def PCA():
#Principal Component Analysis.
q = 2 #latent dimension
d = 3 #observation dimension
N = 200
niters = 200
true_W = np.random.randn(d,q)*10
true_Z = np.random.randn(N,q)
true_mean = np.random.randn(d,1)
true_prec = 100.
X_data = np.dot(true_Z,true_W.T) + true_mean.T + np.random.randn(N,d)*np.sqrt(1./true_prec)
#set up the problem...
Ws = [nodes.Gaussian(d,np.zeros((d,1)),np.eye(d)*1e-3) for i in range(q)]
W = nodes.hstack(Ws)
Mu = nodes.Gaussian(d,np.zeros((d,1)),np.eye(d)*1e-3)
Beta = nodes.Gamma(d,1e-3,1e-3)
Zs = [nodes.Gaussian(q,np.zeros((q,1)),np.eye(q)) for i in range(N)]
Xs = [nodes.Gaussian(d,W*z+Mu,Beta) for z in Zs]
[xnode.observe(xval.reshape(d,1)) for xnode,xval in zip(Xs,X_data)]
#infer!
for i in range(niters):
[w.update() for w in Ws]
Mu.update()
[z.update() for z in Zs]
Beta.update()
print niters-i
#plot
import pylab
pylab.figure();pylab.title('True W')
pylab.imshow( np.linalg.qr(W.pass_down_Ex())[0],interpolation='nearest')
pylab.figure();pylab.title('E[W]')
pylab.imshow( np.linalg.qr(true_W)[0],interpolation='nearest')
pylab.figure();pylab.title('true Z')
pylab.scatter(true_Z[:,0],true_Z[:,1],50,true_Z[:,0])
pylab.figure();pylab.title('learned Z')
learned_Z = np.hstack([z.qmu for z in Zs]).T
pylab.scatter(learned_Z[:,0],learned_Z[:,1],50,true_Z[:,0])
print '\nBeta'
print true_prec,Beta.pass_down_Ex()[0,0]
print '\nMu'
print np.hstack((true_mean,Mu.pass_down_Ex()))
def PCA_missing_data(plot=True):
#Principal Component Analysis, with randomly missing data
q = 2 #latent dimension
d = 5 #observation dimension
N = 200
niters = 200
Nmissing = 100
true_W = np.random.randn(d,q)
true_Z = np.random.randn(N,q)
true_mean = np.random.randn(d,1)
true_prec = 20.
Xdata_full = np.dot(true_Z,true_W.T) + true_mean.T
Xdata_observed = Xdata_full + np.random.randn(N,d)*np.sqrt(1./true_prec)
#erase some data
missing_index_i = np.argsort(np.random.randn(N))[:Nmissing]
missing_index_j = np.random.multinomial(1,np.ones(d)/d,Nmissing).nonzero()[1]
Xdata = Xdata_observed.copy()
Xdata[missing_index_i,missing_index_j] = np.nan
#set up the problem...
Ws = [nodes.Gaussian(d,np.zeros((d,1)),np.eye(d)*1e-3) for i in range(q)]
W = nodes.hstack(Ws)
Mu = nodes.Gaussian(d,np.zeros((d,1)),np.eye(d)*1e-3)
Beta = nodes.Gamma(d,1e-3,1e-3)
Zs = [nodes.Gaussian(q,np.zeros((q,1)),np.eye(q)) for i in range(N)]
Xs = [nodes.Gaussian(d,W*z+Mu,Beta) for z in Zs]
[xnode.observe(xval.reshape(d,1)) for xnode,xval in zip(Xs,Xdata)]
#make a network object
net = Network()
net.addnode(W)
net.fetch_network()# automagically fetches all of the other nodes...
