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gpdm_v1.py
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gpdm_v1.py
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# ---- coding: utf-8 ----
# @author: Ziyang Zhang
# @version: v1, original GPDM with CUDA supported
# - observationGP: x->y, RBF
# - dynamicGP: x0->x1, RBF+Linear
# This work partly uses the code from CIGP and CGPDM.
# @license: (C) Copyright 2023, AMMLE Group Limited.
import torch
import numpy as np
import time
from sklearn.decomposition import PCA
from torch.distributions.normal import Normal
import matplotlib.pyplot as plt
sigma_n_num_Y = 10 ** -3
sigma_n_num_X = 10 ** -3
class GPDM(torch.nn.Module):
def __init__(self, D, Q, dyn_target):
super(GPDM, self).__init__()
self.device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
self.dtype = torch.float64
# observation dimension
self.D = D
# latent dimension
self.Q = Q
# dynamic model target, choose full or delta
self.dyn_target = dyn_target
""" Set Y-kernel parameters (RBF) """
# log_lengthscale in RBF kernel
self.y_log_lengthscales = torch.nn.Parameter(
torch.tensor(np.log(np.abs(np.ones(Q))), dtype=self.dtype, device=self.device))
# log(signal inverse std), an initial scaling vector used for constructing W_y
self.y_log_lambdas = torch.nn.Parameter(
torch.tensor(np.log(np.abs(np.ones(D))), dtype=self.dtype, device=self.device))
# log(noise std), noise in RBF kernel
self.y_log_sigma_n = torch.nn.Parameter(
torch.tensor(np.log(np.abs(np.ones(1))), dtype=self.dtype, device=self.device))
""" Set X-kernel parameters (linear+RBF) """
# log_lengthscale in RBF kernel
self.x_log_lengthscales = torch.nn.Parameter(
torch.tensor(np.log(np.abs(np.ones(Q))), dtype=self.dtype, device=self.device))
# log(signal inverse std), an initial scaling vector used for constructing W_x
self.x_log_lambdas = torch.nn.Parameter(
torch.tensor(np.log(np.abs(np.ones(Q))), dtype=self.dtype, device=self.device))
# # log(noise std), noise in RBF kernel
self.x_log_sigma_n = torch.nn.Parameter(
torch.tensor(np.log(np.abs(np.ones(1))), dtype=self.dtype, device=self.device))
# log_linear_coefficients in linear kernel
self.log_coe_linear = torch.nn.Parameter(torch.zeros(1, dtype=self.dtype, device=self.device))
# additional noise variance for numerical issues
self.sigma_n_num_Y = sigma_n_num_Y
self.sigma_n_num_X = sigma_n_num_X
# Initialize observations
self.observations_list = []
self.num_sequences = 0
def add_data(self, Y):
# Note: each sequence shares the same timestep N and dimension D
if Y.shape[1] != self.D:
raise ValueError('Y must be a N x D matrix collecting observation data!')
