Bayesian Temporal Tensor Factorization (or BTTF for short) is a type of Bayesian tensor decomposition that achieves state-of-the-art results on challenging the missing data imputation problem. In the following, we will discuss:

• What the BTTF is?

• How to implement BTTF mainly using Python Numpy with high efficiency?

• How to make imputations with real-world spatiotemporal datasets?

If you want to understand what is BTMF and its modeling tricks in detail, our paper is for you:

Xinyu Chen, Lijun Sun (2019). Bayesian temporal factorization for multidimensional time series prediction.

## Quick Run¶

This notebook is publicly available for any usage at our data imputation project. Please click transdim.

In [1]:
import numpy as np
from numpy.random import multivariate_normal as mvnrnd
from scipy.stats import wishart
from scipy.stats import invwishart
from numpy.linalg import inv as inv


# Part 1: Matrix Computation Concepts¶

## 1) Kronecker product¶

• Definition:

Given two matrices $A\in\mathbb{R}^{m_1\times n_1}$ and $B\in\mathbb{R}^{m_2\times n_2}$, then, the Kronecker product between these two matrices is defined as

$$A\otimes B=\left[ \begin{array}{cccc} a_{11}B & a_{12}B & \cdots & a_{1m_2}B \\ a_{21}B & a_{22}B & \cdots & a_{2m_2}B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m_11}B & a_{m_12}B & \cdots & a_{m_1m_2}B \\ \end{array} \right]$$

where the symbol $\otimes$ denotes Kronecker product, and the size of resulted $A\otimes B$ is $(m_1m_2)\times (n_1n_2)$ (i.e., $m_1\times m_2$ columns and $n_1\times n_2$ rows).

• Example:

If $A=\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right]$ and $B=\left[ \begin{array}{ccc} 5 & 6 & 7\\ 8 & 9 & 10 \\ \end{array} \right]$, then, we have

$$A\otimes B=\left[ \begin{array}{cc} 1\times \left[ \begin{array}{ccc} 5 & 6 & 7\\ 8 & 9 & 10\\ \end{array} \right] & 2\times \left[ \begin{array}{ccc} 5 & 6 & 7\\ 8 & 9 & 10\\ \end{array} \right] \\ 3\times \left[ \begin{array}{ccc} 5 & 6 & 7\\ 8 & 9 & 10\\ \end{array} \right] & 4\times \left[ \begin{array}{ccc} 5 & 6 & 7\\ 8 & 9 & 10\\ \end{array} \right] \\ \end{array} \right]$$$$=\left[ \begin{array}{cccccc} 5 & 6 & 7 & 10 & 12 & 14 \\ 8 & 9 & 10 & 16 & 18 & 20 \\ 15 & 18 & 21 & 20 & 24 & 28 \\ 24 & 27 & 30 & 32 & 36 & 40 \\ \end{array} \right]\in\mathbb{R}^{4\times 6}.$$

## 2) Khatri-Rao product (kr_prod)¶

• Definition:

Given two matrices $A=\left( \boldsymbol{a}_1,\boldsymbol{a}_2,...,\boldsymbol{a}_r \right)\in\mathbb{R}^{m\times r}$ and $B=\left( \boldsymbol{b}_1,\boldsymbol{b}_2,...,\boldsymbol{b}_r \right)\in\mathbb{R}^{n\times r}$ with same number of columns, then, the Khatri-Rao product (or column-wise Kronecker product) between $A$ and $B$ is given as follows,

$$A\odot B=\left( \boldsymbol{a}_1\otimes \boldsymbol{b}_1,\boldsymbol{a}_2\otimes \boldsymbol{b}_2,...,\boldsymbol{a}_r\otimes \boldsymbol{b}_r \right)\in\mathbb{R}^{(mn)\times r},$$

where the symbol $\odot$ denotes Khatri-Rao product, and $\otimes$ denotes Kronecker product.

