예제 #1
0
def run(M=10,
        N=100,
        D_y=3,
        D=5,
        seed=42,
        rotate=False,
        maxiter=100,
        debug=False,
        plot=True):

    if seed is not None:
        np.random.seed(seed)

    # Generate data
    w = np.random.normal(0, 1, size=(M, 1, D_y))
    x = np.random.normal(0, 1, size=(1, N, D_y))
    f = misc.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.2, size=(M, N))

    # Construct model
    (Y, F, W, X, tau, alpha) = model(M, N, D)

    # Data with missing values
    mask = random.mask(M, N, p=0.5)  # randomly missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha)

    # Initialize some nodes randomly
    X.initialize_from_random()
    W.initialize_from_random()

    # Run inference algorithm
    if rotate:
        # Use rotations to speed up learning
        rotW = transformations.RotateGaussianARD(W, alpha)
        rotX = transformations.RotateGaussianARD(X)
        R = transformations.RotationOptimizer(rotW, rotX, D)
        for ind in range(maxiter):
            Q.update()
            if debug:
                R.rotate(check_bound=True, check_gradient=True)
            else:
                R.rotate()

    else:
        # Use standard VB-EM alone
        Q.update(repeat=maxiter)

    # Plot results
    if plot:
        plt.figure()
        bpplt.timeseries_normal(F, scale=2)
        bpplt.timeseries(f, color='g', linestyle='-')
        bpplt.timeseries(y, color='r', linestyle='None', marker='+')
예제 #2
0
파일: pca.py 프로젝트: viveksck/bayespy
def run(M=10, N=100, D_y=3, D=5, seed=42, rotate=False, maxiter=100, debug=False):

    if seed is not None:
        np.random.seed(seed)
    
    # Generate data
    w = np.random.normal(0, 1, size=(M,1,D_y))
    x = np.random.normal(0, 1, size=(1,N,D_y))
    f = misc.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.2, size=(M,N))

    # Construct model
    (Y, F, W, X, tau, alpha) = model(M, N, D)

    # Data with missing values
    mask = random.mask(M, N, p=0.5) # randomly missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha)

    # Initialize some nodes randomly
    X.initialize_from_random()
    W.initialize_from_random()

    # Run inference algorithm
    if rotate:
        # Use rotations to speed up learning
        rotW = transformations.RotateGaussianARD(W, alpha)
        rotX = transformations.RotateGaussianARD(X)
        R = transformations.RotationOptimizer(rotW, rotX, D)
        for ind in range(maxiter):
            Q.update()
            if debug:
                R.rotate(check_bound=True,
                         check_gradient=True)
            else:
                R.rotate()
            
    else:
        # Use standard VB-EM alone
        Q.update(repeat=maxiter)

    # Plot results
    plt.figure()
    bpplt.timeseries_normal(F, scale=2)
    bpplt.timeseries(f, color='g', linestyle='-')
    bpplt.timeseries(y, color='r', linestyle='None', marker='+')
    plt.show()
예제 #3
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def run(M=10, N=100, D_y=3, D=5):
    seed = 45
    print('seed =', seed)
    np.random.seed(seed)
    
    # Generate data
    w = np.random.normal(0, 1, size=(M,1,D_y))
    x = np.random.normal(0, 1, size=(1,N,D_y))
    f = utils.utils.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.5, size=(M,N))

    # Construct model
    (Y, WX, W, X, tau, alpha) = pca_model(M, N, D)

    # Data with missing values
    mask = utils.random.mask(M, N, p=0.9) # randomly missing
    mask[:,20:40] = False # gap missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha)

    # Initialize some nodes randomly
    X.initialize_from_value(X.random())
    W.initialize_from_value(W.random())

    # Inference loop.
    Q.update(repeat=100)

    # Plot results
    plt.clf()
    WX_params = WX.get_parameters()
    fh = WX_params[0] * np.ones(y.shape)
    err_fh = 2*np.sqrt(WX_params[1] + 1/tau.get_moments()[0]) * np.ones(y.shape)
    for m in range(M):
        plt.subplot(M,1,m+1)
        myplt.errorplot(fh[m], x=np.arange(N), error=err_fh[m])
        plt.plot(np.arange(N), f[m], 'g')
        plt.plot(np.arange(N), y[m], 'r+')

    plt.show()
예제 #4
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def pca():

    np.random.seed(41)

    M = 10
    N = 3000
    D = 5

    # Construct the PCA model
    alpha = Gamma(1e-3, 1e-3, plates=(D, ), name='alpha')
    W = GaussianARD(0, alpha, plates=(M, 1), shape=(D, ), name='W')
    X = GaussianARD(0, 1, plates=(1, N), shape=(D, ), name='X')
    tau = Gamma(1e-3, 1e-3, name='tau')
    W.initialize_from_random()
    F = SumMultiply('d,d->', W, X)
    Y = GaussianARD(F, tau, name='Y')

    # Observe data
    data = np.sum(np.random.randn(M, 1, D - 1) * np.random.randn(1, N, D - 1),
                  axis=-1) + 1e-1 * np.random.randn(M, N)
    Y.observe(data)

    # Initialize VB engine
    Q = VB(Y, X, W, alpha, tau)

    # Take one update step (so phi is ok)
    Q.update(repeat=1)
    Q.save()

    # Run VB-EM
    Q.update(repeat=200)
    bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')

    # Restore the state
    Q.load()

    # Run Riemannian conjugate gradient
    #Q.optimize(X, alpha, maxiter=100, collapsed=[W, tau])
    Q.optimize(W, tau, maxiter=100, collapsed=[X, alpha])
    bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')

    bpplt.pyplot.show()
예제 #5
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def run(M=30, D=5):

    # Generate data
    y = np.random.randint(D, size=(M, ))

    # Construct model
    p = nodes.Dirichlet(1 * np.ones(D), name='p')
    z = nodes.Categorical(p, plates=(M, ), name='z')

    # Observe the data with randomly missing values
    mask = random.mask(M, p=0.5)
    z.observe(y, mask=mask)

    # Run VB-EM
    Q = VB(p, z)
    Q.update()

    # Show results
    z.show()
    p.show()
예제 #6
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def pca():

    np.random.seed(41)

    M = 10
    N = 3000
    D = 5

    # Construct the PCA model
    alpha = Gamma(1e-3, 1e-3, plates=(D,), name='alpha')
    W = GaussianARD(0, alpha, plates=(M,1), shape=(D,), name='W')
    X = GaussianARD(0, 1, plates=(1,N), shape=(D,), name='X')
    tau = Gamma(1e-3, 1e-3, name='tau')
    W.initialize_from_random()
    F = SumMultiply('d,d->', W, X)
    Y = GaussianARD(F, tau, name='Y')

    # Observe data
    data = np.sum(np.random.randn(M,1,D-1) * np.random.randn(1,N,D-1), axis=-1) + 1e-1 * np.random.randn(M,N)
    Y.observe(data)

    # Initialize VB engine
    Q = VB(Y, X, W, alpha, tau)

    # Take one update step (so phi is ok)
    Q.update(repeat=1)
    Q.save()

    # Run VB-EM
    Q.update(repeat=200)
    bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')

    # Restore the state
    Q.load()

    # Run Riemannian conjugate gradient
    #Q.optimize(X, alpha, maxiter=100, collapsed=[W, tau])
    Q.optimize(W, tau, maxiter=100, collapsed=[X, alpha])
    bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')

    bpplt.pyplot.show()
예제 #7
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def run(M=30, D=5):

