201 lines
5.5 KiB
Python
201 lines
5.5 KiB
Python
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"""Module contain algorithms for (burdensome) calculations.
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There is no typechecking of any kind here, just focus on speed
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"""
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from scipy.linalg import svd,norm,inv,pinv,qr
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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
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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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u,s,vt = svd(a, full_matrices=0)
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T = u*s
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T = T[:,:aopt]
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P = vt[:aopt,:].T
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if scale=='loads':
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tnorm = apply_along_axis(norm, 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 pls(a, b, aopt=2, scale='scores', mode='normal', ab=None):
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"""Kernel pls for tall/wide matrices.
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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,l = ab.shape
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else:
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k,l = b.shape
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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
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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/norm(w)
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r = w.copy()
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if i>0:
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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 = norm(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(norm, 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(norm, 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,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/norm(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 = a.shape
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k, l = b.shape
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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(norm, 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))
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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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# 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 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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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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