Package 'tensorTS'

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

xx	$T \times d_1 \times d_2$ matrix-valued time series, $T$ is the length of the series.
method	character string, specifying the method of the estimation to be used. "RRLSE", Least squares. "RRMLE", MLE under a separable cov(vec($E_t$)).
A1.init	initial value of $A_1$. The default is the identity matrix.
A2.init	initial value of $A_2$. The default is the identity matrix.
Sig1.init	only if method=RRMLE, initial value of $\Sigma_1$. The default is the identity matrix.
Sig2.init	only if method=RRMLE, initial value of $\Sigma_2$. The default is the identity matrix.
k1	rank of $A_1$, a positive integer.
k2	rank of $A_2$, a positive integer.
niter	maximum number of iterations if error stays above tol.
tol	relative Frobenius norm error tolerance.

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

A1	left coefficient matrix.
A2	right coefficient matrix.
k1	rank of $A_1$.
k2	rank of $A_2$.
method	character string, specifying the method of the estimation to be used. "RRLSE", Least squares. "RRMLE", MLE under a separable cov(vec($E_t$)).
Sigma.e	only if method = "RRLSE". Cov(vec($E_t$)) = Sigma.e: covariance matrix of dimension $(d_1 d_2) \times (d_1 d_2)$
Sigma1, Sigma2	only if method = "RRMLE". Cov(vec($E_t$)) = $\Sigma_2 \otimes \Sigma_1$. $\Sigma_i$ is $d_i \times d_i$, $i=1,2$.
RU1, RV1, RU2, RV2	orthogonal rotations of $U_1,V_1,U_2,V_2$, e.g., new_U1=U1 RU1.
mpower	truncate the VMA($\infty$) representation of vec($X_t$) at mpower for the purpose of calculating the autocovariances. The default is 100.

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

xx	$T \times d_1 \times d_2$ matrix-valued time series. Note that the number of mode is 3, where the first mode is time.

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

xx	$T \times d_1 \times d_2$ matrix-valued time series. Note that the number of mode is 3, where the first mode is time.

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,
  print.true = TRUE,
  ...
)

Arguments

object	a model object returned by tenAR.est().
n.ahead	prediction horizon.
xx	$T^{\prime} \times d_1 \times \cdots \times d_K$ new tensor time series to be used for prediction. Must have at least n.ahead length.
rolling	TRUE or FALSE, rolling forecast, is FALSE by default.
n0	only if rolling = TRUE, the starting point of rolling forecast.
print.true	only if rolling = TRUE, print the rolling forcast steps.
...	Additional arguments passed to the method.

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

t	length of output series.
print.tar.coef	print coefficients, default FALSE.
print.sig	print covariance matrices, default FALSE.
seed	random seed.

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

t	length of output series.
print.tar.coef	print coefficients used for simulation, default FALSE.
print.loading	print loading matrices used for simulation, default FALSE.
seed	random seed.

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

xx	$T \times d_1 \times \cdots \times d_K$ tensor-valued time series, $T$ is the length of the series.
R	Kronecker rank for each lag, a vector for $P>1$, a positive integer, it assumes same number of terms in each lag.
P	Autoregressive order, a positive integer.
method	character string, specifying the type of the estimation method to be used. "PROJ", Projection method. "LSE", Least squares. "MLE", MLE under a separable cov(vec($E_t$)). "VAR", VAR($P$) model for the $\mathrm{vec}(E_t)$.
init.A	initial values of coefficient matrices $A_k^{(ir)}$ in estimation algorithms, which is a multi-layer list such that 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". By default, we use PROJ estimators as initial values.
init.sig	only if method=MLE, a list of initial values of $\Sigma_1,\ldots,\Sigma_K$. The default are identity matrices.
niter	maximum number of iterations if error stays above tol.
tol	error tolerance in terms of the Frobenius norm.

