Sparse multi-way DMDC for longitudinal classification in high dimension low sample size data

Abstract

Recent advances in data registration techniques have enabled researchers to analyze high-dimensional, low sample size (HDLSS) data, which poses challenges for traditional statistical and machine learning methods—especially in fields like neurodevelopmental disorder classification. Among various proposed solutions, the Data Maximum Dispersion Classifier (DMDC) has demonstrated superior performance. This study introduces two extensions: Sparse DMDC (SDMDC) and Longitudinal Sparse DMDC (LSDMDC), with the latter leveraging multi-way data frameworks. Simulated HDLSS datasets were generated from multivariate normal distributions with sparse structures. Evaluation of SDMDC involved varying class imbalance, predictor count, and correlation structures. Performance was measured via average F-measure and signal-to-noise ratio (SNR). Results showed that SDMDC outperformed other sparse models. Moreover, promising improvement was observed in model classification performance and the selection of effective predictors when the correlation between predictor variables was increased. In longitudinal simulations, LSDMDC maintained performance even as the number of predictors increased along the first dimension, effectively handling rank-1 and rank-2 data. Notably, LSDMDC achieved the best results under autoregressive correlation. In conclusion, SDMDC demonstrated clear advantages over other sparse classifiers, while LSDMDC proved highly effective for complex longitudinal structures.

Introduction

Classification is a powerful tool that can be utilized to provide a solution for a wide variety of problems. It plays a major role in statistics and machine learning and applied in a broad knowledge-based areas such as disease diagnosis, image segregation, audio and speech processing and bioinformatics [1]. In general, machine learning algorithms that works well with statistical methods such as Naïve Bayes Classifier [2], Logistic Regression [3], Neural Networks [4] and Random Forest [5] perform a reasonably acceptable job when there are no special requirements either in data or in algorithm itself. However, when we are dealing with a high dimensional data for which there are not many samples available, these classical algorithms fail to provide a feasible estimation. This special case where the number of samples are negligible compared to the number of dimensions d, i.e. n <<d, is called High-Dimensional Low-Sample-Size in the literature or abbreviated as HDLSS [1]. fMRI data [6] or gene expression [7] are good examples of when there are few samples with high dimension available. While feature selection methods can reduce dimensionality during preprocessing, they can be time-consuming. Therefore, preference is often given to algorithms that can efficiently handle high-dimensional data without requiring explicit dimensionality reduction [8]. Certain classifiers exist for this specific task of HDLSS model training, such as Support Vector Machine (SVM) [9], distance weighted discrimination (DWD) [10] and DMDC [8].

Recent advancements in high-dimensional data classification have introduced novel frameworks to improve accuracy and scalability. Mo and Chen (2023) proposed a classification framework leveraging pairwise distance ranks to identify patterns overlooked by previous methods, demonstrating superior performance across diverse datasets [11]. Zhou et al. (2024) developed a multiobjective genetic programming approach (PS-MOGP) that integrates recursive feature elimination and a progressive Pareto archive strategy to address solution diversity, multiclass imbalance, and large feature spaces. These approaches contribute significantly to tackling the challenges of high-dimensional classification [12].

In 2020 Shen and Yin having simulated a scenario with high dimensional data proved that DWD and SVM face overfitting and so called data-piling problem. In the same study they introduced a new method called Data Maximum Dispersion Classifier (DMDC) which, by maximizing the dispersion of data in the projection space and the gap between classes, overcomes the aforementioned issues of data-piling. For a data set represented by matrix with n data vectors of and predictors and a response variable with values belonging to , DMDC model is defined as:

Here, is the training data average in jth class and is the eigenvector corresponding to the largest eigenvalue of matrix. The proposed model improves training time compared to that of DWD [8].

For computational convenience, the original DMDC formulation can be equivalently expressed in an unconstrained hinge-loss form. Specifically, the slack variables can be incorporated into the objective through the hinge-loss representation, leading to an unconstrained optimization problem of the form:

where is the eigenvector corresponding to the largest eigenvalue of ​, and the first term corresponds to the hinge-loss penalization of misclassified points. This form lays the groundwork for introducing sparsity in the next step.

Classically, DWD, SVM, DMDC and all other applicable methods make use of all variables in the training procedure. Practically however, there may only be few variables playing a decisive role in determining the outcome of the model. Having all variables participated in the training phase would unequivocally introduce noise into the system, unimportant parameters can inject more uncertainty and thus reduce performance [13]. This, on its own, suggests the possibility of omitting a handful of variables and applying a form of feature selection on the model. Committing to this idea will produce a sparse representation of the model while providing a more meaningful interpretation [14].

Zhu et. al in 2004 [15] introduced L1-Norm SVM as an approach to come up with sparse SVM, although many other versions of sparsity in SVM have introduced based on different penalty terms by zhang [16] and wang [17]. Following the work of Zhu et. al, in 2016 Zhang and Lin applied L1-Norm and elastic-net on DWD and presented the first sparse DWD model [18].

In numerous fields of research, often we require a more complex structure than the classic two-way data matrix (samples x variables). Data structure in these cases are best represented in the form of a multi-way array or as is called a tensor. A clarifying example can be fMRI data taken from different regions of brain over different points of time. To present this, a multi-way structure is best suited and defined in the form of samples x regions x time-points. A comprehensive research on multi-way data classifiers was proposed by LYU et. al in 2017 [19] where a discriminant framework is mathematically introduced. Coefficients specific to each way were computed by taking into account the effects of the correlation of predictors in each way. The core of the framework mainly consists of DWD that utilizes rank-1 and rank-r versions of multi-way classification. Sparsity was later on added to framework core by Guo et. al [20] in 2023 and extend the model functionality to work with two-way data and beyond.

In this study, we are tackling data-piling problem and, with regards to the undeniable advantage of introducing sparsity into the model, as heavily studied by aforementioned researchers, proposing a new method that combines sparsity with DMDC to be applied in HDLSS environments.

This study focuses on data types where the second way concerns different time-points which facilitates longitudinal classification. Framework core is based on sparse DMDC which all in all, makes it feasible to handle HDLSS data using Longitudinal Sparse DMDC.

Methods

Sparse DMDC

The data maximum dispersion classifier is an advantageous model for high-dimensional low-sample-size datasets that can perform better than the conventional models by overcoming the problem of data-piling. Similar to the standard SVM and DWD models, which use L2-norm penalty (ridge penalty) on their optimization function, DMDC may also exhibit suboptimal performance for two reasons. First, based on scientific assumptions, in a high-dimensional data environment only a limited number of variables can affect the overall model performance, and a good classifier should be able to identify effective variables and discard irrelevant ones; whereas the standard SVM, DWD and DMDC make use of all the variables in the model building process. Second, having all the variables enlisted and participated in creating such a model may result in classification inefficiency due to the accumulation of errors in the estimation of a large number of noisy parameters. Based on these two considerations, and to compensate for the overall decline in the performance, sparse models are usually utilized for high-dimensional data classification task. The adoption of penalty functions to L1-norm (LASSO penalty), in classification models for sparse modeling, constitutes a prevalent approach in this context. Replacing with in the optimization function of classification models results in the automatic selection of relevant variables. This is due to the fact that the LASSO penalty corresponds to a double-exponential distribution for coefficients, while the ridge penalty corresponds to a Gaussian distribution for obtaining coefficients. As the exponential distribution has heavier tails than the Normal distribution, it tends to estimate larger coefficients for important variables and consequently leave others at zero. In these sparse models, coefficients of less important variables will be exactly zero. Thus, the lasso penalty does a kind of continuous feature selection, while this is not the case for the ridge penalty. To obtain the sparse version of DMDC we reformulated the DMDC model then replaced with in DMDC minimization problem as below:

1

Directly minimizing the fractional regularization term in Eq. (1) leads to a non-convex optimization problem, which is computationally demanding and sensitive to local minima. Under the constraint , the dispersion term in the denominator remains strictly positive and bounded away from zero; thus, a larger consistently corresponds to stronger alignment with the dispersion direction. Instead of optimizing the fractional form directly, we therefore adopt an approximation in the form of a difference type surrogate that penalizes sparsity and rewards dispersion: . This surrogate preserves the qualitative trade-off of the original ratio—encouraging small while favouring large but yields a convex objective that can be solved efficiently.

