Research on identification of server energy consumption characteristics via dirichlet max-margin factor analysis similarity preservation model

Growing server energy consumption is a significant environmental issue, and mitigating it is a key technological challenge. Application-level energy minimization strategies depend on accurate modeling of energy consumption during an application’s execution. This paper presents a theoretical and experimental study of the dpMMSPFA model in the field of server energy consumption identification. The dpMMSPFA for classification of hidden spaces uses latent variable support vector machines (LVSVM) to learn discriminative subspaces with maximal marginal constraints. The factor analysis (FA) model, the similarity preservation (SP) item, the Dirichlet process mixture (DPM) model, and the maximal marginal classifier are jointly learned beneath a unified Bayesian architecture to advance classification of predictive power. The parameters of the proposed model can be inferred by the simple and efficient Gibbs sampling in terms of the conditional conjugate property. Empirical results on various datasets demonstrate that 1) max-margin joint learning can significantly improve the prediction performance of the model implemented by feature extraction and classification separately and meanwhile retain the generative ability; 2) dpMMSPFA is superior to MMFA when employing SP item and Dirichlet process mixture as prior knowledge; 3) the classification of dpMMSPFA model can often achieve better results on benchmark and measured energy server consumption datasets; 4) and the recognition rate can reach as high as 95.79% at 10 components, far better than other models on measured energy server consumption datasets.

1 Introduction

The issue of energy consumption in data centers has increased significantly in recent years due to the rapid growth of ICT technology and infrastructure. Especially since the outbreak of COVID-19 in 2020, the demand for digital services for economic and social development has skyrocketed, more and more consumer and commercial activities have turned to online, and the digital and information technology sectors have experienced tremendous growth. Omdia’s relevant statistics shows that consumer data traffic from cellular networks and fixed broadband will increase between 2018 and 2024 at a compound annual growth rate of 29%, increasing from 1.3 million PB in 2018 to 5.8 million PB in 2024. The current ICT infrastructure, which includes data centers, data center Internet, and Internet access networks, is put to a great deal of pressure by this development rate (Moises, 2021).

To meet the new demand, operators, cloud manufacturers and Internet enterprises have upgraded and expanded their data centers. While processing business load requires a lot of electric energy, the data center also generates a large amount of indirect carbon emissions. Global data centers’ power consumption, including those in China, is expected to rise from 2% in 2020 to over 4% in 2025 (DC Cooling, 2021). The hassle of high energy consumption has a serious impact on social security, climate warming, air quality and reliability of power grid. With the gradual depletion of traditional power sources and the soaring price, the cost of maintaining operational data will exceed the cost of purchasing system hardware. Therefore, the optimization of server energy consumption of a cloud operating system or a data center has become a greater essential problem in the current technical environment.

The ordinary way of measuring energy is to directly measure the electrical parameters of the server through the electrical instrument to achieve the actual energy of the server (Konstantakos et al., 2008; Rotem et al., 2012). However, this physical measurement approach can solely obtain the actual power, and it is impossible to analyze from these statistics what causes the rise, drop or unexpected change of power. Since the alternate of server energy consumption is bound to be accompanied via the change of system resource usage, it is necessary to design an identification model for energy usage, which can accurately classify the server energy consumption level, reflect the relationship between system resource utilization and server energy consumption, analyze the influence of resource utilization on energy consumption.

In brief overview, this paper proposes research on the Dirichlet max-margin factor analysis similarity preservation model (dpMMSPFA) for feature identification of server energy consumption level. To ensure that energy consumption analysis can be determined through high classification accuracy, this research aims to provide a comprehensive energy consumption feature recognition method that can meet the needs of high accuracy. It also aims to provide theoretical support and assist designers to control energy consumption when constructing servers.

1.1 The main contributions of the paper are summarized as follows

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.

2) Extensive experimental analysis is conducted to validate the proposed model using widely adopted UCI benchmarks to evaluate generalization performance of the proposed MMSPFA model.

3) To create and train the model in this study, 17 features relating to server energy consumption are chosen and only 7.5% of all collected server energy consumption data (small training size) is used in training. The proposed approach is adjusted by experiments on altering value of hyper-parameters to achieve the best energy consumption feature recognition performance.