• Example:

If $A=\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right]=\left( \boldsymbol{a}_1,\boldsymbol{a}_2 \right)$ and $B=\left[ \begin{array}{cc} 5 & 6 \\ 7 & 8 \\ 9 & 10 \\ \end{array} \right]=\left( \boldsymbol{b}_1,\boldsymbol{b}_2 \right)$, then, we have

$$A\odot B=\left( \boldsymbol{a}_1\otimes \boldsymbol{b}_1,\boldsymbol{a}_2\otimes \boldsymbol{b}_2 \right)$$$$=\left[ \begin{array}{cc} \left[ \begin{array}{c} 1 \\ 3 \\ \end{array} \right]\otimes \left[ \begin{array}{c} 5 \\ 7 \\ 9 \\ \end{array} \right] & \left[ \begin{array}{c} 2 \\ 4 \\ \end{array} \right]\otimes \left[ \begin{array}{c} 6 \\ 8 \\ 10 \\ \end{array} \right] \\ \end{array} \right]$$$$=\left[ \begin{array}{cc} 5 & 12 \\ 7 & 16 \\ 9 & 20 \\ 15 & 24 \\ 21 & 32 \\ 27 & 40 \\ \end{array} \right]\in\mathbb{R}^{6\times 2}.$$
In [2]:
def kr_prod(a, b):
return np.einsum('ir, jr -> ijr', a, b).reshape(a.shape[0] * b.shape[0], -1)

In [3]:
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8], [9, 10]])
print(kr_prod(A, B))

[[ 5 12]
[ 7 16]
[ 9 20]
[15 24]
[21 32]
[27 40]]


## 3) Computing Covariance Matrix (cov_mat)¶

For any matrix $X\in\mathbb{R}^{m\times n}$, cov_mat can return a $n\times n$ covariance matrix for special use in the following.

In [4]:
def cov_mat(mat):
dim1, dim2 = mat.shape
new_mat = np.zeros((dim2, dim2))
mat_bar = np.mean(mat, axis = 0)
for i in range(dim1):
new_mat += np.einsum('i, j -> ij', mat[i, :] - mat_bar, mat[i, :] - mat_bar)
return new_mat


## 4) CP decomposition (cp_combine)¶

• Definition:

The CP decomposition factorizes a tensor into a sum of outer products of vectors. For example, for a third-order tensor $\mathcal{Y}\in\mathbb{R}^{m\times n\times f}$, the CP decomposition can be written as

$$\hat{\mathcal{Y}}=\sum_{s=1}^{r}\boldsymbol{u}_{s}\circ\boldsymbol{v}_{s}\circ\boldsymbol{x}_{s},$$

or element-wise,

$$\hat{y}_{ijt}=\sum_{s=1}^{r}u_{is}v_{js}x_{ts},\forall (i,j,t),$$

where vectors $\boldsymbol{u}_{s}\in\mathbb{R}^{m},\boldsymbol{v}_{s}\in\mathbb{R}^{n},\boldsymbol{x}_{s}\in\mathbb{R}^{f}$ are columns of factor matrices $U\in\mathbb{R}^{m\times r},V\in\mathbb{R}^{n\times r},X\in\mathbb{R}^{f\times r}$, respectively. The symbol $\circ$ denotes vector outer product.

• Example:

Given matrices $U=\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ \end{array} \right]\in\mathbb{R}^{2\times 2}$, $V=\left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ \end{array} \right]\in\mathbb{R}^{3\times 2}$ and $X=\left[ \begin{array}{cc} 1 & 5 \\ 2 & 6 \\ 3 & 7 \\ 4 & 8 \\ \end{array} \right]\in\mathbb{R}^{4\times 2}$, then if $\hat{\mathcal{Y}}=\sum_{s=1}^{r}\boldsymbol{u}_{s}\circ\boldsymbol{v}_{s}\circ\boldsymbol{x}_{s}$, then, we have

$$\hat{Y}_1=\hat{\mathcal{Y}}(:,:,1)=\left[ \begin{array}{ccc} 31 & 42 & 65 \\ 63 & 86 & 135 \\ \end{array} \right],$$$$\hat{Y}_2=\hat{\mathcal{Y}}(:,:,2)=\left[ \begin{array}{ccc} 38 & 52 & 82 \\ 78 & 108 & 174 \\ \end{array} \right],$$$$\hat{Y}_3=\hat{\mathcal{Y}}(:,:,3)=\left[ \begin{array}{ccc} 45 & 62 & 99 \\ 93 & 130 & 213 \\ \end{array} \right],$$$$\hat{Y}_4=\hat{\mathcal{Y}}(:,:,4)=\left[ \begin{array}{ccc} 52 & 72 & 116 \\ 108 & 152 & 252 \\ \end{array} \right].$$
In [5]:
def cp_combine(U, V, X):
return np.einsum('is, js, ts -> ijt', U, V, X)

