Estimation for Reduced Rank MAR(1) Model

Description

Estimation of the reduced rank MAR(1) model, using least squares (RRLSE) or MLE (RRMLE), as determined by the value of method.

Usage

matAR.RR.est(xx, method, A1.init=NULL, A2.init=NULL,Sig1.init=NULL,Sig2.init=NULL,
k1=NULL, k2=NULL, niter=200,tol=1e-4)

Arguments

Details

The reduced rank MAR(1) model takes the form:

X_t = A_1 X_{t-1} A_2^{^\top} + E_t,

where A_i are d_i \times d_i coefficient matrices of ranks \mathrm{rank}(A_i) = k_i \le d_i, i=1,2. For the MLE method we also assume

\mathrm{Cov}(\mathrm{vec}(E_t))=\Sigma_2 \otimes \Sigma_1

Value

return a list containing the following:

A1

estimator of A_1, a d_1 by d_1 matrix.

A2

estimator of A_2, a d_2 by d_2 matrix.

loading

a list of estimated U_i, V_i, where we write A_i=U_iD_iV_i as the singular value decomposition (SVD) of A_i, i = 1,2.

Sig1

only if method=MLE, when \mathrm{Cov}(\mathrm{vec}(E_t))=\Sigma_2 \otimes \Sigma_1.

Sig2

only if method=MLE, when \mathrm{Cov}(\mathrm{vec}(E_t))=\Sigma_2 \otimes \Sigma_1.

res

residuals.

Sig

sample covariance matrix of the residuals vec(\hat E_t).

cov

a list containing

Sigma

asymptotic covariance matrix of (vec( \hat A_1),vec(\hat A_2^{\top})).

Theta1.u, Theta1.v

asymptotic covariance matrix of vec(\hat U_1), vec(\hat V_1).

Theta2.u, Theta2.v

asymptotic covariance matrix of vec(\hat U_2), vec(\hat V_2).

sd.A1

element-wise standard errors of \hat A_1, aligned with A1.

sd.A2

element-wise standard errors of \hat A_2, aligned with A2.

niter

number of iterations.

BIC

value of the extended Bayesian information criterion.

References

Reduced Rank Autoregressive Models for Matrix Time Series, by Han Xiao, Yuefeng Han, Rong Chen and Chengcheng Liu.

Examples

set.seed(333)
dim <- c(3,3)
xx <- tenAR.sim(t=500, dim, R=2, P=1, rho=0.5, cov='iid')
est <- matAR.RR.est(xx, method="RRLSE", k1=1, k2=1)

Asymptotic Covariance Matrix of One-Term Reduced rank MAR(1) Model

Description

Asymptotic covariance matrix of the reduced rank MAR(1) model. If Sigma1 and Sigma2 is provided in input, we assume a separable covariance matrix, Cov(vec(E_t)) = \Sigma_2 \otimes \Sigma_1.

Usage

matAR.RR.se(A1,A2,k1,k2,method,Sigma.e=NULL,Sigma1=NULL,Sigma2=NULL,RU1=diag(k1),
RV1=diag(k1),RU2=diag(k2),RV2=diag(k2),mpower=100)

Arguments

Value

a list containing the following:

Sigma

asymptotic covariance matrix of (vec(\hat A_1),vec(\hat A_2^T)).

Theta1.u

asymptotic covariance matrix of vec(\hat U_1).

Theta1.v

asymptotic covariance matrix of vec(\hat V_1).

Theta2.u

asymptotic covariance matrix of vec(\hat U_2).

Theta2.v

asymptotic covariance matrix of vec(\hat V_2).

References

Han Xiao, Yuefeng Han, Rong Chen and Chengcheng Liu, Reduced Rank Autoregressive Models for Matrix Time Series.

Plot Matrix-Valued Time Series

Description

Plot matrix-valued time series, can be also used to plot a given slice of a tensor-valued time series.

Usage

mplot(xx)

Arguments

Value

a figure.