4) The proposed model is compared with other five models (including two-stage model and joint model) in the recognition ability of server workload (including “CPU intensive tasks”, “I/O intensive tasks”, “Load intensive tasks” and “Non-Loaded tasks”) under different energy consumption characteristic dimensions.

1.2 The bright structure of the accomplished paper is organized as follows

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

2) Section 2: The existing linear model and nonlinear model (machine learning, deep learning and reinforce learning), research on server energy consumption models and their defects are introduced. The development status of nonparametric Bayesian model is stated in this section.

3) Section 3: Four vital models underneath Bayesian framework are described, along with factor analysis (FA) model, similarity preservation supervision item (SP) model, Dirichlet mixture process (DPM) and latent variable support vector machine (LVSVM).

4) Section 4: The mathematical construction of the algorithm of the model focuses on the Gibbs sampling inference method, which lays an important foundation on the nonparametric Bayesian classification model dpMMSPFA proposed in Section 4. A Dirichlet max-margin factor analysis similarity preservation model (dpMMSPFA classification model) is proposed.

5) Section 5: Through the conducted experiments of dpMMSPFA model on UCI benchmark data and measured server energy consumption data, it indicates that dpMMSPFA model has notable identification performance and generalization performance.

6) Section 6: This section summarizes the primary work of this paper and the future research work of nonparametric Bayesian model and energy consumption research in intelligent hardware is envisaged and prospected.

2 Related work

2.1 Current research on server energy consumption model

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.

FIGURE 1

FIGURE 1. The general layout of the data center energy consumption modeling and forecasting framework.

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.

2.2 The background for dpMMSPFA 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.

3 Preliminaries for dpMMSPFA model

3.1 Factor analysis (FA)

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 ${\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,\cdots ,{\mathit{x}}_N$ and each of them is a $P$ -dimensional vector. Let $\mathit{X}=\left[{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,\cdots ,{\mathit{x}}_N\right]$, and then the original input observations $\mathit{X}$ is generated as a linear transformation of some lower $K$ -dimensional hidden variable $\mathit{S}$ plus additive Gaussian noise $\mathit{\epsilon }$. The transformation matrix $\mathit{D}$ is the load matrix with each column ${\mathit{d}}_k,k=1,\cdots ,K$. 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

${\mathit{X}}_{P\times N}={\mathit{D}}_{P\times K}{\mathit{S}}_{K\times N}+{\mathit{\epsilon }}_{P\times N}(1)$

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.

3.2 Similarity preservation (SP) item

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.

$l_{ij}=\frac{{\mathit{x}}_i^T{\mathit{x}}_j}{‖{\mathit{x}}_i‖*\Vert {\mathit{x}}_j\Vert }(2)$

where $i$ and $j$ denote label content of column vector $\mathit{x}$.

The weight $l$ of the similarity preservation items are defined based on whether the data are in the same classification.

$l_{ij}=\left\{\begin{array}{c}l\left({\mathit{x}}_i,{\mathit{x}}_j\right)i=j\\ 0,i\ne j\end{array}\right.(3)$

An example of a sparse coding matrix ${\mathit{U}}_l$ is as follows.

$\left[\begin{array}{ccccccc}l\left({\mathit{x}}_1,{\mathit{x}}_1\right)& l\left({\mathit{x}}_1,{\mathit{x}}_2\right)& 0& 0& 0& 0& 0\\ l\left({\mathit{x}}_2,{\mathit{x}}_1\right)& l\left({\mathit{x}}_2,{\mathit{x}}_2\right)& 0& 0& 0& 0& 0\\ 0& 0& l\left({\mathit{x}}_3,{\mathit{x}}_3\right)& l\left({\mathit{x}}_3,{\mathit{x}}_4\right)& l\left({\mathit{x}}_3,{\mathit{x}}_5\right)& 0& 0\\ 0& 0& l\left({\mathit{x}}_4,{\mathit{x}}_3\right)& l\left({\mathit{x}}_4,{\mathit{x}}_4\right)& l\left({\mathit{x}}_4,{\mathit{x}}_5\right)& 0& 0\\ 0& 0& l\left({\mathit{x}}_5,{\mathit{x}}_3\right)& l\left({\mathit{x}}_5,{\mathit{x}}_4\right)& l\left({\mathit{x}}_5,{\mathit{x}}_5\right)& 0& 0\\ 0& 0& 0& 0& 0& l\left({\mathit{x}}_6,{\mathit{x}}_6\right)& l\left({\mathit{x}}_{\mathbf{6}},{\mathit{x}}_7\right)\\ 0& 0& 0& 0& 0& l\left({\mathit{x}}_7,{\mathit{x}}_6\right)& l\left({\mathit{x}}_7,{\mathit{x}}_7\right)\end{array}\right](4)$

