Regularized estimation of Kronecker structured covariance matrix using modified Cholesky decomposition

Figures & data

Table 1. Risk functions based on the entropy and quadratic loss for the corresponding estimated covariance matrices $\hat{\mathbf{\Lambda }}$ with $\rho =0.3$.

	Estimator
$n\times p\times q$	MLE	Sh $L_1$	Sm	Sh $L_1$ and Sm
Entropy loss
$50\times 10\times 10$	2.34	1.98	2.27	1.94
$10\times 30\times 30$	105.72	61.14	71.51	55.97
$30\times 30\times 30$	33.26	21.46	31.04	21.26
$10\times 50\times 10$	910.72	123.32	503.20	123.23
Quadratic loss
$50\times 10\times 10$	5.20	4.39	5.08	4.32
$10\times 30\times 30$	235.45	137.07	171.69	131.61
$30\times 30\times 30$	67.80	43.65	65.28	43.13
$10\times 50\times 10$	342.16	137.71	158.92	137.60

Note: MLE: only maximum likelihood estimation is used, Sh $L_1$: shrinkage under $L_1$ norm, Sm: smoothing.

Table 2. Risk functions based on the entropy and quadratic loss for the corresponding estimated covariance matrices $\hat{\mathbf{\Lambda }}$ with $\rho =0.5$.

	Estimator
$n\times p\times q$	MLE	Sh $L_1$	Sm	Sh $L_1$ and Sm
Entropy loss
$50\times 10\times 10$	2.45	1.84	2.33	1.83
$10\times 30\times 30$	107.01	61.23	94.14	58.85
$30\times 30\times 30$	34.26	20.96	30.28	20.74
$10\times 50\times 10$	1858.59	1616.64	1711.29	1530.85
Quadratic loss
$50\times 10\times 10$	5.22	4.08	4.94	4.05
$10\times 30\times 30$	255.23	138.80	225.98	138.73
$30\times 30\times 30$	71.49	42.58	62.35	41.92
$10\times 50\times 10$	325.89	140.51	194.05	137.34

Note: MLE: only maximum likelihood estimation is used, Sh $L_1$: shrinkage under $L_1$ norm, Sm: smoothing.

Table 3. Risk functions based on the entropy and quadratic loss for the corresponding estimated covariance matrices $\hat{\mathbf{\Lambda }}$ with $\rho =0.7$.

	Estimator
$n\times p\times q$	MLE	Sh $L_1$	Sm	Sh $L_1$ and Sm
Entropy loss
$50\times 10\times 10$	2.49	1.75	2.34	1.73
$10\times 30\times 30$	107.07	59.54	102.67	57.57
$30\times 30\times 30$	33.49	20.88	30.31	20.66
$10\times 50\times 10$	1845.29	1273.84	1139.97	940.15
Quadratic loss
$50\times 10\times 10$	5.26	3.87	5.18	3.83
$10\times 30\times 30$	270.99	129.63	228.62	127.92
$30\times 30\times 30$	68.66	42.20	62.30	41.69
$10\times 50\times 10$	355.15	159.73	310.05	150.23

Note: MLE: only maximum likelihood estimation is used, Sh $L_1$: shrinkage under $L_1$ norm, Sm: smoothing.

Figure 1. Presentation of recording on EEG voltages.

Figure 2. Estimated correlation matrices for the alcoholic group using MLE.

Figure 3. Estimated correlation matrices for the control group using MLE.

Figure 4. Estimated correlation matrices for the alcoholic group using the proposed method.

Figure 5. Estimated correlation matrices for the control group using the proposed method.