Package 'IALS'

The distance between the spaces spanned by the column of two matrices.

Description

Calculate the distance between spaces spanned by the column of two matrices. The distance is between 0 and 1. If the two spaces are the same, the distance is 0. if the two spaces are orthogonal, the distance is 1.

Usage

Distance(Z1, Z2)

Arguments

Z1	Input a matrix with $p \times q_1$.
Z2	Input another matrix with $p \times q_2$

Details

Define

$\mathcal{D}(\bold{Q}_1,\bold{Q}_2)=\left(1-\frac{1}{\max{(q_1,q_2)}}\text{Tr}\left(\bold{Q}_1\bold{Q}_1^\top\bold{Q}_2\bold{Q}_2^\top\right)\right)^{1/2}.$

By the definition of $\mathcal{D}(\bold{Q}_1,\bold{Q}_2)$, we can easily see that $0 \leq \mathcal{D}(\bold{Q}_1,\bold{Q}_2)\leq 1$, which measures the distance between the column spaces spanned by two orthogonal matrices $\bold{Q}_1$ and $\bold{Q}_2$, i.e., $\text{span}(\bold{Q}_1)$ and $\text{span}(\bold{Q}_2)$. In particular, $\text{span}(\bold{Q}_1)$ and $\text{span}(\bold{Q}_2)$ are the same when $\mathcal{D}(\bold{Q}_1,\bold{Q}_2)=0$, while $\text{span}(\bold{Q}_1)$ and $\text{span}(\bold{Q}_2)$ are orthogonal when $\mathcal{D}(\bold{Q}_1,\bold{Q}_2)=1$. The Gram-Schmidt orthogonalization can be used to make $\bold{Q}_1$ and $\bold{Q}_2$ column-orthogonal matrices.

Value

Output a number between 0 and 1.

Author(s)

Yong He, Ran Zhao, Wen-Xin Zhou.

References

He, Y., Zhao, R., & Zhou, W. X. (2023). Iterative Least Squares Algorithm for Large-dimensional Matrix Factor Model by Random Projection. <arXiv:2301.00360>.

Examples

set.seed(1111)
A=matrix(rnorm(10),5,2)
B=matrix(rnorm(15),5,3)
Distance(A,B)

---

Iterative Alternating Least Square Estimation for Large-dimensional Matrix Factor Model

Description

This function is designed to fit the matrix factor model using the Iterative Least Squares (IALS) method, rather than Principal Component Analysis (PCA)-based methods. In detail, in the first step, we propose to estimate the latent factor matrices by projecting the matrix observations with two deterministic weight matrices, chosen to diversify away the idiosyncratic components. In the second step, we update the row/column loading matrices by minimizing the squared loss function under the identifiability condition. The estimators of the loading matrices are then treated as the new weight matrices, and the algorithm iteratively performs these two steps until a convergence criterion is reached.

Usage

IALS(X, W1 = NULL, W2 = NULL, m1, m2, max_iter = 100, ep = 1e-06)

Arguments

X	Input an array with $T \times p_1 \times p_2$, where $T$ is the sample size, $p_1$ is the the row dimension of each matrix observation and $p_2$ is the the column dimension of each matrix observation.
W1	The initial value for the row factor loading matrix. The default is NULL, with an initial estimate chosen from $\alpha$-PCA if not provided.
W2	The initial value for the column factor loading matrix. The default is NULL, with an initial estimate chosen from $\alpha$-PCA if not provided.
m1	A positive integer indicating the row factor number.
m2	A positive integer indicating the column factor number.
max_iter	The maximum number of iterations for the algorithm, default is 100.
ep	The stopping criterion in the iteration algorithm, default is $10^{-6} \times Tp_1 p_2$.

Details

Assume we have two weight matrices $\bold{W}_i$ of dimension $p_{i} \times m_{i}$ for $i=1,2$, as substitutes for $\bold{R}$ and $\bold{C}$ respectively. Then it is straightforward to estimate $\bold{F}_t$ simply by

$\hat{\bold{F}}_{t}=\frac{1}{p_{1}p_{2}}\bold{W}_1^\top\bold{X}_t\bold{W}_2.$

Given $\hat{\bold{F}}_t$ and $\bold{W}_1$, we can derive that

$\hat{\bold{R}}=\sqrt{p_{1}}\left(\sum_{t=1}^{T}\bold{X}_t\bold{W}_2\hat{\bold{F}}_t^\top\right)\left[\left(\sum_{t=1}^{T}\hat{\bold{F}}_t\bold{W}_2^\top\bold{X}_t^\top\right)\left(\sum_{t=1}^{T}\bold{X}_t\bold{W}_2\hat{\bold{F}}_t^\top\right)\right]^{-1/2}.$

Similarly, we get the following estimator of the column factor loading matrix

$\hat{\bold{C}}=\sqrt{p_{2}}\left(\sum_{t=1}^{T}\bold{X}_t^\top\hat{\bold{R}}\hat{\bold{F}}_t\right)\left[\left(\sum_{t=1}^{T}\hat{\bold{F}}_t^\top\hat{\bold{R}}^\top\bold{X}_t\right)\left(\sum_{t=1}^{T}\bold{X}_t^\top\hat{\bold{R}}\hat{\bold{F}}_t\right)\right]^{-1/2}.$

Afterwards, we sequentially update $\bold{F}$, $\bold{R}$ and $\bold{C}$. In simulation, the iterative procedure is terminated either when a pre-specified maximum iteration number ($\text{maxiter}=100$) is reached or when

$\sum_{t=1}^{T}\|\hat{\bold{S}}^{(s+1)}_t-\hat{\bold{S}}^{(s)}_t\|_{F} \leq \epsilon \cdot Tp_1p_2,$

where $\hat{\bold{S}}^{(s)}_t$ is the common component estimated at the $s$-th step, $\epsilon$ is a small constant ($10^{-6}$) given in advance.

