Estimation of output measurement variances for error‐in‐variables model parameter estimation

Abstract

Error-in-variables model (EVM) methods require information about variances of input and output measured variables when estimating the parameters in mathematical models for chemical processes. In EVM, using replicate experiments for estimating output measurement variances is complicated because true values of inputs may be different when multiple attempts are made to repeat an experiment. To address this issue, we categorize attempted replicate experiments as: (i) true replicates (TRs) when uncertain inputs are the same in replicated runs and (ii) pseudo replicates (PRs) when measured inputs are the same, but unknown true values of inputs are different. We propose methodologies to obtain output measurement variance estimates and associated parameter estimates for both situations. We also propose bootstrap methods for obtaining joint-confidence information for the resulting parameter estimates.A copolymerization case study is used to illustrate the proposed techniques. We show that different assumptions noticeably affect the uncertainties in the resulting reactivity-ratio estimates.copolymerization, error-in-variables model, Mayo-Lewis equation, reactivity-ratio estimation, replicate experiments | INTRODUCTIONFundamental mathematical models are widely used for chemical process development and improvement. These models usually contain unknown parameters that require estimation. 1,2 In conventional parameter-estimation methodologies such as weighted least-squares (WLS) estimation, the model inputs are assumed to be perfectly known (i.e., measured without error), while random measurement errors are considered for the model outputs. 3,4 Nevertheless, there are important situations where model inputs are not perfectly known, but contain significant uncertainties. The error-in-variables model (EVM) technique was developed to account for measurement uncertainties in both inputs and outputs during parameter estimation. 5,6 In

Parameter estimation and prediction uncertainties for multi-response kinetic models with uncertain inputs

“…In the chemical engineering literature, output measurement variances are also usually assumed to be known a priori in most EVM parameter estimation studies. 4 However, modelers can also use replicate data to estimate the output measurement variances 𝜎 𝑌 𝑘 2 . 2,[4][5][6] The resulting variance estimates should account for the measurement variability for the output responses as well as all other sources of experimental variations (e.g., ambient condition and sample collection ) except variability associated with 𝒖 𝒊 .…”

“…In TR situations, quantifying output measurement variances 𝜎 𝑌 𝑘 2 is relatively straightforward (e.g., using pooled variance estimates from several different experimental conditions). 4 Estimation of output measurement variances from pseudoreplicate data is more difficult because the contribution of uncertain inputs to the overall variability in the outputs should be taken into account. In a previous study, we devised a method to estimate 𝜎 𝑌 𝑘 2 based on pseudo-replicates, but we only considered a simple single-response copolymerization model.…”

“…In a previous study, we devised a method to estimate 𝜎 𝑌 𝑘 2 based on pseudo-replicates, but we only considered a simple single-response copolymerization model. 4 One of the objectives in the current paper is to extend this method for use in multi-response models. We use a n-hexane hydroisomerization model as a case study to illustrate the proposed methodology.…”

“…We also show how to obtain model prediction uncertainties based on the bootstrap parameter uncertainties, input uncertainties and estimated output uncertainties. In general, for a multi-response system with several uncertain inputs, an error-in-variables model of the process can be expressed as: 4 𝒀 𝒊𝒋 = 𝒈(𝒙 𝑖 , 𝒖 𝑖𝑗 , 𝜽) + 𝜺 𝒀 𝒊𝒋 (2) 𝒖 𝒎 𝒊 = 𝒖 𝒊𝒋 + 𝜺 𝑼 𝒊𝒋 (3) where 𝒀 𝒊𝒋 ∈ 𝑅 𝑁 𝑌 is a vector of uncertain measurements for 𝑁 𝑌 dependent variables obtained from the 𝑗 𝑡ℎ…”

“…The diagonal elements in 𝚺 𝐘 (i.e., 𝜎 𝑌 𝑘 2 where 𝑘 = 1, … , 𝑁 𝑌 ) are unknown and need to be estimated from the pseudo-replicate data. 4 In a previous study, we developed a linearization-based approach to obtain estimates of output measurement variances for each of the 𝑁 𝑌 dependent variables based on the pseudo-replicate data obtained at the 𝑖 𝑡ℎ target condition. For the 𝑘 𝑡ℎ dependent variable:…”