Substituting the positive part function with the max function holds:

By transferring the term into the max function, we will get:

And thus:

Following the discussion in [21], optimization function can be formulated into:

Finally, by equating the inequalities in the conditions of the optimization problem, the sparse DMDC model becomes apparent in the following form:

That is a standard linear form of constrained optimization problems, which is often written as:

where is the objective function to be minimized on the n-variable vector , is the constraint coefficient matrix and is a vector of values on the right hand sides of the constraints. The expanded form of inequality can be viewed as follows:

where

Considering a dataset with n samples and p predictors, the constraint coefficient matrix in will contain 2n+3p+1 rows and n+2p+1 columns. This matrix can become very large for a high-dimensional dataset. For instance, with n = 200 and p = 5000, matrix M would encompass with 15,401 rows and 10,201 columns. Therefore, provisioning of any form of solution to the optimization problems such as the one stated in cannot be achieved through the usual simplex method [22] which makes it imperative to employ more sophisticated algorithms such as the interior point [23] for the task of linear programming.

Unlike simplex method, which, in order to reach the optimal point, moves along the edges of a multifaceted space constructed from the constraints to the vertex where the value of the cost function is lower. In the interior point method, the iterations of the algorithm start inside the multifaceted space and regardless of the edges move towards the lowest cost vertex. This approach reduces the number of iterations required to reach that vertex and thus speeds up the computation time deemed necessary to produce a viable solution. In the realm of linear programming optimization the interior point method removes the constraints from an optimization problem and reaches the optimal point using Newton's numerical calculation method (usually in 20 to 100 iterations). This is achieved by obtaining the standard or focal form of the linear problem, changing the form of the initial and dual optimization function and adding the logarithmic Brier function and applying Lagrange in the interior point method.

In this research, the commercial MOSEK (version 10.0.27) solver with an academic license was employed to optimize the sparse DMDC model, and the optimal weights of the coefficients were estimated using the interior point method. After estimating the optimal coefficients, to classify the new observation x, is defined by:

Then the new observation is assigned to class 1 given , or class −1 otherwise.

Longitudinal sparse DMDC; rank-1

Multi-way model, as an extension of linear classification models, is a framework presented for high-dimensional multi-way data. Considering time as a dimension (way), an important application of these models is the classification of longitudinal data with a large number of predictors. In these models, it is assumed that the matrix of coefficients can be decomposed into patterns specific to each dimension. Multi-way structure improves the interpretation of the coefficients and hence, outperforms the standard linear classification models that consider the predictor variables for each sample as a vector and not a matrix.

Suppose the classification of samples with array data . If is the measurement of the jth predictor from the first dimension and the kth predictor from the second dimension for the ith subject, where , and . For each subject i, these measurements are naturally arranged in a matrix

whose th entry is .

In standard linear classification models, the matrix is vectorized as

and the classifier is formulated based on this vector representation. Accordingly, standard linear classifiers estimate coefficients for each pair of predictors corresponding to dim1 and dim2 . The separating hyperplane for the i th subject is given by

where is the th element of the coefficient matrix .

Standard linear classifiers estimate different coefficients for the same jth predictor from the first dimension under different characteristics in the second dimension. For example, if the first dimension deals with different brain regions and the second dimension handles time, for a specific region in the brain at two different time points, two coefficients are estimated independently regardless of the correlation between these two predictors. These models do not contemplate the correlation amongst variables in each dimension and this issue in turn, can reduce the classification performance. On the other hand, multi-way ranking models perform the exact opposite by taking these correlations into account. In the simplest case, the matrix of coefficients in multi-way models can be presented in the following decomposed form:

where and represent the vector of coefficients of the first and second dimensions respectively. With this assumption, the separating hyperplane in multi-way classification models for the ith sample is as follows:

where, denotes the jth predictor coefficient in the first dimension and , represents the kth predictor coefficient in the second dimension. Because the coefficients have been estimated specific to each dimension, larger values in each dimension can be considered as the greater influence of its corresponding predictor in classification. While in standard classification models, one cannot interpret a coefficient of one dimension without considering another dimension.

In the present study, multi-way classification structure [19] was utilized to classify longitudinal data. Also, the sparse data maximum dispersion classifier is implemented as the core of proposed methodology. Steps to estimate the coefficients of the data classification model with the rank-1 SDMDC using the alternating convex search (ACS) algorithm is depicted in Algorithm 1:

Algorithm 1 Coefficients Estimation Steps for Rank-1 SDMDC

Longitudinal sparse DMDC; rank-r

To compute the matrix of coefficients in the longitudinal sparse DMDC, rank-1 is assumed for decomposition. However, in reality the rank one structure may not be able to express all the information of the coefficient matrix. For instance, if in multi-way data, the first dimension deals with different brain regions and time is considered for second dimension, rank-1 produces only one vector g of coefficients reflecting only that specific region, while the influence of other regions which are inflected over time may be ignored in the resulting matrix. Realistically, there may be more than one vector of coefficients, such as , decisively participated in the determination of the final effect on a certain brain region over a course of time. In such cases, the r-rank model for the coefficient matrix can be considered as a compromise between the rank-1 and a standard linear ranking model. In multi-way with rank of r, the matrix of coefficients is considered as follows:

Steps to estimate the coefficients of the data classification model with the rank-r SDMDC using the alternating convex search (ACS) algorithm is as follows:

Algorithm 2 Coefficients Estimation Steps for Rank-r SDMDC

Simulation

Two separate simulation studies for each of the presented models, namely SDMDC and longitudinal SDMDC, were performed, details of which are descriptively presented in this section.

Sparse DMDC

Simulation scenarios for sparse DMDC

Data for the predictor variables in two categories were randomly generated from the multivariate Normal distribution such that a p-dimensional vector of elements equal to one. In order to create sparse data in , the first 10 elements were considered non-zero and the rest equal to zero. The total number of samples of two classes in the training data was considered to be 200. The scenarios were defined based on the combination of different states of the number of predictor variables, the balance state of response variable categories and the structure of the correlation matrix of predictor variables Σ.

Number of predictor variables is initially set to p = 100 and then increased to 1000, 5000 and 10,000, respectively. The samples were then divided into two categories, −1 and 1, in three different states: balanced 0.5, unbalanced 0.7 and severely unbalanced 0.9. We also considered three different structures for Σ including:

• Independent: ( is an p× p identity matrix).

• First-Order Autoregressive: where

  

• Exchangeable:  where  

For each of the 36 above-mentioned scenarios, 100 different random data sets with a sample size of 200 were generated. Then SVM, DWD and DMDC models and their respective sparse versions (SSVM, SDWD, SDMDC) were fitted on each data set. The optimal parameter of each model was obtained separately using a tenfold cross-validation. Finally, the classification performance of each model was evaluated against 500 test data in two classes based on average f-measure and signal to noise ratio. The f-measure formula is provided in Table 1.

Table 1.

