A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model.

This paper introduces a new family of matrix variate distributions based on the mean-mixture of normal (MMN) models. The properties of the new matrix variate family, namely stochastic representation, moments and characteristic function, linear and quadratic forms as well as marginal and conditional distributions are investigated. Three special cases including the restricted skew-normal, exponentiated MMN and the mixed-Weibull MMN matrix variate distributions are presented and studied. Based on the specific presentation of the proposed model, an EM-type algorithm can be directly implemented for obtaining maximum likelihood estimate of the parameters. The usefulness and practical utility of the proposed methodology are illustrated through two conducted simulation studies and through the Landsat satellite dataset analysis.

1 Introduction

The skew-normal (SN) distribution, initially introduced by Azzalini [1], has received considerable attention in both theoretical and applied statistics in the past two decades. Various extensions, forms and properties of the SN distribution in the multivariate case were derived in [2-5], and the acknowledged articles therein. An interesting form of the SN distribution was presented by Pyne et al. [3] who named it the restricted multivariate SN (rSN) model. Generally, the rSN distribution can be expressed as a linear transformation of the multivariate normally distributed random vector and the univariate truncated normal distribution. Although the rSN model, like the original SN one, can describe the skewness of data, it still is not robust in dealing with the outlying observations. To cover this drawback, Negarestani et al. [6] used the rSN transformation to introduce the family of multivariate mean mixture of normal (MMN) model. Specifically, a p-dimension random vector is in the family of MMN distributions if where ‘’ stands for the equality in distribution, follows the multivariate normal model with zero mean and covariance matrix Σ, and W is an arbitrary random variable independent of . It is clear that the rSN distribution is a special case of (1) where the mixing variable W is followed by the truncated standard normal distribution lying within a truncated interval (0, ∞), denoted by . It is shown by Negarestani et al. [6] that the family of MMN may provide a new model with wider range of skewness and kurtosis than the rSN, skew-t [4] and skew Student-t-normal [7] distributions. From (1), the probability distribution function (pdf) of random vector can be presented as where ϕ(⋅;⋅) denotes the pdf of multivariate normal distribution and h(·; ) is the pdf of W parameterized by the vector parameter . The notation will be used to indicate that has pdf (2). Depending on the random variable W that can take values on the real line, the pdf (2) can be both symmetric and asymmetric. However, a more flexible and skewed version of the MMN model can be obtain if W has any asymmetric distribution or any positive support model like the truncated-normal, exponential and gamma distributions. Moreover, the pdf (2) can include skew-elliptical models, as the rSN distribution, or can result in skew non-elliptically contoured models if, for example, W is distributed as the exponential, Weibull and gamma models. From Fig 2 in Appendix A, it is observed that the family of MMN distributions offers different orientation compared with the family of mean-variance mixture of normal (MVMN) distributions [8].

Fig 2

Contour plots comparison of special cases of the MMN and MVMN families.

Matrix variate distribution finds its genesis in modeling dependent multivariate observations in the normal case [9]. The recent use of the matrix variate normal (MVN) distribution can be found in modeling a wide variety of three-way data appearing in studies including control theory, stochastic systems, image recognition, repeated vector measurements, multivariate time series, spatial data, among others [10, 11]. The MVN distribution not only inherits some appealing properties, features as well as widespread applications from the multivariate normal model, but also it is still not stable and robust against non-normal features such as asymmetry and heavy tails. To deal with the heavy tailed data, Kshirsagar and Bartlett [12] proposed the matrix variate t distribution by showing that the estimator of the parameter matrix of regression coefficients unconditionally follows matrix variate t model. Bulut and Arslan [13] proposed the matrix variate slash distribution as a scale mixture of the matrix variate normal and the uniform distributions. Moreover, in accommodating skewness and kurtosis, the interest of skew distributions provides a platform for robust extension of matrix variate distribution. For instance, works on the matrix variate versions of SN distribution can be found in [14-17]. Even though the matrix variate SN distribution has many attractive properties, it suffers from robustness in dealing with heavy tailed data and from parameter estimation. Regarding these drawbacks of the matrix variate SN model and considering the aforementioned properties of the MMN family of distributions, the objective of this paper is to propose a family of matrix variate mean-mixture of normal (MVMMN) distributions. Some properties and features of our introduced family such as moments, the characteristic function, marginal and conditional distributions are studied. The maximum likelihood (ML) estimate of model parameters are computed by applying expectation-maximization (EM) type algorithm [18].

The contribution of this work can be broken down into six parts. We will begin the usual procedure with the model formulation of the MVMMN distribution in Section 2. Properties and characteristics of the MVMMN distribution are also studied in Section 3. The parameter estimation procedure using the EM-type algorithm and some computational strategies of implementation are given in Section 4. To examine the performance of the methodology into practice, simulation and real-world data analyses are presented in Sections 5 and 6. Finally, Section 7 gives some concluding remarks and future extensions.

2 Proposed family

To start the whole process, we begin with some notations and definitions. A random matrix variable defined as follows a MVN distribution if its pdf is given as where etr{A} = exp(tr(A)), tr(⋅) is the trace operator of a matrix, (X, M, Ψ, Σ) = Σ−1(X − M)Ψ−1(X − M)⊤ denotes the matrix variate Mahalanobis squared distance, and the mean matrix M and two dispersion matrices , are defined as

We shall use notation if has pdf (3). The following definition is a new result from the representation (1) in the matrix format.