#infer!
net.learn(100)
#plot
if plot:
import pylab
import hinton
#compare true and learned W
Qtrue,Rtrue = np.linalg.qr(true_W)
Qlearn,Rlearn = np.linalg.qr(W.pass_down_Ex())
pylab.figure();pylab.title('True W')
hinton.hinton(Qtrue)
pylab.figure();pylab.title('E[W]')
hinton.hinton(Qlearn)
if q==2:#plot the latent variables
pylab.figure();pylab.title('true Z')
pylab.scatter(true_Z[:,0],true_Z[:,1],50,true_Z[:,0])
pylab.figure();pylab.title('learned Z')
learned_Z = np.hstack([z.pass_down_Ex() for z in Zs]).T
pylab.scatter(learned_Z[:,0],learned_Z[:,1],50,true_Z[:,0])
#recovered X mean
X_rec = np.hstack([x.pass_down_Ex() for x in Xs]).T
#Recovered X Variance
#slight hack here - set q variance of observed nodes to zeros (it should be random...)
for x in Xs:
if x.observed:
x.qcov *=0
var_rec = np.vstack([np.diag(x.qcov) for x in Xs]) + 1./np.diag(Beta.pass_down_Ex())
#plot each recovered signal in a separate figure
for i in range(d):
pylab.figure();pylab.title('recovered_signal '+str(i))
pylab.plot(Xdata_full[:,i],'g',marker='.',label='True') # 'true' values of missing data (without noise)
pylab.plot(X_rec[:,i],'b',label='Recovered') # recovered mising data values
pylab.plot(Xdata[:,i],'k',marker='o',linewidth=2,label='Observed') # with noise, and holes where we took out values
pylab.legend()
volume_x = np.hstack((np.arange(len(Xs)),np.arange(len(Xs))[::-1]))
volume_y = np.hstack((X_rec[:,i]+2*np.sqrt(var_rec[:,i]), X_rec[:,i][::-1]-2*np.sqrt(var_rec[:,i])[::-1]))
pylab.fill(volume_x,volume_y,'b',alpha=0.3)
print '\nBeta'
print true_prec,Beta.pass_down_Ex()[0,0]
print '\nMu'
print np.hstack((true_mean,Mu.pass_down_Ex()))
pylab.show()
#true_Q = np.dot(q_temp.T,q_temp)/10000 true_Q = np.diag(np.random.rand(q))*0.1 true_Q_chol = np.linalg.cholesky(true_Q) #simulate the system. TODO check I've not got cholesky the wrong way around... (I think so...) true_X = np.zeros((T,q)) Y_data = np.zeros((T,d)) true_X[0] = np.random.randn(q) Y_data[0] = np.dot(true_C,true_X[0].reshape(q,1)).flatten() + np.dot(true_R_chol,np.random.randn(d,1)).flatten() for xlag,xnow,ynow in zip(true_X[:-1],true_X[1:],Y_data[1:]): xnow[:] = np.dot(true_A,xlag).flatten() + np.dot(true_Q_chol,np.random.randn(q,1)).flatten() ynow[:] = np.dot(true_C,xnow) + np.dot(true_R_chol,np.random.randn(d,1)).flatten() #set up the problem... As = [nodes.Gaussian(q,np.zeros((q,1)),np.eye(q)*1e-3) for i in range(q)] A = nodes.hstack(As) #node to represent A Cs = [nodes.Gaussian(d,np.zeros((d,1)),np.eye(d)*1e-3) for i in range(q)] C = nodes.hstack(Cs) #node to represent #Q = nodes.Gamma(q,1e-3,1e-3) #R = nodes.Gamma(d,1e-3,1e-3) Q = nodes.DiagonalGamma(q,np.ones(q)*1e-3,np.ones(q)*1e-3) R = nodes.DiagonalGamma(d,np.ones(d)*1e-3,np.ones(d)*1e-3) #Q = nodes.Wishart(q,1e-3,np.eye(q)*1e-3) #R = nodes.Wishart(d,1e-3,np.eye(d)*1e-3) X0 = nodes.Gaussian(q,np.zeros((q,1)),np.eye(q)) Y0 = nodes.Gaussian(d,C*X0,R) Y0.observe(Y_data[0].reshape(d,1)) Xs = [X0] Ys = [Y0] for t in range(1,T):