To find the best solution for the similarity preservation items, we employed singular vector decomposition (SVD) of the matrix ${\mathit{U}}_l$. The generating phrase for similarity preservation can therefore be stated as

${\mathit{U}}_{N\times N}={\mathit{A}}_{N\times K}{\mathit{S}}_{K\times N}+{\mathit{\phi }}_{N\times N}(5)$

3.3 Dirichlet process mixture model (DPM)

A Dirichlet process $DP\left(G_0,\alpha \right)$ is the distribution in the distribution (Ferguson, 1973). It is characterized by a baseline distribution $G_0$ and a positive scalar parameter $\alpha$. Suppose that we randomly draw a sample distribution $G$ from a DP. Subsequently, $N$ random variables $\left\{{\Theta }_n{\}}_{n=1}^N\right.$ are independently sampled from $G$.

$G|\left\{G_0,\alpha \right\}\sim DP\left(G_0,\alpha \right)$

${\Theta }_n|G\sim G,n=1,...,N(6)$

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 ${\Theta }_n$ can parameterize the distribution of the input data ${\mathit{x}}_n$.

$G|\left\{G_0,\alpha \right\}\sim DP\left(G_0,\alpha \right)$

${\Theta }_n\left|G\sim G;{\mathit{x}}_n\right|{\Theta }_n\sim p\left(\mathit{x}|{\Theta }_n\right),n=1...N(7)$

To describe the distribution of the stochastic variable $G$ in $\mathrm{D}\mathrm{P}\left(G_0,\alpha \right)$, 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: ${\left\{{\Theta }_c\right\}}_{c=1}^{\infty }$ and ${\left\{{\mathit{\upsilon }}_c\right\}}_{c=1}^{\infty }$ , in which ${\Theta }_c\sim G_0$ and${\mathit{\upsilon }}_c\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}\left(1,\alpha \right)$. The stick-breaking description of $G$ can be expressed as

$G={\displaystyle \sum }_{c=1}^{\infty }{\pi }_c\left(\mathit{\upsilon }\right){\delta }_{{\Theta }_c}$

${\pi }_c\left(\mathit{\upsilon }\right)={\mathit{\upsilon }}_c{\displaystyle \prod }_{j=1}^{c-1}\left(1-{\mathit{\upsilon }}_j\right)\in \left[\mathrm{0,1}\right],{\displaystyle \sum }_{c=1}^{\infty }{\pi }_c\left(\mathit{\upsilon }\right)=1(8)$

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 ${\Theta }_c$. At the same time, the stick-breaking hyper-parameter $\alpha$ determines the average value of the stick-breaking variables ${\upsilon }_c$ 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

${\mathit{\upsilon }}_c|\alpha \sim Beta\left(1,\alpha \right)$

${\Theta }_c|G_0\sim G_0,c=1,...,\infty$

$z_n|\mathit{\pi }\left(\mathit{\upsilon }\right)\sim Mult\left(\mathit{\pi }\left(\mathit{\upsilon }\right)\right)$

${\mathit{x}}_n|z_n=c;{\Theta }_c\sim p\left(\mathit{x}|{\Theta }_c\right),n=1,...,N(9)$

where $z_n$ is the index of the cluster of the observation data ${\mathit{x}}_n$, $\mathit{\pi }\left(\mathit{\upsilon }\right)=\left({\pi }_c\right(\mathit{\upsilon }){)}_{c=1}^{\infty }$ can be accquired by Eq. 8, and $Mult\left(\mathit{\pi }\left(\mathit{\upsilon }\right)\right)$ is a multinomial distribution over the mixture proportions $\mathit{\pi }\left(\mathit{\upsilon }\right)$.

