1) A novel Dirichlet maximum marginal similarity preservation factor analysis model (dpMMSPFA) that considers the FA model, the SP item, the Dirichlet process mixture model, and the LVSVMs is designed in a united Bayesian framework.

1) Section 1: This section explains the essential research significance of server energy consumption classification.

The general layout of the data center energy consumption modeling and forecasting framework is shown in Figure 1. The parts of the data center can generally be separated into three levels: hardware, software, and applications. The server energy consumption that will be studied in this research is not only influenced by its hardware configuration, but also affected by its operating system and various application types. There are two types of server energy consumption models, linear and nonlinear, which have been applied to the research of server energy consumption estimation.

Preliminary studies of prediction model for server energy consumption were based on the linear model (Pöss and Nambiar, 2010; Davis et al., 2012; Davis et al., 2014). Subramaniam and Feng, 2014 used the SPEC power benchmark to conduct tests on seven heterogeneous servers and evaluated the accuracy of the linear regression model based on CPU utilization (Kilper et al., 2011). The findings demonstrated that not all servers exhibit a linear relationship between power usage and server utilization characteristics. Therefore, researchers started to think about utilizing machine learning and deep learning nonlinear models to develop server energy consumption models to boost the precision of energy consumption forecasting and refine management of energy consumption control.

Real-world data frequently have a multimodal distribution, making it impossible for a straightforward classification/regression algorithm to provide an acceptable criterion. The dimensionality reduction techniques are categorized according to whether or not they incorporate supervised content, and can be essentially divided into two types, unsupervised and supervised dimensionality reduction. Unsupervised downscaling methods normally extract this characteristic information that maintains the statistics structure, while supervised downscaling techniques not solely extract low-dimensional information representations, but also carry certain a priori records (e.g., category information). In the discipline of unsupervised dimensionality reduction, FA (Chen et al., 2010; Shi et al., 2011; Du et al., 2012), PCA (Tharwat, 2016), and other techniques have a vital and far-reaching status, and for supervised dimensionality reduction methods, LDA strategies are widely valued and appreciated via researchers, experts and scholars. Zhou et al. (2018) incorporated PCA dimension reduction prior to applying the typical machine learning energy consumption model. While this somewhat alleviates the risk of instability and overfitting of prediction, it has certain restrictions on the processing of energy consumption data in general.

An energy consumption model FSDL based on feature selection and deep learning was proposed (Liang et al., 2020). In order to maximize the forecast accuracy of the energy consumption model, this model combines feature selection and deep learning techniques. This model is vulnerable to overfitting, though. Lin et al. (2020) developed three power consumption models based on BP neural networks, LSTM neural networks, and Elman neural networks, respectively. The prediction accuracy of the three power consumption models under various task loads is compared. When the cost of training and prediction accuracy are taken into account, ENN-PM outperforms TW_BP_PM and ML_STM-PM. As is common knowledge, humans are adept at recognizing a novel object from a relatively limited number of examples. In contrast, the deep learning technology requires a large amount of data to train a proper model. Especially when the number of neural network layers increases, the model becomes more complex. It will take more time and computing power to train the model to the convergence stage as the number of parameters to be optimized rises. The server energy consumption was forecasted via Q-learning, B-ANN, MLP and other reinforcement learning techniques (Shen et al., 2013; Li et al., 2010; Islam et al., 2012; Moreno and Xu, 2012; Caglar and Gokhale, 2014; et al., Tesauro et al., 2017). However, before the training effect may truly improve, reinforcement learning necessitates experience accumulation to a significant level. Additionally, it can be easier to fall into local optimization and not really achieve global optimization if the training object receives rewards from the environment in an untimely manner and the reward setting is unreasonable. This presents another challenge for reinforcement learning in the field of server energy consumption model.

In light of the aforementioned restrictions and the fact that the characteristics of server energy consumption datasets have not been dealt with in depth, we made the decision to develop a novel machine learning model.