    # Generate data
    y = np.random.randint(D, size=(M,))

    # Construct model
    p = nodes.Dirichlet(1*np.ones(D),
                        name='p')
    z = nodes.Categorical(p, 
                          plates=(M,), 
                          name='z')

    # Observe the data with randomly missing values
    mask = random.mask(M, p=0.5)
    z.observe(y, mask=mask)

    # Run VB-EM
    Q = VB(p, z)
    Q.update()

    # Show results
    z.show()
    p.show()
예제 #8
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def run(M=40, N=100, D_y=6, D=8, seed=42, rotate=False, maxiter=1000, debug=False, plot=True):
    """
    Run pattern search demo for PCA.
    """

    if seed is not None:
        np.random.seed(seed)
    
    # Generate data
    w = np.random.normal(0, 1, size=(M,1,D_y))
    x = np.random.normal(0, 1, size=(1,N,D_y))
    f = misc.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.2, size=(M,N))

    # Construct model
    Q = VB(*(pca.model(M, N, D)))

    # Data with missing values
    mask = random.mask(M, N, p=0.5) # randomly missing
    y[~mask] = np.nan
    Q['Y'].observe(y, mask=mask)

    # Initialize some nodes randomly
    Q['X'].initialize_from_random()
    Q['W'].initialize_from_random()

    # Use a few VB-EM updates at the beginning
    Q.update(repeat=10)
    Q.save()

    # Standard VB-EM as a baseline
    Q.update(repeat=maxiter)
    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')

    # Restore initial state
    Q.load()

    # Pattern search method for comparison
    for n in range(maxiter):

        Q.pattern_search('W', 'tau', maxiter=3, collapsed=['X', 'alpha'])
        Q.update(repeat=20)

        if Q.has_converged():
            break

    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')

        bpplt.pyplot.xlabel('CPU time (in seconds)')
        bpplt.pyplot.ylabel('VB lower bound')
        bpplt.pyplot.legend(['VB-EM', 'Pattern search'], loc='lower right')
예제 #9
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def run(M=50, N=200, D_y=10, D=20, maxiter=100):
    seed = 45
    print("seed =", seed)
    np.random.seed(seed)

    # Generate data (covariance eigenvalues: 1,1,...,1,2^2,3^2,...,(D_y+1)^2
    (q, r) = scipy.linalg.qr(np.random.randn(M, M))
    C = np.diag(np.arange(2, 2 + D_y))
    C = np.ones(M)
    C[:D_y] += np.arange(1, 1 + D_y)
    y = C[:, np.newaxis] * np.random.randn(M, N)
    y = np.dot(q, y)

    # Construct model
    (Y, WX, W, X, tau, alpha) = pca_model(M, N, D)

    # Data with missing values
    mask = utils.random.mask(M, N, p=0.9)  # randomly missing
    mask[:, 20:40] = False  # gap missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha, autosave_filename=utils.utils.tempfile())

    # Initialize nodes (from prior and randomly)
    X.initialize_from_value(X.random())
    W.initialize_from_value(W.random())

    Q.update(repeat=1)
    Q.save()

    #
    # Run inference with rotations.
    #
    rotX = transformations.RotateGaussian(X)
    rotW = transformations.RotateGaussianARD(W, alpha)
    R = transformations.RotationOptimizer(rotX, rotW, D)

    for ind in range(maxiter):
        Q.update()
        R.rotate(
            check_gradient=False,
            maxiter=10,
            verbose=False,
            check_bound=Q.compute_lowerbound,
            check_bound_terms=Q.compute_lowerbound_terms,
        )

    L_rot = Q.L

    #
    # Re-run inference without rotations.
    #
    Q.load()
    Q.update(repeat=maxiter)
    L_norot = Q.L

    #
    # Plot comparison
    #
    plt.plot(L_rot)
    plt.plot(L_norot)
    plt.legend(["With rotations", "Without rotations"], loc="lower right")
    plt.show()
예제 #10
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def run(maxiter=100):

    seed = 496#np.random.randint(1000)
    print("seed = ", seed)
    np.random.seed(seed)

    # Simulate some data
    D = 3
    M = 6
    N = 200
    c = np.random.randn(M,D)
    w = 0.3
    a = np.array([[np.cos(w), -np.sin(w), 0], 
                  [np.sin(w), np.cos(w),  0], 
                  [0,         0,          1]])
    x = np.empty((N,D))
    f = np.empty((M,N))
    y = np.empty((M,N))
    x[0] = 10*np.random.randn(D)
    f[:,0] = np.dot(c,x[0])
    y[:,0] = f[:,0] + 3*np.random.randn(M)
    for n in range(N-1):
        x[n+1] = np.dot(a,x[n]) + np.random.randn(D)
        f[:,n+1] = np.dot(c,x[n+1])
        y[:,n+1] = f[:,n+1] + 3*np.random.randn(M)

    # Create the model
    (Y, CX, X, tau, C, gamma, A, alpha) = linear_state_space_model(D, N, M)
    
    # Add missing values randomly
    mask = random.mask(M, N, p=0.3)
    # Add missing values to a period of time
    mask[:,30:80] = False
    y[~mask] = np.nan # BayesPy doesn't require this. Just for plotting.
    # Observe the data
    Y.observe(y, mask=mask)
    

    # Initialize nodes (must use some randomness for C)
    C.initialize_from_random()

    # Run inference
    Q = VB(Y, X, C, gamma, A, alpha, tau)

    #
    # Run inference with rotations.
    #
    rotA = transformations.RotateGaussianARD(A, alpha)
    rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
    rotC = transformations.RotateGaussianARD(C, gamma)
    R = transformations.RotationOptimizer(rotX, rotC, D)

    #maxiter = 84
    for ind in range(maxiter):
        Q.update()
        #print('C term', C.lower_bound_contribution())
        R.rotate(maxiter=10, 
                 check_gradient=True,
                 verbose=False,
                 check_bound=Q.compute_lowerbound,
        #check_bound=None,
                 check_bound_terms=Q.compute_lowerbound_terms)
        #check_bound_terms=None)

    X_vb = X.u[0]
    varX_vb = utils.diagonal(X.u[1] - X_vb[...,np.newaxis,:] * X_vb[...,:,np.newaxis])

    u_CX = CX.get_moments()
    CX_vb = u_CX[0]
    varCX_vb = u_CX[1] - CX_vb**2

    # Show results
    plt.figure(3)
    plt.clf()
    for m in range(M):
        plt.subplot(M,1,m+1)
        plt.plot(y[m,:], 'r.')
        plt.plot(f[m,:], 'b-')
        bpplt.errorplot(y=CX_vb[m,:],
                        error=2*np.sqrt(varCX_vb[m,:]))

    plt.figure()
    Q.plot_iteration_by_nodes()
예제 #11
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def run(N=100000, N_batch=50, seed=42, maxiter=100, plot=True):
    """
    Run deterministic annealing demo for 1-D Gaussian mixture.
    """

    if seed is not None:
        np.random.seed(seed)

    # Number of clusters in the model
    K = 20

    # Dimensionality of the data
    D = 5

    # Generate data
    K_true = 10
    spread = 5
    means = spread * np.random.randn(K_true, D)
    z = random.categorical(np.ones(K_true), size=N)
    data = np.empty((N,D))
    for n in range(N):
        data[n] = means[z[n]] + np.random.randn(D)

    #
    # Standard VB-EM algorithm
    #

    # Full model
    mu = Gaussian(np.zeros(D), np.identity(D),
                  plates=(K,),
                  name='means')
    alpha = Dirichlet(np.ones(K),
                      name='class probabilities')
    Z = Categorical(alpha,
                    plates=(N,),
                    name='classes')
    Y = Mixture(Z, Gaussian, mu, np.identity(D),
                name='observations')