self.observations_list.append(Y)
self.num_sequences = self.num_sequences + 1
print('Num. of sequences = ' + str(self.num_sequences) + ' [Data points = ' + str(
np.concatenate(self.observations_list, 0).shape[0]) + ']')
def observationGP_kernel(self, X1, X2, flg_noise=True):
"""
currently use RBF
"""
return self.kernel_rbf(X1, X2, self.y_log_lengthscales, self.y_log_sigma_n, self.sigma_n_num_Y, flg_noise)
def dynamicGP_kernel(self, X1, X2, flg_noise=True):
"""
currently use RBF+linear
"""
return self.kernel_rbf(X1, X2, self.x_log_lengthscales, self.x_log_sigma_n, self.sigma_n_num_X, flg_noise) + \
self.kernel_lin(X1, X2)
def kernel_rbf(self, X1, X2, log_lengthscales, log_sigma_n, sigma_n_num=0, flg_noise=True):
N = X1.shape[0]
X1 = X1 / log_lengthscales.exp()
X2 = X2 / log_lengthscales.exp()
X1_norm2 = torch.sum(X1 * X1, dim=1).view(-1, 1)
X2_norm2 = torch.sum(X2 * X2, dim=1).view(-1, 1)
if flg_noise:
K = -2.0 * X1 @ X2.t() + X1_norm2.expand(X1.size(0), X2.size(0)) + X2_norm2.t().expand(X1.size(0),
X2.size(0))
return torch.exp(-0.5 * K) + torch.exp(log_sigma_n) ** 2 * torch.eye(N, dtype=self.dtype,
device=self.device) + \
sigma_n_num ** 2 * torch.eye(N, dtype=self.dtype, device=self.device)
else:
K = -2.0 * X1 @ X2.t() + X1_norm2.expand(X1.size(0), X2.size(0)) + X2_norm2.t().expand(X1.size(0),
X2.size(0))
return torch.exp(-0.5 * K)
def kernel_lin(self, x1, x2):
k_linear = self.log_coe_linear.exp() * (x1 @ x2.t())
return k_linear
def get_y_neg_log_likelihood(self, Y, X, N):
"""
L_y = D/2*log(|K_y(X,X)|) + 1/2*trace(K_y^-1*Y*W_y^2*Y) - N*log(|W_y|)
"""
K_y = self.observationGP_kernel(X, X)
U, info = torch.linalg.cholesky_ex(K_y, upper=True)
U_inv = torch.inverse(U)
Ky_inv = U_inv @ U_inv.t()
log_det_K_y = 2 * torch.sum(torch.log(torch.diag(U)))
W2 = torch.diag(torch.exp(self.y_log_lambdas) ** 2)
log_det_W = 2 * torch.sum(self.y_log_lambdas)
return self.D / 2 * log_det_K_y + 1 / 2 * \
torch.trace(torch.linalg.multi_dot([Ky_inv, Y, W2, Y.t()])) \
- N * log_det_W
def get_x_neg_log_likelihood(self, Xout, Xin, flg_W=True):
"""
If use scaling matrix W, choose flg_W = True
L_x = d/2*log(|K_x(Xin,Xin)|) + 1/2*trace(K_x^-1*Xout*W_x^2*Xout) - (N-dyn_back_step)*log(|W_x|)
If use original GPDM, choose flg_W = False
L_x = d/2*log(|K_x(Xin,Xin)|) + 1/2*trace(K_x^-1*Xout*Xout)
ISSUE: W_x is necessary
"""
K_x = self.dynamicGP_kernel(Xin, Xin)
U, info = torch.linalg.cholesky_ex(K_x, upper=True)
U_inv = torch.inverse(U)
Kx_inv = U_inv @ U_inv.t()
log_det_K_x = 2 * torch.sum(torch.log(torch.diag(U)))
if flg_W:
W2 = torch.diag(torch.exp(self.x_log_lambdas) ** 2)
log_det_W = 2 * torch.sum(self.x_log_lambdas)
return self.Q / 2 * log_det_K_x + 1 / 2 * \
torch.trace(torch.linalg.multi_dot([Kx_inv, Xout, W2, Xout.t()])) \
- Xin.shape[0] * log_det_W
else:
return self.Q / 2 * log_det_K_x + 1 / 2 * \
torch.trace(torch.linalg.multi_dot([Kx_inv, Xout, Xout.t()]))
def get_Xin_Xout(self):
X_list = []
x_start_index = 0
start_indeces = []
for j in range(len(self.observations_list)):
sequence_length = self.observations_list[j].shape[0]
X_list.append(self.X[x_start_index:x_start_index + sequence_length, :])
start_indeces.append(x_start_index)