Classification evaluation metrics

Evaluation Metrics	Formula
Recall+ (recall for class 1)
Recall− (recall for class −1)
Precision+ (precision for class 1)
Precision− (precision for class −1)
F-measure+ (f-measure for class 1)
F-measure− (f-measure for class −1)
Average of F-measure

To evaluate sparse models in simulated data, two indices—signal and noise—were used. The signal index represents the number of nonzero coefficients estimated by the model for predictor variables whose mean was considered nonzero in the simulation. The noise index refers to the number of nonzero coefficients estimated by the model for predictor variables whose mean was considered zero in the simulation. To combine these indices, the signal-to-noise ratio (SNR) was used. A higher SNR (greater than one) indicates a better ability of the sparse model to select important predictors.

Simulation result: SDMDC in comparison with SDWD and SSVM

Detailed description and numerical evaluation results can be accessed in Appendix. To easily compare the results line-charts were preferred.

As seen from Fig. 1, SDMDC does not appear promising with the number of predictors fixed at 100, as it falls below SDWD in all different balance states of the response variable. The key point is with the sample size of 200, the data does not satisfy the HDLSS criteria and hence, SDMDC seems to be not a suitable candidate for data classification. However, as the number of predictors grows, SDWD is no longer able to estimate the model coefficients and assigns all samples to class −1.

Fig. 1.

The average F-score for SSVM, SDWD, SDMDC models by the number of predictor variables, the balance status of response variable categories, and the correlation structure of predictor variables

A more noticeable observation to be made is the superiority of SDMDC when it comes to autoregressive and exchangeable categories in all balance levels.

In general, when number of predictors fall in the range of [100, 1000], DMDC and DWD outperform SVM. As the correlation among predictors arises, classification performance of these two models plunges arguably. When the sample size exceeds 5000, particularly under imbalanced scenarios, DMDC also fails to perform proper classification, and overall DWD achieves the best average F-score (Fig. 2).

Fig. 2.

The average of F-score for SVM, DWD, DMDC by the number of predictor variables, the balance status of response classes, and the structure of correlation amongst predictors

As seen in Fig. 3, sparse methods (SDMDC, SDWD, SSVM) tend to maintain higher F-scores under high imbalance levels (0.7 and 0.9), while non-sparse methods (DMDC, DWD, SVM) drop more noticeably. For the balanced case (imbalance 0.5), both sparse and non-sparse methods perform well, although sparse methods sometimes show slightly lower F-scores than DWD or SVM, reflecting a minor bias introduced by feature selection when imbalance is low.

Fig. 3.

The average of F-score for SVM, DWD, DMDC by the number of predictor variables, the balance status of response classes, and the structure of correlation amongst predictors

Comparing the three investigated sparse models, SDMDC proved to have a higher signal-to-noise ratio in HDLSS conditions. The performance of the SDMDC was the same in different balance states of the response variable. In all scenarios, increasing the correlation between the predictors resulted in a better signal-to-noise ratio (Fig. 4).

Fig. 4.

The signal-to-noise ratio for SSVM, SDWD, SDMDC by the number of predictors, the balance status of response classes, and the correlation structure of predictors

The runtime analysis reveals distinct patterns among sparse and non-sparse models as the number of predictors increases. For high-dimensional settings (over 100 predictors), SDWD struggles to estimate model coefficients correctly, assigning all samples to class −1, which affects its runtime interpretation. In contrast, SDMDC, the proposed model, remains stable and scalable, showing competitive efficiency against SDWD and SSVM. While SSVM is the fastest sparse model for lower dimensions, its runtime grows significantly for large predictor sizes. Compared to non-sparse models, SDMDC maintains comparable performance to DMDC and SVM but is more efficient than DWD, which experiences drastic runtime increases. This suggests that SDMDC balances computational feasibility and predictive reliability in high-dimensional settings. The average model runtimes based on the number of predictors are provided in Appendix Tables 9 and 10.

Longitudinal SDMDC

Simulation scenarios for sparse DMDC

The simulation data of multi-way predictors for two classes were randomly generated from the multivariate Normal distribution and then converted into a multi-way array with two dimensions, such that the first and second dimensions include p and m time-related variables.

To construct the vector in rank-1, vector h and g were randomly generated from the multivariate Normal distributions and correspondingly. was then computed and then transformed into a p× m vector. Finally, was calculated using where and are Mahalanobis distances of and vectors. This setting corresponds to a difficulty level for classification using the naïve Bayes model and in accordance with the literature at [24, 25]. Also, is defined as .

To construct the vector in the data with rank-2, each of the two rows of the random matrix was generated from the multivariate Normal distribution . Also, each of the two rows of the random matrix was generated from the multivariate Normal distribution . The value of was then obtained with and then converted into a p× m vector. Similar to what was discussed, μ was considered equal to where . Calculation of Σ is exactly the same as the previous paragraph.

To create sparsity in , 33% of predictors in the first dimension were considered non-zero. The predictors of the second dimension were considered non-sparse. 100 training samples used in total for two classes and the response variable was balanced at 50 percent for each category. Different scenarios were defined based on the combination of different states of the number of predictor variables, rank of the data matrix and the structure of the correlation matrix of the predictor variables () in the first dimension.

Considering first dimension, 200 and 500 predictors were used for simulation purposes. Also the three independent, first-order autoregressive and exchangeable structures were applied for . This setting however differs for the second dimension as only 5 time-points were introduced and first-order autoregressive was selected among the three for structure.

To avoid ambiguity in terminology, we note that the rank-1, rank-2, and more generally rank-r models considered in the simulations belong to the class of multi-way DMDC models, in which a low-rank structure is imposed on the coefficient matrix to exploit the multi-way nature of the predictors. In contrast, the full DMDC refers to the full-rank formulation, where no rank constraint is imposed on the coefficient matrix and its rank is equal to . This full-rank setting is equivalent to the standard linear classification model in which all coefficients are estimated independently. Regarding sparsity, sparse versions of the models are indicated by adding the prefix S to the corresponding method name. For example, SDMDC denotes the sparse version of DMDC, Accordingly, a rank-1 DMDC represents a multi-way model with rank 1 whose core classifier is the non-sparse DMDC, while its sparse counterpart is denoted by rank-1 SDMDC.

For each of the 12 mentioned scenarios, full-rank SDMDC (standard linear classification model), rank-1 and rank-2 were fitted on each of the 100 different randomly generated datasets. Also, the non-sparse versions of these three models were also used for comparison purposes. The optimal parameter of each model was obtained separately using a tenfold cross-validation and finally, the classification performance of the models was evaluated against 500 test data.

Model development and simulation were coded in python (version 3.10.1). DWD and SDWD simulations were scripted in R using packages DWDLargeR (version 0.1.0) and sdwd (1.0.5).

Simulation result: longitudinal SDMDC performance

Figure 5 proves that as the data rank gets higher, the corresponding average F-score of the same rank in the model outperform the others with a clear margin.

Fig. 5.

The average of F-score for rank-1 SDMDC, rank-2 SDMDC and full SDMDC by the number of predictors, data structure, and the structure of correlation amongst predictors. Rank-1 and Rank-2 on the horizontal axis refer to the data structure in the simulation, while Rank-1 and Rank-2 in the legends refer to the modeling structure

In Fig. 6 all three non-sparse models under investigation produce almost same results in all data and correlation structures. Having correlation increased among predictors, average F-score generally plummets accordingly whereas in sparse models the figure stayed almost steady.

Fig. 6.