Examples

dim <- c(3,3,3)
xx <- tenAR.sim(t=50, dim, R=2, P=1, rho=0.5, cov='iid')
mplot(xx[1:30,,,1])

Plot ACF of Matrix-Valued Time Series

Description

Plot ACF of matrix-valued time series, can be also used to plot ACF of a given slice of a tensor-valued time series.

Usage

mplot.acf(xx)

Arguments

Value

a figure.

Examples

dim <- c(3,3,3)
xx <- tenAR.sim(t=50, dim, R=2, P=1, rho=0.5, cov='iid')
mplot.acf(xx[1:30,,,1])

Predict funcions for Tensor Autoregressive Models

Description

S3 method for the 'tenAR' class using the generic predict function. Prediction based on the tensor autoregressive model or reduced rank MAR(1) model. If rolling = TRUE, returns the rolling forecasts.

Usage

## S3 method for class 'tenAR'
predict(object, n.ahead = 1, xx = NULL, rolling = FALSE, n0 = NULL, ...)

Arguments

Value

a tensor time series of length n.ahead if rolling = FALSE;

a tensor time series of length T^{\prime} - n_0 - n.ahead + 1 if rolling = TRUE.

See Also

'predict.ar' or 'predict.arima'

Examples

set.seed(333)
dim <- c(2,2,2)
t = 20
xx <- tenAR.sim(t, dim, R=2, P=1, rho=0.5, cov='iid')
est <- tenAR.est(xx, R=1, P=1, method="LSE")
pred <- predict(est, n.ahead = 1)
# rolling forcast
n0 = t - min(50,t/2)
pred.rolling <- predict(est, n.ahead = 5, xx = xx, rolling=TRUE, n0)

# prediction for reduced rank MAR(1) model
dim <- c(2,2)
t = 20
xx <- tenAR.sim(t, dim, R=1, P=1, rho=0.5, cov='iid')
est <- matAR.RR.est(xx, method="RRLSE", k1=1, k2=1)
pred <- predict(est, n.ahead = 1)
# rolling forcast
n0 = t - min(50,t/2)
pred.rolling <- predict(est, n.ahead = 5, rolling=TRUE, n0=n0)

Simulate taxi data by tenAR models

Description

Simulate tensor time series by autoregressive models, using estimated coefficients by taxi data.

Usage

taxi.sim.AR(t, print.tar.coef=FALSE, print.sig=FALSE, seed=123)

Arguments

Value

A tensor-valued time series of dimension (5,5,7).

See Also

taxi.sim.FM

Examples

xx = taxi.sim.AR(t=753)

Simulate taxi data by factor models

Description

Simulate tensor time series by factor models, using estimated coefficients by taxi data.

Usage

taxi.sim.FM(t, print.tar.coef=FALSE, print.loading=FALSE, seed=216)

Arguments

Value

A tensor-valued time series of dimension (4,4,3).

See Also

taxi.sim.AR

Examples

xx = taxi.sim.FM(t=252)

Estimation for Autoregressive Model of Tensor-Valued Time Series

Description

Estimation function for tensor autoregressive models. Methods include projection (PROJ), Least Squares (LSE), maximum likelihood estimation (MLE) and vector autoregressive model (VAR), as determined by the value of method.

Usage

tenAR.est(xx,R=1,P=1,method="LSE",init.A=NULL,init.sig=NULL,niter=150,tol=1e-6)

Arguments

Details

Tensor autoregressive model (of autoregressive order one) has the form:

X_t = \sum_{r=1}^R X_{t-1} \times_{1} A_1^{(r)} \times_{2} \cdots \times_{K} A_K^{(r)} + E_t,

where A_k^{(r)} are d_k \times d_k coefficient matrices, k=1,\cdots,K, and E_t is a tensor white noise. R is the Kronecker rank. The model of autoregressive order P takes the form

X_t = \sum_{i=1}^{P} \sum_{r=1}^{R_i} X_{t-i} \times_{1} A_{1}^{(ir)} \times_{2} \cdots \times_{K} A_{K}^{(ir)} + E_t.