x_start_index = x_start_index + sequence_length
if self.dyn_target == 'full':
# in: x(t)
Xin = X_list[0][:-1, :]
# out: x(t+1)
Xout = X_list[0][1:, :]
for j in range(1, len(self.observations_list)):
Xin = torch.cat((Xin, X_list[j][:-1, :]), 0)
Xout = torch.cat((Xout, X_list[j][1:, :]), 0)
elif self.dyn_target == 'delta':
# in: x(t)
Xin = X_list[0][:-1, :]
# out: x(t+1)-x(t)
Xout = X_list[0][1:, :] - X_list[0][:-1, :]
for j in range(1, len(self.observations_list)):
Xin = torch.cat((Xin, X_list[j][:-1, :]), 0)
Xout = torch.cat((Xout, X_list[j][1:, :] - X_list[j][0:-1, :]), 0)
else:
raise ValueError('target must be either \'full\' or \'delta\'')
return Xin, Xout
def gpdm_loss(self, Y, N, balance=1):
"""
loss = Ly + B*Lx
"""
Xin, Xout = self.get_Xin_Xout()
lossY = self.get_y_neg_log_likelihood(Y, self.X, N)
lossX = self.get_x_neg_log_likelihood(Xout, Xin)
loss = lossY + balance * lossX
return loss
def init_X(self):
"""
initialize latent embeddings X with PCA
return the latent trajectories associated to each observation sequence recorded
"""
Y = np.concatenate(self.observations_list, 0) # (M*N, D)
pca = PCA(n_components=self.Q)
X0 = pca.fit_transform(Y)
# set latent variables as parameters
self.X = torch.nn.Parameter(torch.tensor(X0, dtype=self.dtype, device=self.device), requires_grad=True)
# init inverse kernel matrices
Ky = self.observationGP_kernel(self.X, self.X)
U, info = torch.linalg.cholesky_ex(Ky, upper=True)
U_inv = torch.inverse(U)
self.Ky_inv = U_inv @ U_inv.t()
Xin, Xout = self.get_Xin_Xout()
Kx = self.dynamicGP_kernel(Xin, Xin)
U, info = torch.linalg.cholesky_ex(Kx, upper=True)
U_inv = torch.inverse(U)
self.Kx_inv = U_inv @ U_inv.t()
return self.get_latent_sequences()
def get_latent_sequences(self):
X_np = self.X.clone().detach().cpu().numpy()
X_np = X_np.reshape(len(self.observations_list), -1, self.X.shape[1])
return [x for x in X_np]
def train_lbfgs(self, num_opt_steps, lr=0.01, balance=1):
print('\n### START TRAINING WITH L-BFGS ###')
Y = torch.tensor(np.concatenate(self.observations_list, 0), dtype=self.dtype, device=self.device) # (M*N, D)
N = Y.shape[0]
# optimizer
optimizer = torch.optim.LBFGS(self.parameters(), lr=lr, max_iter=20, history_size=7,line_search_fn='strong_wolfe')
# training
losses = []
t_start = time.time()
for epoch in range(num_opt_steps):
def closure():
optimizer.zero_grad()
loss = self.gpdm_loss(Y, N, balance)
if loss.requires_grad:
loss.backward()
return loss
losses.append(closure().item())
optimizer.step(closure)
print('\nEpoch:' + str(epoch+1) + '/' + str(num_opt_steps))
print('Running loss:', "{:.4e}".format(losses[-1]))
t_stop = time.time()
print('Used time:', t_stop - t_start)
t_start = t_stop
# save inverse kernel matrices of dynamicGP after training
# calculate in forward_dynamicGP() will be extremely slow
Xin, Xout = self.get_Xin_Xout()
U, _ = torch.linalg.cholesky_ex(self.dynamicGP_kernel(Xin, Xin), upper=True)
U_inv = torch.inverse(U)
self.Kx_inv = U_inv @ U_inv.t()
return losses
def forward_observationGP(self, Xstar, flg_noise=False):
U, _ = torch.linalg.cholesky_ex(self.observationGP_kernel(self.X, self.X), upper=True)