The average of F-score for rank-1 DMDC, rank-2 DMDC and full DMDC by the number of predictors, data structure, and the structure of correlation amongst predictors. Rank-1 and Rank-2 on the horizontal axis refer to the data structure in the simulation, while Rank-1 and Rank-2 in the legends refer to the modeling structure

An obvious soar in signal-to-noise ratio of all sparse longitudinal models is observable as the correlation among predictors arises, as Fig. 7 suggests. It can also be viewed that with exchangeable structure selected, the ratio jumps evidently compared to the other cases. All in all, rank-1 and rank-2 SDMDC accomplishes a far better variable selection in comparison with full SDMDC.

Fig. 7.

The signal-to-noise ratio for rank-1 SDMDC, rank-2 SDMDC and full SDMDC by the number of predictors, data structure, and the structure of correlation amongst predictors. Rank-1 and Rank-2 on the horizontal axis refer to the data structure in the simulation, while Rank-1 and Rank-2 in the legends refer to the modeling structure

Tables 11 and 12 (in Appendix) compare the runtime performance of the proposed longitudinal models (full SDMDC, Rank-1 SDMDC, and Rank-2 SDMDC) with their DMDC counterparts. For SDMDC variants, the Rank-1 model is notably faster—completing in about 47 s at 500 predictors—compared to the full and Rank-2 versions, which take roughly 96 and 57 s, respectively. In contrast, while the full DMDC model remains efficient, its Rank-1 and Rank-2 variants are significantly slower, with runtimes exceeding 900 and 1200 s at 500 predictors. This clearly demonstrates the computational advantages of the SDMDC approaches in handling high-dimensional longitudinal data.

Real data analysis

The proposed methods, SDMDC and LSDMDC, were applied to baseline and longitudinal real data obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database and compared to corresponding versions of SVM and DWD.

Data source

Data for this study were obtained from the ADNI database, which was initiated in 2003 as a public–private partnership led by Principal Investigator Michael W. Weiner, MD. The ADNI aims to combine serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessments to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). For more information, see adni.loni.usc.edu.

Considering the data requirements for this study, all individuals with longitudinal fMRI images from at least three consecutive time points were selected as samples in two groups: healthy and AD (Alzheimer's Disease). Accordingly, a total of 153 individuals were included, comprising 66 AD individuals (43.2%) and 87 healthy individuals (56.8%), each having fMRI images taken at three time points (at diagnosis, three months after diagnosis, and six months after diagnosis). The data were downloaded in August 2022.

Data preprocessing

After downloading the research data, preprocessing of fMRI images was conducted using the CONN toolbox in MATLAB. This process involved multiple steps. First, skull-stripping was performed to remove non-brain tissue from the MRI images. Motion correction and slice-timing correction were then applied, with the interleaved acquisition method used for the latter. Spatial smoothing with a full-width at half maximum (FWHM) of 8 mm was conducted to improve signal quality. Group registration and normalization were performed to align individual brain images to a standard template. Finally, the brain was segmented into 132 regions using the Automated Anatomical Labeling (AAL) atlas. These preprocessing steps ensured that the data were standardized and suitable for further analysis.

Analytical steps

Evaluation framework

Due to the limited number of observations (87 healthy and 66 Alzheimer’s patients), a fivefold cross-validation strategy was adopted. Observations were evenly distributed across folds to maintain the same response variable distribution. In each fold, four subsets were used for training the models, and one subset was reserved for testing.

Correlation matrix calculation

Correlation matrices (132 × 132) were computed for each individual, representing the connectivity between brain regions. Each matrix element reflected the correlation between the activities of two brain regions.

Feature engineering for classification models

Upper triangular elements of the correlation matrices (excluding the diagonal) were extracted, resulting in 8,646 features per individual. This feature set was augmented with demographic variables (gender, age, marital status, handedness, education, retirement status, and race) and clinical statuses. To standardize the data, Min–Max scaling was applied, normalizing all features to the range [−1, 1].

Parameter optimization

Using the training data, penalty parameters for each model were optimized via a tenfold cross-validation within the training set. For each iteration, independent t-tests were performed on values for healthy and Alzheimer’s groups, and the parameter maximizing the t-statistic was selected.

For the SDMDC and SSVM models implemented in Python, the sparse penalty parameter λ was selected from a linear grid . Larger values of λ were not considered, as they resulted in an insufficient number of non-zero coefficients. For the corresponding multi-way sparse models, including rank-1 and rank-2 SSVM and SDMDC, a wider linear grid was used to account for the increased flexibility of the multi-way structure.

For the SDWD model implemented in R using the package, the default-tuning scheme was adopted, with the lambda sequence automatically generated using and set according to the package defaults. For the multi-way SDWD models (rank-1 and rank-2), the predefined penalty values were used. In all cases, the optimal parameter was selected using cross-validation.

Model fitting

The SDMDC, SSVM, and SDWD models were trained using the optimized parameters on baseline data (time = 1). Additionally, the rank-1 and rank-2 variants of these models were trained with the optimized parameters on longitudinal data spanning multiple time points (time = 1, 2, 3). Each model underwent evaluation five times based on the initial fivefold cross-validation split.

Model testing

Testing data were classified using the trained models. Predictions were compared with actual classifications to calculate evaluation metrics, including the mean F-score and the number of non-zero coefficients. Final metrics were averaged across the five folds to provide robust performance measures. To assess whether the observed differences in model performance were statistically significant, we employed a bootstrap-based pairwise comparison. For each pair of models, the F-scores average obtained across cross-validation folds were resampled with replacement for a large number of bootstrap iterations. At each iteration, the mean difference in F-score average between the two models was computed, and a 95% confidence interval was constructed from the empirical distribution of these differences. Performance differences were considered statistically significant if the confidence interval did not include zero.

Results

The study evaluates the performance of sparse models for classifying Alzheimer's patients and healthy controls, focusing on both baseline and longitudinal data. The proposed Sparse DMDC (SDMDC) and its longitudinal variant (LSDMDC) are compared with Sparse SVM (SSVM) and Sparse DWD (SDWD) and their longitudinal variants. Detailed results across folds for each model are provided in the Appendix.

Results of baseline analysis

In baseline data classification, according to Table 2, SDMDC outperformed competing models, achieving the highest average F-measure (0.51). This indicates its superior ability to balance precision and recall compared to SSVM and SDWD, which yielded average F-measures of 0.47 and 0.38, respectively. SDMDC also demonstrated effective sparsity with an average of 17 non-zero coefficients, making it more interpretable than SSVM with 39.80 coefficients. Despite its strong performance, SDWD lagged in F-measure and required higher modeling time, underscoring its trade-off between sparsity and computational cost. The lower standard deviation of SDMDC in Table 2 suggests comparatively more stable performance across folds.

Table 2.

Performance of sparse models in fivefold cross-validation

Model	F-measure average	Non-zero coefficients	Modeling time (s)	SDWD	SDMDC
SSVM	0.47 (0.21*)	39.80 (3.03)	0.05 (0.01)	(−0.12,0.30)	(−0.36,0.11)
SDWD	0.38 (0.24)	14.40 (5.86)	5.37 (1.04)		(−0.07,−0.005**)
SDMDC	0.51 (0.14)	17.00 (14.09)	2.27 (0.05)

^*Standard deviation

^**Intervals excluding zero, indicating significant differences

As shown in the last two columns of Table 2, the bootstrap comparisons indicate that the performance difference between SDMDC and SDWD is statistically significant, since the 95% confidence interval (− 0.07, − 0.005) does not include zero, favoring SDMDC. In contrast, the confidence intervals for SSVM vs. SDWD (− 0.12, 0.30) and SSVM vs. SDMDC (− 0.36, 0.11) include zero, indicating no statistically significant differences in those comparisons.