Data interrelationship evaluation, data dimensionality reduction, pattern classification, and characteristic description are all frequently carried out using the method of factor analysis (FA) (Chen et al., 2010; Shi et al., 2011; Du et al., 2012). In FA models, implicit factors serve as representations of low-dimensional observations of data in the hidden space. Even though FA is an unsupervised dimensionality reduction technique, it has the ability to not only reduce dimensions but also to represent how subspace and original space are reconfigured, with Bayesian inference used to implement the FA model. Since FA is an unsupervised model and without a priori data like label content, it can only characterize low-dimensional observations (Chen et al., 2010; Shi et al., 2011; Du et al., 2012). Many experts and academics have been interested in the topic of how to introduce supervised content of latent elements recently. Attempts have been made to include discriminative supervised content as properly part of the input elements (Lacoste-Julien et al., 2008; Jiang et al., 2011; Zhu et al., 2012; Zhu et al., 2013), and a supervised K-SVD approach of label consistency is proposed to train discriminative dictionaries for dispersed coding (Jiang et al., 2011). K-SVD assembly characterizes with each dictionary item to carry out the discriminative supervised approach for dispersed encoding during concordance learning (Jiang et al., 2011).

Thus, it seems that supervised content is very vital to boost the predictive ability for classification model. Against this background, our model determines to introduce label content material of the original input data, referred to as similarity preservation (SP) item. By introducing supervised content material into FA, the proposed model not only keeps the finest data description and characteristic extraction capabilities, but also maximizes the priori predictive potential of characterization content.

DP mixture (DPM) models have been introduced as nonparametric Bayesian clustering algorithms for ME models (Rasmussen and Ghahramani, 2001; Shahbaba and Neal, 2009; Zhang et al., 2014). As an illustration, Shahbaba and Neal, 2009 created the dpMNL, a multi-metric Logit (MNL) nonlinear model based on DP mixtures.

The foundation of recognition accuracy is the classification model. The most classical representatives are support vector machines and random forest (RF). Zoubin et al. (2015) transformed the random forest classification model into β- Bayesian posterior framework presents a new idea for creating classifiers in Bayesian framework. Although β- Bayesian posterior is not an actual Bayesian theoretical framework, there is still promising. Support vector machines (SVMs), as a traditional representative of classifiers (Upadhyay et al., 2021), are capable to maximize the margins between different classes of data. Data augmentation techniques characterize the latent variables of SVMs as LVSVMs (Polson and Scott, 2011), hence successfully inferring the further proposed Gibbs’ maximum marginal topic model (Zhu, et al., 2014) and fast maximum marginal matrix decomposition (Xu, et al., 2013).

In this study, we accordingly construct the Dirichlet maximum marginal similarity preservation factor analysis model (dpMMSPFA) with LVSVM, which mutually learns discriminative subspaces, supervised content, clustering and maximum marginal classifiers underneath a Bayesian architecture. Therefore, hidden representations are authentic and reasonable for supervised predictive recognition tasks. Gibbs MedLDA (Zhu et al., 2012; Zhu et al., 2013) was inferred via an efficient estimation algorithm, Gibbs sampling. Similarly, the ambit of dpMMSPFA can acquire desirable covariance underneath the motion of augmented variables. Thus, the model dpMMSPFA can be estimated with simple and effective Gibbs sampling for all parameters.

The precise inner structural links between high dimension observations and low dimension hidden variables can be represented by factor analysis models (FA). By projecting the high-dimensional observations into the low-dimensional space, the factor analysis model is able to determine the potential low-dimensional effective features of the data. Suppose there are N column vectors x 1 , x 2 , x 3 , ⋯ , x N and each of them is a P -dimensional vector. Let X = x 1 , x 2 , x 3 , ⋯ , x N , and then the original input observations X is generated as a linear transformation of some lower K -dimensional hidden variable S plus additive Gaussian noise ε . The transformation matrix D is the load matrix with each column d k ,   k = 1 , ⋯ , K . However, the number of factors K is not known in advance and needs to be determined earlier, therefore, the factor analysis generating expression can be given as X P × N = D P × K S K × N + ε P × N (1)

FA is employed as an unsupervised archetypal model that does not make use of implicit features with label content to describe the initial observations of the data (Chen et al., 2010; Shi et al., 2011; Du et al., 2012). In context of this, we provide an approach for supervised data representation that, while simultaneously using FA to represent original observations, successfully increases the classifier’s predictive power by labeling content.