For the "MLE" method, we also assume,

\mathrm{Cov}(\mathrm{vec}(E_t))= \Sigma_K \otimes \Sigma_{K-1} \otimes \cdots \otimes \Sigma_1,

Value

return a list containing the following:

A

a list of estimated coefficient matrices A_k^{(ir)}. It is a multi-layer list, the first layer for the lag 1 \le i \le P, the second the term 1 \le r \le R, and the third the mode 1 \le k \le K. See "Details".

SIGMA

only if method=MLE, a list of estimated \Sigma_1,\ldots,\Sigma_K.

res

residuals

Sig

sample covariance matrix of the residuals vec(\hat E_t).

cov

grand covariance matrix of all estimated entries of A_k^{(ir)}

sd

standard errors of the coefficient matrices A_k^{(ir)}, returned as a list aligned with A.

niter

number of iterations.

BIC

value of extended Bayesian information criterion.

References

Rong Chen, Han Xiao, and Dan Yang. "Autoregressive models for matrix-valued time series". Journal of Econometrics, 2020.

Zebang Li, Han Xiao. "Multi-linear tensor autoregressive models". arxiv preprint arxiv:2110.00928 (2021).

Examples

set.seed(333)

# case 1: tensor-valued time series

dim <- c(2,2,2)
xx <- tenAR.sim(t=100, dim, R=2, P=1, rho=0.5, cov='iid')
est <- tenAR.est(xx, R=2, P=1, method="LSE") # two-term tenAR(1) model
A <- est$A # A is a multi-layer list

length(A) == 1 # TRUE, since the order P = 1
length(A[[1]]) == 2 # TRUE, since the number of terms R = 2
length(A[[1]][[1]]) == 3 # TRUE, since the mode K = 3

# est <- tenAR.est(xx, R=c(1,2), P=2, method="LSE") # tenAR(2) model with R1=1, R2=2

# case 2: matrix-valued time series

dim <- c(2,2)
xx <- tenAR.sim(t=100, dim, R=2, P=1, rho=0.5, cov='iid')
est <- tenAR.est(xx, R=2, P=1, method="LSE") # two-term MAR(1) model 
A <- est$A # A is a multi-layer list

length(A) == 1 # TRUE, since the order P = 1
length(A[[1]]) == 2 # TRUE, since the number of terms R = 2
length(A[[1]][[1]]) == 2 # TRUE, since the mode K = 2

Generate TenAR(p) tensor time series

Description

Simulate from the TenAR(p) model.

Usage

tenAR.sim(t, dim, R, P, rho, cov, A = NULL, Sig = NULL)

Arguments

Value

A tensor-valued time series generated by the TenAR(p) model.

See Also

tenFM.sim

Examples

set.seed(123)
dim <- c(3,3,3)
xx <- tenAR.sim(t=500, dim, R=2, P=1, rho=0.5, cov='iid')

Estimation for Tucker structure Factor Models of Tensor-Valued Time Series

Description

Estimation function for Tucker structure factor models of tensor-valued time series. Two unfolding methods of the auto-covariance tensor, Time series Outer-Product Unfolding Procedure (TOPUP), Time series Inner-Product Unfolding Procedure (TIPUP), are included, as determined by the value of method.

Usage

tenFM.est(x,r,h0=1,method='TIPUP',iter=TRUE,tol=1e-4,maxiter=100)

Arguments

Details

Tensor factor model with Tucker structure has the following form,

X_t = F_t \times_{1} A_1 \times_{2} \cdots \times_{K} A_k + E_t,

where A_k is the deterministic loading matrix of size d_k \times r_k and r_k \ll d_k, the core tensor F_t itself is a latent tensor factor process of dimension r_1 \times \cdots \times r_K, and the idiosyncratic noise tensor E_t is uncorrelated (white) across time. Two estimation approaches, named TOPUP and TIPUP, are studied. Time series Outer-Product Unfolding Procedure (TOPUP) are based on