Definition 1 A random matrix variable is said to have a MVMMN distribution if it can be generated by the stochastic representation where , W is a random variable, independent of , distributed by h(w; is a skewness matrix defined as

It can be easily seen that the hierarchical representation of MVMMN model is

Hence, the pdf of can be given as

Applying the well-known property of the MVN distribution, we have where vec(B) denotes the vectorization operator of matrix B, and ⊗ stands for the Kronecker product.

Remark 1 Referring to representation (4), it is clear that the mean of is + Λ E(W), showing the assumption that the mean of MVMMN distribution is not fixed for all members of the population. We would like to emphasize that the family of matrix variate normal mean-variance mixture (MVNMVM) models [19, 20], assumes that both the mean and variance of the population member are not fixed. Therefore, an interesting extension of the MVMMN distribution can be introduced by considering the family of scale mixture of MVMMN distributions.

Restricted matrix variate skew-normal: If in (4), then restricted matrix variate SN (RMVSN) distribution is arisen. The resulting pdf of directly obtained by integrating out (6), is where 2 = tr(Ψ−1 Λ⊤ Σ−1 Λ) +1, A = −1 [tr(Ψ−1 Λ⊤ Σ−1(Y − M))], and Φ(⋅) denotes the cumulative distribution function of standard normal model.

Lemma 1 If , then where ϕ(⋅) is the pdf of standard normal distribution.

Proposition 1 Let and . Then, W conditionally on , denoted by W .

Proof. Using the hierarchical representation (5), the pdf of RMVSN model (8), and the Bayes’ rule, we have which completes the proof after using some matrix factorizations.

Convolution with exponential model: The exponentiated MVMMN (MVMMNE) distribution, say , is derived as another special case of (4) if , where denotes the exponential distribution with mean 1. This leads to obtain the pdf of form (6) as where , .

Proposition 2 Let and . Then, .

Proof. In a similar manner as Proposition 1, the proof can be completed.

Convolution with Weibull model: The mixed-Weibull MVMMN (MVMMNW) distribution, denoted by , is arisen when W in (4) follows the Weibull model respectively with shape and scale parameters α = 2 and β = 1, . Hence, the associated pdf of obtained by (6) is where , .

Proposition 3 Let and . Then, W has the pdf

Moreover, for r = 1, 2, …, where .

Proof. Results can be obtained from the Bayes’ rule and some matrix factorizations.

Theorem 1 The MVMMN distribution is log-concave if W has log-concave pdf.

Proof. Based on [21], if f(x) and g(y) are log-concave functions, then their convolution, i.e., is also a log-concave function. Hence, the property of vectorization operator of the MVMMN distribution (7) and the fact that the MVN is log-concave completes the proof if W has a log-concave pdf.

Corollary 1 The RMVSN, MVMMNE and MVMMNW distributions are log-concave.

Proof. Since the truncated normal, exponential and Weibull (if the shape parameter is ≥1) distributions are log-concave, their associated matrix variate models are, using Theorem 1.

3 Characteristics

This section provides some substantial statistical properties of the MVMMN distribution.

Theorem 2 If , then the mean and the characteristic function of , respectively, are where φ(⋅) is the characteristic function of W ∼ h(w; ).

Proof. The proof of theorem can be completed by using the presented representations in Definition 1. Taking expectation on both sides of the stochastic representation (4) the first part is proved. Moreover for the second part, recall that the characteristic function of the matrix variate is given as

Hence, through the hierarchical representation (5), the characteristic function of is obtained by .

Theorem 3 Let , and = (m), Λ = (λ), Σ = (σ), Ψ = (ψ). Then, we have

For 1 < i1, i2 < p, and 1 < j1, j2 < n,

If M = 0,

Proof. (i) follows by using the hierarchical representation (5) and applying theorems 2.3.3 of [22]. For M = 0, it is clear from part (i) that

Therefore, we have which completes the proof.

Theorem 4 The family of MVMMN distributions is closed under the transpose operator, i.e.,

Proof. Based on theorem 2.3.1 of [22], we have

Now, applying this transpose property of the MVN distribution into the hierarchical representation (5) results in

Theorem 5 Let , and × p matrix of rank q ≤ p and × m matrix of rank m ≤ n. Then,

Proof. The proof of the theorem is completed through obtaining the characteristic function of : where 1 = ⊤ . Now, by applying Theorem 2, we have which is the characteristic function of .

Theorem 6 Let , and partition , , Λ, Σ, and Ψ as where , and . Then,

Similarly, the marginal distribution of , and can be obtained.

Proof. The proof follows by applying Theorem 5 with considering = ( 0) and = ( 0)⊤, where denotes the unit matrix of order d.

Theorem 7 Let , and partition Ψ, Σ as Theorem 6, and , , Λ as follows where , and . Then,

, and .

, and , where , , and .

Proof. The proof of (i) is completed by considering proper matrices and in Theorem 5. Using the hierarchical representation (5) and applying theorem 2.3.12 of [22], the second part of the theorem is proven.