The incorporation of supervised content to model learning can significantly enhance the classifier’s overall performance. We added the SP item to FA so that the extracted hidden variables could best characterize the original input data. This can maximize the a priori prediction ability of labelling content. Commonly, similarity is denoted by any symmetric positive definite matrix. Jiang et al. (2011) successfully improved the classifier’s recognition performance by relying it on label content. In a similar vein, the similarity preservation item (SP) in our suggested model yields label content illustration.

As shown in Eq. 2, we start from the cosine distance matrix. l i j = x i T x j x i * ‖ x j ‖ (2) where i and j denote label content of column vector x .

The weight l of the similarity preservation items are defined based on whether the data are in the same classification. l i j = l x i , x j i = j 0 , i ≠ j (3)

An example of a sparse coding matrix U l is as follows. l x 1 , x 1 l x 1 , x 2 0 0 0 0 0 l x 2 , x 1 l x 2 , x 2 0 0 0 0 0 0 0 l x 3 , x 3 l x 3 , x 4 l x 3 , x 5 0 0 0 0 l x 4 , x 3 l x 4 , x 4 l x 4 , x 5 0 0 0 0 l x 5 , x 3 l x 5 , x 4 l x 5 , x 5 0 0 0 0 0 0 0 l x 6 , x 6 l x 6 , x 7 0 0 0 0 0 l x 7 , x 6 l x 7 , x 7 (4)

To find the best solution for the similarity preservation items, we employed singular vector decomposition (SVD) of the matrix U l . The generating phrase for similarity preservation can therefore be stated as U N × N = A N × K S K × N + φ N × N (5)

A Dirichlet process D P G 0 , α is the distribution in the distribution (Ferguson, 1973). It is characterized by a baseline distribution G 0 and a positive scalar parameter α . Suppose that we randomly draw a sample distribution G from a DP. Subsequently, N random variables Θ n } n = 1 N are independently sampled from G . G | G 0 , α ∼ D P G 0 , α Θ n | G ∼ G ,   n = 1 , . . . , N (6)

DP mixtures (DPM) can be used in cluster community problems where the kinds of clusters are uncertain (Rasmussen and Ghahramani, 2001; Shahbaba and Neal, 2009; Zhang et al., 2014). The DPM model can be expressed as Eq. 7, in which Θ n can parameterize the distribution of the input data x n . G | G 0 , α ∼ D P G 0 , α Θ n G ∼ G ;   x n Θ n ∼ p x | Θ n ,   n = 1 . . . N (7)

To describe the distribution of the stochastic variable G in D P G 0 , α , Antoniak, 1974 constructs a pioneering method that is called “stick-breaking”. In the stick-breaking description, there are two infinite kinds of independent stochastic variables: Θ c c = 1 ∞ and υ c c = 1 ∞ , in which Θ c ∼ G 0 and υ c ∼ B e t a 1 , α . The stick-breaking description of G can be expressed as G = ∑ c = 1 ∞ π c υ δ Θ c π c υ = υ c ∏ j = 1 c − 1 1 − υ j ∈ 0,1 ,   ∑ c = 1 ∞ π c υ = 1 (8)

DP, as a representative of the stick-breaking, has an apparently discrete stochastic variable G consisting of an infinite but countable number of independently sampled atoms from G 0 in Θ c . At the same time, the stick-breaking hyper-parameter α determines the average value of the stick-breaking variables υ c and then adjust the effective number of different parameter values.

Based on the DP’s stick-breaking expression in Eq. 8, the DPM is then be given as υ c | α ∼ B e t a 1 , α Θ c | G 0 ∼ G 0 , c = 1 , . . . , ∞ z n | π υ ∼ M u l t π υ x n | z n = c ; Θ c ∼ p x | Θ c , n = 1 , . . . , N (9) where z n is the index of the cluster of the observation data x n , π ( υ ) = ( π c ( υ ) ) c = 1 ∞ can be accquired by Eq. 8, and M u l t π υ is a multinomial distribution over the mixture proportions π υ .

To make for use of the advantage of the nonparametric Bayesian method, as a supervised cluster model, dpMMSPFA is developed on the basis of the truncated stick-breaking DPM in this study.