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

t	length of output series, a strictly positive integer.
dim	dimension of the tensor at each time.
R	Kronecker rank for each lag.
P	autoregressive order.
rho	spectral radius of coefficient matrix $\Phi$.
cov	covariance matrix of the error term: diagonal ("iid"), separable ("mle"), random ("svd").
A	coefficient matrices. If not provided, they are randomly generated according to given dim, R, P and rho. 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" of tenAR.est.
Sig	only if cov=mle, a list of initial values of $\Sigma_1,\ldots,\Sigma_K$. The default are identity matrices.

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

x	$T \times d_1 \times \cdots \times d_K$ tensor-valued time series.
r	input rank of factor tensor.
h0	the number of lags used in auto-covariance tensor. If h0=0, covariance tensor is used.
method	character string, specifying the type of the estimation method to be used. "TIPUP", TIPUP method. "TOPUP", TOPUP method.
iter	boolean, specifying using an iterative approach or an non-iterative approach.
tol	tolerance in terms of the Frobenius norm.
maxiter	maximum number of iterations if error stays above tol.

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

x	$T \times d_1 \times \cdots \times d_K$ tensor-valued time series.
r	initial guess of the rank of factor tensor.
h0	the number of lags used in auto-covariance tensor. If h0=0, covariance tensor is used.
rank	character string, specifying the type of the rank determination method to be used. "IC", information criterion. "ER", eigen ratio criterion.
method	character string, specifying the type of the factor estimation method to be used. "TIPUP", TIPUP method. "TOPUP", TOPUP method.
inputr	boolean, if TRUE, always use initial guess rank r in each iteration; if FLASE, the rank will be updated in each iteration.
iter	boolean, specifying using an iterative approach or a non-iterative approach.
penalty	takes value in $1,2,3,4,5$, decides which penalty function to use for each tensor mode $k$. Here $\nu$ is a tuning parameter defined in the argument "delta1", and $d=\prod_{i=1}^{K} d_k$. When rank= 'IC': if penalty=1, $g_1= \frac{h_0 d^{2-2\nu}}{T}\log(\frac{dT}{d+T})$; if penalty=2, $g_2= h_0 d^{2-2\nu}(\frac{1}{T}+\frac{1}{d})\log(\frac{dT}{d+T})$; if penalty=3, $g_3= \frac{h_0 d^{2-2\nu}}{T} \log(\min{(d,T)})$; if penalty=4, $g_4= h_0 d^{2-2\nu}(\frac{1}{T}+\frac{1}{d})\log(\min{(d,T)})$; if penalty=5, $g_5= h_0 d^{2-2\nu}(\frac{1}{T}+\frac{1}{d})\log(\min{(d_k,T)})$. When rank= 'ER': if penalty=1, $h_1= c_0 h_0$; if penalty=2, $h_2= \frac{h_0 d^2}{T^2}$; if penalty=3, $h_3= \frac{h_0 d^2}{T^2 d_k^2}$; if penalty=4, $h_4= \frac{h_0 d^2}{T^2 d_k^2} + \frac{h_0 d_k^2}{T^2}$; if penalty=5, $h_5= \frac{h_0 d^2}{T^2 d_k^2} + \frac{h_0 dd_k^2}{T^2}$.
delta1	weakest factor strength, a tuning parameter used for IC method only
tol	tolerance in terms of the Frobenius norm.
maxiter	maximum number of iterations if error stays above tol.

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

Ft	input of the factor process, of dimension $T \times r_1 \times r_2 \times \cdots \times r_k$. It can be TenAR(p) tensor time series generated by the function tenAR.sim.
dims	dimensions of the output tensor at each time, $d_1\times d_2\cdots\times d_K$.
lambda	signal strength parameter of the tensor factor models, see Details section for more information.
A	a list of the factor loading matrices $A_1, A_2, \cdots, A_K$. The default is random orthogonal matrices $A_k$ of dimension $d_k \times r_k$.
cov	covariance matrix of the error tensor: identity ("iid"), separable Kronecker structure ("separable"), random ("random").
rho	a parameter only for "separable" covariance matrix of the error tensor. It is the off-diagonal element of the error matrices, with the diagonal being 1.

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)

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