Results of longitudinal analysis

In longitudinal data classification, as shown in Table 3, both Rank-1 and Rank-2 SDMDC achieved the highest average F-measure (0.55). These models maintained sparsity, with Rank-2 showing an impressive reduction in non-zero coefficients (33) while preserving classification performance. SDWD, while competitive in terms of F-measure, required substantially more non-zero coefficients, particularly in Rank-1 (813.4) and Rank-2 (992.6), suggesting a higher level of complexity. SSVM demonstrated lower performance overall, with F-measures of 0.27 and 0.39 for Rank-1 and Rank-2, respectively. The box plots in Fig. 8 for longitudinal models further underscores SDMDC's stability across folds, with a narrower range of F-measure values compared to SSVM and SDWD.

Table 3.

Performance of rank-1 and rank-2 sparse models in fivefold cross-validation

Model	F-measure average	Non-zero coefficients	Modeling time (s)
Rank-1 SSVM	0.27 (0.12*)	40.2 (1.3)	1.53 (0.14)
Rank-2 SSVM	0.39 (0.28)	38.6 (2.19)	38.6 (0.76)
Rank-1 SDWD	0.55 (0.09)	813.4 (647.59)	255.42 (51.24)
Rank-2 SDWD	0.47 (0.28)	992.6 (833.46)	302.6 (66.3)
Rank-1 SDMDC	0.55 (0.08)	60.2 (0.49)	44.84 (37.63)
Rank-2 SDMDC	0.55 (0,16)	33 (0.28)	75.46 (47.82)

^*Standard deviation

Fig. 8.

Comparison of F-measure Averages across Models on Longitudinal Data

Table 4 presents the 95% bootstrap confidence intervals for pairwise differences in F-measure. Intervals that do not include zero indicate statistically significant differences. The results show that both Rank-1 and Rank-2 SDMDC, as well as Rank-1 SDWD, achieve significantly higher F-measure average compared to Rank-1 SSVM, since the corresponding confidence intervals are entirely below zero (A–B < 0). In contrast, most other pairwise comparisons include zero within their intervals, suggesting that those performance differences are not statistically significant at the 95% confidence level.

Table 4.

95% Bootstrap confidence intervals for pairwise differences in F-measure between models

Model	Rank-2 SSVM	Rank-1 SDMDC	Rank-2 SDMDC	Rank-1 SDWD	Rank-2 SDWD
Rank-1 SSVM	(−0.36,0.09)	(−0.46,−0.21)*	(−0.34,−0.16)*	(−0.39,−0.11)*	(−0.42,0.15)
Rank-2 SSVM		(−0.41,0.06)	(−0.33,0.08)	(−0.34,0.16)	(−0.38,0.28)
Rank-1 SDMDC			(−0.1,0.24)	(−0.05,0.15)	(−0.01,0.44)
Rank-2 SDMDC				(−0.14,0.12)	(−0.19,0.44)
Rank-1 SDWD					(−0.13,0.36)

^*Intervals excluding zero, indicating significant differences

When comparing baseline and longitudinal models, the longitudinal analyses exhibited notable improvements in classification performance. The longitudinal SDMDC demonstrated enhanced stability and accuracy over the baseline SDMDC while maintaining sparsity. These results affirm the adaptability and robustness of the proposed model, particularly in handling the complexities of longitudinal data.

We incorporated a direct comparison with the original DMDC framework to explicitly evaluate the impact of introducing sparsity and longitudinal structure (Table 5).

Table 5.

Performance of DMDC models in fivefold cross-validation

Model	F-measure Average	Modeling Time (s)	SDMDC	Rank-1 SDMDC	Rank-2 SDMDC
DMDC	0.38 (0.2*)	2.83 (0.24)	(−0.34,0.12)	(−0.33,0.09)	(−0.35,0.02)
Rank-1 DMDC	0.47 (0.24)	5.37 (27.69)	(−0.15,0.06)	(−0.23,0.07)	(−0.32,0.14)
Rank-2 DMDC	0.3 (0.17)	2.27 (17.39)	(−0.3,−0.12)**	(−0.35,−0.14)**	(−0.46,−0.04)**

^*Standard deviation

^**Intervals excluding zero, indicating significant differences

The results indicate that adding structural constraints leads to improved performance. Specifically, the Rank-1 DMDC model (F = 0.47) improves upon the baseline DMDC (F = 0.38), highlighting the benefit of the longitudinal extension.

More importantly, the sparse and longitudinal SDMDC variants consistently achieve higher F-measure average than the original DMDC, with Rank-2 DMDC showing statistically significant differences in favor of all SDMDC variants (confidence intervals excluding zero). These findings suggest that incorporating sparsity enhances feature selection stability in this high-dimensional setting, while the longitudinal formulation better captures temporal progression patterns in ADNI data. Together, these modifications contribute to more robust and improved classification performance compared with the original non-sparse, non-longitudinal formulation.

Discussion

In this article, we have proposed a sparse DMDC data model to be utilized as the core for the framework of classifying multi-way data. Here, we considered time as the second way to reach longitudinal SDMDC to handle classification of HDLSS data. Performance-wise, SDMDC confirmed superior in classifying high dimensional data and in general, proved to be the best choice in selecting effective predictors. Additionally, promising improvement was observed in model classification performance and the selection of effective predictors when the correlation between predictor variables was increased. Moving towards longitudinal model, LSDMDC showed a great deal of robustness against increasing the number of predictors in the first way.

LSDMDC estimated class labels best when it dealt with autoregressive structure and similar to SDMDC, presented higher accuracy in predictor selection when correlation in predictors was likely to increase. Finally yet importantly, this study demonstrated that our sparse models have a propensity towards facilitating and simplifying interpretation.

Since second way contained time, and usually the number of time-points is limited, LSDMDC rank may not go beyond a certain figure.

In the real data analysis, SDMDC demonstrated competitive and generally strong performance in both baseline and longitudinal classification tasks. Bootstrap-based significance analyses indicate that SDMDC performs significantly better than SDWD. In the longitudinal setting, both Rank-1 and Rank-2 SDMDC, as well as Rank-1 SDWD, achieve significantly higher F-measure compared to Rank-1 SSVM in distinguishing Alzheimer's patients from healthy controls.

This study has several limitations. The computational complexity of the simulation and real data analyses was substantial, and the use of alternative optimization tools due to solver and hardware constraints may have affected computational efficiency. In addition, in the real data application—particularly for the ADNI dataset—the sample size was relatively limited, which may restrict the stability and generalizability of the results. Future studies with larger cohorts and additional clinically relevant variables with fewer missing values may further improve model performance.

A possible future direction to extend this work could involve considering other types of penalty instead of l1-Norm such as elastic-net. In addition, one could go beyond binary classes and encompass multi-category classification, which is worthwhile idea for research.

Appendix

In Tables 6 to 21, the numbers in parentheses represent the standard deviation

Table 6.