Value

The return value is a list. In this list, it contains the following:

R	The estimated row loading matrix of dimension $p_1\times m_1$, satisfying $\bold{R}^\top\bold{R}=p_1\bold{I}_{m_1}$.
C	The estimated column loading matrix of dimension $p_2\times m_2$, satisfying $\bold{C}^\top\bold{C}=p_2\bold{I}_{m_2}$.
F	The estimated factor matrix of dimension $T \times m_1\times m_2$.
iter	The number of iterations when the stopping criterion is met.

Author(s)

Yong He, Ran Zhao, Wen-Xin Zhou.

References

He, Y., Zhao, R., & Zhou, W. X. (2023). Iterative Alternating Least Square Estimation for Large-dimensional Matrix Factor Model. <arXiv:2301.00360>.

Examples

set.seed(11111)
T=20;p1=20;p2=20
k1=3;k2=3

R=matrix(runif(p1*k1,min=-1,max=1),p1,k1)
C=matrix(runif(p2*k2,min=-1,max=1),p2,k2)

X=E=array(0,c(T,p1,p2))
F=array(0,c(T,k1,k2))

for(t in 1:T){
  F[t,,]=matrix(rnorm(k1*k2),k1,k2)
  E[t,,]=matrix(rnorm(p1*p2),p1,p2)
}
for(t in 1:T){
X[t,,]=R%*%F[t,,]%*%t(C)+E[t,,]
}

#Estimating the matrix factor model using the default initial values
fit1 = IALS(X, W1 = NULL, W2 = NULL,k1, k2, max_iter = 100, ep = 1e-06) 
Distance(fit1$R,R);Distance(fit1$C,C)

#Estimating the matrix factor model using one-step iteration
fit2 = IALS(X, W1 = NULL , W2 = NULL, k1, k2, max_iter = 1, ep = 1e-06) 
Distance(fit2$R,R);Distance(fit2$C,C)

---

Estimating the Pair of Factor Numbers via Eigenvalue Ratios Corresponding to IALS

Description

The function is to estimate the pair of factor numbers via eigenvalue ratios corresponding to IALS method.

Usage

KIALS(X, W1 = NULL, W2 = NULL, kmax, max_iter = 100, ep = 1e-06)

Arguments

X	Input an array with $T \times p_1 \times p_2$, where $T$ is the sample size, $p_1$ is the the row dimension of each matrix observation and $p_2$ is the the column dimension of each matrix observation.
W1	The initial value for the row factor loading matrix. The default is NULL, with an initial estimate chosen from $\alpha$-PCA if not provided.
W2	The initial value for the column factor loading matrix. The default is NULL, with an initial estimate chosen from $\alpha$-PCA if not provided.
kmax	The user-supplied maximum factor numbers. Here it means the upper bound of the number of row factors and column factors.
max_iter	The maximum number of iterations for the algorithm, default is 100. See in IALS.
ep	The stopping criterion in the iteration algorithm, default is $10^{-6} \times Tp_1 p_2$. See in IALS.

Details

In detail, we first set $k_{\max}$ is a predetermined upper bound for $k_1,k_2$ and thus by IALS method, we can obtain the estimate of $\bold{F}_t$, denote as $\hat{\bold{F}}_t$, which is of dimension $k_{\max}\times k_{\max}$. Then the dimensions $k_1$ and $k_2$ are further determined as follows:

$\hat{k}_{1}=\arg\max_{j \leq k_{\max}}\frac{\lambda_{j}\left(\dfrac{1}{T}\sum_{t=1}^{T}\hat{\bold{F}}_t\hat{\bold{F}}_t^\top\right)}{\lambda_{j+1}\left(\frac{1}{T}\sum_{t=1}^{T}\hat{\bold{F}}_t\hat{\bold{F}}_t^\top\right)},$

$\hat{k}_{2}=\arg\max_{j \leq k_{\max}}\frac{\lambda_{j}\left(\dfrac{1}{T}\sum_{t=1}^{T}\hat{\bold{F}}_t^\top\hat{\bold{F}}_t\right)}{\lambda_{j+1}\left(\frac{1}{T}\sum_{t=1}^{T}\hat{\bold{F}}_t^\top\hat{\bold{F}}_t\right)}.$

Value

\eqn{k_1}	The estimated row factor number.
\eqn{k_2}	The estimated column factor number.

Author(s)

Yong He, Ran Zhao, Wen-Xin Zhou.

References

He, Y., Zhao, R., & Zhou, W. X. (2023). Iterative Alternating Least Square Estimation for Large-dimensional Matrix Factor Model. <arXiv:2301.00360>.

Examples

set.seed(11111)
T=20;p1=20;p2=20
k1=3;k2=3

R=matrix(runif(p1*k1,min=-1,max=1),p1,k1)
C=matrix(runif(p2*k2,min=-1,max=1),p2,k2)

X=E=array(0,c(T,p1,p2))
F=array(0,c(T,k1,k2))

for(t in 1:T){
  F[t,,]=matrix(rnorm(k1*k2),k1,k2)
  E[t,,]=matrix(rnorm(p1*p2),p1,p2)
}

for(t in 1:T){
X[t,,]=R%*%F[t,,]%*%t(C)+E[t,,]
}

kmax=8
K=KIALS(X, W1 = NULL, W2 = NULL, kmax, max_iter = 100, ep = 1e-06);K

---