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.

3.4 Latent variable support vector machine (LVSVM)

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 $\mathit{X}=\left[{\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,\cdots ,{\mathit{x}}_N\right],{\mathit{x}}_n\in R^P$, and their labels $\mathit{y}=\left(y_1,y_2,y_3,\cdots ,y_N\right),\in \left\{-1,+1\right\}$. It is defined as

$\underset{\mathit{\eta },{\xi }_n}{\mathit{min}}\frac{1}{2}{\Vert \mathit{\eta }\Vert }_2^2+C_0{\displaystyle \sum }_{n=1}^N{\xi }_{n}s.t.y_n{\mathit{\eta }}^T{\stackrel{\sim }{\mathit{x}}}_n\ge 1-{\xi }_n,{\xi }_n\ge 0,n=1,...,N(10)$

where ${\stackrel{\sim }{\mathit{x}}}_n=\left[{\mathit{x}}_n;1\right]$ is the augmented feature vector, $\mathit{\eta }$ is the weighted coefficient and $C_0$ is a positive tuning parameter. The underlying discriminative objective is a linear hinge loss function, $\mathit{max}\left(1-y_n{\mathit{\eta }}^T{\stackrel{\sim }{\mathit{x}}}_n,0\right)$, 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

$p\left(\mathit{\eta },\mathit{\lambda }|\mathit{y}\right)$

$\propto {\displaystyle \prod }_{n=1}^N{\lambda }_n^{-\frac{1}{2}}\mathit{exp}\left(-\frac{({\lambda }_n-C_0(1-y_n{\mathit{\eta }}^T{\stackrel{\sim }{\mathit{x}}}_n){)}^2}{2{\lambda }_n}\right)\mathcal{N}\left(\mathit{\eta };0,\mathit{I}\right)(11)$

4 Methodology

4.1 dpMMSPFA model

As is mentioned above, for each column vector $\mathit{x}$ of FA model, it can be written as

${\mathit{x}}_{P\times 1}={\mathit{D}}_{P\times K}{\mathit{s}}_{K\times 1}+{\mathit{\epsilon }}_{P\times 1}(12)$

and for each column vector $\mathit{u}$ of SP item, it can be written as

${\mathit{u}}_{N\times 1}={\mathit{A}}_{N\times K}{\mathit{s}}_{K\times 1}+{\mathit{\phi }}_{N\times 1}(13)$

In dpMMSPFA, the hidden variable $\left\{\mathit{s}{\}}_{n=1}^N\right.$ is clustered and it can be supposed that each cluster of hidden variables are subordinated to the Gaussian distribution:

${\mathit{s}}_n|z_n=c,{\Theta }_c\sim \mathcal{N}\mathcal{W}\left({\mathit{s}}_n;{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right)(14)$

where ${\Theta }_c$ represents the Gaussian distribution parameter of the $c\mathrm{t}\mathrm{h}$ cluster, $\left\{{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right\}$. A joint normal-Wishart distribution is employed as the conjugate prior of ${\Theta }_c$, then it can be expressed as

$G_0=p\left({\Theta }_c\right)=\mathcal{N}\mathcal{W}\left({\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c|{\mathbf{\mu }}_0,{\mathbf{W}}_0,{\beta }_0,{\upsilon }_0\right)(15)$

On the basis of the DPM’s stick-breaking expression, FA model, SP term and LVSVM, our the hierarchical dpMMSPFA model is described as

${\mathit{d}}_k\sim \mathcal{N}\left({\mathit{d}}_k;0,{\mathbf{\Lambda }}_{{\mathit{d}}_k}\right);{\mathit{x}}_n|{\mathit{s}}_n,\mathit{D}\sim \mathcal{N}\left({\mathit{x}}_n;\mathit{D}{\mathit{s}}_n,{\mathbf{\Lambda }}_{\mathit{\phi }}\right)$