SVM is a powerful machine learning tool that has been widely used in the field of pattern recognition, mainly due to its great generalization ability. Given a labelled training set with data vectors X = x 1 , x 2 , x 3 , ⋯ , x N , x n ∈ R P , and their labels y = y 1 , y 2 , y 3 , ⋯ , y N , ∈ − 1 , + 1 . It is defined as min η , ξ n 1 2 ‖ η ‖ 2 2 + C 0 ∑ n = 1 N ξ n s . t . y n η T x ∼ n ≥ 1 − ξ n , ξ n ≥ 0 , n = 1 , . . . , N (10) where x ∼ n = x n ; 1 is the augmented feature vector, η is the weighted coefficient and C 0 is a positive tuning parameter. The underlying discriminative objective is a linear hinge loss function, max 1 − y n η T x ∼ n , 0 , which seems to lead to the difficulty of traditional Bayesian analysis.

In contrast to the conventional method, Polson and Scott, 2011 suggested a hidden variable method based on data augmentation technology to describe SVMs. With the assistance of the augmentation technology, new parameter information can be extracted from data samples and expanded into the original data samples. Any type of data can be expanded using data augmentation technologies to deepen the knowledge of the original information while also enhancing the data’s quality. By including auxiliary variables, the approach can be dealt with more easily. As a result of its widespread use in dealing with non-conjugate models, data augmentation has grown to be a highly powerful technique for resolving non-conjugate issues.

The pseudo-posterior distribution of an SVM can be expressed as a marginal distribution of a high-dimensional distribution with augmented variables. Thus, the complete pseudo-posterior distribution of the data can be written as p η , λ | y ∝ ∏ n = 1 N λ n − 1 2 exp − ( λ n − C 0 ( 1 − y n η T x ∼ n ) ) 2 2 λ n N η ; 0 , I (11)

As is mentioned above, for each column vector x of FA model, it can be written as x P × 1 = D P × K s K × 1 + ε P × 1 (12)

and for each column vector u of SP item, it can be written as u N × 1 = A N × K s K × 1 + φ N × 1 (13)

In dpMMSPFA, the hidden variable s } n = 1 N is clustered and it can be supposed that each cluster of hidden variables are subordinated to the Gaussian distribution: s n | z n = c , Θ c ∼ N W s n ; μ c , Σ c (14) where Θ c represents the Gaussian distribution parameter of the c t h cluster, μ c , Σ c . A joint normal-Wishart distribution is employed as the conjugate prior of Θ c , then it can be expressed as G 0 = p Θ c = N W μ c , Σ c | μ 0 , W 0 , β 0 , υ 0 (15)

On the basis of the DPM’s stick-breaking expression, FA model, SP term and LVSVM, our the hierarchical dpMMSPFA model is described as d k ∼ N d k ; 0 , Λ d k ; x n | s n , D ∼ N x n ; D s n , Λ φ a k ∼ N a k ; 0 , Λ a k ; u n | s n , A ∼ N x n ; Α s n , Λ ε s n | z n = c , Θ c ∼ p s n | Θ c , n = 1 , . . . , N υ c | α ∼ B e t a 1 , α ; Θ c G 0 ∼ G 0 ; z n π υ ∼ M u l t π υ η c , { λ n } c | { s n , y n , z n = c } n = 1 N ∼ p η c , { λ n } c | − , c = 1 , . . . , ∞ (16)