    # Break symmetry with random initialization of the means
    mu.initialize_from_random()

    # Put the data in
    Y.observe(data)

    # Run inference
    Q = VB(Y, Z, mu, alpha)
    Q.save(mu)
    Q.update(repeat=maxiter)
    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
    max_cputime = np.sum(Q.cputime[~np.isnan(Q.cputime)])


    #
    # Stochastic variational inference
    #

    # Construct smaller model (size of the mini-batch)
    mu = Gaussian(np.zeros(D), np.identity(D),
                  plates=(K,),
                  name='means')
    alpha = Dirichlet(np.ones(K),
                      name='class probabilities')
    Z = Categorical(alpha,
                    plates=(N_batch,),
                    plates_multiplier=(N/N_batch,),
                    name='classes')
    Y = Mixture(Z, Gaussian, mu, np.identity(D),
                name='observations')

    # Break symmetry with random initialization of the means
    mu.initialize_from_random()

    # Inference engine
    Q = VB(Y, Z, mu, alpha, autosave_filename=Q.autosave_filename)
    Q.load(mu)

    # Because using mini-batches, messages need to be multiplied appropriately
    print("Stochastic variational inference...")
    Q.ignore_bound_checks = True

    maxiter *= int(N/N_batch)
    delay = 1
    forgetting_rate = 0.7
    for n in range(maxiter):

        # Observe a mini-batch
        subset = np.random.choice(N, N_batch)
        Y.observe(data[subset,:])

        # Learn intermediate variables
        Q.update(Z)

        # Set step length
        step = (n + delay) ** (-forgetting_rate)

        # Stochastic gradient for the global variables
        Q.gradient_step(mu, alpha, scale=step)

        if np.sum(Q.cputime[:n]) > max_cputime:
            break
    
    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')

        bpplt.pyplot.xlabel('CPU time (in seconds)')
        bpplt.pyplot.ylabel('VB lower bound')
        bpplt.pyplot.legend(['VB-EM', 'Stochastic inference'], loc='lower right')
        bpplt.pyplot.title('VB for Gaussian mixture model')

    return
예제 #12
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def run_dlssm(y, f, mask, D, K, maxiter):
    """
    Run VB inference for linear state space model with drifting dynamics.
    """
        
    (M, N) = np.shape(y)

    # Dynamics matrix with ARD
    # alpha : (D) x ()
    alpha = Gamma(1e-5,
                  1e-5,
                  plates=(K,),
                  name='alpha')
    # A : (K) x (K)
    A = Gaussian(np.zeros(K),
    #np.identity(K),
                 diagonal(alpha),
                 plates=(K,),
                 name='A_S')
    A.initialize_from_value(np.identity(K))

    # rho
    ## rho = Gamma(1e-5,
    ##             1e-5,
    ##             plates=(K,),
    ##             name="rho")

    # S : () x (N-1,K)
    S = GaussianMarkovChain(np.ones(K),
                            1e-6*np.identity(K),
                            A,
                            np.ones(K),
                            n=N-1,
                            name='S')
    S.initialize_from_value(1*np.ones((N-1,K)))

    # Projection matrix of the dynamics matrix
    # beta : (K) x ()
    beta = Gamma(1e-5,
                 1e-5,
                 plates=(K,),
                 name='beta')
    # B : (D) x (D*K)
    B = Gaussian(np.zeros(D*K),
                 diagonal(tile(beta, D)),
                 plates=(D,),
                 name='B')
    b = np.zeros((D,D,K))
    b[np.arange(D),np.arange(D),np.zeros(D,dtype=int)] = 1
    B.initialize_from_value(np.reshape(1*b, (D,D*K)))

    # A : (N-1,D) x (D)
    BS = MatrixDot(B, 
                   S.as_gaussian().add_plate_axis(-1), 
                   name='BS')

    # Latent states with dynamics
    # X : () x (N,D)
    X = GaussianMarkovChain(np.zeros(D),         # mean of x0
                            1e-3*np.identity(D), # prec of x0
                            BS,                   # dynamics
                            np.ones(D),          # innovation
                            n=N,                 # time instances
                            name='X',
                            initialize=False)
    X.initialize_from_value(np.random.randn(N,D))

    # Mixing matrix from latent space to observation space using ARD
    # gamma : (D) x ()
    gamma = Gamma(1e-5,
                  1e-5,
                  plates=(D,),
                  name='gamma')
    # C : (M,1) x (D)
    C = Gaussian(np.zeros(D),
                 diagonal(gamma),
                 plates=(M,1),
                 name='C')
    C.initialize_from_value(np.random.randn(M,1,D))

    # Observation noise
    # tau : () x ()
    tau = Gamma(1e-5,
                1e-5,
                name='tau')

    # Observations
    # Y : (M,N) x ()
    CX = Dot(C, X.as_gaussian())
    Y = Normal(CX,
               tau,
               name='Y')

    #
    # RUN INFERENCE
    #

    # Observe data
    Y.observe(y, mask=mask)
    # Construct inference machine
    Q = VB(Y, X, S, A, alpha, B, beta, C, gamma, tau)

    #
    # Run inference with rotations.
    #

    # Rotate the D-dimensional state space (C, X)
    rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='rows')
    rotX = transformations.RotateDriftingMarkovChain(X, B, S, rotB)
    rotC = transformations.RotateGaussianARD(C, gamma)
    R_X = transformations.RotationOptimizer(rotX, rotC, D)

    # Rotate the K-dimensional latent dynamics space (B, S)
    rotA = transformations.RotateGaussianARD(A, alpha)
    rotS = transformations.RotateGaussianMarkovChain(S, A, rotA)
    rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='cols')
    R_S = transformations.RotationOptimizer(rotS, rotB, K)

    # Iterate
    for ind in range(int(maxiter/5)):
        Q.update(repeat=5)
        #Q.update(X, S, A, alpha, rho, B, beta, C, gamma, tau, repeat=maxiter)
        R_X.rotate()
        R_S.rotate()
        ## R_X.rotate(
        ## check_bound=Q.compute_lowerbound,
        ## check_bound_terms=Q.compute_lowerbound_terms,
        ## check_gradient=True
        ##     )
        ## R_S.rotate(
        ## check_bound=Q.compute_lowerbound,
        ## check_bound_terms=Q.compute_lowerbound_terms,
        ## check_gradient=True
        ##     )

    #
    # SHOW RESULTS
    #

    # Mean and standard deviation of the posterior
    (f_mean, f_squared) = CX.get_moments()
    f_std = np.sqrt(f_squared - f_mean**2)

    # Plot observations space
    for m in range(M):
        plt.subplot(M,1,m+1)
        plt.plot(y[m,:], 'r.')
        plt.plot(f[m,:], 'b-')
        bpplt.errorplot(y=f_mean[m,:], error=2*f_std[m,:])
예제 #13
0
def run_dlssm(y, f, mask, D, K, maxiter):
    """
    Run VB inference for linear state space model with drifting dynamics.
    """
        
    (M, N) = np.shape(y)

    # Dynamics matrix with ARD
    # alpha : (K) x ()
    alpha = Gamma(1e-5,
                  1e-5,
                  plates=(K,),
                  name='alpha')
    # A : (K) x (K)
    A = GaussianArrayARD(np.identity(K),
                         alpha,
                         shape=(K,),
                         plates=(K,),
                         name='A_S',
                         initialize=False)
    A.initialize_from_value(np.identity(K))