Corollary 2 If and under partition of Theorem 7, we have

where , and .

, where , , and .

Corollary 3 If and under partition of Theorem (7), we have

where , , and .

, where , , and .

The presentation of distribution of the matrix quadratic form, done by [23], can also be implemented in the context of the MVMMN family of distributions. Referring to theorem 2.2 of [23], they defined the distribution of quadratic form to be where A is a n × n symmetric real matrix of rank r, and .

Theorem 8 Let and W ∼ h(w;) and any n × n symmetric matrix of rank r. Then, conditionally on W = w, are identically distributed, where δ are the non-zero eigenvalues of and are independent non-central Wishart distribution for j = 1, …, r, where = and are the corresponding orthogonal eigenvectors ().

Proof. Using hierarchical representation (5) of the MVMMN model, we have . Consequently, the property of the matrix variate normal distribution leads to . Now, by definition 2.1 of [23], we have

On the other hand, through theorem 2.2 of [23], we have

Therefore, the random matrices and have identical distributions.

4 Parameter estimation

Suppose N matrix observations Y1, …, Y of dimension p × n are drawn independently and identically from the . Therefore, the log-likelihood function of based on the observed data is

To obtain ML estimate of Θ, an EM-type algorithm is implemented as a powerful estimation approach in dealing with the unobserved (missing and/or censored) data and latent variables [18]. The computations of EM algorithm are based on two iterative E- and M-steps. In E-step, the expected value of the complete-data log-likelihood function, the likelihood of the observed and missing data the latent variable, is computed, while in M-step, parameter estimates are updated by maximizing this expected value.

Through the hierarchical representation (5), the complete-data log-likelihood function of Θ, obtained by introducing latent variables = (w1, …, w) and omitting additive constants, is

ML estimation of Θ is performed by using the expectation-conditional maximization (ECM; [24]) algorithm as follows.

Initialization: Set the number of iteration to k = 0 and choose a relative starting point Θ( = (M(, Λ(), Σ(, Ψ(, (). We point out that in our data examples the parameters are initialized by , Λ(0) = 1, Σ(0) = c1 , Ψ(0) = c2 . Here, 1 is a matrix of dimension p × n with unit elements. Moreover, the elements of two vectors c1 and c2 are computed, respectively, as

E-step: The expected value of the complete-data log-likelihood function (10), called Q-function, is computed as where , , and depending on h(w; ) .

First CM-step: Maximizing Q-function with respect to M and Λ give the following update where and .

Second CM-step: Update Σ and Ψ, respectively,

Third CM-step: The additional parameter depending on the distribution of W is updated by

Remark 2 The conditional expectations and involved in the Q-function (11) can be obtained by Lemma 1 and Propositions 1, 2 and 3 for our three considered models. Furthermore, we note that in all special cases considered in Section 4, the distribution of mixing random variable W, is parameter free. Therefore, the last step of the ECM algorithm is not necessary.

4.1 Computational aspects

4.1.1 Convergence

The process of the EM algorithm can be iterated until a suitable convergence rule, like or , is satisfied where ε is a user specified tolerance and ℓ(⋅) is defined in (9). An alternative approach to determine convergence of the EM algorithm is the Aitken acceleration method [25]. To apply this approach, the asymptotic estimate of the log-likelihood at iteration k + 1, following [26], can be obtained as where the Aitken acceleration of iteration k is

Therefore, the algorithm can be considered to have converged at iteration k + 1 when , [27]. In our study, the tolerance ϵ is considered as 10−5.

4.1.2 Model selection

The models in competition in our data analysis are compared using the most commonly used measures Akaike information criterion (AIC; [28]) and Bayesian information criterion (BIC; [29]) defined as where m is the number of free parameters and ℓmax is the maximized log-likelihood value. Models with lower values of AIC or BIC are considered more preferable.

5 Simulation studies

In this section, the performance of our model and its computational method is illustrated by conducting two simulation studies. The first simulation study aims at comparing the special cases of MVMMN model in dealing with skewed and leptokurtic simulated data. The second simulation study demonstrates whether our proposed ECM algorithm can provide good asymptotic properties.

Example 1 Model performance

In this experiment, simulated data are generated from a matrix variate normal inverse Gaussian (MVNIG; [20]) distribution with sample sizes N = 50, 100, 500, 1000 and 2000, to compare the performance of three special cases of MVMMN model. The MVNIG distribution belongs to the family of MVMVM models where the mixing random variable follows the , such that denotes the generalized inverse Gaussian distribution with parameter (κ, χ, ψ) [30]. We consider this matrix variate distribution to generate non-normal data as it offers the desired level of asymmetry and leptokurtosis. Let χ = ψ = 3 and

Table 1 summarizes the average (ℓ) and standard deviation (Std.) of the maximized log-likelihood together with the frequencies (out of 200 replications) of the particular model chosen based on the biggest ℓ value. The results depicted in Table 1 reveal that the MVMMNE distribution provides a better fit than the other two MVMMN-based models. It is clear that the outperformance of MVMMNE distribution is improved by increasing the sample size, N.

Table 1

Mean and standard deviation for the maximized log-likelihood and frequency of model outperformance in 200 replications for various sample sizes.

