OLCPM (version 0.1.1)

gen.psi.tau.flat: calculate eigenvalue series by ``flat'' method

Description

This function calculates the rolling eigenvalue series for the monitoring process, based on the ``flat'' version of sample covanriance matrix.

Usage

gen.psi.tau.flat(Y, k, m, delta, r)

Value

a \((T-m)\times 3\) matrix, whose three columns are the original, rescaled, and transformed eigenvalue series, respectively.

Arguments

Y

the observed \(T\times p1\times p2\) array. \(T\) is the sample size, \(p1\) and \(p2\) are the row and column dimensions, respectively.

k

a positive integer determining which eigenvalue to monitor. \(k=1\) for the largest eigenvalue.

m

a positive integer (\(>1\)) indicating the bandwidth of the rolling windom.

delta

a number in \((0,1)\) indicating the rescaling parameter for the eigenvalue. The default approach to calcualte delta is in the paper He et al. (2021).

r

a positive integer indicating the order of the transformation function \(g(x)=|x|^r\). Motivated by the paper, \(r\) should be chosen according to the moments of the data; see more details in He et al. (2021).

Author

Yong He, Xinbing Kong, Lorenzo Trapani, Long Yu

Details

The rolling eigenvalue series will start at the stage \(m+1\), with length \(T-m\).

References

He Y, Kong X, Trapani L, & Yu L(2021). Online change-point detection for matrix-valued time series with latent two-way factor structure. arXiv preprint, arXiv:2112.13479.

Examples

## generate data
k1=3
k2=3
epsilon=0.05
Sample_T=50
p1=40
p2=20
kmax=8
r=8
m=p2

# generate data
Y=gen.data(Sample_T,p1,p2,k1,k2,tau=0.5,change=1,pp=0.3)

# calculate delta
temp=log(p1)/log(m*p2)
delta=epsilon*(temp<=0.5)+(epsilon+1-1/(2*temp))*(temp>0.5)

# calculate psi.tau
psi2=gen.psi.tau.flat(Y,k1+1,m,delta,r)
print(psi2)