    # State of the drift
    # S : () x (N,K)
    S = GaussianMarkovChain(np.ones(K),
                            1e-6*np.identity(K),
                            A,
                            np.ones(K),
                            n=N,
                            name='S',
                            initialize=False)
    S.initialize_from_value(np.ones((N,K)))

    # Projection matrix of the dynamics matrix
    # Initialize S and B such that BS is identity matrix
    # beta : (K) x ()
    beta = Gamma(1e-5,
                 1e-5,
                 plates=(D,K),
                 name='beta')
    # B : (D) x (D,K)
    b = np.zeros((D,D,K))
    b[np.arange(D),np.arange(D),np.zeros(D,dtype=int)] = 1
    B = GaussianArrayARD(0,
                         beta,
                         plates=(D,),
                         name='B',
                         initialize=False)
    B.initialize_from_value(np.reshape(1*b, (D,D,K)))
    # BS : (N-1,D) x (D)
    # TODO/FIXME: Implement __getitem__ method
    BS = SumMultiply('dk,k->d',
                     B, 
                     S.as_gaussian()[...,np.newaxis],
    #                     iterator_axis=0,
                     name='BS')

    # Latent states with dynamics
    # X : () x (N,D)
    X = GaussianMarkovChain(np.zeros(D),         # mean of x0
                            1e-3*np.identity(D), # prec of x0
                            BS,                  # dynamics
                            np.ones(D),          # innovation
                            n=N+1,               # time instances
                            name='X',
                            initialize=False)
    X.initialize_from_value(np.random.randn(N+1,D))

    # Mixing matrix from latent space to observation space using ARD
    # gamma : (D) x ()
    gamma = Gamma(1e-5,
                  1e-5,
                  plates=(D,K),
                  name='gamma')
    # C : (M,1) x (D,K)
    C = GaussianArrayARD(0,
                         gamma,
                         plates=(M,1),
                         name='C',
                         initialize=False)
    C.initialize_from_random()

    # Observation noise
    # tau : () x ()
    tau = Gamma(1e-5,
                1e-5,
                name='tau')

    # Observations
    # Y : (M,N) x ()
    F = SumMultiply('dk,d,k',
                    C,
                    X.as_gaussian()[1:],
                    S.as_gaussian(),
                    name='F')
                  
    Y = GaussianArrayARD(F,
                         tau,
                         name='Y')

    #
    # RUN INFERENCE
    #

    # Observe data
    Y.observe(y, mask=mask)
    # Construct inference machine
    Q = VB(Y, X, S, A, alpha, B, beta, C, gamma, tau)

    #
    # Run inference with rotations.
    #

    rotate = False
    if rotate:
        # Rotate the D-dimensional state space (C, X)
        rotB = transformations.RotateGaussianMatrixARD(B, beta, D, K, axis='rows')
        rotX = transformations.RotateDriftingMarkovChain(X, B, S, rotB)
        rotC = transformations.RotateGaussianARD(C, gamma)
        R_X = transformations.RotationOptimizer(rotX, rotC, D)

        # Rotate the K-dimensional latent dynamics space (B, S)
        rotA = transformations.RotateGaussianARD(A, alpha)
        rotS = transformations.RotateGaussianMarkovChain(S, A, rotA)
        rotB = transformations.RotateGaussianMatrixARD(B, beta, D, K, axis='cols')
        R_S = transformations.RotationOptimizer(rotS, rotB, K)

    # Iterate
    for ind in range(maxiter):
        print("X update")
        Q.update(X)
        print("S update")
        Q.update(S)
        print("A update")
        Q.update(A)
        print("alpha update")
        Q.update(alpha)
        print("B update")
        Q.update(B)
        print("beta update")
        Q.update(beta)
        print("C update")
        Q.update(C)
        print("gamma update")
        Q.update(gamma)
        print("tau update")
        Q.update(tau)
        if rotate:
            if ind >= 0:
                R_X.rotate()
            if ind >= 0:
                R_S.rotate()

    Q.plot_iteration_by_nodes()
    
    #
    # SHOW RESULTS
    #

    # Plot observations space
    plt.figure()
    bpplt.timeseries_normal(F)
    bpplt.timeseries(f, 'b-')
    bpplt.timeseries(y, 'r.')
    
    # Plot latent space
    plt.figure()
    bpplt.timeseries_gaussian_mc(X, scale=2)
    
    # Plot drift space
    plt.figure()
    bpplt.timeseries_gaussian_mc(S, scale=2)
예제 #14
0
파일: saving.py 프로젝트: zehsilva/bayespy
def run(M=10, N=100, D_y=3, D=5):
    seed = 45
    print('seed =', seed)
    np.random.seed(seed)

    # Check HDF5 version.
    if h5py.version.hdf5_version_tuple < (1, 8, 7):
        print(
            "WARNING! Your HDF5 version is %s. HDF5 versions <1.8.7 are not "
            "able to save empty arrays, thus you may experience problems if "
            "you for instance try to save before running any iteration steps."
            % str(h5py.version.hdf5_version_tuple))

    # Generate data
    w = np.random.normal(0, 1, size=(M, 1, D_y))
    x = np.random.normal(0, 1, size=(1, N, D_y))
    f = misc.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.5, size=(M, N))

    # Construct model
    (Y, WX, W, X, tau, alpha) = pca_model(M, N, D)

    # Data with missing values
    mask = random.mask(M, N, p=0.9)  # randomly missing
    mask[:, 20:40] = False  # gap missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha, autosave_iterations=5)

    # Initialize some nodes randomly
    X.initialize_from_value(X.random())
    W.initialize_from_value(W.random())

    # Save the state into a HDF5 file
    filename = tempfile.NamedTemporaryFile(suffix='hdf5').name
    Q.update(X, W, alpha, tau, repeat=1)
    Q.save(filename=filename)

    # Inference loop.
    Q.update(X, W, alpha, tau, repeat=10)

    # Reload the state from the HDF5 file
    Q.load(filename=filename)

    # Inference loop again.
    Q.update(X, W, alpha, tau, repeat=10)

    # NOTE: Saving and loading requires that you have the model
    # constructed. "Save" does not store the model structure nor does "load"
    # read it. They are just used for reading and writing the contents of the
    # nodes. Thus, if you want to load, you first need to construct the same
    # model that was used for saving and then use load to set the states of the
    # nodes.

    plt.clf()
    WX_params = WX.get_parameters()
    fh = WX_params[0] * np.ones(y.shape)
    err_fh = 2 * np.sqrt(WX_params[1] + 1 / tau.get_moments()[0]) * np.ones(
        y.shape)
    for m in range(M):
        plt.subplot(M, 1, m + 1)
        #errorplot(y, error=None, x=None, lower=None, upper=None):
        bpplt.errorplot(fh[m], x=np.arange(N), error=err_fh[m])
        plt.plot(np.arange(N), f[m], 'g')
        plt.plot(np.arange(N), y[m], 'r+')

    plt.figure()
    Q.plot_iteration_by_nodes()

    plt.figure()
    plt.subplot(2, 2, 1)
    bpplt.binary_matrix(W.mask)
    plt.subplot(2, 2, 2)
    bpplt.binary_matrix(X.mask)
    plt.subplot(2, 2, 3)
    #bpplt.binary_matrix(WX.get_mask())
    plt.subplot(2, 2, 4)
    bpplt.binary_matrix(Y.mask)
예제 #15
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def run(M=40,
        N=100,
        D_y=6,
        D=8,
        seed=42,
        rotate=False,
        maxiter=1000,
        debug=False,
        plot=True):
    """
    Run pattern search demo for PCA.
    """

    if seed is not None:
        np.random.seed(seed)