In [6]:
U = np.array([[1, 2], [3, 4]])
V = np.array([[1, 3], [2, 4], [5, 6]])
X = np.array([[1, 5], [2, 6], [3, 7], [4, 8]])
print(cp_combine(U, V, X))
print()
print('tensor size:')
print(cp_combine(U, V, X).shape)

[[[ 31  38  45  52]
[ 42  52  62  72]
[ 65  82  99 116]]

[[ 63  78  93 108]
[ 86 108 130 152]
[135 174 213 252]]]

tensor size:
(2, 3, 4)


## 5) Tensor Unfolding (ten2mat) and Matrix Folding (mat2ten)¶

Using numpy reshape to perform 3rd rank tensor unfold operation. [link]

In [7]:
import numpy as np
def ten2mat(tensor, mode):
return np.reshape(np.moveaxis(tensor, mode, 0), (tensor.shape[mode], -1), order = 'F')

In [8]:
X = np.array([[[1, 2, 3, 4], [3, 4, 5, 6]],
[[5, 6, 7, 8], [7, 8, 9, 10]],
[[9, 10, 11, 12], [11, 12, 13, 14]]])
print('tensor size:')
print(X.shape)
print('original tensor:')
print(X)
print()
print('(1) mode-1 tensor unfolding:')
print(ten2mat(X, 0))
print()
print('(2) mode-2 tensor unfolding:')
print(ten2mat(X, 1))
print()
print('(3) mode-3 tensor unfolding:')
print(ten2mat(X, 2))

tensor size:
(3, 2, 4)
original tensor:
[[[ 1  2  3  4]
[ 3  4  5  6]]

[[ 5  6  7  8]
[ 7  8  9 10]]

[[ 9 10 11 12]
[11 12 13 14]]]

(1) mode-1 tensor unfolding:
[[ 1  3  2  4  3  5  4  6]
[ 5  7  6  8  7  9  8 10]
[ 9 11 10 12 11 13 12 14]]

(2) mode-2 tensor unfolding:
[[ 1  5  9  2  6 10  3  7 11  4  8 12]
[ 3  7 11  4  8 12  5  9 13  6 10 14]]

(3) mode-3 tensor unfolding:
[[ 1  5  9  3  7 11]
[ 2  6 10  4  8 12]
[ 3  7 11  5  9 13]
[ 4  8 12  6 10 14]]

In [9]:
def mat2ten(mat, tensor_size, mode):
index = list()
index.append(mode)
for i in range(tensor_size.shape[0]):
if i != mode:
index.append(i)
return np.moveaxis(np.reshape(mat, list(tensor_size[index]), order = 'F'), 0, mode)


## 6) Generating Matrix Normal Distributed Random Matrix¶

In [10]:
def mnrnd(M, U, V):
"""
Generate matrix normal distributed random matrix.
M is a m-by-n matrix, U is a m-by-m matrix, and V is a n-by-n matrix.
"""
dim1, dim2 = M.shape
X0 = np.random.rand(dim1, dim2)
P = np.linalg.cholesky(U)
Q = np.linalg.cholesky(V)
return M + np.matmul(np.matmul(P, X0), Q.T)


# Part 2: Bayesian Temporal Tensor Factorization (BTTF)¶

In [11]:
def BTTF(dense_tensor, sparse_tensor, init, rank, time_lags, maxiter1, maxiter2):
"""Bayesian Temporal Tensor Factorization, BTTF."""
U = init["U"]
V = init["V"]
X = init["X"]

d = time_lags.shape[0]
dim1, dim2, dim3 = sparse_tensor.shape
dim = np.array([dim1, dim2, dim3])
pos = np.where((dense_tensor != 0) & (sparse_tensor == 0))
position = np.where(sparse_tensor != 0)
binary_tensor = np.zeros((dim1, dim2, dim3))
binary_tensor[position] = 1

beta0 = 1
nu0 = rank
mu0 = np.zeros((rank))
W0 = np.eye(rank)
tau = 1
alpha = 1e-6
beta = 1e-6
S0 = np.eye(rank)
Psi0 = np.eye(rank * d)
M0 = np.zeros((rank * d, rank))