{\rm{TOPUP}}_{k}(X_{1:T}) = \left(\sum_{t=h+1}^T \frac{{\rm{mat}}_{k}( X_{t-h}) \otimes {\rm{mat}}_k(X_t)} {T-h}, \ h=1,...,h_0 \right),

where h_0 is a predetermined positive integer, \otimes is tensor product. Note that {\rm{TOPUP}}_k(\cdot) is a function mapping a tensor time series to an order-5 tensor. Time series Inner-Product Unfolding Procedure (TIPUP) replaces the tensor product in TOPUP with the inner product:

{\rm{TIPUP}}_k(X_{1:T})={\rm{mat}}_1\left(\sum_{t=h+1}^T \frac{{\rm{mat}}_k(X_{t-h}) {\rm{mat}}_k^\top(X_t)} {T-h}, \ h=1,...,h_0 \right).

Value

returns a list containing the following:

Ft

estimated factor processes of dimension T \times r_1 \times r_2 \times \cdots \times r_k.

Ft.all

Summation of factor processes over time, of dimension r_1,r_2,\cdots,r_k.

Q

a list of estimated factor loading matrices Q_1,Q_2,\cdots,Q_K.

x.hat

fitted signal tensor, of dimension T \times d_1 \times d_2 \times \cdots \times d_k.

niter

number of iterations.

fnorm.resid

Frobenius norm of residuals, divide the Frobenius norm of the original tensor.

References

Chen, Rong, Dan Yang, and Cun-Hui Zhang. "Factor models for high-dimensional tensor time series." Journal of the American Statistical Association (2021): 1-59.

Han, Yuefeng, Rong Chen, Dan Yang, and Cun-Hui Zhang. "Tensor factor model estimation by iterative projection." arXiv preprint arXiv:2006.02611 (2020).

Examples

set.seed(333)
dims <- c(16,18,20) # dimensions of tensor time series
r <- c(3,3,3)  # dimensions of factor series
Ft <- tenAR.sim(t=100, dim=r, R=1, P=1, rho=0.9, cov='iid')
lambda <- sqrt(prod(dims))
x <- tenFM.sim(Ft,dims=dims,lambda=lambda,A=NULL,cov='iid') # generate t*dims tensor time series
result <- tenFM.est(x,r,h0=1,iter=TRUE,method='TIPUP')  # Estimation
Ft <- result$Ft

Rank Determination for Tensor Factor Models with Tucker Structure

Description

Function for rank determination of tensor factor models with Tucker Structure. Two unfolding methods of the auto-covariance tensor, Time series Outer-Product Unfolding Procedure (TOPUP), Time series Inner-Product Unfolding Procedure (TIPUP), are included, as determined by the value of method. Different penalty functions for the information criterion (IC) and the eigen ratio criterion (ER) can be used, which should be specified by the value of rank and penalty. The information criterion resembles BIC in the vector factor model literature. And the eigen ratio criterion is similar to the eigenvalue ratio based methods in the vector factor model literature.

Usage

tenFM.rank(x,r=NULL,h0=1,rank='IC',method='TIPUP',inputr=FALSE,iter=TRUE,penalty=1,
delta1=0,tol=1e-4,maxiter=100)

Arguments

Details

Let W be a p\times p symmetric and non-negative definite matrix and \widehat{W} be its sample version, {\hat\lambda}_j be the eigenvalues of \widehat{W} such that {\hat\lambda}_1\geq {\hat\lambda}_2 \geq \cdots \hat{\lambda}_p. The rank determination methods using the information criterion ("IC") and the eigen ratio criterion ("ER") are defined as follows:

IC(\widehat{W}) = \mathrm{argmin}_{0\leq m \leq m^{*}} \left\{ \sum_{j=m+1}^{p} {\hat\lambda}_j + mg(\widehat{W}) \right\},