	RMVSN			MVMMNE			MVMMNW
N	ℓAV	Std.	Freq.	ℓAV	Std.	Freq.	ℓAV	Std.	Freq.
50	-731.14	35.60	55	-729.90	35.54	141	-733.99	35.82	4
100	-1503.40	43.96	46	-1501.06	43.65	154	-1508.61	43.98	0
500	-7622.68	114.60	14	-7610.95	113.25	186	-7649.34	115.98	0
1000	-15278.40	147.93	2	-15254.32	145.88	197	-15329.81	148.41	1
2000	-30574.13	206.84	1	-30528.32	205.33	198	-30680.56	206.61	1

In order to compare the accuracy of parameter estimates to the real values, the Frobenius (Frob.) norm is adopted. For a given d × m matrix = [a], the Frob. norm is defined as the square root of the sum of the squares of its elements, i.e. . Table 2 shows the average Frob. norm of and , where and are the ML estimates of the fitted model in the ith replication. It is observe that the Frob. norm decreases when the sample size increases. We can also see that the Frob. norm for Σ and Ψ for all models are very close to each other while the MVMME model has the furthest estimates of and Λ.

Table 2

Mean of Frob. norm for parameter estimates of the candidate distributions for various sample sizes.

parameter→	M-M^			Λ-Λ^
N↓	RMVSN	MVMMNE	MVMMNW	RMVSN	MVMMNE	MVMMNW
50	1.7135	2.0228	1.1425	1.3717	2.1097	1.5439
100	1.6714	2.0197	0.8540	1.0385	2.0324	1.0986
500	1.5130	1.8976	0.3955	0.6317	1.9081	0.6722
1000	1.4822	1.8772	0.3033	0.5559	1.8806	0.6163
2000	1.4796	1.864 1	0.2460	0.5268	1.8752	0.5931
parameter→	Σ-Σ^			Ψ-Ψ^
N↓	RMVSN	MVMMNE	MVMMNW	RMVSN	MVMMNE	MVMMNW
50	0.3536	0.3546	0.3520	0.4378	0.4370	0.4361
100	0.2311	0.2314	0.2300	0.2997	0.3000	0.2986
500	0.1043	0.1039	0.1044	0.1300	0.1291	0.1303
1000	0.0744	0.0739	0.0741	0.0981	0.0976	0.0979
2000	0.0515	0.0511	0.0513	0.0707	0.0701	0.0702

Example 2 Performance of the model under AR(1) dependent structure

In order to investigate the effect of auto-regressive (AR(1)) dependent structure in Σ and Λ to the parameter estimates, we conduct another Monte Carlo simulation. In this experiment, we set = 0 and Ψ−1 = 4 and where λ = 0.5, 2 and ρ = 0.5, 0.8. For generating a random sample from the MVMMN model, the value 0.001 is added to the diagonal elements of Σ to ensure that it is a positive definite matrix.

In each replication of 200 trials, the we generate data from the MVNIG distribution with true parameter values displayed above and χ = ψ = 3 for the sample sizes N = 100 and 1000. By fitting the RMVSN, MVMMNE and MVMMNW distributions to the generated data, the Frob. norm of and are obtained. Table 3 summarizes the average Frob. norm of the ML estimates of the fitted models. As expected, the Frob. norm of the parameters decreases as the sample size increases. It can also be observed that the MVMMNW distribution has the smallest Frob. norm of and for the selected combinations of λ and ρ.

Table 3

Mean of Frob. norm for parameter estimates of the candidate distributions for some selected values of λ and ρ.

		M-M^						Λ-Λ^
		RMVSN		MVMMNE		MVMMNW		RMVSN		MVMMNE		MVMMNW
λ	ρ	100	1000	100	1000	100	1000	100	1000	100	1000	100	1000
0.5	0.5	1.1167	0.5698	1.2671	0.9176	1.2919	0.4306	1.2240	0.3709	1.2753	0.9257	1.5625	0.6880
	0.8	1.0401	0.6036	1.2095	0.9326	1.1386	0.3661	1.1023	0.3522	1.2184	0.9407	1.3944	0.6036
2	0.5	3.5017	2.9992	4.1028	3.5957	1.2340	0.7310	1.8416	1.3392	4.2938	3.7512	1.3229	0.6188
	0.8	3.3566	1.1863	3.8494	1.4686	1.2609	0.3373	1.8779	0.6136	4.0888	1.5914	1.1736	0.2349
		Σ-Σ^						Ψ-Ψ^
		RMVSN		MVMMNE		MVMMNW		RMVSN		MVMMNE		MVMMNW
λ	ρ	100	1000	100	1000	100	1000	100	1000	100	1000	100	1000
0.5	0.5	0.2829	0.1004	0.2754	0.0963	0.2821	0.0943	0.3376	0.1028	0.3319	0.1026	0.3384	0.1000
	0.8	0.2931	0.0916	0.2889	0.0903	0.2906	0.0890	0.3345	0.0995	0.3310	0.1002	0.3333	0.0979
2	0.5	0.2412	0.0768	0.2395	0.0747	0.2411	0.0779	0.3041	0.0917	0.3048	0.0931	0.3039	0.0912
	0.8	0.2489	0.0470	0.2463	0.0464	0.2480	0.0477	0.3064	0.0536	0.3070	0.0550	0.3056	0.0532

