Bayesian Sparse Regression for Mixed Multi-Responses with Application to Runtime Metrics Prediction in Fog Manufacturing

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Bayesian Sparse Regression for Mixed

Multi-Responses with Application to Runtime

Metrics Prediction in Fog Manufacturing

Xiaoyu Chen1, Xiaoning Kang2, Ran Jin3and Xinwei Deng4

1Department of Industrial Engineering, University of Louisville, USA.

2International Business College and Institute of Supply Chain Analytics,

Dongbei University of Finance and Economics, China

3Grado Department of Industrial and Systems Engineering,

Virginia Tech, USA

4Department of Statistics, Virginia Tech, USA

Abstract

Fog manufacturing can greatly enhance traditional manufacturing systems through

distributed Fog computation units, which are governed by predictive computational

workload oﬄoading methods under diﬀerent Industrial Internet architectures. It is

known that the predictive oﬄoading methods highly depend on accurate prediction

and uncertainty quantiﬁcation of runtime performance metrics, containing multivariate

mixed-type responses (i.e., continuous, counting, binary). In this work, we propose a

Bayesian sparse regression for multivariate mixed responses to enhance the prediction

of runtime performance metrics and to enable the statistical inferences. The proposed

method considers both group and individual variable selection to jointly model the

mixed types of runtime performance metrics. The conditional dependency among

multiple responses is described by a graphical model using the precision matrix, where a

arXiv:2210.04811v2 [stat.ME] 11 Oct 2022

spike-and-slab prior is used to enable the sparse estimation of the graph. The proposed

method not only achieves accurate prediction, but also makes the predictive model more

interpretable with statistical inferences on model parameters and prediction in the Fog

manufacturing. A simulation study and a real case example in a Fog manufacturing

are conducted to demonstrate the merits of the proposed model.

Keywords: Graphical model, Mixed responses, Spike-and-slab prior, Variable Selection.

1 Introduction

Fog computing (also referred as Edge computing) techniques have served as an important

role in Industrial Internet of things (IIoT) for smart manufacturing systems. It provides

local and distributed computation capabilities. The concept of Fog manufacturing is de-

ﬁned on integrating a Fog computing network with interconnected manufacturing processes,

facilitates, and systems. With local computation units (i.e., Fog units) close to the man-

ufacturing processes, the Cloud-based centralized computation architecture can be evolved

to a Cloud-Fog collaborative computation to provide higher responsiveness and signiﬁcantly

lower time latency (Wu et al. 2017; Zhang et al. 2019). There is a trade-oﬀ between the

local computing eﬃciency on a Fog unit and the global collaborative eﬃciency of the cen-

tralized Cloud. Speciﬁcally, the speciality of Fog units can signiﬁcantly speedup the local

computations, but it can pose signiﬁcant challenges for the Cloud to assign the computation

tasks and supervise the heterogeneous Fog units. Besides, ﬂuctuated computation capability

of the Fog units and intermittent communication conditions among the Fog units and the

Cloud make it even harder for the collaboration (Zhang et al. 2015). Therefore, computation

oﬄoading methods have been widely investigated to enable eﬃcient collaboration between

the Fog units and the Cloud with the consideration of constraints on resources.

In Fog manufacturing, the runtime performance metrics are often multivariate with mixed

types (Chen et al. 2018). These metrics include the CPU utilization (i.e., continuous re-

sponse), temperature of the CPU (i.e., continuous response), the number of computation

tasks executed within a certain time period (i.e., counting response), and whether the mem-

ory utilization exceeds certain thresholds (i.e., binary response). Prediction and uncertainty

quantiﬁcation of these metrics are essential to support the computation in the Fog manu-

facturing, advancing analytics and optimization for high responsiveness and reliability (Wu

et al. 2017; Zhang et al. 2019). Based on the runtime performance metrics of these Fog

nodes, the Fog computing can dynamically assign computation tasks to diﬀerent Fog nodes

(Chen et al. 2018). The manufacturing must provide responsive and reliable computation

services by meeting all requirements in runtime performance metrics. It is thus of great

importance to accurately predict runtime performance metrics of Fog nodes and quantify

the uncertainty of prediction in task assignment and oﬄoading problems.