${\mathit{a}}_k\sim \mathcal{N}\left({\mathit{a}}_k;0,{\mathbf{\Lambda }}_{{\mathit{a}}_k}\right);{\mathbf{u}}_n|{\mathit{s}}_n,\mathit{A}\sim \mathcal{N}\left({\mathit{x}}_n;\mathit{{\rm A}}{\mathit{s}}_n,{\mathbf{\Lambda }}_{\mathit{\epsilon }}\right)$

${\mathit{s}}_n|z_n=c,{\Theta }_c\sim p\left({\mathit{s}}_n|{\Theta }_c\right),n=1,...,N$

${\mathit{\upsilon }}_c|\alpha \sim Beta\left(1,\alpha \right);{\Theta }_c\left|G_0\sim G_0;z_n\right|\mathit{\pi }\left(\mathit{\upsilon }\right)\sim Mult\left(\mathit{\pi }\left(\mathit{\upsilon }\right)\right)$

${\mathit{\eta }}_c,\left\{{\lambda }_n{\}}_c\right|\{{\mathit{s}}_n,{\mathit{y}}_n,z_n=c{\}}_{n=1}^N\sim p\left({\mathit{\eta }}_c,\left\{{\lambda }_n{\}}_c\right|-\right),c=1,...,\infty (16)$

We continue to add expressions and descriptions for proposed dpMMSPFA model:

${\mathit{d}}_k\sim N\left({\mathit{d}}_k;0,{\mathbf{\Lambda }}_{{\mathit{d}}_k}\right),{\mathbf{\Lambda }}_{{\mathit{d}}_k}=diag\left({\alpha }_{k1}^{-1},\dots ,{\alpha }_{kP}^{-1}\right),{\alpha }_{kp}\sim Gamma\left(a_0,b_0\right)$

${\mathit{a}}_k\sim N\left({\mathit{a}}_k;0,{\mathbf{\Lambda }}_{{\mathit{a}}_k}\right),{\mathbf{\Lambda }}_{{\mathit{a}}_k}=diag\left({\sigma }_{k1}^{-1},\dots ,{\sigma }_{kN}^{-1}\right),{\sigma }_{kn}\sim Gamma\left({\stackrel{\sim }{a}}_0,{\stackrel{\sim }{b}}_0\right)$

$\mathit{s}\sim N\left(\mathit{s};0,{\mathbf{\Lambda }}_{\mathit{s}}\right),{\mathbf{\Lambda }}_{\mathit{s}}=diag\left({\beta }_1^{-1},\dots ,{\beta }_K^{-1}\right),{\beta }_k\sim Gamma\left(c_0,d_0\right)$

$\mathit{\epsilon }\sim N\left(\mathit{\epsilon };0,{\mathbf{\Lambda }}_{\mathit{\epsilon }}\right),{\mathbf{\Lambda }}_{\mathit{\epsilon }}=diag\left({\gamma }_1^{-1},\dots ,{\gamma }_P^{-1}\right),{\gamma }_p\sim Gamma\left(e_0,f_0\right)$

$\mathit{\phi }\sim N\left(\mathit{\phi };0,{\mathbf{\Lambda }}_{\mathit{\phi }}\right),{\mathbf{\Lambda }}_{\mathit{\phi }}=diag\left({\omega }_1^{-1},...,{\omega }_N^{-1}\right),{\omega }_n\sim Gamma\left({\stackrel{\sim }{e}}_0,{\stackrel{\sim }{f}}_0\right)(17)$

where hyper-priors ${\alpha }_{kp}\sim Gamma\left(a_0,b_0\right)$, ${\sigma }_{kn}\sim Gamma\left({\stackrel{\sim }{a}}_0,{\stackrel{\sim }{b}}_0\right)$ and ${\omega }_n\sim Gamma\left({\stackrel{\sim }{e}}_0,{\stackrel{\sim }{f}}_0\right)$ are employed with $\left(a_0,b_0\right)$, $\left(c_0,d_0\right)$, $\left(e_0,f_0\right)$, $\left({\stackrel{\sim }{a}}_0,{\stackrel{\sim }{b}}_0\right)$ and $\left({\stackrel{\sim }{e}}_0,{\stackrel{\sim }{f}}_0\right)$ as the corresponding hyper-parameters.where ${\mathit{\eta }}_c$ and $\left\{{\lambda }_n{\}}_c\right.$ represent the weighted coefficient and the augmented variables of the LVSVM classifier related to the $c\text{th}$ cluster. The complete pseudo-posterior of parameters can be expressed as