Example 3 Finite sample properties of the ML estimates

The second simulation study aims at investigating the finite-sample properties of ML estimators obtained by using the ECM algorithm. We consider the situation where Monte Carlo samples of sizes N = 100 and 500 are generated for each of the three special cases of MVMMN distribution. The presumed parameters for all distributions are same as used in Example 1. Fig 1 shows the marginal distributions of the columns, labeled by V1, V2, V3, and V4, for the RMVSN, MVMMNE and MVMMNW distributions of a typical dataset with size 100. The solid red line highlights the marginal mean. In each replication of 1000 trials, the synthetic dataset was fitted with the true generator model via ECM algorithm. To investigate the estimation accuracies, we calculate the bias and the mean squared error (MSE), defined as where denotes the ML estimate of θ (a specific parameter) at the kth replication.

Fig 1

Marginals of a typical simulated data form the RMVSN, MVMMNE and MVMMNW distributions if the drawing has been lengthwise stretched.

The detailed numerical results are reported in Table 4. It can be observed that the bias and MSE for all three special cases of MVMMN distribution tend to decrease toward zero by increasing the sample size, showing empirically the consistency of the ML estimates obtained via the ECM algorithm.

Table 4

Simulation results for assessing the consistency of ML parameter estimates with two sample sizes.

Model	N	Measure	M	Λ	Σ	Ψ
RMVSN	100	Bias	[0.0150.003-0.0020.015-0.008-0.0140.005-0.018-0.007-0.0070.002-0.010]	[-0.019-0.0010.001-0.0150.0120.014-0.0090.0150.0090.006-0.0070.008]	[-0.0230.010-0.0030.010-0.024-0.014-0.003-0.014-0.025]	[-0.001-0.0020.004-0.002-0.0020.007-0.0030.0040.004-0.0030.003-0.002-0.0020.004-0.0020.003]
		MSE	[0.0300.0300.0300.0280.0350.0300.0310.0370.0300.0290.0300.042]	[0.0360.0360.0300.0340.0550.0370.0310.0550.0290.0350.0310.083]	[0.0060.0030.0030.0030.0060.0040.0030.0040.007]	[0.0070.0040.0040.0050.0040.0060.0040.0050.0040.0040.0050.0040.0050.0050.0040.007]
	500	Bias	[0.0040.001-0.0010.001-0.0020.001-0.001-0.0030.0000.0040.000-0.001]	[-0.0050.0000.0000.0000.0040.0000.0000.0010.000-0.0040.003-0.001]	[-0.0040.0020.0010.002-0.004-0.0020.001-0.002-0.003]	[-0.001-0.0010.0010.000-0.0010.002-0.0010.0030.001-0.0010.000-0.0010.0000.003-0.0010.002]
		MSE	[0.0060.0060.0060.0060.0070.0060.0060.0070.0060.0060.0060.009]	[0.0070.0070.0060.0070.0110.0070.0060.0100.0060.0070.0060.016]	[0.0010.0010.0010.0010.0010.0010.0010.0010.001]	[0.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.001]
MVMMNE	100	Bias	[0.015-0.010-0.0040.0040.0110.0000.003-0.0090.0050.003-0.002-0.016]	[-0.0220.0210.002-0.021-0.0390.018-0.0030.0360.0020.016-0.0030.057]	[-0.0170.0090.0040.009-0.017-0.0110.004-0.011-0.018]	[0.001-0.0030.0050.000-0.0030.008-0.0060.0030.005-0.0060.005-0.0030.0000.003-0.003-0.001]
		MSE	[0.0210.0200.0200.0190.0230.0200.0190.0240.0200.0180.0180.030]	[0.0200.0200.0110.0180.0470.0190.0110.0450.0110.0180.0110.092]	[0.0050.0030.0030.0030.0060.0040.0030.0040.006]	[0.0060.0040.0040.0040.0040.0060.0050.0040.0040.0050.0060.0030.0040.0040.0030.006]
	500	Bias	[0.0000.000-0.0030.0010.0030.0000.0020.0020.0010.002-0.0020.003]	[-0.0040.0020.002-0.005-0.0070.004-0.0010.0070.0020.0020.0030.011]	[-0.0020.0010.0010.001-0.003-0.0030.001-0.003-0.003]	[-0.001-0.0020.0020.000-0.0020.003-0.0020.0010.002-0.0020.002-0.0010.0000.001-0.001-0.001]
		MSE	[0.0030.0020.0020.0030.0030.0030.0030.0030.0020.0030.0020.003]	[0.0030.0030.0020.0020.0040.0030.0020.0040.0020.0020.0020.007]	[0.0010.0010.0010.0010.0010.0010.0010.0010.001]	[0.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.001]
MVMMNW	100	Bias	[0.005-0.0100.0010.0040.0080.001-0.003-0.0150.011-0.0070.004-0.021]	[-0.0060.014-0.007-0.006-0.0090.0030.0050.022-0.0060.012-0.0030.028]	[-0.0230.011-0.0040.011-0.023-0.014-0.004-0.014-0.030]	[0.002-0.0010.003-0.001-0.0010.006-0.0040.0040.003-0.0040.003-0.002-0.0010.004-0.0020.003]
		MSE	[0.0530.0560.0500.0560.0620.0550.0500.0670.0530.0580.0550.085]	[0.0560.0610.0510.0580.0710.0580.0510.0730.0570.0630.0550.104]	[0.0060.0040.0030.0040.0060.0040.0030.0040.007]	[0.0070.0050.0040.0050.0050.0060.0050.0050.0040.0050.0060.0040.0050.0050.0040.008]
	500	Bias	[-0.0040.0010.0010.0020.0030.000-0.001-0.0060.001-0.0010.002-0.003]	[0.0020.004-0.003-0.006-0.0070.0010.0030.0100.0000.002-0.0030.006]	[-0.0070.0030.0000.003-0.004-0.0020.000-0.002-0.005]	[0.002-0.0010.002-0.001-0.0010.001-0.0010.0010.002-0.0010.000-0.002-0.0010.001-0.0020.000]
		MSE	[0.0110.0110.010.0110.0130.0110.010.0130.0100.0110.010.017]	[0.0120.0110.010.0120.0140.0110.010.0140.0100.0120.010.020]	[0.0010.0010.0010.0010.0010.0010.0010.0010.001]	[0.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.0010.001]

