laydi/fluents/lib/engines.py

"""Module contain algorithms for  (burdensome) calculations.

There is no typechecking of any kind here, just focus on speed
"""

from scipy.linalg import svd,norm,inv,pinv,qr
from scipy import dot,empty,eye,newaxis,zeros,sqrt,diag,\
     apply_along_axis,mean,ones,randn,empty_like,outer,c_,\
     rand,sum,cumsum

def pca(a, aopt, scale='scores', mode='normal'):
    """ Principal Component Analysis model
    mode:
         -- fast : returns smallest dim scaled (T for n<=m, P for n>m )
         -- normal : returns all model params and residuals after aopt comp
         -- detailed    : returns all model params and all residuals
    """
    
    m,n = a.shape
    u,s,vt = svd(a, full_matrices=0)
    T = u*s
    T = T[:,:aopt]
    P = vt[:aopt,:].T
    
    if scale=='loads':
        tnorm = apply_along_axis(norm, 0, T)
        T = T/tnorm
        P = P*tnorm

    if mode == 'fast':
        return {'T':T, 'P':P}
    
    if mode=='detailed':
        """Detailed mode returns residual matrix for all comp.
        That is E, is a three-mode matrix: (amax, m, n) """
        E = empty((aopt,  m,  n))
        for ai in range(aopt):
            e = a - dot(T[:,:ai+1], P[:,:ai+1].T)
            E[ai,:,:] = e.copy()
    else:
        E = a - dot(T,P.T)
            
    return {'T':T, 'P':P, 'E':E}

def pls(a, b, aopt=2, scale='scores', mode='normal', ab=None):
    """Kernel pls for tall/wide matrices.

    Fast pls for calibration. Only inefficient for many Y-vars.
    
    """
    m,n = a.shape
    if ab!=None:
        mm,l = ab.shape
    else:
        k,l = b.shape

    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))

    if ab==None: 
        ab = dot(a.T, b)
    for i in range(aopt):
        if ab.shape[1]==1:
            w = ab
        else:
            u,s,vh = svd(dot(ab.T, ab))
            w = dot(ab,u[:,:1])
    
        w = w/norm(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(a, r)
        tt = norm(t)**2
        p  = dot(a.T, t)/tt
        q = dot(r.T, ab).T/tt
        ab = ab - dot(p, q.T)*tt
        T[:,i] = t.ravel()
        W[:,i] = w.ravel()
        P[:,i] = p.ravel()
        R[:,i] = r.ravel()

        if mode=='fast' and i==aopt-1:
            if scale=='loads':
                tnorm = apply_along_axis(norm, 0, T)
                T = T/tnorm
                W = W*tnorm
            return {'T':T, 'W':W}

        Q[:,i] = q.ravel()
        B[i] = dot(R[:,:i+1], Q[:,:i+1].T)
    
    if mode=='detailed':
        E = empty((aopt, m, n))
        F = empty((aopt, k, l))
        for i in range(1,aopt+1,1):
            E[i-1] = a - dot(T[:,:i],P[:,:i].T)
            F[i-1] = b - dot(T[:,:i],Q[:,:i].T)
    else:
        E = a - dot(T[:,:aopt], P[:,:aopt].T)
        F = b - dot(T[:,:aopt], Q[:,:aopt].T)

    if scale=='loads':
        tnorm = apply_along_axis(norm, 0, T)
        T = T/tnorm
        W = W*tnorm
        Q = Q*tnorm
        P = P*tnorm
        
    return {'B':B, 'Q':Q, 'P':P, 'T':T, 'W':W, 'R':R, 'E':E, 'F':F}

def w_simpls(aat, b, aopt):
    """ Simpls for wide matrices.
    Fast pls for crossval, used in calc rmsep for wide X
    There is no P,W.  T is normalised
    """
    bb = b.copy()
    m,m = aat.shape
    U = empty((m, aopt))
    T = empty((m, aopt))
    H = empty((m, aopt)) #just like W in simpls
    PROJ = empty((m, aopt)) #just like R in simpls

    for i in range(aopt):
        u,s,vh = svd(dot(dot(b.T, aat), b), full_matrices=0)
        u = dot(b, u[:,:1]) #y-factor scores
        U[:,i] = u.ravel()
        t =dot(aat, u)
        t = t/norm(t)
        T[:,i] = t.ravel()
        h = dot(aat, t) #score-weights
        H[:,i] = h.ravel()
        PROJ[:,:i+1] = dot(T[:,:i+1], inv(dot(T[:,:i+1].T, H[:,:i+1])) )
        if i<aopt:
            b = b - dot(PROJ[:,:i+1], dot(H[:,:i+1].T,b) )
    C = dot(bb.T, T)

    return {'T':T,'U':U,'Q':C,'H':H}

def bridge(a, b, aopt, scale='scores', mode='normal', r=0):
    """Undeflated Ridged svd(X'Y)
    """
    m, n = a.shape
    k, l = b.shape
    u,s,vt = svd(b, full_matrices=0)
    g0 = dot(u*s, u.T)
    g = (1 - r)*g0 + r*eye(m)
    ag = dot(a.T, g)
    
    u,s,vt = svd(ag, full_matrices=0)
    W = u[:,:aopt]
    K = vt[:aopt,:].T
    T = dot(a, W)
    tnorm = apply_along_axis(norm, 0, T) # norm of T-columns

    if mode == 'fast':
        if scale=='loads':
            T = T/tnorm
            W = W*tnorm
        return {'T':T, 'W':W}

    U = dot(g0, K) #fixme check this 
    Q = dot(b.T, dot(T, inv(dot(T.T,T)) ))
    B = zeros((aopt, n, l))
    for i in range(aopt):
        B[i] = dot(W[:,:i+1], Q[:,:i+1].T)
    # leverages
    # fixme: probably need an orthogonal basis for row-space leverage
    #        T (scores) are not orthogonal
    #        Using a qr decomp to get an orthonormal basis for row-space
    #Tq = qr(T)[0]
    #s_lev,v_lev = leverage(aopt,Tq,W)
    # explained variance
    #var_x, exp_var_x = variances(a,T,W)
    #qnorm = apply_along_axis(norm, 0, Q)
    #var_y, exp_var_y = variances(b,U,Q/qnorm)
    
    if mode == 'detailed':
        E = empty((aopt, m, n))
        F = empty((aopt, k, l))
        for i in range(aopt):
            E[i] = a - dot(T[:,:i+1], W[:,:i+1].T)
            F[i] = b - dot(a, B[i])
    else: #normal
        F = b - dot(a, B[-1])
        E = a - dot(T, W.T)

    if scale=='loads':
        T = T/tnorm
        W = W*tnorm
        Q = Q*tnorm
        
    return {'B':B, 'W':W, 'T':T, 'Q':Q, 'E':E, 'F':F, 'U':U, 'P':W}