The average F-score for SDMDC, SDWD, SSVM models by the number of predictor variables, the balance status of response variable categories, and the correlation structure of predictor variables

			Models
No. Predictors	Balance Ratio	Correlation str	SDMDC	SDWD	SSVM
100	0.5	Independent	0.558 (0.125)	0.993 (0.003)	0.848 (0)
		Autoregressive	0.851 (0.019)	0.924 (0.007)	0.788 (0.085)
		Exchangeable	0.858 (0.013)	0.88 (0.01)	0.698 (0.183)
	0.7	Independent	0.558 (0.111)	0.991 (0.005)	0.835 (0.002)
		Autoregressive	0.838 (0.014)	0.893 (0.013)	0.749 (0.133)
		Exchangeable	0.806 (0.016)	0.818 (0.014)	0.696 (0.185)
	0.9	Independent	0.523 (0.071)	0.948 (0.026)	0.678 (0.002)
		Autoregressive	0.786 (0.024)	0.825 (0.04)	0.665 (0.091)
		Exchangeable	0.778 (0.025)	0.734 (0.046)	0.598 (0.135)
1000	0.5	Independent	0.525 (0.076)	0.333 (0)	0.616 (0)
		Autoregressive	0.853 (0.015)	0.333 (0)	0.654 (0)
		Exchangeable	0.836 (0.01)	0.333 (0)	0.607 (0.022)
	0.7	Independent	0.516 (0.067)	0.412 (0)	0.574 (0)
		Autoregressive	0.818 (0.011)	0.412 (0)	0.587 (0.031)
		Exchangeable	0.815 (0.01)	0.412 (0)	0.564 (0.044)
	0.9	Independent	0.503 (0.027)	0.474 (0)	0.49 (0.001)
		Autoregressive	0.821 (0.025)	0.474 (0)	0.511 (0.019)
		Exchangeable	0.805 (0.019)	0.474 (0)	0.496 (0.058)
5000	0.5	Independent	0.503 (0.029)	0.333 (0)	0.576 (0)
		Autoregressive	0.645 (0.165)	0.333 (0)	0.568 (0)
		Exchangeable	0.822 (0.067)	0.333 (0)	0.553 (0.023)
	0.7	Independent	0.502 (0.033)	0.412 (0)	0.546 (0)
		Autoregressive	0.649 (0.154)	0.412 (0)	0.507 (0.001)
		Exchangeable	0.827 (0.073)	0.412 (0)	0.481 (0.004)
	0.9	Independent	0.497 (0.022)	0.474 (0)	0.455 (0)
		Autoregressive	0.585 (0.109)	0.474 (0)	0.454 (0.002)
		Exchangeable	0.796 (0.062)	0.474 (0)	0.456 (0.021)
10000	0.5	Independent	0.503 (0.02)	0.333 (0)	0.542 (0)
		Autoregressive	0.553 (0.107)	0.333 (0)	0.502 (0)
		Exchangeable	0.675 (0.162)	0.333 (0)	0.557 (0.037)
	0.7	Independent	0.503 (0.026)	0.412 (0)	0.522 (0)
		Autoregressive	0.557 (0.119)	0.412 (0)	0.531 (0.001)
		Exchangeable	0.684 (0.157)	0.412 (0)	0.524 (0.027)
	0.9	Independent	0.499 (0.023)	0.474 (0)	0.411 (0)
		Autoregressive	0.525 (0.071)	0.474 (0)	0.414 (0.005)
		Exchangeable	0.647 (0.134)	0.474 (0)	0.421 (0.012)

Table 7.

The average of F-score for DMDC, DWD, SVM by the number of predictor variables, the balance status of response classes, and the structure of correlation amongst predictors

			Models
No. Predictors	Balance Ratio	Correlation str	DMDC	DWD	SVM
100	0.5	Independent	0.985 (0.007)	0.995 (0.002)	0.847 (0.001)
		Autoregressive	0.908 (0.008)	0.904 (0.012)	0.526 (0.348)
		Exchangeable	0.866 (0.008)	0.829 (0.023)	0.706 (0.176)
	0.7	Independent	0.977 (0.011)	0.999 (0.001)	0.837 (0.001)
		Autoregressive	0.89 (0.006)	0.878 (0.014)	0.567 (0.264)
		Exchangeable	0.822 (0.011)	0.799 (0.021)	0.707 (0.175)
	0.9	Independent	0.934 (0.034)	0.988 (0.009)	0.678 (0.002)
		Autoregressive	0.862 (0.018)	0.841 (0.026)	0.603 (0.161)
		Exchangeable	0.806 (0.017)	0.753 (0.031)	0.607 (0.125)
1000	0.5	Independent	0.949 (0.009)	0.991 (0.004)	0.616 (0)
		Autoregressive	0.895 (0.011)	0.893 (0.01)	0.427 (0.323)
		Exchangeable	0.844 (0.011)	0.847 (0.011)	0.585 (0.069)
	0.7	Independent	0.874 (0.038)	0.983 (0.006)	0.574 (0)
		Autoregressive	0.844 (0.012)	0.861 (0.015)	0.63 (0.269)
		Exchangeable	0.817 (0.01)	0.806 (0.015)	0.554 (0.057)
	0.9	Independent	0.527 (0.039)	0.622 (0.046)	0.49 (0.001)
		Autoregressive	0.783 (0.046)	0.561 (0.035)	0.641 (0.176)
		Exchangeable	0.778 (0.028)	0.541 (0.031)	0.487 (0.061)
5000	0.5	Independent	0.849 (0.023)	0.945 (0.01)	0.576 (0)
		Autoregressive	0.812 (0.028)	0.852 (0.015)	0.552 (0.214)
		Exchangeable	0.814 (0.016)	0.812 (0.015)	0.548 (0.033)
	0.7	Independent	0.515 (0.049)	0.912 (0.016)	0.542 (0.023)
		Autoregressive	0.566 (0.074)	0.829 (0.017)	0.542 (0.173)
		Exchangeable	0.678 (0.066)	0.789 (0.016)	0.48 (0.003)
	0.9	Independent	0.474 (0)	0.665 (0.045)	0.451 (0.018)
		Autoregressive	0.474 (0)	0.64 (0.043)	0.548 (0.109)
		Exchangeable	0.475 (0.004)	0.625 (0.042)	0.447 (0.029)
10000	0.5	Independent	0.77 (0.03)	0.902 (0.013)	0.533 (0.026)
		Autoregressive	0.747 (0.033)	0.823 (0.017)	0.51 (0.093)
		Exchangeable	0.75 (0.042)	0.793 (0.016)	0.549 (0.047)
	0.7	Independent	0.421 (0.012)	0.787 (0.027)	0.517 (0.021)
		Autoregressive	0.429 (0.022)	0.768 (0.024)	0.534 (0.107)
		Exchangeable	0.454 (0.045)	0.748 (0.026)	0.519 (0.034)
	0.9	Independent	0.474 (0)	0.487 (0.016)	0.41 (0.004)
		Autoregressive	0.474 (0)	0.485 (0.015)	0.511 (0.057)
		Exchangeable	0.474 (0)	0.488 (0.035)	0.42 (0.013)

Table 8.

The signal-to-noise ratio for SDMDC, SDWD, SSVM by the number of predictors, the balance status of response classes, and the correlation structure of predictors