6 Analysis of Landsat data

To investigate the performance of the developed model in real-world data analysis, we consider Landsat satellite data (LSD) originally obtained by NASA and available at Irvine machine learning repository (http://archive.ics.uci.edu/ml). Each line of the LSD contains of four spectral values of nine pixel neighborhoods in a satellite image. In other words, the lines of LSD are related to a matrix of observations of 4 × 9 dimension. Moreover, each of the LSD matrix of observations belongs to one of six different classes, namely red soil, cotton crop, grey soil, damp grey soil, soil with vegetation stubble, and very damp grey soil. In our analysis, we focus on two classes, the red soil and cotton crop, with size 461 and 224, respectively, for illustrative purposes.

We fitted RMMVSN, MVMMNE and MVMMNW distributions by implementing the ECM algorithm. Table 5 shows a summary of ML fitting results, including the parameter estimates, maximized log-likelihood values, AIC and BIC of the three fitted models. It is observed that the MVMMNW and MVMMNE distributions respectively outperform the others for the red soil and cotton crop data. Based on the values of the shape matrix Λ, it is clear that the estimated skewness parameters are moderately to highly significant, showing that the distribution of matrix observation is skewed. Moreover, the estimated scale matrices Σ and Ψ highlight the covariance structure in the data.

Table 5

Parameters estimates and the performance summary of three matrix models on the LSD subsets.