$p\left(\mathit{D},\mathit{A},\mathit{S},{\left\{{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right\}}_{c=1}^C,{\left\{{\mathbf{\Lambda }}_{{\mathit{d}}_k}\right\}}_{k=1}^K,{\left\{{\mathbf{\Lambda }}_{{\mathit{a}}_k}\right\}}_{k=1}^K,Z,\mathit{v},{\left\{\mathit{\eta }\right\}}_{c=1}^C,\lambda ,{\mathbf{\Lambda }}_{\mathit{\epsilon }},{\mathbf{\Lambda }}_{\mathit{\phi }}|\mathit{X},\mathit{y}\right)$

$={\displaystyle \prod }_{c=1}^C{\displaystyle \prod }_{n=1,z_n=c}^{N}p\left(z_n|\mathit{\upsilon }\right)p\left({\mathit{x}}_n|\mathit{D},\mathit{S},{\mathbf{\Lambda }}_{\mathit{\epsilon }}\right)p\left({\mathit{u}}_n|\mathit{A},\mathit{S},{\mathbf{\Lambda }}_{\mathit{\phi }}\right)$

$\times {\displaystyle \prod }_{c=1}^C{\displaystyle \prod }_{n=1,z_n=c}^{N}p\left({\mathit{s}}_n|{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right)p\left({\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right)p\left({\mathit{\eta }}_c\right)\varphi \left(y_n,{\lambda }_n|{\mathit{s}}_n,{\mathit{\eta }}_c\right)p\left(\mathit{\upsilon }|\alpha \right)p\left(\alpha \right)$

$\times {\displaystyle \prod }_{k=1}^{K}p\left({\mathit{d}}_k|{\mathbf{\Lambda }}_{{\mathit{d}}_k}\right)p\left({\mathbf{\Lambda }}_{{\mathit{d}}_k}\right)p\left({\mathbf{\Lambda }}_{\mathit{\epsilon }}\right)p\left({\mathit{a}}_k|{\mathbf{\Lambda }}_{{\mathit{a}}_k}\right)p\left({\mathbf{\Lambda }}_{{\mathit{a}}_k}\right)p\left({\mathbf{\Lambda }}_{\mathit{\phi }}\right)(18)$

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 ${\mathit{s}}_n$: the conditional distribution of ${\mathit{s}}_{\mathit{n}}$ is

$p\left({\mathit{s}}_n|-\right)\propto \mathcal{N}\left({\mathit{x}}_n;\mathit{D}{\mathit{s}}_n,{\mathbf{\Lambda }}_{\mathit{\epsilon }}\right)\mathcal{N}\left({\mathit{u}}_n;\mathit{A}{\mathit{s}}_n,{\mathbf{\Lambda }}_{\mathit{\phi }}\right)\mathcal{N}\left({\mathit{s}}_n;{\mathbf{\mu }}_{z_n},{\mathbf{\Sigma }}_{z_n}\right)$

$\times \frac{1}{\sqrt{2\pi {\lambda }_n}}\mathit{exp}-\frac{({\lambda }_n+C_0(l-y_n{\mathit{\eta }}_{z_n}^T{\stackrel{\sim }{\mathit{s}}}_n){)}^2}{2{\lambda }_n}$

$\sim \mathcal{N}\left({\mathit{s}}_n;{\mathit{\mu }}_{{\mathit{s}}_{\mathit{n}}},{\mathbf{\Lambda }}_{{\mathit{s}}_n}\right)(19)$