			Models
No. Predictors	Balance Ratio	Correlation str	SDMDC	SDWD	SSVM
100	0.5	Independent	0.135 (0.253)	10.6 (0.885)	1.034 (0.512)
		Autoregressive	8.933 (1.587)	5.012 (2.812)	0.134 (0.038)
		Exchangeable	11.111 (0)	3.7 (2.452)	0.092 (0.025)
	0.7	Independent	0.189 (0.326)	10.7 (0.625)	1.091 (0.309)
		Autoregressive	8.933 (1.394)	4.16 (2.92)	0.137 (0.035)
		Exchangeable	11.111 (0)	3.701 (2.717)	0.095 (0.026)
	0.9	Independent	0.18 (0.317)	9.634 (1.16)	1.404 (0.738)
		Autoregressive	8.491 (1.219)	3.388 (2.538)	0.166 (0.052)
		Exchangeable	11.089 (0.156)	2.331 (2.154)	0.105 (0.035)
1000	0.5	Independent	0.017 (0.056)	0 (0)	0.333 (0.074)
		Autoregressive	8.4 (1.754)	0 (0)	0.07 (0.016)
		Exchangeable	11.111 (0)	0 (0)	0.049 (0.013)
	0.7	Independent	0.013 (0.048)	0 (0)	0.364 (0.065)
		Autoregressive	8.767 (1.385)	0 (0)	0.074 (0.018)
		Exchangeable	11.111 (0)	0 (0)	0.05 (0.014)
	0.9	Independent	0.01 (0.041)	0 (0)	0.447 (0.119)
		Autoregressive	7.844 (1.913)	0 (0)	0.088 (0.027)
		Exchangeable	10.967 (0.376)	0 (0)	0.056 (0.018)
5000	0.5	Independent	0.024 (0.092)	0 (0)	0.218 (0.037)
		Autoregressive	2.827 (3.802)	0 (0)	0.058 (0.012)
		Exchangeable	10.646 (2.18)	0 (0)	0.04 (0.01)
	0.7	Independent	0.02 (0.088)	0 (0)	0.241 (0.041)
		Autoregressive	2.795 (3.75)	0 (0)	0.058 (0.014)
		Exchangeable	10.647 (2.178)	0 (0)	0.041 (0.012)
	0.9	Independent	0.016 (0.064)	0 (0)	0.29 (0.05)
		Autoregressive	2.265 (3.207)	0 (0)	0.071 (0.018)
		Exchangeable	9.689 (3.045)	0 (0)	0.047 (0.015)
10000	0.5	Independent	0.018 (0.086)	0 (0)	0.186 (0.026)
		Autoregressive	0.361 (1.107)	0 (0)	0.054 (0.011)
		Exchangeable	4.921 (5.133)	0 (0)	0.04 (0.01)
	0.7	Independent	0.008 (0.043)	0 (0)	0.206 (0.032)
		Autoregressive	0.609 (1.513)	0 (0)	0.055 (0.012)
		Exchangeable	5.222 (5.238)	0 (0)	0.04 (0.011)
	0.9	Independent	0.002 (0.01)	0 (0)	0.254 (0.04)
		Autoregressive	0.433 (1.309)	0 (0)	0.064 (0.016)
		Exchangeable	3.656 (4.361)	0 (0)	0.044 (0.013)

Table 9.

The Model Runtime of SDMDC, SDWD, and SSVM Based on the Number of Predictors

	Models
No. Predictors	SDMDC	SDWD	SSVM
100	19.041 (1.925)	151.8 (119.4)	0.08 (0.014)
1000	74.643 (9.689)	26.04 (1.14)	2.754 (0.054)
5000	371.666 (11.045)	124.86 (4.92)	100.959 (8.713)
10000	795.824 (26.348)	386.64 (59.82)	376.318 (58.953)

Table 10.

The Model Runtime of DMDC, DWD, and SVM Based on the Number of Predictors

	Models
No. Predictors	DMDC	DWD	SVM
100	13.816 (1.309)	11.94 (5.64)	0.07 (0.009)
1000	86.663 (7.992)	582.36 (394.38)	2.726 (0.053)
5000	356.566 (9.876)	624.12 (445.92)	100.904 (8.629)
10000	763.089 (21.625)	311.04 (154.44)	376.467 (58.469)

Table 11.

The Model Runtime of full SDMDC, Rank-1 SDMDC and Rank-2 SDMDC Based on the Number of Predictors

	Models
No. Predictors	Full SDMDC	Rank-1 SDMDC	Rank-2 SDMDC
200	45.914 (22.532)	38.754 (21.086)	43.797 (23.674)
500	95.649 (28.236)	46.86 (25.003)	56.753 (26.271)

Table 12.

The Model Runtime of full DMDC, Rank-1 DMDC and Rank-2 DMDC Based on the Number of Predictors

	Models
No. Predictors	Full DMDC	Rank-1 DMDC	Rank-2 DMDC
200	42.114 (17.268)	685.835 (387.075)	822.462 (449.41)
500	92.543 (24.193)	905.029 (441.866)	1218.938 (516.366)

Table 13.

The average of F-score for full SDMDC, rank-1 SDMDC and rank-2 SDMDC by the number of predictors, data structure, and the structure of correlation amongst predictors

			Models
No. Predictors	Data Rank	Correlation str	Full SDMDC	Rank-1 SDMDC	Rank-2 SDMDC
200	Rank-1	Independent	0.545 (0.126)	0.617 (0.156)	0.618 (0.166)
		Autoregressive	0.708 (0.281)	0.689 (0.285)	0.683 (0.293)
		Exchangeable	0.401 (0.248)	0.402 (0.212)	0.381 (0.206)
	Rank-2	Independent	0.6 (0.135)	0.698 (0.158)	0.681 (0.149)
		Autoregressive	0.701 (0.245)	0.675 (0.216)	0.76 (0.252)
		Exchangeable	0.693 (0.191)	0.651 (0.203)	0.667 (0.194)
500	Rank-1	Independent	0.534 (0.091)	0.563 (0.124)	0.579 (0.126)
		Autoregressive	0.584 (0.253)	0.596 (0.233)	0.576 (0.237)
		Exchangeable	0.583 (0.179)	0.568 (0.156)	0.573 (0.149)
	Rank-2	Independent	0.564 (0.086)	0.651 (0.129)	0.664 (0.119)
		Autoregressive	0.645 (0.148)	0.68 (0.17)	0.702 (0.189)
		Exchangeable	0.587 (0.096)	0.574 (0.093)	0.549 (0.111)

Table 14.

The average of F-score for full DMDC, rank-1 DMDC and rank-2 DMDC by the number of predictors, data structure, and the structure of correlation amongst predictors

			Models
No. Predictors	Data Rank	Correlation str	Full  DMDC	Rank-1 DMDC	Rank-2 DMDC
200	Rank-1	Independent	0.771 (0.189)	0.86 (0.136)	0.783 (0.189)
		Autoregressive	0.8 (0.311)	0.813 (0.298)	0.802 (0.31)
		Exchangeable	0.412 (0.263)	0.413 (0.263)	0.413 (0.263)
	Rank-2	Independent	0.889 (0.117)	0.943 (0.064)	0.902 (0.095)
		Autoregressive	0.873 (0.231)	0.889 (0.218)	0.876 (0.226)
		Exchangeable	0.68 (0.194)	0.681 (0.204)	0.683 (0.192)
500	Rank-1	Independent	0.743 (0.198)	0.767 (0.155)	0.727 (0.186)
		Autoregressive	0.683 (0.33)	0.698 (0.316)	0.685 (0.331)
		Exchangeable	0.586 (0.169)	0.583 (0.159)	0.583 (0.168)
	Rank-2	Independent	0.933 (0.086)	0.934 (0.153)	0.941 (0.086)
		Autoregressive	0.808 (0.186)	0.803 (0.188)	0.801 (0.19)
		Exchangeable	0.601 (0.082)	0.596 (0.071)	0.602 (0.08)

Table 15.

The signal-to-noise ratio for full SDMDC, rank-1 SDMDC and rank-2 SDMDC by the number of predictors, data structure, and the structure of correlation amongst predictors

			Models
No. Predictors	Data Rank	Correlation str	Full  SDMDC	Rank-1 SDMDC	Rank-2 SDMDC
200	Rank-1	Independent	0.853 (0.143)	1.131 (0.577)	1.241 (1.448)
		Autoregressive	4.73 (9.523)	3.781 (8.56)	4.132 (9.901)
		Exchangeable	13.768 (37.834)	18.63 (29.791)	17.755 (28.978)
	Rank-2	Independent	0.904 (0.408)	1.297 (1.231)	1.489 (2.182)
		Autoregressive	4.107 (6.428)	4.73 (9.619)	3.468 (7.939)
		Exchangeable	8.73 (15.893)	17.566 (29.315)	18.298 (29.792)
500	Rank-1	Independent	0.736 (0.106)	0.85 (0.302)	0.985 (0.587)
		Autoregressive	3.615 (2.299)	1.11 (0.341)	1.615 (3.544)
		Exchangeable	11.01 (23.183)	13.947 (44.328)	22.62 (55.979)
	Rank-2	Independent	0.72 (0.201)	0.854 (0.188)	1.276 (1.503)
		Autoregressive	3.822 (1.633)	1.64 (2.944)	3.105 (8.941)
		Exchangeable	13.546 (76.77)	15.847 (44.094)	13.523 (44.437)

Table 16.

