A Hidden-Gamma Model-Based Filtering and Prediction Approach for Monotonic Health Factors in Manufacturing Gian Antonio Sustoa,∗, Andrea Schirrub, Simone Pampurib, Alessandro Beghia, Giuseppe De Nicolaoc aDepartment of Information Engineering, University of Padova, via G. Gradenigo 6/B, 35131 Padova, Italy. bStatwolf LTD, 38-39 Baggot Street Lower, Dublin, Ireland. cDepartment of Computer Engineering and Systems Science, University of Pavia, via Ferrata 1, 27100 Pavia, Italy. Abstract In the context of Smart Monitoring and Fault Detection and Isolation in industrial systems, the aim of Predictive Maintenance technologies is to predict the happening of process or equipment faults. In order for a Predictive Main- tenance technology to be effective, its predictions have to be both accurate and timely for taking strategic decisions on maintenance scheduling, in a cost-minimization perspective. A number of Predictive Maintenance technologies are based on the use of "health factors", quantitative indicators associated with the equipment wear that exhibit a monotone evolution. In real industrial environment, such indicators are usually affected by measurement noise and non-uniform sampling time. In this work we present a methodology, formulated as a stochastic filtering problem, to optimally predict the evolution of the aforementioned health factors based on noisy and irregularly sampled observations. In particular, a hidden Gamma process model is proposed to capture the nonnegativity and nonnegativity of the derivative of the health factor. As such filtering problem is not amenable to a closed form solution, a numerical Monte Carlo approach based on particle filtering is here employed. An adaptive parameter identification procedure is proposed to achieve the best trade-off between promptness and low noise sensitivity. Furthermore, a methodology to identify the risk function associated to the observed equipment based on previous maintenance data is proposed. The present study is motivated and tested on a real industrial Predictive Maintenance problem in semiconductor manufacturing, with reference to a dry etching equipment. Keywords: Decision Support System, Gamma Distribution, Industry 4.0, Monte Carlo Methods, Particle Filters, Predictive Maintenance, Prognostic and Health Management, Semiconductor Manufacturing 1. Introduction or, more generally, to failures that can be predicted in ad- vance [8, 9]. Examples of such type of faults are the break- Advanced monitoring is a fundamental activity in the ing of the source in ion-implantation processes in semicon- Industry 4.0 scenario to implement control, maintenance, ductor manufacturing [7], the flute wear in cutting tool quality, reliability, and safety policies [1, 2, 3]. In particu- equipment [10], and the lifespan of lithium-ion batteries lar, Fault Detection and Isolation (FDI) [3] and Predictive [11]. Maintenance (PdM) [4] technologies have proliferated in In this work we focus on the so-called 'Health Fac- the past recent years for diagnosis and prognosis of pro- tors' (HFs), an important concept in prognostic1. HFs cess/tool failures [5]. While the aim of such technologies is are quantitative indexes used to define the current sta- similar and partly overlapped, PdM technologies are more tus of a tool/process and to assess the future statuses of focused on prognosis. Prognosis can be defined as the ca- the system under exam (or of one of its components/sub- pability to provide early detection of the precursor and/or systems), and its Remaining Useful Life (RUL) [14, 16, 17], incipient fault condition of a component, and to design so that strategic decisions regarding maintenance schedul- tools for managing and predicting the progression of such ing and dynamic sampling plans can be taken [4]. Be- fault condition to component failure [6]. Given their goal, ing in direct relationship with wear, usage or stress of an PdM technologies are typically applied to failures that are equipment/component or system, HFs generally have a associated with wear and usage of the system/process [7], monotone evolution. A HF can be of very different na- ture: in its simplest form, HFs can be observable param- ∗Corresponding author Email addresses: [email protected] (Gian Antonio Susto), [email protected] (Andrea Schirru), 1Health Factors are also indicated as 'Component Health' [5], [email protected] (Simone Pampuri), 'State of Health'/'Health State' [12, 13] or as 'Health Indicators' [email protected] (Alessandro Beghi), [10, 14] by different authors and they are closely in relation with the [email protected]. (Giuseppe De Nicolao) concept of 'degradation data' [15]. Preprint submitted to Elsevier March 7, 2019 eters that, thanks to specific domain expertise, can be as- proach proposed in this work considers the HF as a mono- sociated with equipment/process health status. Example tonic Gamma process (with time-varying shape parame- of health factors as quantities that are directly related to ter) corrupted by Gaussian