Dataset	Parameter	red soil			cotton crop
MVRSN	M	[54.5053.7053.8853.6353.3953.7553.6453.8854.0376.0275.0575.0574.4073.9674.0973.9773.9574.1191.9291.2191.4690.9889.8890.3190.7690.4489.6577.1076.8577.2976.2875.7876.2776.6576.1376.04]			[46.0042.4540.7048.5445.9243.2350.4349.2647.5434.0228.3525.2938.6333.8529.3342.7640.1436.56115.13116.99116.38113.82115.33116.39111.31112.05112.91123.46128.03127.46119.52124.22126.47114.94116.70119.61]
	Λ	[11.0111.6210.9811.5111.4610.7010.7710.3210.2422.9723.9723.0524.4925.1724.0923.9924.4323.8418.8820.0918.9719.9221.2420.2719.7020.6820.9513.4313.8713.0714.4415.1514.2013.9714.5814.13]			[4.288.6211.760.463.437.05-0.900.132.528.5216.0321.881.496.4613.16-2.070.435.53-1.98-3.77-3.370.36-1.19-1.982.681.780.47-7.44-12.73-13.18-1.24-6.06-9.473.271.34-3.06]
	Σ	[10.128.075.323.978.0726.5616.3310.515.3216.3323.7011.043.9710.5111.0412.46]			[17.4324.37-13.00-27.1224.3750.79-25.58-53.50-13.00-25.5844.7953.03-27.12-53.5053.0398.70]
	Ψ	[2.161.360.761.120.800.430.660.460.221.362.031.390.840.880.890.430.450.450.761.392.300.560.811.300.260.460.741.120.840.561.950.990.491.210.900.470.800.880.810.991.510.970.690.770.720.430.891.300.490.972.070.450.721.150.660.430.261.210.690.452.531.440.720.460.450.460.900.770.721.441.861.120.220.450.740.470.721.150.721.121.91]			[2.781.881.312.001.600.971.351.220.971.882.181.621.471.351.120.960.810.791.311.622.201.181.061.110.830.690.642.001.471.182.711.801.082.061.831.321.601.351.061.802.141.451.511.591.540.971.121.111.081.451.961.221.311.531.350.960.832.061.511.223.232.441.811.220.810.691.831.591.312.442.832.190.970.790.641.321.541.531.812.192.76]
MVMMNE	M	[55.3754.6054.7354.5354.2854.5854.4854.6954.8177.5976.7176.6476.0775.7075.7375.5875.6075.7193.1792.5892.7992.3391.3491.7292.1091.8591.0677.9977.8278.2277.2676.8277.2677.6277.1376.99]			[46.8244.0342.8448.4646.3944.3549.9448.9547.6735.6031.3429.3438.6334.7931.4541.7439.6237.01114.94116.50115.98114.24115.51116.47112.39112.94113.59122.24125.79125.21119.77123.59125.29116.45117.85119.93]
	Λ	[8.689.198.699.099.068.478.518.168.1118.3819.1618.4319.5920.1219.2919.2219.5719.1015.1416.0815.1415.9416.9916.1915.7716.5616.7810.7711.0810.4211.5612.1211.3411.1611.6611.32]			[2.264.616.290.411.993.93-0.150.401.684.538.5111.641.063.707.32-0.460.833.51-1.23-2.21-2.02-0.16-1.03-1.500.840.38-0.34-4.12-6.89-7.20-1.14-3.72-5.610.84-0.19-2.51]
	Σ	[10.558.535.664.228.5327.6317.0410.985.6617.0424.6111.504.2210.9811.5012.93]			[18.1325.28-13.55-28.2725.2852.87-26.70-55.89-13.55-26.7047.2955.88-28.27-55.8955.88103.84]
	Ψ	[2.111.340.771.110.800.440.670.470.241.342.001.370.850.890.900.450.470.460.771.372.250.570.821.290.280.480.741.110.850.571.910.990.501.200.900.480.800.890.820.991.490.970.700.770.730.440.901.290.500.972.040.470.721.140.670.450.281.200.700.472.471.410.720.470.470.480.900.770.721.411.821.110.240.460.740.480.731.140.721.111.87]			[2.651.781.241.901.510.911.271.150.901.782.051.511.391.261.020.880.740.711.241.512.041.100.981.000.740.610.561.901.391.102.581.711.021.971.741.251.511.260.981.712.021.351.431.511.440.911.021.001.021.351.821.131.221.421.270.880.741.971.431.133.092.331.721.150.740.611.741.511.222.332.702.080.900.710.561.251.441.421.722.082.61]
MVMMNW	M	[51.5250.5750.9350.5350.3250.8950.7451.0951.2770.1468.8669.1068.1467.5167.9167.9367.7268.0087.1586.0386.5285.9084.4285.0485.7585.1384.2673.7073.2373.8672.6271.9072.5873.0672.3772.40]			[45.0139.4836.2849.3545.1940.7551.8650.0947.1632.0022.7417.1839.8132.3324.5945.6241.5535.42114.99117.62116.95112.69114.92116.43109.29110.53111.96124.78131.89131.66118.19125.29129.24111.68114.63119.67]
	Λ	[13.2113.9313.1613.7913.7312.8112.9112.3912.2727.2128.4627.3729.0029.8328.5628.3228.9328.2622.3023.8422.5723.5825.1924.1023.3124.5224.8515.8816.5115.5717.0817.9516.8816.5717.3116.76]			[4.7710.7415.06-0.573.748.84-2.43-0.862.579.5420.0627.91-0.137.2116.62-5.09-1.295.99-1.51-3.90-3.501.62-0.51-1.714.623.271.51-7.81-15.23-16.000.52-6.36-11.226.573.55-2.64]
	Σ	[10.047.895.173.867.8926.4216.2110.425.1716.2123.6010.973.8610.4210.9712.41]			[17.5524.60-13.16-27.4024.6051.26-25.85-54.03-13.16-25.8544.8053.22-27.40-54.0353.2299.23]
	Ψ	[2.141.340.731.100.770.400.640.430.191.342.001.360.820.850.860.410.420.420.731.362.270.530.781.270.230.440.711.100.820.531.930.970.461.190.880.440.770.850.780.971.480.950.670.740.700.400.861.270.460.952.050.430.691.130.640.410.231.190.670.432.521.420.700.430.420.440.880.740.691.421.831.100.190.420.710.440.701.130.701.101.89]			[2.821.941.392.021.631.021.371.241.001.942.241.691.531.391.161.020.860.831.391.692.281.261.121.160.910.770.712.021.531.262.701.821.132.051.821.341.631.391.121.822.161.481.521.611.561.021.161.161.131.481.991.261.351.561.371.020.912.051.521.263.192.421.811.240.860.771.821.611.352.422.822.201.000.830.711.341.561.561.812.202.78]
		red soil			cotton crop
Model	Criterion →	ℓmax	AIC	BIC	ℓmax	AIC	BIC
RMVSN		-46110.78	92475.55	93000.49	-24169.20	48592.41	49025.69
MVMMNE		-46167.80	92589.60	93114.54	-24137.68	48529.37	48962.65
MVMMNW		-46079.34	92412.68	92937.62	-24183.09	48620.18	49053.46

7 Conclusion

This paper has introduced a new family of matrix variate distributions whose component pdfs arise from the mean-mixture of matrix variate normal model. Some properties and characteristics as well as three special cases of the new model are derived. We have developed a computationally EM-based algorithm for calibrating the matrix type parameters to the data. It is shown that the MVMMN distribution is closed under the formation of marginal and conditional distributions and under affine transformation which make it flexible to use in the various fields of three-variate data analysis, such as multivariate time series, image processing and longitudinal data analysis. Simulation results show that the ML estimates obtained via the ECM algorithm are empirically consistent. Moreover, numerical results from application to real dataset reveal that the proposed model is well suited in dealing with the skewed matrix variate experimental data.