where the posterior mean is ${\mathbf{\mu }}_{{\mathit{s}}_n}={\mathbf{\Lambda }}_{s_n}\left({\mathit{D}}_k^T{\mathbf{\Lambda }}_{\mathit{\epsilon }}^{-1}{\mathit{x}}_n+{\mathit{A}}_k^T{\mathbf{\Lambda }}_{\mathit{\phi }}^{-1}{\mathit{u}}_n+{\mathbf{\Sigma }}_{z_n}^{-1}{\mathbf{\mu }}_{z_n}+C_0\frac{\left({\lambda }_n+C_0-C_0{y}_n{\eta }_0\right)}{{\lambda }_n}{y}_n{\mathit{\eta }}_{z_n}\right)$ and the covariance matrix is ${\mathbf{\Lambda }}_{{\mathit{s}}_n}=\left({\mathit{D}}^T{\mathbf{\Lambda }}_{\mathit{\epsilon }}^{-1}\mathit{D}+{\mathit{A}}^T{\mathbf{\Lambda }}_{\mathit{\phi }}^{-1}\mathit{A}+{\mathbf{\Sigma }}_{z_n}^{-1}+C_0^2{\mathit{\eta }}_{z_n}{\mathit{\eta }}_{z_n}^T/{\lambda }_n\right)$.

For $z_n$: the conditional distribution of $z_n$ is

$z_n\sim Mult\left({\mathbf{\kappa }}_n\right),{\mathit{\kappa }}_n=\left[{\kappa }_{n1},\dots ,{\kappa }_{nC}\right]$

${\kappa }_{nc}=p\left(z_n=c|{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c,\mathit{v},{\mathit{\eta }}_c,{\lambda }_n\right)$

$\propto {\pi }_c\mathcal{N}\left({\mathit{s}}_n;{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right)\mathit{exp}(-\frac{\left({\lambda }_n+C_0(l-y_n{\mathit{\eta }}_c^T{\stackrel{\sim }{\mathit{s}}}_n){)}^2\right.}{2{\lambda }_n})(20)$

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.

FIGURE 2

FIGURE 2. The classification system frame for dpMMSPFA.

4.2 Model learning

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.

TABLE 1

TABLE 1. The training stage of dpMMSPFA model.

4.3 Model prediction

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.

TABLE 2

TABLE 2. The prediction stage of dpMMSPFA model.

The test data is pre-processed first, and the hidden variable $\mathit{s}$ and the cluster index $z$ of $\mathit{x}$ are sampled unsupervised on the basis of their related posterior Eq. 21 and Eq. 22, respectively.

$p\left(\mathit{s}|\mathit{D},z,{\mathbf{\mu }}_z,{\mathbf{\Sigma }}_z\right)\propto \mathcal{N}\left(\mathit{x};\mathit{D}\mathit{s},{\mathbf{\Lambda }}_{\mathit{\epsilon }}\right)\mathcal{N}\left(\mathit{s};{\mathbf{\mu }}_z,{\mathbf{\Sigma }}_z\right)(21)$

$z\sim Mult\left(\mathbf{\kappa }\right),\mathit{\kappa }=\left[{\kappa }_1,\dots ,{\kappa }_C\right]$

${\kappa }_c=p\left(z=c|{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c,\mathit{v},{\mathit{\eta }}_c,{\lambda }_n\right)\propto {\pi }_c\mathcal{N}\left(\mathit{s};{\mathbf{\mu }}_c,{\mathbf{\Sigma }}_c\right)(22)$

All the collected samples are averaged from Gibbs sampler to predict the label $y$ of $\mathit{x}$ as

$\widehat{y}=sign\left(\frac{1}{T_0}{\displaystyle \sum }_{t=1}^{T_0}{\mathit{\eta }}_{z_t}^T{\stackrel{\sim }{\mathit{s}}}_t\right)(23)$

where ${\mathit{s}}_t$ represents the $t\mathrm{t}\mathrm{h}$ sample of the hidden variable, which is sampled with the $t\mathrm{t}\mathrm{h}$ gathered samples, $z_t$ represents the $t\mathrm{t}\mathrm{h}$ sample of the cluster index which the test data belongs to and ${\mathit{\eta }}_{z_t}$ represents the $t\mathrm{t}\mathrm{h}$ gathered SVM coefficients of cluster $z_t$.

5 Experiments

5.1 Classifiers and parameters setting

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 $\left\{\mathrm{0.001,0.01,0.1,1,10}\right\}$.

5.2 UCI benchmark data

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.

TABLE 3

TABLE 3. Characteristics of datasets used in experiments.

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.

FIGURE 3

FIGURE 3. Test accuracies of different models (A) Breast-cancer; (B) German; (C) Heart; (D) Waveform; (E) Diabetis; (F) Splice.