U_inv = torch.inverse(U)
Ky_inv = U_inv @ U_inv.t()
Y_obs = torch.tensor(np.concatenate(self.observations_list, 0), dtype=self.dtype,
device=self.device) # (M*N, D)
Ky_star = self.observationGP_kernel(self.X, Xstar, False)
mean_Y_pred = torch.linalg.multi_dot([Y_obs.t(), Ky_inv, Ky_star]).t()
if flg_noise:
diag_var_Y_pred_common = torch.ones(Xstar.shape[0], dtype=self.dtype, device=self.device) + \
torch.exp(self.y_log_sigma_n) ** 2 + self.sigma_n_num_Y ** 2
else:
diag_var_Y_pred_common = torch.ones(Xstar.shape[0], dtype=self.dtype, device=self.device)
diag_var_Y_pred_common = diag_var_Y_pred_common - torch.sum(Ky_star.t() @ Ky_inv * Ky_star.t(),
dim=1)
y_log_lambdas = torch.exp(self.y_log_lambdas) ** -2
diag_var_Y_pred = diag_var_Y_pred_common.unsqueeze(1) * y_log_lambdas.unsqueeze(0)
return mean_Y_pred, diag_var_Y_pred
def forward_dynamicGP(self, Xstar, flg_noise=False):
Xin, Xout = self.get_Xin_Xout()
n = Xstar.shape[0]
Kx_star = self.dynamicGP_kernel(Xin, Xstar, False)
mean_Xout_pred = torch.linalg.multi_dot([Xout.t(), self.Kx_inv, Kx_star]).t()
if flg_noise:
diag_var_Xout_pred_common = torch.ones(n, dtype=self.dtype, device=self.device) + \
torch.exp(self.x_log_sigma_n) ** 2 + self.sigma_n_num_X ** 2 + \
torch.sum(self.log_coe_linear.exp() * Xstar * Xstar) - \
torch.sum(Kx_star.t() @ self.Kx_inv * Kx_star.t(), dim=1)
else:
diag_var_Xout_pred_common = torch.ones(n, dtype=self.dtype, device=self.device) + \
torch.sum(self.log_coe_linear.exp() * Xstar * Xstar) - \
torch.sum(Kx_star.t() @ self.Kx_inv * Kx_star.t(), dim=1)
x_log_lambdas = torch.exp(self.x_log_lambdas) ** -2
diag_var_Xout_pred = diag_var_Xout_pred_common.unsqueeze(1) * x_log_lambdas.unsqueeze(0)
return mean_Xout_pred, diag_var_Xout_pred
def forward(self, num_steps, X0, num_sample=10, flg_noise=False):
print('\n ### START SAMPLING & PREDICTING... ###')
with torch.no_grad():
X_hat = torch.zeros((num_steps, self.Q), dtype=self.dtype, device=self.device)
# init latent variables
X_hat[0, :] = torch.tensor(X0, dtype=self.dtype, device=self.device)
t_start = 0
# sample dynamic GP
sample_list = torch.zeros((num_sample, num_steps, self.Q), dtype=self.dtype, device=self.device)
# generate latent rollout
for t in range(t_start, num_steps):
Xin = X_hat[t:t + 1, :]
mean_Xout_pred, var_Xout_pred = self.forward_dynamicGP(Xin, flg_noise)
# generate distribution to sample
distribution = Normal(mean_Xout_pred, torch.sqrt(var_Xout_pred))
sample_list[:, t_start, :] = distribution.sample((num_sample,)).squeeze(1)
if self.dyn_target == 'full':
X_hat[t + 1:t + 2, :] = mean_Xout_pred
elif self.dyn_target == 'delta':
X_hat[t + 1:t + 2, :] = X_hat[t:, :] + mean_Xout_pred
# map X mean to observation space to get Y mean
mean_Y_pred, _ = self.forward_observationGP(X_hat, flg_noise)
# map X samples to observation space to get Y var
var_list = torch.zeros((num_sample, num_steps, self.D), dtype=self.dtype, device=self.device)
for s in range(num_sample):
X_mean_sample = sample_list[s, :, :]
_, var_Y_sample = self.forward_observationGP(X_mean_sample, flg_noise)
var_list[s, :, :] = var_Y_sample
# get samples mean
var_Y_pred = torch.mean(var_list, dim=0)