noise (hidden-Gamma model). system health, such as the thermal index of a polymeric Such assumptions lead to the lack of closed-form solutions material [18], the scar width in sliding metal wear [19], and for the estimation of model parameters in the proposed the temperature difference in semiconductor manufactur- approach. However, it will be shown that the prediction ing epitaxy processes [20]. HFs can also be the output of problem can be efficiently solved by resorting to particle fil- Soft Sensor modules [21, 22], where the status health is im- tering methods [43, 44, 45], employing Monte Carlo (MC) possible/costly to be monitored. Moreover, HFs can be the simulations to derive the target posterior distributions. Fi- residual of first principle FDI models [23]. In fact, in many nally, a recursive procedure to estimate the time-varying practical examples [1, 17, 23, 24], residuals have a mono- shape parameter is proposed. Such procedure allows to tonic behaviour and threshold-based policies to mainte- optimize a trade-off between the need for promptness and nance management are implemented on such quantities. noise insensitivity/outlier rejection. HFs are therefore relevant quantities in both model-based The paper is organized as follows. In Section 2 the [24, 25, 26] and model-free [1, 3, 27, 28] prognostic ap- hidden-Gamma model is presented. In Section 3.1 the proaches. principles of Particle Filtering (PF) are briefly summa- In the present paper, the problem of designing a HF rized and adapted to Gamma processes. In Section 4 an for Predictive Maintenance (PdM) purposes is considered adaptive recursive scheme for estimation of monotone HFs [7, 29, 30]. In particular, the issue of assessing the probabi- is presented. Section 5 is dedicated to the definition and lity distribution of the HF future values given its past mea- estimation of an appropriate Risk Function for the pro- surements is addressed, under the following assumptions: posed model. In Section 6 some experimental results on (i) the HF is monotonically increasing; (ii) its measure- synthetic datasets are reported, whereas in Section 7 a real ments are subject to random noise that may conceal its PdM semiconductor manufacturing problem related to dry monotonic nature; (iii) measurements are non-uniformly etching is tackled. 2 sampled over time. The aforementioned features are typi- cal traits of HF signals [17, 20, 31, 32, 33], but they are gen- erally not simultaneously accounted for in the related liter- 2. The Hidden Gamma Process ature. Non-stochastic models (see [12] for a broad review 2.1. Gamma Probability Distribution on RUL estimation) for HFs have been presented in litera- ture, as well as inspection and intervention approaches for The most notable property of Gamma distributions is increasing maintenance actions effectiveness and decreas- their non-negative support. We consider a random vari- ing the associated costs. However, such methodologies are able x with Gamma distribution Γ(a; θ), where a is the well suited for noise-free scenarios and, given the aforemen- shape parameter and θ is the scale factor. The first two tioned assumptions on the HF signals, it is here proposed statistical moments of x are E[x] = aθ and V ar[x] = aθ2 to adopt a stochastic filtering paradigm [34]. With the and the probability density function (PDF) is p(x) = − x proposed approach, the HF is treated as a stochastic pro- xa−1e θ Γ(a)θa . Gamma distributed random variables enjoy the cess, with the possibility to combine prior knowledge on following property: the HF with statistical information regarding the observed Property 1 (Infinite Divisibility): if x1 ∼ Γ(a1; θ) and noisy data. A simple approach to deal with the problem x2 ∼ Γ(a2; θ), then the sum x = x1 + x2 has a Gamma at hand is provided by the Wiener and Kalman predictors distribution with shape a1 + a2 and scale factor θ. [20, 35, 36, 37], which are statistically optimal for linear The shape of the Gamma probability distribution for Gaussian models. However, such classical approaches may different values of a and θ is shown in Figure 1. be considered suboptimal for signals with the character- istics given in assumptions (i)-(iii). As a matter of fact, 2.2. The Hidden-Gamma Model far from being Gaussian, the HF derivative is in this work considered to be a nonnegative random variable. In the following, the HF is denoted by x(·). Measure- Given such premises, a framework for HF filtering and ments x(t1); x(t2); : : : ; x(tk) are available for time instants prediction based on the Gamma distribution is here pro- t1 ≤ t2 ≤ ::: ≤ tk. We indicate with t0 ≤ t1 the ini- posed. PdM applications employing Gamma distributions tial time instant, and with tk+1 ≥ tk the instant points in has been developed since the 1970s [38], especially in me- which predictions of the HF are desired. chanical and civil engineering applications [39, 40, 41] and, In the following, we adopt a Bayesian paradigm by as- recently, in industrial environments [42].