mat_hat = np.zeros((dim1, dim2, dim3 + 1))
U_plus = np.zeros((dim1, rank))
V_plus = np.zeros((dim2, rank))
X_plus = np.zeros((dim3, rank))
X_new = np.zeros((dim3 + 1, rank))
X_new_plus = np.zeros((dim3 + 1, rank))
A_plus = np.zeros((rank, rank, d))
tensor_hat_plus = np.zeros((dim1, dim2, dim3 + 1))
for iters in range(maxiter1):
for order in range(2):
if order == 0:
mat = U.copy()
elif order == 1:
mat = V.copy()
mat_bar = np.mean(mat, axis = 0)
var_mu_hyper = (dim[order] * mat_bar + beta0 * mu0)/(dim[order] + beta0)
var_W_hyper = inv(inv(W0) + cov_mat(mat) + dim[order] * beta0/(dim[order] + beta0)
* np.outer(mat_bar - mu0, mat_bar - mu0))
var_Lambda_hyper = wishart(df = dim[order] + nu0, scale = var_W_hyper, seed = None).rvs()
var_mu_hyper = mvnrnd(var_mu_hyper, inv((dim[order] + beta0) * var_Lambda_hyper))

if order == 0:
var1 = kr_prod(X, V).T
elif order == 1:
var1 = kr_prod(X, U).T
var2 = kr_prod(var1, var1)
var3 = (tau * np.matmul(var2, ten2mat(binary_tensor, order).T).reshape([rank, rank, dim[order]])
+ np.dstack([var_Lambda_hyper] * dim[order]))
var4 = (tau * np.matmul(var1, ten2mat(sparse_tensor, order).T)
+ np.dstack([np.matmul(var_Lambda_hyper, var_mu_hyper)] * dim[order])[0, :, :])
for i in range(dim[order]):
inv_var_Lambda = inv(var3[ :, :, i])
vec = mvnrnd(np.matmul(inv_var_Lambda, var4[:, i]), inv_var_Lambda)
if order == 0:
U[i, :] = vec.copy()
elif order == 1:
V[i, :] = vec.copy()

Z_mat = X[np.max(time_lags) : dim3, :]
Q_mat = np.zeros((dim3 - np.max(time_lags), rank * d))
for t in range(np.max(time_lags), dim3):
Q_mat[t - np.max(time_lags), :] = X[t - time_lags, :].reshape([rank * d])
var_Psi = inv(inv(Psi0) + np.matmul(Q_mat.T, Q_mat))
var_M = np.matmul(var_Psi, np.matmul(inv(Psi0), M0) + np.matmul(Q_mat.T, Z_mat))
var_S = (S0 + np.matmul(Z_mat.T, Z_mat) + np.matmul(np.matmul(M0.T, inv(Psi0)), M0)
- np.matmul(np.matmul(var_M.T, inv(var_Psi)), var_M))
Sigma = invwishart(df = nu0 + dim3 - np.max(time_lags), scale = var_S, seed = None).rvs()
A = mat2ten(mnrnd(var_M, var_Psi, Sigma).T, np.array([rank, rank, d]), 0)
if iters + 1 > maxiter1 - maxiter2:
A_plus += A

Lambda_x = inv(Sigma)
var1 = kr_prod(V, U).T
var2 = kr_prod(var1, var1)
var3 = (tau * np.matmul(var2, ten2mat(binary_tensor, 2).T).reshape([rank, rank, dim3])
+ np.dstack([Lambda_x] * dim3))
var4 = tau * np.matmul(var1, ten2mat(sparse_tensor, 2).T)
for t in range(dim3):
Mt = np.zeros((rank, rank))
Nt = np.zeros(rank)
if t < np.max(time_lags):
Qt = np.zeros(rank)
else:
Qt = np.matmul(Lambda_x, np.matmul(ten2mat(A, 0), X[t - time_lags, :].reshape([rank * d])))
if t < dim3 - np.min(time_lags):
if t >= np.max(time_lags) and t < dim3 - np.max(time_lags):
index = list(range(0, d))
else:
index = list(np.where((t + time_lags >= np.max(time_lags)) & (t + time_lags < dim3)))[0]
for k in index:
Ak = A[:, :, k]
Mt += np.matmul(np.matmul(Ak.T, Lambda_x), Ak)
A0 = A.copy()
A0[:, :, k] = 0
var5 = (X[t + time_lags[k], :]
- np.matmul(ten2mat(A0, 0), X[t + time_lags[k] - time_lags, :].reshape([rank * d])))
Nt += np.matmul(np.matmul(Ak.T, Lambda_x), var5)
var_mu = var4[:, t] + Nt + Qt
if t < np.max(time_lags):
inv_var_Lambda = inv(var3[:, :, t] + Mt - Lambda_x + np.eye(rank))
else:
inv_var_Lambda = inv(var3[:, :, t] + Mt)
X[t, :] = mvnrnd(np.matmul(inv_var_Lambda, var_mu), inv_var_Lambda)

