Semi‐supervised dynamic latent variable modeling: I/O probabilistic slow feature analysis approach

Abstract

Modeling of high dimensional dynamic data is a challenging task. The high dimensionality problem in process data is usually accounted for using latent variable models. Probabilistic slow feature analysis (PSFA) is an example of such an approach that accounts for high dimensionality while simultaneously capturing the process dynamics. However, PSFA also suffers from a drawback that it cannot use output information when determining the latent slow features. To address this lacunae, extension of the PSFA by incorporating outputs, resulting in Input-Output PSFA (IOPSFA) is proposed. IOPSFA can use both input and output information for extracting latent variables. Hence, inferential models based on IOPSFA are expected to have better predictive ability. The efficacy of the proposed approach with an industrial and a laboratory scale soft sensing case studies that have both complete and incomplete output measurements is evaluated, respectively.

A generalized probabilistic monitoring model with both random and sequential data

“…where, [0, 1] i   is a parameter which is used to determine the similarity between the i-th variable in t s and t z . In recent years, an input-output probabilistic SFA (I/O PSFA) is proposed with both input and output information (Fan et al, 2019). It is noted that the formulations of GPMM and I/O PSFA are different, which finally results in different estimations of model parameters.…”

Nonlinear Semi-supervised Inference Networks for the Extraction of Slow Oscillating Features

“…Further, z z z k is formed by stacking the oscillatory slow feature s s s k ∈ R ms and drift-type nonstationary feature h h h k ∈ R m h . The generative model for the oscillating slow feature inference network is shown in (19). p θ (y y y 1:N , t t t 1:N T ) = p θ (y y y 1:N , t t t 1:N −T , z z z 0:N )dz z z 0:N (19) where…”

Nonlinear Semi-supervised Inference Networks for the Extraction of Slow Oscillating Features

“…Unlike SFA, the parameters are estimated using a procedure called expectation-maximization. Several deterministic [18] and probabilistic [19] extensions were proposed to extract quality-relevant slow features. Recently, complex probabilistic slow feature analysis [20] was proposed to model the stationary oscillatory patterns explicitly in the feature space.…”