Gaussian process based modeling and experimental design for sensor calibration in drifting environments

Abstract

It remains a challenge to accurately calibrate a sensor subject to environmental drift. The calibration task for such a sensor is to quantify the relationship between the sensor's response and its exposure condition, which is specified by not only the analyte concentration but also the environmental factors such as temperature and humidity. This work developed a Gaussian Process (GP)-based procedure for the efficient calibration of sensors in drifting environments. Adopted as the calibration model, GP is not only able to capture the possibly nonlinear relationship between the sensor responses and the various exposure-condition factors, but also able to provide valid statistical inference for uncertainty quantification of the target estimates (e.g., the estimated analyte concentration of an unknown environment). Built on GP's inference ability, an experimental design method was developed to achieve efficient sampling of calibration data in a batch sequential manner. The resulting calibration procedure, which integrates the GP-based modeling and experimental design, was applied on a simulated chemiresistor sensor to demonstrate its effectiveness and its efficiency over the traditional method.

Introduction

Chemical sensors have been widely used in indoor and outdoor environment monitoring, vehicle exhaust measurement, human breath detection, etc [1], [2], [3]. It has been long recognized that the responses of chemical sensors, especially chemiresistors, are affected by the drift of environmental factors such as temperature and humidity [4], [5], [6], [7]. To reduce detection errors and false alarm, it is important to accurately calibrate a sensor in a drifting environment, which primarily motivated this work. The environmental factors are denoted as the vector x, and the task of sensor calibration is to establish the functional dependence of the sensor response r upon the analyte concentration c as well as x.

Quantifying the c − x − r relationship is challenging due to two main reasons: First, the variables (c, x) may affect the response r in a nonlinear fashion and also interact nonlinearly with each other. The underlying mechanism is complicated [8], [9], [10], [11] and difficult to be adequately captured by traditional regression analysis [6]. Second, to estimate a calibration model of high dimension, an extremely large sample size is typically required by the classic design of experiments (DOE) [6], [12]. Thus, there is a need to develop new modeling and DOE methods for the efficient calibration of sensors subject to environmental drift.

While focusing on calibrating sensors with environmental drift, this work falls into the research efforts to calibrate sensors with general drifting behaviors, which can be classified into two categories [13], [14]: external (i.e. environmental) and internal drifts. The latter is caused by the physical and/or chemical changes of the sensor itself, and examples of such changes include re-organization of the sensing materials and irreversible interaction with analytes. When calibrating drifting sensors, most of the literature used a reference-based linear compensation or linear regression to quantify the drifting effects [15], [16], [17], [18]. Recognizing the possible nonlinear nature of sensor drifts, powerful nonlinear models have also been employed, such as neural network [6], [19], kernel ridge regression [20] and nonlinear supporting vector machine [21]. However, in this stream of nonlinear modeling work, no effort was ever made to quantify the uncertainty of the target estimates (e.g., the analyte concentration estimated by the calibration model from an observed sensor response). This is at least partly due to the difficulties in deriving valid statistical inference (i.e., quantifying model uncertainty) based on those models [22], [23]. It is known that statistical inference lays the basis for optimum DOE: Experiments are designed to minimize the uncertainty on the model estimates of interest [24], [25], [26]. Thus, optimum DOE is a research issue that has barely been touched in the nonlinear model-based sensor calibration.

In light of the discussions above, our objective is to develop a statistical procedure, which leads to a calibration model of the highest quality by using the least experimental effort. In this work, the calibration model assumes the form of a Gaussian process (GP), which is highly flexible and able to capture practically any continuous functional relationships. GP is chosen over other powerful nonlinear models because of its statistical inference capability [27], which allows for uncertainty quantification and provides the necessary basis for optimum DOE. For sensor calibration, the inference issues are further complicated by the coexistence of forward modeling and inverse estimation (as will become clearer in Section 2.1), and hence a GP-based bootstrap resampling method is developed in this work. The DOE is performed in a batch sequential manner to circumvent the dilemma that the optimum DOE depends on the true c − x − r relationship, which however, is unknown at the stage of designing experiments [28], [25], [24]. A learning process is allowed in such a sequential procedure: For the design of a new batch of experiments to be performed, all the information derived from the experimental data already collected is utilized to search for the optimum DOE of that new batch; and the DOE optimization seeks to minimize the calibration model uncertainty with a given batch size.

The remainder of the paper is organized as follows: Section 2 presents the formulation of the calibration model, which takes the form of a GP. The GP-based model fitting and statistical inference issues are discussed in Section 3. The batch sequential procedure for sensor calibration is described in Section 4. Section 5 is devoted to an empirical study to evaluate the effectiveness and efficiency of the calibration procedure. A brief summary is given in Section 6.