return X_hat.detach().cpu().numpy(), mean_Y_pred.detach().cpu().numpy(), var_Y_pred.detach().cpu().numpy()
if __name__ == "__main__":
""" hyper-parameters """
Q = 3 # latent dim
epochs = 3 # optimization steps (max epochs)
lr = 0.01 # learning rate
print('\n *********** TRAIN GPDM *********** :')
print(' - latent dimension: ' + str(Q))
print(' - optimization steps: ' + str(epochs))
print(' - learning rate: ' + str(lr))
print(' - device: ' + "cuda" if torch.cuda.is_available() else "cpu")
print('\n')
""" prepare data """
# generate periodic data (observation data)
# here we generate 5 sequences, each with (N=200, D=5)
Y_data = []
for i in range(5):
i += 1
y1 = np.sin(np.arange(0, 20, 0.1)) * i
y2 = np.cos(np.arange(0, 20, 0.1)) * 5 / i
y3 = np.sin(np.arange(0, 20, 0.1) + np.pi / 4 * i / 2) * 4
y4 = np.cos(np.arange(0, 20, 0.1) + np.pi / 4 * i / 3) * 3
y5 = np.sin(np.arange(0, 20, 0.1) + np.pi / 2 * i / 2) * 2 * i
Y_data.append(np.concatenate(
(y1.reshape(-1, 1), y2.reshape(-1, 1), y3.reshape(-1, 1), y4.reshape(-1, 1), y5.reshape(-1, 1)), axis=1))
# plot training data
fig, axs = plt.subplots(5, 1, figsize=(8, 8))
fig.suptitle('Training Data, 5 seqs with shape (200x5)', fontsize=16)
for i in range(5):
for j in range(5):
axs[i].plot([i for i in range(200)], Y_data[i][:,j])
plt.show()
""" init GPDM """
D = Y_data[0].shape[1]
dyn_target = 'full' # choose full or delta, see Higher-order Features in the GPDM paper
model = GPDM(D=D, Q=Q, dyn_target=dyn_target)
# add training data
for i in Y_data:
model.add_data(i)
# get initial X by PCA
X_list_pca = model.init_X()
""" train GPDM """
start_time = time.time()
loss = model.train_lbfgs(num_opt_steps=epochs, lr=lr, balance=1)
end_time = time.time()
train_time = end_time - start_time
print("\nTotal Training Time: " + str(train_time) + " s")
""" plot results """
## plot loss
plt.figure()
plt.plot(loss)
plt.grid()
plt.title('Loss')
plt.xlabel('Optimization steps')
plt.show()
## latent trajectories
X_list = model.get_latent_sequences()
plt.figure()
plt.suptitle('Latent trajectories')
for j in range(Q):
plt.subplot(Q,1,j+1)
plt.xlabel('Time [s]')
plt.ylabel(r'$x_{'+str(j+1)+'}$')
for i in range(len(X_list)):
plt.plot(X_list[i][:,j])
plt.grid()
plt.show()
## latent trajectories in 3D (only when Q=3)
if Q == 3:
fig = plt.figure()
ax = fig.gca(projection='3d')
X = X_list[0]
x = X[:, 0]
y = X[:, 1]
z = X[:, 2]
ax.plot(x, y, z, label='trajectory')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.set_title('3D trajectory')
plt.show()
## test and plot one of the dimensions
X_list = model.get_latent_sequences()
Y_list = model.observations_list
# choose the end of the sequences
X = X_list[4]
Y = Y_list[4]
N = Y.shape[0] # timestep
forward_steps = 100 # how many steps to inference
_, Ypred, Ystd = model(num_steps=forward_steps, num_sample=100, X0=X[-1, :], flg_noise=True)
plt.figure()
plt.plot([i for i in range(N)], Y[:, 0]) # original seq
plt.plot([i + N for i in range(forward_steps)], Ypred[:, 0]) # inference part
plt.fill_between([i + N for i in range(forward_steps)], # confidence
Ypred[:, 0] + np.sqrt(Ystd[:, 0]),
Ypred[:, 0] - np.sqrt(Ystd[:, 0]), alpha=0.2)
plt.show()