if iters + 1 > maxiter1 - maxiter2:
U_plus += U
V_plus += V
X_plus += X

tensor_hat = cp_combine(U, V, X)
if iters + 1 > maxiter1 - maxiter2:
X_new[0 : dim3, :] = X.copy()
X_new[dim3, :] = np.matmul(ten2mat(A, 0), X_new[dim3 - time_lags, :].reshape([rank * d]))
X_new_plus += X_new
tensor_hat_plus += cp_combine(U, V, X_new)

tau = np.random.gamma(alpha + 0.5 * sparse_tensor[position].shape[0],
1/(beta + 0.5 * np.sum((sparse_tensor - tensor_hat)[position] ** 2)))
rmse = np.sqrt(np.sum((dense_tensor[pos] - tensor_hat[pos]) ** 2)/dense_tensor[pos].shape[0])
if (iters + 1) % 200 == 0 and iters < maxiter1 - maxiter2:
print('Iter: {}'.format(iters + 1))
print('RMSE: {:.6}'.format(rmse))
print()

U = U_plus/maxiter2
V = V_plus/maxiter2
X = X_plus/maxiter2
X_new = X_new_plus/maxiter2
A = A_plus/maxiter2
tensor_hat = tensor_hat_plus/maxiter2
if maxiter1 >= 100:
final_mape = np.sum(np.abs(dense_tensor[pos]
- tensor_hat[pos])/dense_tensor[pos])/dense_tensor[pos].shape[0]
final_rmse = np.sqrt(np.sum((dense_tensor[pos] - tensor_hat[pos]) ** 2)/dense_tensor[pos].shape[0])
print('Imputation MAPE: {:.6}'.format(final_mape))
print('Imputation RMSE: {:.6}'.format(final_rmse))
print()

return tensor_hat, U, V, X_new, A


How to transform a data set into something we can use for missing data imputation?

In [12]:
import scipy.io

dense_tensor = tensor['tensor']
rm_tensor = rm_tensor['rm_tensor']
nm_tensor = nm_tensor['nm_tensor']

missing_rate = 0.1

# =============================================================================
### Random missing (RM) scenario
### Set the RM scenario by:
binary_tensor = np.round(rm_tensor + 0.5 - missing_rate)
# =============================================================================

sparse_tensor = np.multiply(dense_tensor, binary_tensor)


Question: Given only the partially observed data $\mathcal{Y}\in\mathbb{R}^{m\times n\times f}$, how can we impute the unknown missing values?

The main influential factors for such imputation model are:

• rank.

• maxiter1.

• maxiter2.

In [13]:
import time
start = time.time()
dim1, dim2, dim3 = sparse_tensor.shape
rank = 30
time_lags = np.array([1, 2, 24])
d = time_lags.shape[0]
init = {"U": 0.1 * np.random.rand(dim1, rank), "V": 0.1 * np.random.rand(dim2, rank),
"X": 0.1 * np.random.rand(dim3, rank)}
maxiter1 = 1100
maxiter2 = 100
BTTF(dense_tensor, sparse_tensor, init, rank, time_lags, maxiter1, maxiter2)
end = time.time()
print('Running time: %d seconds'%(end - start))

Iter: 200
RMSE: 5.03257

Iter: 400
RMSE: 4.9825

Iter: 600
RMSE: 4.94821

Iter: 800
RMSE: 4.88249

Iter: 1000
RMSE: 4.85344

Imputation MAPE: 0.519817
Imputation RMSE: 4.65973

Running time: 1841 seconds

In [14]:
import scipy.io

dense_tensor = tensor['tensor']
rm_tensor = rm_tensor['rm_tensor']
nm_tensor = nm_tensor['nm_tensor']

missing_rate = 0.3

# =============================================================================
### Random missing (RM) scenario
### Set the RM scenario by:
binary_tensor = np.round(rm_tensor + 0.5 - missing_rate)
# =============================================================================

sparse_tensor = np.multiply(dense_tensor, binary_tensor)

In [15]:
import time
start = time.time()
dim1, dim2, dim3 = sparse_tensor.shape
rank = 30
time_lags = np.array([1, 2, 24])
d = time_lags.shape[0]
init = {"U": 0.1 * np.random.rand(dim1, rank),
"V": 0.1 * np.random.rand(dim2, rank),
"X": 0.1 * np.random.rand(dim3, rank),
"theta": 0.1 * np.random.rand(d, rank)}
maxiter1 = 1100
maxiter2 = 100
BTTF(dense_tensor, sparse_tensor, init, rank, time_lags, maxiter1, maxiter2)
end = time.time()
print('Running time: %d seconds'%(end - start))

