':W}
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U = dot(g0, K) #fixme check this
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Q = dot(b.T, dot(T, inv(dot(T.T, T)) ))
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B = zeros((aopt, n, l), dtype='f')
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for i in range(aopt):
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B[i] = dot(W[:,:i+1], Q[:,:i+1].T)
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if mode == 'detailed':
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E = empty((aopt, m, n))
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F = empty((aopt, k, l))
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for i in range(aopt):
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E[i] = a - dot(T[:,:i+1], W[:,:i+1].T)
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F[i] = b - dot(a, B[i])
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else: #normal
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F = b - dot(a, B[-1])
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E = a - dot(T, W.T)
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# leverages
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# fixme: probably need an orthogonal basis for row-space leverage
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# T (scores) are not orthogonal
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# Using a qr decomp to get an orthonormal basis for row-space
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#Tq = qr(T)[0]
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#s_lev,v_lev = leverage(aopt,Tq,W)
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# explained variance
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#var_x, exp_var_x = variances(a,T,W)
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#qnorm = apply_along_axis(norm, 0, Q)
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#var_y, exp_var_y = variances(b,U,Q/qnorm)
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if scale=='loads':
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T = T/tnorm
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W = W*tnorm
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Q = Q*tnorm
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return {'B':B, 'W':W, 'T':T, 'Q':Q, 'E':E, 'F':F, 'U':U, 'P':W}
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def nipals_lpls(X, Y, Z, a_max, alpha=.7, mean_ctr=[2, 0, 1], mode='normal', scale='scores', verbose=False):
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""" L-shaped Partial Least Sqaures Regression by the nipals algorithm.
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(X!Z)->Y
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:input:
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X : data matrix (m, n)
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Y : data matrix (m, l)
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Z : data matrix (n, o)
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:output:
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T : X-scores
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W : X-weights/Z-weights
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P : X-loadings
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Q : Y-loadings
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U : X-Y relation
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L : Z-scores
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K : Z-loads
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B : Regression coefficients X->Y
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b0: Regression coefficient intercept
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evx : X-explained variance
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evy : Y-explained variance
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evz : Z-explained variance
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:Notes:
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"""
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if mean_ctr!=None:
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xctr, yctr, zctr = mean_ctr
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X, mnX = center(X, xctr)
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Y, mnY = center(Y, xctr)
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Z, mnZ = center(Z, zctr)
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print Z.mean(1)
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varX = pow(X, 2).sum()
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varY = pow(Y, 2).sum()
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varZ = pow(Z, 2).sum()
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m, n = X.shape
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k, l = Y.shape
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u, o = Z.shape
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# initialize
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U = empty((k, a_max))
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Q = empty((l, a_max))
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T = empty((m, a_max))
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W = empty((n, a_max))
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P = empty((n, a_max))
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K = empty((o, a_max))
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L = empty((u, a_max))
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var_x = empty((a_max,))
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var_y = empty((a_max,))
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var_z = empty((a_max,))
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for a in range(a_max):
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if verbose:
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print "\n Working on comp. %s" %a
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u = Y[:,:1]
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diff = 1
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MAX_ITER = 100
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lim = 1e-5
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niter = 0
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while (diff>lim and niter<MAX_ITER):
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niter += 1
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u1 = u.copy()
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w = dot(X.T, u)
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w = w/sqrt(dot(w.T, w))
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l = dot(Z, w)
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k = dot(Z.T, l)
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k = k/sqrt(dot(k.T, k))
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w = alpha*k + (1-alpha)*w
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w = w/sqrt(dot(w.T, w))
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t = dot(X, w)
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c = dot(Y.T, t)
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c = c/sqrt(dot(c.T, c))
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u = dot(Y, c)
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diff = abs(u1 - u).max()
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if verbose:
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print "Converged after %s iterations" %niter
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tt = dot(t.T, t)
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p = dot(X.T, t)/tt
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q = dot(Y.T, t)/tt
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l = dot(Z, w)
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U[:,a] = u.ravel()
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W[:,a] = w.ravel()
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P[:,a] = p.ravel()
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T[:,a] = t.ravel()
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Q[:,a] = q.ravel()
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L[:,a] = l.ravel()
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K[:,a] = k.ravel()
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X = X - dot(t, p.T)
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Y = Y - dot(t, q.T)
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Z = (Z.T - dot(w, l.T)).T
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var_x[a] = pow(X, 2).sum()
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var_y[a] = pow(Y, 2).sum()
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var_z[a] = pow(Z, 2).sum()
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B = dot(dot(W, inv(dot(P.T, W))), Q.T)
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b0 = mnY - dot(mnX, B)
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# variance explained
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evx = 100.0*(1 - var_x/varX)
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evy = 100.0*(1 - var_y/varY)
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evz = 100.0*(1 - var_z/varZ)
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if scale=='loads':
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tnorm = apply_along_axis(vnorm, 0, T)
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T = T/tnorm
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W = W*tnorm
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Q = Q*tnorm
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knorm = apply_along_axis(vnorm, 0, K)
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L = L*knorm
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K = K/knorm
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return {'T':T, 'W':W, 'P':P, 'Q':Q, 'U':U, 'L':L, 'K':K, 'B':B, 'b0':b0, 'evx':evx, 'evy':evy, 'evz':evz}
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########### Helper routines #########
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def m_shape(array):
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return matrix(array).shape
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def esvd(data):
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"""SVD with the option of economy sized calculation
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Calculate subspaces of X'X or XX' depending on the shape
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of the matrix.
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Good for extreme fat or thin matrices
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:notes:
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Numpy supports this by setting full_matrices=0
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"""
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m, n = data.shape
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if m>=n:
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kernel = dot(data.T, data)
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u, s, vt = svd(kernel)
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u = dot(data, vt.T)
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v = vt.T
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for i in xrange(n):
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s[i] = vnorm(u[:,i])
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u[:,i] = u[:,i]/s[i]
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else:
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kernel = dot(data, data.T)
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#data = (data + data.T)/2.0
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u, s, vt = svd(kernel)
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v = dot(u.T, data)
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for i in xrange(m):
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s[i] = vnorm(v[i,:])
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v[i,:] = v[i,:]/s[i]
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return u, s, v.T
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def vnorm(x):
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# assume column arrays (or vectors)
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return math.sqrt(dot(x.T, x))
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def center(a, axis):
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# 0 = col center, 1 = row center, 2 = double center
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# -1 = nothing
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if axis==-1:
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mn = zeros((a.shape[1],))
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elif axis==0:
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mn = a.mean(0)
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elif axis==1:
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mn = a.mean(1)[:,newaxis]
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elif axis==2:
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mn = a.mean(0) + a.mean(1)[:,newaxis] - a.mean()
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else:
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raise IOError("input error: axis must be in [-1,0,1,2]")
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return a - mn, mn
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