ER(\widehat{W}) = \mathrm{argmin}_{0\leq m \leq m^{*}} \left\{ \frac{{\hat\lambda}_{m+1}+h(\widehat{W})}{ {\hat\lambda}_m +h(\widehat{W})} \right\},

where m^{*} is a predefined upper bound, g and h are some appropriate positive penalty functions. We have provided 5 choices for g and h; see more details in the argument "penalty". For non-iterative TOPUP and TIPUP methods, \widehat{W} is {\rm mat}_1({\rm{TOPUP}}_{k}(X_{1:T})) {\rm mat}_1({\rm{TOPUP}}_{k}(X_{1:T}))^\top or ({\rm{TIPUP}}_{k}(X_{1:T})) ({\rm{TIPUP}}_{k}(X_{1:T}))^\top , for each tensor mode k, 1\leq k \leq K, where {\rm{TOPUP}}_{k}(X_{1:T}) and {\rm{TIPUP}}_{k}(X_{1:T}) are defined in the Details section of the function tenFM.est. For iterative TOPUP and TIPUP methods, we refer to the literature in the References section for more information.

Value

return a list containing the following:

path

a K \times (\rm{niter}+1) matrix of the estimated Tucker rank of the factor process as a path of the maximum number of iteration (\rm{niter}) used. The first row is the estimated rank under non-iterative approach, the i+1-th row is the estimated rank \hat r_1, \hat r_2, \cdots, \hat r_K at (i)-th iteration.

factor.num

final solution of the estimated Tucker rank of the factor process \hat r_1, \hat r_2, \cdots, \hat r_K.

References

Han, Yuefeng, Cun-Hui Zhang, and Rong Chen. "Rank Determination in Tensor Factor Model." Available at SSRN 3730305 (2020).

Examples

set.seed(333)
dims <- c(16,18,20) # dimensions of tensor time series
r <- c(3,3,3)  # dimensions of factor series
Ft <- tenAR.sim(t=100, dim=r, R=1, P=1, rho=0.9, cov='iid')
lambda <- sqrt(prod(dims))
x <- tenFM.sim(Ft,dims=dims,lambda=lambda,A=NULL,cov='iid') # generate t*dims tensor time series
rank <- tenFM.rank(x,r=c(4,4,4),h0=1,rank='IC',iter=TRUE,method='TIPUP')  # Estimate the rank

Generate Tensor Time series using given Factor Process and Factor Loading Matrices

Description

Simulate tensor time series X_t using a given factor process F_t. The factor process F_t can be generated by the function tenAR.sim.

Usage

tenFM.sim(Ft,dims=NULL,lambda=1,A=NULL,cov='iid',rho=0.2)

Arguments

Details

Simulate from the model :

X_t = \lambda F_t \times_{1} A_1 \times_{2} \cdots \times_{K} A_k + E_t,

where A_k is the deterministic loading matrix of size d_k \times r_k and r_k \ll d_k, the core tensor F_t itself is a latent tensor factor process of dimension r_1 \times \cdots \times r_K, \lambda is an additional signal strength parameter, and the idiosyncratic noise tensor E_t is uncorrelated (white) across time. In this function, by default A_k are orthogonal matrices.

Value

A tensor-valued time series of dimension T\times d_1\times d_2\cdots\times d_K.

See Also

tenAR.sim

Examples

set.seed(333)
dims <- c(16,18,20) # dimensions of tensor time series
r <- c(3,3,3)  # dimensions of factor series
Ft <- tenAR.sim(t=100, dim=r, R=1, P=1, rho=0.9, cov='iid')
lambda <- sqrt(prod(dims))
# generate t*dims tensor time series with iid error covaraince structure
x <- tenFM.sim(Ft,dims=dims,lambda=lambda,A=NULL,cov='iid')
# generate t*dims tensor time series with separable error covaraince structure
x <- tenFM.sim(Ft,dims=dims,lambda=lambda,A=NULL,cov='separable',rho=0.2)