We continue to add expressions and descriptions for proposed dpMMSPFA model: d k ∼ N d k ; 0 , Λ d k ,   Λ d k = d i a g α k 1 − 1 , … , α k P − 1 , α k p ∼ G a m m a a 0 , b 0 a k ∼ N a k ; 0 , Λ a k ,   Λ a k = d i a g σ k 1 − 1 , … , σ k N − 1 , σ k n ∼ G a m m a a ∼ 0 , b ∼ 0 s ∼ N s ; 0 , Λ s   , Λ s = d i a g β 1 − 1 , … , β K − 1 , β k ∼ G a m m a c 0 , d 0 ε ∼ N ε ; 0 , Λ ε   , Λ ε = d i a g γ 1 − 1 , … , γ P − 1 , γ p ∼ G a m m a e 0 , f 0 φ ∼ N φ ; 0 , Λ φ , Λ φ = d i a g ω 1 − 1 , . . . , ω N − 1 , ω n ∼ G a m m a e ∼ 0 , f ∼ 0 (17) where hyper-priors α k p ∼ G a m m a a 0 , b 0 , σ k n ∼ G a m m a a ∼ 0 , b ∼ 0 and ω n ∼ G a m m a e ∼ 0 , f ∼ 0 are employed with a 0 , b 0 , c 0 , d 0 , e 0 , f 0 , a ∼ 0 , b ∼ 0 and e ∼ 0 , f ∼ 0 as the corresponding hyper-parameters.where η c and λ n } c represent the weighted coefficient and the augmented variables of the LVSVM classifier related to the c th cluster. The complete pseudo-posterior of parameters can be expressed as p D , A , S , μ c , Σ c c = 1 C , Λ d k k = 1 K , Λ a k k = 1 K , Z , v , η c = 1 C , λ , Λ ε , Λ φ | X , y = ∏ c = 1 C ∏ n = 1 , z n = c N p z n | υ p x n | D , S , Λ ε p u n | A , S , Λ φ × ∏ c = 1 C ∏ n = 1 , z n = c N p s n | μ c , Σ c p μ c , Σ c p η c ϕ y n , λ n | s n , η c p υ | α p α × ∏ k = 1 K p d k | Λ d k p Λ d k p Λ ε p a k | Λ a k p Λ a k p Λ φ (18)

Here the posterior computation is implemented by a Markov chain Monte Carlo (MCMC) algorithm based on Gibbs sampling, where the posterior distribution is approximated by a sufficient number of samples. Then, the conditional distributions used in Gibbs sampling are shown as follows.

For s n : the conditional distribution of s n is p s n | − ∝ N x n ; D s n , Λ ε N u n ; A s n , Λ φ N s n ; μ z n , Σ z n × 1 2 π λ n exp − ( λ n + C 0 ( l − y n η z n T s ∼ n ) ) 2 2 λ n ∼ N s n ; μ s n , Λ s n (19) where the posterior mean is μ s n = Λ s n D k T Λ ε − 1 x n + A k T Λ φ − 1 u n + Σ z n − 1 μ z n + C 0 λ n + C 0 − C 0 y n η 0 λ n y n η z n and the covariance matrix is Λ s n = D T Λ ε − 1 D + A T Λ φ − 1 A + Σ z n − 1 + C 0 2 η z n η z n T / λ n .

For z n : the conditional distribution of z n is z n ∼ M u l t κ n ,   κ n = κ n 1 , … , κ n C κ n c = p z n = c | μ c , Σ c , v , η c , λ n ∝ π c N s n ; μ c , Σ c exp ( − λ n + C 0 ( l − y n η c T s ∼ n ) ) 2 2 λ n ) (20)

A Markov chain can be successfully created using the conditional distribution given above. Samples from the aforementioned conditional posterior distribution can be taken from each iteration in the expressions (19) and (20) depending on the initial conditions. The Markov chain is worked to run to finish the burn-in phase during the training phase. And with regard to Figure 2's architecture diagram for the integrated recognition system based on dpMMSPFA. The dpMMSPFA integrated recognition scheme comprises two stages: the MMSPFA model establishment phase, represented by the blue box in the diagram, and the usage phase of dpMMSPFA model, represented by the red box in the diagram. In the next section, we’ll go into more depth about these two phases.

The dpMMSPFA model establishment is a supervised process, in which Gibbs sampling is used to infer model parameters summarized in Table 1. We run this Markov chain to finish the burn-in phase during the model establishment phase.

While the dpMMSPFA model’s training phase is supervised, the model’s prediction process is unsupervised shown in Table 2. In the aforementioned procedures, it is essential to set the data sample so that all parameters are sampled T 0 times and in simultaneously.

The test data is pre-processed first, and the hidden variable s and the cluster index z of x are sampled unsupervised on the basis of their related posterior Eq. 21 and Eq. 22, respectively. p s | D , z , μ z , Σ z ∝ N x ; D s , Λ ε N s ; μ z , Σ z (21) z ∼ M u l t κ ,   κ = κ 1 , … , κ C κ c = p z = c | μ c , Σ c , v , η c , λ n ∝ π c N s ; μ c , Σ c (22)

All the collected samples are averaged from Gibbs sampler to predict the label y of x as y ^ = s i g n 1 T 0   ∑ t = 1 T 0   η z t T s ∼ t (23) where s t represents the t t h sample of the hidden variable, which is sampled with the t t h gathered samples, z t represents the t t h sample of the cluster index which the test data belongs to and η z t represents the t t h gathered SVM coefficients of cluster z t .