Iter: 200
RMSE: 5.40704

Iter: 400
RMSE: 5.16461

Iter: 600
RMSE: 5.11729

Iter: 800
RMSE: 5.0672

Iter: 1000
RMSE: 5.04901

Imputation MAPE: 0.517833
Imputation RMSE: 4.77444

Running time: 1788 seconds

In [16]:
import scipy.io

dense_tensor = tensor['tensor']
rm_tensor = rm_tensor['rm_tensor']
nm_tensor = nm_tensor['nm_tensor']

missing_rate = 0.1

# =============================================================================
### Non-random missing (NM) scenario
### Set the NM scenario by:
binary_tensor = np.zeros(dense_tensor.shape)
for i1 in range(dense_tensor.shape[0]):
for i2 in range(dense_tensor.shape[1]):
for i3 in range(61):
binary_tensor[i1, i2, i3 * 24 : (i3 + 1) * 24] = np.round(nm_tensor[i1, i2, i3] + 0.5 - missing_rate)
# =============================================================================

sparse_tensor = np.multiply(dense_tensor, binary_tensor)

In [17]:
import time
start = time.time()
dim1, dim2, dim3 = sparse_tensor.shape
rank = 30
time_lags = np.array([1, 2, 24])
d = time_lags.shape[0]
init = {"U": 0.1 * np.random.rand(dim1, rank),
"V": 0.1 * np.random.rand(dim2, rank),
"X": 0.1 * np.random.rand(dim3, rank),
"theta": 0.1 * np.random.rand(d, rank)}
maxiter1 = 1100
maxiter2 = 100
BTTF(dense_tensor, sparse_tensor, init, rank, time_lags, maxiter1, maxiter2)
end = time.time()
print('Running time: %d seconds'%(end - start))

Iter: 200
RMSE: 5.05047

Iter: 400
RMSE: 5.02066

Iter: 600
RMSE: 4.98022

Iter: 800
RMSE: 4.94399

Iter: 1000
RMSE: 4.93455

Imputation MAPE: 0.526488
Imputation RMSE: 4.74862

Running time: 1639 seconds

In [18]:
import scipy.io

dense_tensor = tensor['tensor']
rm_tensor = rm_tensor['rm_tensor']
nm_tensor = nm_tensor['nm_tensor']

missing_rate = 0.3

# =============================================================================
### Non-random missing (NM) scenario
### Set the NM scenario by:
binary_tensor = np.zeros(dense_tensor.shape)
for i1 in range(dense_tensor.shape[0]):
for i2 in range(dense_tensor.shape[1]):
for i3 in range(61):
binary_tensor[i1, i2, i3 * 24 : (i3 + 1) * 24] = np.round(nm_tensor[i1, i2, i3] + 0.5 - missing_rate)
# =============================================================================

sparse_tensor = np.multiply(dense_tensor, binary_tensor)

In [19]:
import time
start = time.time()
dim1, dim2, dim3 = sparse_tensor.shape
rank = 30
time_lags = np.array([1, 2, 24])
d = time_lags.shape[0]
init = {"U": 0.1 * np.random.rand(dim1, rank),
"V": 0.1 * np.random.rand(dim2, rank),
"X": 0.1 * np.random.rand(dim3, rank),
"theta": 0.1 * np.random.rand(d, rank)}
maxiter1 = 1100
maxiter2 = 100
BTTF(dense_tensor, sparse_tensor, init, rank, time_lags, maxiter1, maxiter2)
end = time.time()
print('Running time: %d seconds'%(end - start))

Iter: 200
RMSE: 5.36823

Iter: 400
RMSE: 5.23173

Iter: 600
RMSE: 5.191

Iter: 800
RMSE: 5.1764

Iter: 1000
RMSE: 5.15288

Imputation MAPE: 0.527149
Imputation RMSE: 4.89655

Running time: 1668 seconds


Experiment results of missing data imputation using Bayesian Temporal Tensor Factorization (BTTF):

scenario rank maxiter1 maxiter2 mape rmse
0.1, RM 30 1100 100 0.5198 4.66
0.3, RM 30 1100 100 0.5178 4.77
0.1, NM 30 1100 100 0.5265 4.75
0.3, NM 30 1100 100 0.5271 4.90