    # Generate data
    w = np.random.normal(0, 1, size=(M, 1, D_y))
    x = np.random.normal(0, 1, size=(1, N, D_y))
    f = misc.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.2, size=(M, N))

    # Construct model
    Q = VB(*(pca.model(M, N, D)))

    # Data with missing values
    mask = random.mask(M, N, p=0.5)  # randomly missing
    y[~mask] = np.nan
    Q['Y'].observe(y, mask=mask)

    # Initialize some nodes randomly
    Q['X'].initialize_from_random()
    Q['W'].initialize_from_random()

    # Use a few VB-EM updates at the beginning
    Q.update(repeat=10)
    Q.save()

    # Standard VB-EM as a baseline
    Q.update(repeat=maxiter)
    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')

    # Restore initial state
    Q.load()

    # Pattern search method for comparison
    for n in range(maxiter):

        Q.pattern_search('W', 'tau', maxiter=3, collapsed=['X', 'alpha'])
        Q.update(repeat=20)

        if Q.has_converged():
            break

    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')

        bpplt.pyplot.xlabel('CPU time (in seconds)')
        bpplt.pyplot.ylabel('VB lower bound')
        bpplt.pyplot.legend(['VB-EM', 'Pattern search'], loc='lower right')
예제 #16
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def run(M=10, N=100, D_y=3, D=5):
    seed = 45
    print('seed =', seed)
    np.random.seed(seed)

    # Check HDF5 version.
    if h5py.version.hdf5_version_tuple < (1,8,7): 
        print("WARNING! Your HDF5 version is %s. HDF5 versions <1.8.7 are not "
              "able to save empty arrays, thus you may experience problems if "
              "you for instance try to save before running any iteration steps."
              % str(h5py.version.hdf5_version_tuple))
    
    # Generate data
    w = np.random.normal(0, 1, size=(M,1,D_y))
    x = np.random.normal(0, 1, size=(1,N,D_y))
    f = misc.sum_product(w, x, axes_to_sum=[-1])
    y = f + np.random.normal(0, 0.5, size=(M,N))

    # Construct model
    (Y, WX, W, X, tau, alpha) = pca_model(M, N, D)

    # Data with missing values
    mask = random.mask(M, N, p=0.9) # randomly missing
    mask[:,20:40] = False # gap missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha, autosave_iterations=5)

    # Initialize some nodes randomly
    X.initialize_from_value(X.random())
    W.initialize_from_value(W.random())

    # Save the state into a HDF5 file
    filename = tempfile.NamedTemporaryFile(suffix='hdf5').name
    Q.update(X, W, alpha, tau, repeat=1)
    Q.save(filename=filename)

    # Inference loop.
    Q.update(X, W, alpha, tau, repeat=10)

    # Reload the state from the HDF5 file
    Q.load(filename=filename)

    # Inference loop again.
    Q.update(X, W, alpha, tau, repeat=10)

    # NOTE: Saving and loading requires that you have the model
    # constructed. "Save" does not store the model structure nor does "load"
    # read it. They are just used for reading and writing the contents of the
    # nodes. Thus, if you want to load, you first need to construct the same
    # model that was used for saving and then use load to set the states of the
    # nodes.

    plt.clf()
    WX_params = WX.get_parameters()
    fh = WX_params[0] * np.ones(y.shape)
    err_fh = 2*np.sqrt(WX_params[1] + 1/tau.get_moments()[0]) * np.ones(y.shape)
    for m in range(M):
        plt.subplot(M,1,m+1)
        #errorplot(y, error=None, x=None, lower=None, upper=None):
        bpplt.errorplot(fh[m], x=np.arange(N), error=err_fh[m])
        plt.plot(np.arange(N), f[m], 'g')
        plt.plot(np.arange(N), y[m], 'r+')

    plt.figure()
    Q.plot_iteration_by_nodes()

    plt.figure()
    plt.subplot(2,2,1)
    bpplt.binary_matrix(W.mask)
    plt.subplot(2,2,2)
    bpplt.binary_matrix(X.mask)
    plt.subplot(2,2,3)
    #bpplt.binary_matrix(WX.get_mask())
    plt.subplot(2,2,4)
    bpplt.binary_matrix(Y.mask)
예제 #17
0
def run_dlssm(y, f, mask, D, K, maxiter):
    """
    Run VB inference for linear state space model with drifting dynamics.
    """

    (M, N) = np.shape(y)

    # Dynamics matrix with ARD
    # alpha : (D) x ()
    alpha = Gamma(1e-5, 1e-5, plates=(K, ), name='alpha')
    # A : (K) x (K)
    A = Gaussian(
        np.zeros(K),
        #np.identity(K),
        diagonal(alpha),
        plates=(K, ),
        name='A_S')
    A.initialize_from_value(np.identity(K))

    # rho
    ## rho = Gamma(1e-5,
    ##             1e-5,
    ##             plates=(K,),
    ##             name="rho")

    # S : () x (N-1,K)
    S = GaussianMarkovChain(np.ones(K),
                            1e-6 * np.identity(K),
                            A,
                            np.ones(K),
                            n=N - 1,
                            name='S')
    S.initialize_from_value(1 * np.ones((N - 1, K)))

    # Projection matrix of the dynamics matrix
    # beta : (K) x ()
    beta = Gamma(1e-5, 1e-5, plates=(K, ), name='beta')
    # B : (D) x (D*K)
    B = Gaussian(np.zeros(D * K),
                 diagonal(tile(beta, D)),
                 plates=(D, ),
                 name='B')
    b = np.zeros((D, D, K))
    b[np.arange(D), np.arange(D), np.zeros(D, dtype=int)] = 1
    B.initialize_from_value(np.reshape(1 * b, (D, D * K)))

    # A : (N-1,D) x (D)
    BS = MatrixDot(B, S.as_gaussian().add_plate_axis(-1), name='BS')

    # Latent states with dynamics
    # X : () x (N,D)
    X = GaussianMarkovChain(
        np.zeros(D),  # mean of x0
        1e-3 * np.identity(D),  # prec of x0
        BS,  # dynamics
        np.ones(D),  # innovation
        n=N,  # time instances
        name='X',
        initialize=False)
    X.initialize_from_value(np.random.randn(N, D))

    # Mixing matrix from latent space to observation space using ARD
    # gamma : (D) x ()
    gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
    # C : (M,1) x (D)
    C = Gaussian(np.zeros(D), diagonal(gamma), plates=(M, 1), name='C')
    C.initialize_from_value(np.random.randn(M, 1, D))

    # Observation noise
    # tau : () x ()
    tau = Gamma(1e-5, 1e-5, name='tau')

    # Observations
    # Y : (M,N) x ()
    CX = Dot(C, X.as_gaussian())
    Y = Normal(CX, tau, name='Y')

    #
    # RUN INFERENCE
    #

    # Observe data
    Y.observe(y, mask=mask)
    # Construct inference machine
    Q = VB(Y, X, S, A, alpha, B, beta, C, gamma, tau)

    #
    # Run inference with rotations.
    #

    # Rotate the D-dimensional state space (C, X)
    rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='rows')
    rotX = transformations.RotateDriftingMarkovChain(X, B, S, rotB)
    rotC = transformations.RotateGaussianARD(C, gamma)
    R_X = transformations.RotationOptimizer(rotX, rotC, D)