Performance of sparse SVM Model in 5-fold cross-validation

Fold	TN	FN	FP	TP	F-measure Average	Non-Zero Coefficients	Modeling Time (s)
1	5	5	1	0	0.31	39	0.06
2	4	2	2	2	0.58	37	0.05
3	4	2	2	2	0.58	45	0.06
4	6	3	0	1	0.60	39	0.05
5	0	1	5	4	0.29	39	0.05
Mean	3.80	2.60	2.00	1.80	0.47	39.80	0.05

Table 17.

Performance of sparse DWD model in 5-fold cross-validation

Fold	TN	FN	FP	TP	F-measure Average	Non-Zero Coefficients	Modeling Time (s)
1	3	3	3	2	0.00	15	4.23
2	3	1	3	3	0.49	8	5.73
3	4	3	2	1	0.58	16	5.39
4	4	1	2	3	0.29	10	6.48
5	5	5	0	0	0.52	23	5.01
Mean	3.80	2.60	2.00	1.80	0.38	14.40	5.37

Table 18.

Performance of sparse DMDC model in 5-fold cross-validation

Fold	TN	FN	FP	TP	F-measure Average	Non-Zero Coefficients	Modeling Time (s)
1	3	3	3	2	0.45	36	2.28
2	3	1	3	3	0.60	22	2.29
3	4	3	2	1	0.45	3	2.21
4	4	1	2	3	0.70	3	2.33
5	5	5	0	0	0.33	21	2.24
Mean	3.80	2.60	2.00	1.80	0.51	17.00	2.27

Table 19.

Performance of rank-1 and eank-2 sparse SVM models in 5-fold cross-validation

Fold	Rank	TN	FN	FP	TP	F-measure Average	Non-Zero Coefficients	Modeling Time (s)
1	1	4	4	2	1	0.41	42	1.61
	2	2	4	4	1	0.27	36	2.98
2	1	2	3	4	1	0.29	39	1.32
	2	6	1	0	3	0.89	38	1.37
3	1	0	3	6	1	0.09	39	1.47
	2	1	3	5	1	0.20	38	1.93
4	1	4	4	2	0	0.29	40	1.68
	2	4	4	2	0	0.29	42	3.02
5	1	1	3	4	2	0.29	41	1.59
	2	2	4	3	1	0.29	39	2.98
Mean	1	2.2	3.4	3.6	1	0.27	40.2	1.53
	2	3	3.2	2.8	1.2	0.39	38.6	38.6

Table 20.

Performance of Rank-1 and Rank-2 Sparse DWD models in 5-fold cross-validation

Fold	Rank	TN	FN	FP	TP	F-measure Average	Non-Zero Coefficients	Modeling Time (s)
1	1	5	3	2	1	0.48	1606	245.21
	2	6	2	1	2	0.69	1921	274.72
2	1	4	4	1	1	0.45	231	291.69
	2	5	4	0	1	0.52	374	320.36
3	1	4	1	3	2	0.58	794	238.70
	2	5	1	2	2	0.67	395	201.24
4	1	2	3	1	4	0.58	130	316.95
	2	3	7	0	0	0.00	383	356.74
5	1	5	1	2	2	0.67	1306	184.57
	2	4	2	3	1	0.45	1890	359.92
Mean	1	4	2.4	1.8	2	0.55	813.4	255.42
	2	4.6	3.2	1.2	1.2	0.47	992.6	302.6

Table 21.

Performance of Rank-1 and Rank-2 sparse DMDC models in 5-fold cross-validation

Fold	Rank	TN	FN	FP	TP	F-measure Average	Non-Zero Coefficients	Modeling Time (s)
1	1	6	4	0	1	0.54	99	7.16
	2	4	3	2	2	0.53	12	121.16
2	1	5	3	1	1	0.52	25	91.03
	2	3	1	3	3	0.60	123	119.47
3	1	3	0	3	4	0.70	1	62.66
	2	3	4	3	0	0.23	3	81.84
4	1	5	2	1	2	0.67	57	58.57
	2	4	1	2	3	0.70	3	42.71
5	1	5	5	0	0	0.33	119	4.75
	2	4	2	1	3	0.70	24	12.13
Mean	1	4.8	2.8	1	1.6	0.55	60.2	44.84
	2	3.6	2.2	2.2	2.2	0.55	33	75.46

Authors’ contributions

Z.Z. conceptualized the study, performed data preprocessing, implemented the proposed models, and drafted the manuscript. H.A.M. contributed to study design, supervised the methodological framework, and revised the manuscript critically for intellectual content. M.R. assisted in data management, simulation experiments, and technical validation. Y.M. provided overall supervision, contributed to methodological refinement, and served as corresponding author. A.E.M. contributed to fMRI data interpretation and domain-specific insights in neuroimaging. S.S. provided neuroscience expertise, interpreted results in the context of Alzheimer’s disease, and contributed to manuscript revisions. All authors reviewed and approved the final manuscript.

Funding

The authors did not receive any specific funding for this work. (For the data source: The Alzheimer's Disease Neuroimaging Initiative (ADNI) is supported through funding from the National Institute on Aging via the National Institutes of Health (Grant U19AG024904). Oversight of the grant is managed by the Northern California Institute for Research and Education. Historically, ADNI has also received support from the National Institute of Biomedical Imaging and Bioengineering, the Canadian Institutes of Health Research, and a variety of private sector organizations. These private contributions were facilitated by the Foundation for the National Institutes of Health (FNIH) and include support from entities such as AbbVie, the Alzheimer’s Association, the Alzheimer’s Drug Discovery Foundation, Araclon Biotech, BioClinica, Biogen, Bristol-Myers Squibb, CereSpir, Cogstate, Eisai, Elan Pharmaceuticals, Eli Lilly and Company, EuroImmun, F. Hoffmann-La Roche and Genentech, Fujirebio, GE Healthcare, IXICO, Janssen Alzheimer Immunotherapy, Johnson & Johnson Pharmaceutical R&D, Lumosity, Lundbeck, Merck, Meso Scale Diagnostics, NeuroRx Research, Neurotrack Technologies, Novartis, Pfizer, Piramal Imaging, Servier, Takeda, and Transition Therapeutics.

Data availability

The data used in this study were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The datasets are publicly available upon application from the ADNI website.

Declarations

Ethics approval and consent to participate

This study is a secondary analysis of data obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI). The original ADNI study was conducted in full compliance with ethical standards. As per the ADNI protocol (Section D.5.c), informed consent was obtained from all participants, their guardians, or authorized representatives prior to enrollment. This process adhered to the principles of the Declaration of Helsinki and was documented using written consent forms approved by the Institutional Review Boards (IRBs) of all participating sites. The secondary analysis of the de-identified data for the current manuscript was approved by the Ethics Committee of Shahid Beheshti University of Medical Sciences (code: IR.SBMU.PHNS.REC.1402.032).

Consent for publication

Not applicable. (This study uses a publicly available dataset with pre-existing consent for data sharing and publication).

Competing interests

The authors declare no competing interests.