This section compares our proposed dpMMSPFA model to the following classification techniques and uses experimental data to demonstrate how effective and predictive it is on certain datasets. 1) PCA + SVM (Tharwat, 2016); 2) Kmeans + SVM (Wu and Peng, 2017); 3) LVSVM (SVM) (Polson, 2011), 4) dpMNL (Shahbaba and Neal, 2009), 5) MMFA, and 6) dpMMSPFA. In the experiments, the LVSVM classifier is used for PCA and Kmeans. The feature chosen and the classifier are different since models (1)–(2) are two-stage models. In joint approaches (5) and (6), LVSVM is employed, and the tuning parameter C 0 is picked from 0.001,0.01,0.1,1,10 .

In this section, we perform experiments on Benchmark data sets of varying size and difficulty, and for each we average the accuracy over ten random splits. The benchmark data sets can be found at either the University of California at Irvine (UCI) or the Machine Learning Dataset Repository (Dua and Graff, 2022). Table 3 summarizes the data information.

The average testing accuracy is displayed in Figure 3, with the best results throughout many components explicitly shown for each approach. The low-dimensional features that are extracted using unsupervised approach PCA may not be sufficient for the subsequent prediction task. Another two-stage model Kmeans, which build SVMs with an ensemble of clusters, performs well merely in some datasets. These two-stage models are unable to produce adequate outcomes as a result. The robustness of MNL’s classification performance is insufficient, making dpMNL less effective than the unsupervised joint model MMFA. In contrast to dpMNL and LVSVM, dpMMSPFA employs FA, which pulls out more beneficial features from cluster, label, and classification content using a unified Bayesian framework. MMFA is a particular instance of dpMMSPFA that does not entirely utilize the label content when there is just one cluster. As shown in Figure 3, the suggested model dpMMSPFA works well in the experiment and gets the maximum accuracy, especially when working with multimodal datasets like the German, Heart, and other similar cases. Figure 4 depicts the estimated posterior distributions of the number of clusters by dpMMSPFA. We can observe that, regarding the distribution of the data, the number of clusters detected by the dpMMSPFA is more logical.

The energy consumption grade is more valuable than the specific energy consumption data. It influences both the choice to purchase enterprise server assets and the decision to implement energy-saving measures at various levels of energy consumption. The 5 categories of energy consumption in this study, which range from low to high, are represented by the numbers [1, 2, 3, 4, 5] listed in Table 4.

In this section, we introduce dpMMSPFA to the server energy consumption level classification community. Results of the experiment are presented in this subsection chiefly based on measured Inspur NF5280M5 server. Table 5 displays the Inspur NF5280M5 server configuration used in this study. The power consumption is measured using an energy consumption tool through which “CPU intensive tasks”, “I/O intensive tasks”, “Load intensive tasks” and “Non-loaded tasks,” are produced, accordingly. Underneath Linux, there is a testing tool named “Stress”, a tool imposing a configurable amount of load on system, which is designed primarily for people who would like to test load systems and monitor on how these devices are operating (Ulianytskyi, 2022).

Load characteristic parameters of the Inspur NF5280M5 while it executes tasks under various loads are summarized in Table 6.

Thermodynamic chart is a graphical representation of pattern similarity, with each component value in the matrix representing a different colour. The heatmap, the visualized ″ U l ″ introduced in Section 3, corresponding to 7.5% of the collected server energy consumption data (training samples) is used as supervised information to characterize the similarity and distinction of data in the matrix, as shown in Figure 5. The similarity between the training data gradually changes in accordance with darkish black, darkish blue, blue, lake blue, bluish green, green, yellowish green and mild yellow. When the colour is darkish black on the diagonal of the matrix, the similarity is 1. Furthermore, the comparison data of two pairs are from different categories visualized with bluish green, and the similarity is 0. We incorporate the matrix category and similarity information represented by this heatmap into the proposed dpMMSPFA model, which aims to improve the model’s overall discriminant performance.