    # Rotate the K-dimensional latent dynamics space (B, S)
    rotA = transformations.RotateGaussianARD(A, alpha)
    rotS = transformations.RotateGaussianMarkovChain(S, A, rotA)
    rotB = transformations.RotateGaussianMatrixARD(B, beta, axis='cols')
    R_S = transformations.RotationOptimizer(rotS, rotB, K)

    # Iterate
    for ind in range(int(maxiter / 5)):
        Q.update(repeat=5)
        #Q.update(X, S, A, alpha, rho, B, beta, C, gamma, tau, repeat=maxiter)
        R_X.rotate()
        R_S.rotate()
        ## R_X.rotate(
        ## check_bound=Q.compute_lowerbound,
        ## check_bound_terms=Q.compute_lowerbound_terms,
        ## check_gradient=True
        ##     )
        ## R_S.rotate(
        ## check_bound=Q.compute_lowerbound,
        ## check_bound_terms=Q.compute_lowerbound_terms,
        ## check_gradient=True
        ##     )

    #
    # SHOW RESULTS
    #

    # Mean and standard deviation of the posterior
    (f_mean, f_squared) = CX.get_moments()
    f_std = np.sqrt(f_squared - f_mean**2)

    # Plot observations space
    for m in range(M):
        plt.subplot(M, 1, m + 1)
        plt.plot(y[m, :], 'r.')
        plt.plot(f[m, :], 'b-')
        bpplt.errorplot(y=f_mean[m, :], error=2 * f_std[m, :])
예제 #18
0
def run_lssm(y, f, mask, D, maxiter):
    """
    Run VB inference for linear state space model.
    """

    (M, N) = np.shape(y)

    #
    # CONSTRUCT THE MODEL
    #

    # Dynamic matrix
    # alpha: (D) x ()
    alpha = Gamma(1e-5, 1e-5, plates=(D, ), name='alpha')
    # A : (D) x (D)
    A = Gaussian(np.zeros(D), diagonal(alpha), plates=(D, ), name='A')
    A.initialize_from_value(np.identity(D))

    # Latent states with dynamics
    # X : () x (N,D)
    X = GaussianMarkovChain(
        np.zeros(D),  # mean of x0
        1e-3 * np.identity(D),  # prec of x0
        A,  # dynamics
        np.ones(D),  # innovation
        n=N,  # time instances
        name='X',
        initialize=False)
    X.initialize_from_value(np.random.randn(N, D))

    # Mixing matrix from latent space to observation space using ARD
    # gamma : (D) x ()
    gamma = Gamma(1e-5, 1e-5, plates=(D, ), name='gamma')
    # C : (M,1) x (D)
    C = Gaussian(np.zeros(D), diagonal(gamma), plates=(M, 1), name='C')
    C.initialize_from_value(np.random.randn(M, 1, D))

    # Observation noise
    # tau : () x ()
    tau = Gamma(1e-5, 1e-5, name='tau')

    # Observations
    # Y : (M,N) x ()
    CX = Dot(C, X.as_gaussian())
    Y = Normal(CX, tau, name='Y')

    #
    # RUN INFERENCE
    #

    # Observe data
    Y.observe(y, mask=mask)
    # Construct inference machine
    Q = VB(Y, X, A, alpha, C, gamma, tau)
    # Iterate
    Q.update(X, A, alpha, C, gamma, tau, repeat=maxiter)

    #
    # SHOW RESULTS
    #

    # Mean and standard deviation of the posterior
    (f_mean, f_squared) = CX.get_moments()
    f_std = np.sqrt(f_squared - f_mean**2)

    # Plot observations space
    #plt.figure()
    for m in range(M):
        plt.subplot(M, 1, m + 1)
        plt.plot(y[m, :], 'r.')
        plt.plot(f[m, :], 'b-')
        bpplt.errorplot(y=f_mean[m, :], error=2 * f_std[m, :])
예제 #19
0
def run(N=100000, N_batch=50, seed=42, maxiter=100, plot=True):
    """
    Run deterministic annealing demo for 1-D Gaussian mixture.
    """

    if seed is not None:
        np.random.seed(seed)

    # Number of clusters in the model
    K = 20

    # Dimensionality of the data
    D = 5

    # Generate data
    K_true = 10
    spread = 5
    means = spread * np.random.randn(K_true, D)
    z = random.categorical(np.ones(K_true), size=N)
    data = np.empty((N, D))
    for n in range(N):
        data[n] = means[z[n]] + np.random.randn(D)

    #
    # Standard VB-EM algorithm
    #

    # Full model
    mu = Gaussian(np.zeros(D), np.identity(D), plates=(K, ), name='means')
    alpha = Dirichlet(np.ones(K), name='class probabilities')
    Z = Categorical(alpha, plates=(N, ), name='classes')
    Y = Mixture(Z, Gaussian, mu, np.identity(D), name='observations')

    # Break symmetry with random initialization of the means
    mu.initialize_from_random()

    # Put the data in
    Y.observe(data)

    # Run inference
    Q = VB(Y, Z, mu, alpha)
    Q.save(mu)
    Q.update(repeat=maxiter)
    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'k-')
    max_cputime = np.sum(Q.cputime[~np.isnan(Q.cputime)])

    #
    # Stochastic variational inference
    #

    # Construct smaller model (size of the mini-batch)
    mu = Gaussian(np.zeros(D), np.identity(D), plates=(K, ), name='means')
    alpha = Dirichlet(np.ones(K), name='class probabilities')
    Z = Categorical(alpha,
                    plates=(N_batch, ),
                    plates_multiplier=(N / N_batch, ),
                    name='classes')
    Y = Mixture(Z, Gaussian, mu, np.identity(D), name='observations')

    # Break symmetry with random initialization of the means
    mu.initialize_from_random()

    # Inference engine
    Q = VB(Y, Z, mu, alpha, autosave_filename=Q.autosave_filename)
    Q.load(mu)

    # Because using mini-batches, messages need to be multiplied appropriately
    print("Stochastic variational inference...")
    Q.ignore_bound_checks = True

    maxiter *= int(N / N_batch)
    delay = 1
    forgetting_rate = 0.7
    for n in range(maxiter):

        # Observe a mini-batch
        subset = np.random.choice(N, N_batch)
        Y.observe(data[subset, :])

        # Learn intermediate variables
        Q.update(Z)

        # Set step length
        step = (n + delay)**(-forgetting_rate)

        # Stochastic gradient for the global variables
        Q.gradient_step(mu, alpha, scale=step)

        if np.sum(Q.cputime[:n]) > max_cputime:
            break

    if plot:
        bpplt.pyplot.plot(np.cumsum(Q.cputime), Q.L, 'r:')

        bpplt.pyplot.xlabel('CPU time (in seconds)')
        bpplt.pyplot.ylabel('VB lower bound')
        bpplt.pyplot.legend(['VB-EM', 'Stochastic inference'],
                            loc='lower right')
        bpplt.pyplot.title('VB for Gaussian mixture model')

    return
예제 #20
0
파일: annealing.py 프로젝트: chagge/bayespy
def run(N=500, seed=42, maxiter=100, plot=True):
    """
    Run deterministic annealing demo for 1-D Gaussian mixture.
    """

    if seed is not None:
        np.random.seed(seed)

    mu = GaussianARD(0, 1,
                     plates=(2,),
                     name='means')
    Z = Categorical([0.3, 0.7],
                    plates=(N,),
                    name='classes')
    Y = Mixture(Z, GaussianARD, mu, 1,
                name='observations')

    # Generate data
    z = Z.random()
    data = np.empty(N)
    for n in range(N):
        data[n] = [4, -4][z[n]]

    Y.observe(data)