As the runtime performance metrics are multivariate with mixed types, a simple method

is to model each individual metric separately. Clearly, such an approach overlooks the de-

pendency relationship among the metrics, resulting in inaccurate prediction associated with

high uncertainty. For example, as the increment in the executed number of computation

tasks per minute (i.e., counting response), the CPU utilization and temperature (i.e., con-

tinuous responses) will increase. Quantifying such dependency among mixed responses is

expected to improve the prediction accuracy. Moreover, by only providing point estimation

of mixed responses, the model prediction may not be trustworthy for those with high predic-

tion variance. Therefore, it calls for a joint model for the mixed responses with uncertainty

quantiﬁcation. Towards predictive oﬄoading, the objective is to jointly ﬁt the mixed runtime

performance metrics with the capability of statistical inferences to quantify uncertainties of

the predicted metrics in Fog manufacturing.

In this work, we propose a Bayesian sparse multivariate regression for mixed responses

(BS-MRMR) to achieve accurate model prediction and, more importantly, to obtain proper

statistical inferences of the responses. The use of Bayesian estimation naturally enables

uncertainty quantiﬁcation of model prediction. Both group sparsity and individual sparsity

are imposed on regression coeﬃcients via proper spike-and-slab priors. The group structures

often occur in the runtime performance metrics prediction problem when the metrics at the

next time instance are regressed on two groups of predictors: the features extracted from the

current and previous metrics (i.e., Group 1) and the covariates of the computation tasks (i.e.,

Group 2). On the other hand, not all predictors are important within each group. Hence

the individual sparsity is also induced for better estimation of model coeﬃcients. More-

over, the proposed method considers the conditional dependency among multiple responses

by a graphical model using the precision matrix, where a spike-and-slab prior is used to

enable the sparse estimation of the graph. A Gibbs sampling scheme is then developed to

eﬃciently conduct model estimation and inferences for the proposed BS-MRMR method.

The proposed BS-MRMR model not only achieves accurate prediction, but also makes the

predictive model more interpretable in the Fog manufacturing. Note that one can consider a

two-step Bayesian method to model the multivariate mixed responses Bradley (2022), where

the ﬁrst step transforms the multivariate mixed-responses to continuous responses, and the

second step models the transformed responses. However, the obtained model coeﬃcients are

less interpretable since the transformation typically change the scale of the original responses.

Diﬀerent from the recent work of Kang et al. (2021) on a penalized regression for multi-

variate mixed responses, the proposed BS-MRMR is a Bayesian approach with the following

key novelty. First, Kang et al. (2021) only imposes individual sparsity while the BS-MRMR

model takes into account of both the group and individual sparsity. Second, the model intro-

duced by Kang et al. (2021) cannot provide statistical inferences, such as prediction intervals

for the responses due to their complicated parameter estimation procedure. In contrast, the

proposed BS-MRMR model is able to quantify the uncertainty of the estimated parameters

and predicted responses within the Bayesian framework. It provides a comprehensive infor-

mation of prediction and uncertainty quantiﬁcation to support the predictive oﬄoading in

Fog manufacturing. Third, a careful investigation of the posterior distribution makes the

computation of the Gibbs sampling eﬃcient for model estimation and inference.

The remainder of this work is organized as follows. The proposed BS-MRMR model and

the Gibbs sampling scheme are detailed in Section 3. A simulation study is conducted to

validate the BS-MRMR model in Section 4. Section 5 describes a real case study in Fog

manufacturing. Section 6 concludes this work with some discussions of future directions.

2 Literature Review

The joint modeling of mixed responses has attracted great attention in the literature. Various

existing studies focused on the bivariate responses. For example, Fitzmaurice and Laird

(1995) considered a bivariate linear regression model with a continuous and a binary response

via joint likelihood estimation. Yang et al. (2007) proposed to jointly ﬁt a continuous and a

counting response, and evaluated the correlation between the bivariate responses varying over

time through a likelihood ratio test. These methods usually factorize the joint distribution

of two responses as the product of a marginal and a conditional distribution (Cox and

Wermuth 1992), which cannot be easily generalized for multivariate mixed responses in real

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BayesianSparseRegressionforMixedMulti-ResponseswithApplicationtoRuntimeMetricsPredictioninFogManufacturingXiaoyuChen1,XiaoningKang2,RanJin3andXinweiDeng41DepartmentofIndustrialEngineering,UniversityofLouisville,USA.2InternationalBusinessCollegeandInstituteofSupplyChainAnalytics,DongbeiUniversityofFi...

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Bayesian Sparse Regression for Mixed Multi-Responses with Application to Runtime Metrics Prediction in Fog Manufacturing

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