The utility of our current approach can be extended to accommodate censored data based on a recent work studied in the multivariate case by [31, 32]. It may also be interesting to propose a family of scale mixture of MVMMN distribution to deal with heavy tailed three-way data. Another possible extension of the work herein is to consider finite mixture model based on the MVMMN distribution as a promising tools in classification and clustering heterogeneous matrix-valued asymmetric data [19, 33]. It would be of interest the distributions of the associated eigenvalues of the quadratic form (Theorem 8; for the complex form) to compute the channel capacity in wireless communication systems, since experimental data do not follow necessarily a normal distribution (see [34, 35]). All computations were carried out by R language and the computer program is available from the first author upon request.

Appendix A: Comparison of contour plots of the MMN and MVMN families

Fig 2 illustrate the contour plots of the bivariate rSN and bivariate exponentiated MMN (MMNE) distributions as special cases of MMN family as well as the contour plots of the bivariate generalized hyperbolic skew-t (GHST) and bivariate normal inverse Gaussian (NIG) distributions as special cases of MVMN family. O

11 Dec 2019

PONE-D-19-24001

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

PLOS ONE

Dear Dr Naderi,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

We would appreciate receiving your revised manuscript by Jan 25 2020 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter.

To enhance the reproducibility of your results, we recommend that if applicable you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). This letter should be uploaded as separate file and labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. This file should be uploaded as separate file and labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. This file should be uploaded as separate file and labeled 'Manuscript'.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

We look forward to receiving your revised manuscript.

Kind regards,

Daniel Capella Zanotta

Academic Editor

PLOS ONE

Journal Requirements:

When submitting your revision, we need you to address these additional requirements.

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at http://www.journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and http://www.journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2.  Thank you for stating the following in your Competing Interests section:

'No'

Please complete your Competing Interests on the online submission form to state any Competing Interests. If you have no competing interests, please state "The authors have declared that no competing interests exist.", as detailed online in our guide for authors at http://journals.plos.org/plosone/s/submit-now

This information should be included in your cover letter; we will change the online submission form on your behalf.

Please know it is PLOS ONE policy for corresponding authors to declare, on behalf of all authors, all potential competing interests for the purposes of transparency. PLOS defines a competing interest as anything that interferes with, or could reasonably be perceived as interfering with, the full and objective presentation, peer review, editorial decision-making, or publication of research or non-research articles submitted to one of the journals. Competing interests can be financial or non-financial, professional, or personal. Competing interests can arise in relationship to an organization or another person. Please follow this link to our website for more details on competing interests: http://journals.plos.org/plosone/s/competing-interests

Additional Editor Comments (if provided):

Dear author, we are now sending you the results of the reviewing by one referee. The remaing ones never materialized, so we decided, since it pointed minor revision, to proceed with this only one.

Please, improve the theoretical framework of the process putting most of the things in an easier way. Please, add coments clarifying each of the mathematical steps allowing future readers to better undertanding your intentions. As suggested by the reviewer, try to put auxiliar figures and schemes like flowcharts.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The author's research is valuable and significant. In this paper, authors show a strong foundation in mathematics and physics, mathematical logic is rigorous. This research is innovative, however, the theoretical process would be easier to understand if more graphics were used.

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org. Please note that Supporting Information files do not need this step.

Submitted filename: A theoretical framework for Landsat data modeling based on the matrix variate meanmixture of normal model Cai.pdf

Click here for additional data file.

27 Jan 2020

We highly appreciate the reviewer for encouraging words and nice suggestions. The paper has been revised along the lines suggested.

Submitted filename: Response to Reviewers.pdf

Click here for additional data file.

10 Mar 2020

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

PONE-D-19-24001R1

Dear Dr. Naderi,

We are pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it complies with all outstanding technical requirements.

Within one week, you will receive an e-mail containing information on the amendments required prior to publication. When all required modifications have been addressed, you will receive a formal acceptance letter and your manuscript will proceed to our production department and be scheduled for publication.

Shortly after the formal acceptance letter is sent, an invoice for payment will follow. To ensure an efficient production and billing process, please log into Editorial Manager at https://www.editorialmanager.com/pone/, click the "Update My Information" link at the top of the page, and update your user information. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, you must inform our press team as soon as possible and no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

With kind regards,

Daniel Capella Zanotta

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Dear Dr. Naderi,

Given the delay presented by the referee, I checked your corrections by myself and I have found your paper ready for publication.

Please, follow the instructions above.

Reviewers' comments:

Made directly by academic editor.

20 Mar 2020

PONE-D-19-24001R1

A theoretical framework for Landsat data modeling based on the matrix variate mean-mixture of normal model

Dear Dr. Naderi:

I am pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please notify them about your upcoming paper at this point, to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

For any other questions or concerns, please email plosone@plos.org.

Thank you for submitting your work to PLOS ONE.

With kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Daniel Capella Zanotta

Academic Editor

PLOS ONE