    # Initialize means closer to the inferior local optimum in which the
    # cluster means are swapped
    mu.initialize_from_value([0, 6])

    Q = VB(Y, Z, mu)
    Q.save()

    #
    # Standard VB-EM algorithm
    #
    Q.update(repeat=maxiter)

    mu_vbem = mu.u[0].copy()
    L_vbem = Q.compute_lowerbound()

    #
    # VB-EM with deterministic annealing
    #
    Q.load()
    beta = 0.01
    while beta < 1.0:
        beta = min(beta*1.2, 1.0)
        print("Set annealing to %.2f" % beta)
        Q.set_annealing(beta)
        Q.update(repeat=maxiter, tol=1e-4)

    mu_anneal = mu.u[0].copy()
    L_anneal = Q.compute_lowerbound()

    print("==============================")
    print("RESULTS FOR VB-EM vs ANNEALING")
    print("Fixed component probabilities:", np.array([0.3, 0.7]))
    print("True component means:", np.array([4, -4]))
    print("VB-EM component means:", mu_vbem)
    print("VB-EM lower bound:", L_vbem)
    print("Annealed VB-EM component means:", mu_anneal)
    print("Annealed VB-EM lower bound:", L_anneal)
    
    return
예제 #21
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def run_lssm(y, f, mask, D, maxiter):
    """
    Run VB inference for linear state space model.
    """

    (M, N) = np.shape(y)

    #
    # CONSTRUCT THE MODEL
    #

    # Dynamic matrix
    # alpha: (D) x ()
    alpha = Gamma(1e-5,
                  1e-5,
                  plates=(D,),
                  name='alpha')
    # A : (D) x (D)
    A = Gaussian(np.zeros(D),
                 diagonal(alpha),
                 plates=(D,),
                 name='A')
    A.initialize_from_value(np.identity(D))

    # Latent states with dynamics
    # X : () x (N,D)
    X = GaussianMarkovChain(np.zeros(D),         # mean of x0
                            1e-3*np.identity(D), # prec of x0
                            A,                   # dynamics
                            np.ones(D),          # innovation
                            n=N,                 # time instances
                            name='X',
                            initialize=False)
    X.initialize_from_value(np.random.randn(N,D))

    # Mixing matrix from latent space to observation space using ARD
    # gamma : (D) x ()
    gamma = Gamma(1e-5,
                  1e-5,
                  plates=(D,),
                  name='gamma')
    # C : (M,1) x (D)
    C = Gaussian(np.zeros(D),
                 diagonal(gamma),
                 plates=(M,1),
                 name='C')
    C.initialize_from_value(np.random.randn(M,1,D))

    # Observation noise
    # tau : () x ()
    tau = Gamma(1e-5,
                1e-5,
                name='tau')

    # Observations
    # Y : (M,N) x ()
    CX = Dot(C, X.as_gaussian())
    Y = Normal(CX,
               tau,
               name='Y')

    # Rotate the D-dimensional latent space
    rotA = transformations.RotateGaussianARD(A, alpha)
    rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
    rotC = transformations.RotateGaussianARD(C, gamma)
    R = transformations.RotationOptimizer(rotX, rotC, D)

    #
    # RUN INFERENCE
    #

    # Observe data
    Y.observe(y, mask=mask)
    # Construct inference machine
    Q = VB(Y, X, A, alpha, C, gamma, tau)

    # Iterate
    for ind in range(maxiter):
        Q.update(X, A, alpha, C, gamma, tau)
        R.rotate()

    #
    # SHOW RESULTS
    #

    plt.figure()
    bpplt.timeseries_normal(CX)
    bpplt.timeseries(f, 'b-')
    bpplt.timeseries(y, 'r.')
예제 #22
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def run(maxiter=100):

    seed = 496  #np.random.randint(1000)
    print("seed = ", seed)
    np.random.seed(seed)

    # Simulate some data
    D = 3
    M = 6
    N = 200
    c = np.random.randn(M, D)
    w = 0.3
    a = np.array([[np.cos(w), -np.sin(w), 0], [np.sin(w),
                                               np.cos(w), 0], [0, 0, 1]])
    x = np.empty((N, D))
    f = np.empty((M, N))
    y = np.empty((M, N))
    x[0] = 10 * np.random.randn(D)
    f[:, 0] = np.dot(c, x[0])
    y[:, 0] = f[:, 0] + 3 * np.random.randn(M)
    for n in range(N - 1):
        x[n + 1] = np.dot(a, x[n]) + np.random.randn(D)
        f[:, n + 1] = np.dot(c, x[n + 1])
        y[:, n + 1] = f[:, n + 1] + 3 * np.random.randn(M)

    # Create the model
    (Y, CX, X, tau, C, gamma, A, alpha) = linear_state_space_model(D, N, M)

    # Add missing values randomly
    mask = random.mask(M, N, p=0.3)
    # Add missing values to a period of time
    mask[:, 30:80] = False
    y[~mask] = np.nan  # BayesPy doesn't require this. Just for plotting.
    # Observe the data
    Y.observe(y, mask=mask)

    # Initialize nodes (must use some randomness for C)
    C.initialize_from_random()

    # Run inference
    Q = VB(Y, X, C, gamma, A, alpha, tau)

    #
    # Run inference with rotations.
    #
    rotA = transformations.RotateGaussianARD(A, alpha)
    rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
    rotC = transformations.RotateGaussianARD(C, gamma)
    R = transformations.RotationOptimizer(rotX, rotC, D)

    #maxiter = 84
    for ind in range(maxiter):
        Q.update()
        #print('C term', C.lower_bound_contribution())
        R.rotate(
            maxiter=10,
            check_gradient=True,
            verbose=False,
            check_bound=Q.compute_lowerbound,
            #check_bound=None,
            check_bound_terms=Q.compute_lowerbound_terms)
        #check_bound_terms=None)

    X_vb = X.u[0]
    varX_vb = utils.diagonal(X.u[1] - X_vb[..., np.newaxis, :] *
                             X_vb[..., :, np.newaxis])

    u_CX = CX.get_moments()
    CX_vb = u_CX[0]
    varCX_vb = u_CX[1] - CX_vb**2

    # Show results
    plt.figure(3)
    plt.clf()
    for m in range(M):
        plt.subplot(M, 1, m + 1)
        plt.plot(y[m, :], 'r.')
        plt.plot(f[m, :], 'b-')
        bpplt.errorplot(y=CX_vb[m, :], error=2 * np.sqrt(varCX_vb[m, :]))

    plt.figure()
    Q.plot_iteration_by_nodes()
예제 #23
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def run(M=50, N=200, D_y=10, D=20, maxiter=100):
    seed = 45
    print('seed =', seed)
    np.random.seed(seed)

    # Generate data (covariance eigenvalues: 1,1,...,1,2^2,3^2,...,(D_y+1)^2
    (q, r) = scipy.linalg.qr(np.random.randn(M, M))
    C = np.diag(np.arange(2, 2 + D_y))
    C = np.ones(M)
    C[:D_y] += np.arange(1, 1 + D_y)
    y = C[:, np.newaxis] * np.random.randn(M, N)
    y = np.dot(q, y)

    # Construct model
    (Y, WX, W, X, tau, alpha) = pca_model(M, N, D)

    # Data with missing values
    mask = utils.random.mask(M, N, p=0.9)  # randomly missing
    mask[:, 20:40] = False  # gap missing
    y[~mask] = np.nan
    Y.observe(y, mask=mask)