FIGURE 4

FIGURE 4. Approximate posterior distribution on the number of clusters by dpMMSPFA (A) Breast-cancer; (B) German; (C) Heart; (D) Waveform; (E) Diabetis; (F) Splice.

5.3 Server energy consumption data

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.

TABLE 4

TABLE 4. Server energy consumption level and corresponding range.

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).

TABLE 5

TABLE 5. Configuration of Inspur NF5280M5 server.

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

TABLE 6

TABLE 6. Characteristic parameters.

Thermodynamic chart is a graphical representation of pattern similarity, with each component value in the matrix representing a different colour. The heatmap, the visualized ${{″\mathit{U}}_l}^{″}$ 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.

FIGURE 5

FIGURE 5. Similarity heatmap of sparse coding matrix represented by cosine function.

Since the Gamma hyper-parameters, ${a_0,b}_0,{\stackrel{\sim }{a}}_0,{\stackrel{\sim }{b}}_0,e_0,f_0,$ and ${\stackrel{\sim }{e}}_0,{\stackrel{\sim }{f}}_0$ may indirectly influence the inference of corresponding parameters, the transformation matrix $\mathit{D}$ and $\mathit{A}$, and the noise matrix $\mathit{\epsilon }$ and $\mathit{\phi }$, 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, ${\stackrel{\sim }{a}}_0$ =10⁻², ${\stackrel{\sim }{b}}_0$ =10⁻³, $e_0$ =10⁻¹, $f_0$ =10⁻², ${\stackrel{\sim }{e}}_0$ =10⁻², ${\stackrel{\sim }{f}}_0$ =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.

FIGURE 6

FIGURE 6. The recognition effects on the hyper-parameters.

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.

FIGURE 7

FIGURE 7. Clustering results by dpMMSPFA.

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 8

FIGURE 8. Boxplot of clustering effectiveness comparison.

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.

FIGURE 9

FIGURE 9. Test accuracy on energy consumption classification under different components.

6 Conclusion and future work

For the classification of energy consumption, we develop the Dirichlet maximum marginal similarity preservation factor analysis (dpMMSPFA) model. The Bayesian statistical method and the maximal marginal criterion for classifiers are merged into a single framework by the data augmentation community. In the hidden space that FA extracts, dpMMSPFA concurrently learns the underlying structure of observation data, similarity preservation items, clusters, and classifiers. In summing up it can be stated that the usefulness and efficiency of our suggested dpMMSPFA model have been verified by experiments on datasets of measured server energy.

The Beta process is a crucial model in the nonparametric Bayesian field in addition to the Dirichlet model. When the classifier is given the original samples, it can nevertheless produce stunning recognition results for highly complex, multimodal, and dimensional data without the application of any dimension reduction techniques. Imagine that in the future research on energy consumption modeling, we unify the factor analysis (FA), Beta process and hidden variable support vector machine (LVSVM) beneath the framework of Bayesian theory. The hope is that it will be remarkable potential to achieve to produce enormously excellent recognition performance. For the classifier underneath the nonparametric Bayesian framework, except for the hidden variable support vector machine (LVSVM), there is no nonparametric Bayesian classifier with robust generalization ability. In fact, in the classification model, the overall performance of random forest (RF) is almost equal to that of SVM classifier, and it can even produce higher classification performance for low dimensional data. It deserves to be further studied the transformation of the random forest classification model into an actual random forest model underneath the nonparametric Bayesian framework, which will be a huge leap for nonparametric Bayesian models.

Additionally, as hardware has improved, a popular research topic is the energy consumption model for offloading server power to smart network interface cards (smart NIC), distributed processing units (DPU), graphics processing units (GPU), field programmable gate arrays (FPGA), and other hardware resources. In the future, we shall find out more about and explore the creation of an energy consumption model for intelligent hardware units.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

BC: Project design, Methodology, Software, Simulation, Writing-Original draft preparation, Content revision. HL: Data collection, Experiments discussion, Content revision. CS: Project discussion. BS: Discussion on experimental environment. KL: Supervision.

Funding

This work received the funding from the Science and Technology Project of Research Institute of China Telecom.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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