Since the Gamma hyper-parameters, a 0 , b 0 , a ∼ 0 , b ∼ 0 , e 0 , f 0 , and e ∼ 0 , f ∼ 0 may indirectly influence the inference of corresponding parameters, the transformation matrix D and A , and the noise matrix ε and φ , to evaluate the effect of those parameters, we varied model hyper-parameters from {10⁻⁵,10⁻⁴,10⁻³,10⁻²,10⁻¹, 10⁰, 10¹,10²} to evaluate their effects on recognition rate. In order to achieve highest recognition ability on server energy consumption dataset, we fixed them to suitable values ( a 0 =100, b 0 =10⁻¹, a ∼ 0 =10⁻², b ∼ 0 =10⁻³, e 0 =10⁻¹, f 0 =10⁻², e ∼ 0 =10⁻², f ∼ 0 =1) based on the results provided in Figure 6. At the same time, we discovered that high performance identification rate can always be achieved when the dpMMSPFA model automatically learns 6 or 7 clusters. This finding also demonstrates the usefulness and rationality of introducing the DPM model to cluster and group in our proposed dpMMSPFA.

Here, we provided the samples’ clustering results with a clustering number of 7 (samples have been processed by z-score). Figure 7 depicts that the sample proportion of each cluster is fairly uniform, and there are no instances whenever the proportion of a particular partition is too high or low, showing that the number of clusters that were automatically learned is appropriate and avoids the flaws with conventional clustering, such as overfitting. These results provide evidence to explain why 7 partitions can give remarkable energy consumption feature recognition performance aforementioned in the experiments in Figure 6.

The commonly used CH (Calinski-Harabaz) and SC (Silhouette Coefficient) indexes (Zhang et al., 2021) were used to evaluate the clustering effectiveness and are compared with the clustering methods of Kmeans (Wu and Peng, 2017) and FCM (Bezdek et al.,1984) through experiments. This was done to further confirm the clustering effectiveness of dpMMSPFA after the DPM model is introduced in this paper. The clustering effect is often better the larger the CH and SC. When the automatic clustering results are 6 and 7, we continued to run the dpMMSPFA clustering effectiveness experiment in accordance with the experiment in Figure 6. We set the number of clusters from 2–10 at an interval of 1, conducted cluster effectiveness evaluation statistics, and ensured that the number of clusters is appropriate for Kmeans and FCM. According to the experimental findings in Figure 8, the value of dpMMSPFA is closer to 1 than that of Kmeans and FCM, which means the effect is better and the volatility is lower than that of FCM when using CH to assess the efficacy of clustering. The effectiveness of clustering is assessed for SC. Overall, the dpMMSPFA clustering results have a larger ratio of inter group to intra group dispersion, and the effect is superior to that of Kmeans and FCM. Despite the fact that under the SC index, the volatility of dpMMSPFA is higher than that of FCM, most SC indexes of dpMMSPFA exhibit superior clustering outcomes than FCM. The clustering evaluation experiment also proves that the excellent recognition performance of 6 and 7 clusters in the experiment in Figure 6 is inseparable from the effectiveness of the dpMMSPFA automatically clustering.

Figure 9 depicts the relationship between classification performance and the number of features in the aforementioned models. The experimental findings confirm the continued effectiveness of our suggested dpMMSPFA. The highest accuracy of 95.79% is achieved by dpMMSPFA when 10 extracted components are used. Due to the modest advantage of dimensionality reduction, PCA + LVSVM outperforms all LVSVM model in classification, but as dimensionality increases, it falls short of dpMMSPFA’s classification performance. On the other hand, while LVSVM, Kmeans + LVSVM and dpMNL employ the original data as input features and the server-consumed data contains some redundant material, they both perform less effectively in terms of classification than dpMMSPFA. Additionally, dpMMSPFA performs better than two-stage separation models and has a relatively robust ability for energy consumption classification. According to Figure 9, MMFA has a greater classification prediction performance than other classifiers but a poorer prediction performance than dpMMSPFA most of the time. Joint learning used in a joint architecture can therefore significantly enhance categorization performance. Furthermore, it is crucial to include SP elements and clustering in the joint architecture. The outcomes also demonstrate that using additional characteristics enhances classification performance which is consistent with the theoretical dpMMSPFA framework. The introduction of more and more redundant feature content, however, led to a decline in classification performance, which is why accuracy did not always rise as the number of features increased.