    # Construct inference machine
    Q = VB(Y, W, X, tau, alpha, autosave_filename=utils.utils.tempfile())

    # Initialize nodes (from prior and randomly)
    X.initialize_from_value(X.random())
    W.initialize_from_value(W.random())

    Q.update(repeat=1)
    Q.save()

    #
    # Run inference with rotations.
    #
    rotX = transformations.RotateGaussian(X)
    rotW = transformations.RotateGaussianARD(W, alpha)
    R = transformations.RotationOptimizer(rotX, rotW, D)

    for ind in range(maxiter):
        Q.update()
        R.rotate(check_gradient=False,
                 maxiter=10,
                 verbose=False,
                 check_bound=Q.compute_lowerbound,
                 check_bound_terms=Q.compute_lowerbound_terms)

    L_rot = Q.L

    #
    # Re-run inference without rotations.
    #
    Q.load()
    Q.update(repeat=maxiter)
    L_norot = Q.L

    #
    # Plot comparison
    #
    plt.plot(L_rot)
    plt.plot(L_norot)
    plt.legend(['With rotations', 'Without rotations'], loc='lower right')
    plt.show()
예제 #24
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파일: lssm.py 프로젝트: Sandy4321/bayespy
# 

# In[4]:

from bayespy.utils import random

# Add missing values randomly (keep only 20%)
mask = random.mask(M, N, p=0.2)
Y.observe(y, mask=mask)


# Then inference (100 iterations) can be run simply as

# In[5]:

Q.update(repeat=10)


### Speeding up with parameter expansion

# VB inference can converge extremely slowly if the variables are strongly coupled.  Because VMP updates one variable at a time, it may lead to slow zigzagging.  This can be solved by using parameter expansion which reduces the coupling. In state-space models, the states $\mathbf{x}_n$ and the loadings $\mathbf{C}$ are coupled through a dot product $\mathbf{Cx}_n$, which is unaltered if the latent space is rotated arbitrarily:
# 
# $$
# \mathbf{y}_n = \mathbf{C}\mathbf{x}_n = \mathbf{C}\mathbf{R}^{-1}\mathbf{R}\mathbf{x}_n \,.
# $$
# 
# Thus, one intuitive transformation would be $\mathbf{C}\rightarrow\mathbf{C}\mathbf{R}^{-1}$ and $\mathbf{X}\rightarrow\mathbf{R}\mathbf{X}$.  In order to keep the dynamics of the latent states unaffected by the transformation, the state dynamics matrix $\mathbf{A}$ must be transformed accordingly:
# 
# $$
# \mathbf{R}\mathbf{x}_n = \mathbf{R}\mathbf{A}\mathbf{R}^{-1} \mathbf{R}\mathbf{x}_{n-1} \,,
# $$
예제 #25
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파일: demo_lssm.py 프로젝트: vlall/bayespy
def run(M=6, N=200, D=3, maxiter=100, debug=False, seed=42, rotate=False, precompute=False):

    # Use deterministic random numbers
    if seed is not None:
        np.random.seed(seed)

    # Simulate some data
    K = 3
    c = np.random.randn(M,K)
    w = 0.3
    a = np.array([[np.cos(w), -np.sin(w), 0], 
                  [np.sin(w), np.cos(w),  0], 
                  [0,         0,          1]])
    x = np.empty((N,K))
    f = np.empty((M,N))
    y = np.empty((M,N))
    x[0] = 10*np.random.randn(K)
    f[:,0] = np.dot(c,x[0])
    y[:,0] = f[:,0] + 3*np.random.randn(M)
    for n in range(N-1):
        x[n+1] = np.dot(a,x[n]) + np.random.randn(K)
        f[:,n+1] = np.dot(c,x[n+1])
        y[:,n+1] = f[:,n+1] + 3*np.random.randn(M)

    # Create the model
    (Y, CX, X, tau, C, gamma, A, alpha) = linear_state_space_model(D=D, 
                                                                   N=N,
                                                                   M=M)
    
    # Add missing values randomly
    mask = random.mask(M, N, p=0.3)
    # Add missing values to a period of time
    mask[:,30:80] = False
    y[~mask] = np.nan # BayesPy doesn't require this. Just for plotting.
    # Observe the data
    Y.observe(y, mask=mask)

    # Initialize nodes (must use some randomness for C)
    C.initialize_from_random()

    # Run inference
    Q = VB(Y, X, C, gamma, A, alpha, tau)

    #
    # Run inference with rotations.
    #
    if rotate:
        rotA = transformations.RotateGaussianArrayARD(A, alpha, precompute=precompute)
        rotX = transformations.RotateGaussianMarkovChain(X, A, rotA)
        rotC = transformations.RotateGaussianArrayARD(C, gamma)
        R = transformations.RotationOptimizer(rotX, rotC, D)

        for ind in range(maxiter):
            Q.update()
            if debug:
                R.rotate(maxiter=10, 
                         check_gradient=True,
                         check_bound=True)
            else:
                R.rotate()

    else:
        Q.update(repeat=maxiter)
        
    # Show results
    plt.figure()
    bpplt.timeseries_normal(CX, scale=2)
    bpplt.timeseries(f, 'b-')
    bpplt.timeseries(y, 'r.')
    plt.show()
예제 #26
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def run_lssm(y, f, mask, D, maxiter):
    """
    Run VB inference for linear state space model.
    """

    (M, N) = np.shape(y)

    #
    # CONSTRUCT THE MODEL
    #

    # Dynamic matrix
    # alpha: (D) x ()
    alpha = Gamma(1e-5,
                  1e-5,
                  plates=(D,),
                  name='alpha')
    # A : (D) x (D)
    A = Gaussian(np.zeros(D),
                 diagonal(alpha),
                 plates=(D,),
                 name='A')
    A.initialize_from_value(np.identity(D))

    # Latent states with dynamics
    # X : () x (N,D)
    X = GaussianMarkovChain(np.zeros(D),         # mean of x0
                            1e-3*np.identity(D), # prec of x0
                            A,                   # dynamics
                            np.ones(D),          # innovation
                            n=N,                 # time instances
                            name='X',
                            initialize=False)
    X.initialize_from_value(np.random.randn(N,D))

    # Mixing matrix from latent space to observation space using ARD
    # gamma : (D) x ()
    gamma = Gamma(1e-5,
                  1e-5,
                  plates=(D,),
                  name='gamma')
    # C : (M,1) x (D)
    C = Gaussian(np.zeros(D),
                 diagonal(gamma),
                 plates=(M,1),
                 name='C')
    C.initialize_from_value(np.random.randn(M,1,D))

    # Observation noise
    # tau : () x ()
    tau = Gamma(1e-5,
                1e-5,
                name='tau')

    # Observations
    # Y : (M,N) x ()
    CX = Dot(C, X.as_gaussian())
    Y = Normal(CX,
               tau,
               name='Y')

    #
    # RUN INFERENCE
    #

    # Observe data
    Y.observe(y, mask=mask)
    # Construct inference machine
    Q = VB(Y, X, A, alpha, C, gamma, tau)
    # Iterate
    Q.update(X, A, alpha, C, gamma, tau, repeat=maxiter)

    #
    # SHOW RESULTS
    #

    # Mean and standard deviation of the posterior
    (f_mean, f_squared) = CX.get_moments()
    f_std = np.sqrt(f_squared - f_mean**2)

    # Plot observations space
    #plt.figure()
    for m in range(M):
        plt.subplot(M,1,m+1)
        plt.plot(y[m,:], 'r.')
        plt.plot(f[m,:], 'b-')
        bpplt.errorplot(y=f_mean[m,:], error=2*f_std[m,:])