Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2147935, 12 pages

https://doi.org/10.1155/2017/2147935

## Temperature Distribution Measurement Using the Gaussian Process Regression Method

^{1}School of Energy, Power and Mechanical Engineering, North China Electric Power University, Changping District, Beijing 102206, China^{2}Institute of Engineering Thermophysics, Chinese Academy of Sciences, Haidian District, Beijing 100190, China

Correspondence should be addressed to Huaiping Mu

Received 28 February 2017; Accepted 31 July 2017; Published 29 August 2017

Academic Editor: Carmen Castillo

Copyright © 2017 Huaiping Mu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The temperature distribution in real-world industrial environments is often in a three-dimensional space, and developing a reliable method to predict such volumetric information is beneficial for the combustion diagnosis, the understandings of the complicated physical and chemical mechanisms behind the combustion process, the increase of the system efficiency, and the reduction of the pollutant emission. In accordance with the machine learning theory, in this paper, a new methodology is proposed to predict three-dimensional temperature distribution from the limited number of the scattered measurement data. The proposed prediction method includes two key phases. In the first phase, traditional technologies are employed to measure the scattered temperature data in a large-scale three-dimensional area. In the second phase, the Gaussian process regression method, with obvious superiorities, including satisfactory generalization ability, high robustness, and low computational complexity, is developed to predict three-dimensional temperature distributions. Numerical simulations and experimental results from a real-world three-dimensional combustion process indicate that the proposed prediction method is effective and robust, holds a good adaptability to cope with complicated, nonlinear, and high-dimensional problems, and can accurately predict three-dimensional temperature distributions under a relatively low sampling ratio. As a result, a practicable and effective method is introduced for three-dimensional temperature distribution.

#### 1. Introduction

Three-dimensional (3D) temperature distribution plays an important role in combustion diagnosis tasks. The acquisition of the rich and accurate combustion temperature distribution details is of a great significance to the combustion adjustment and control. Currently, two kinds of approaches are available for achieving such task, for example, the numerical simulation technique and the experiment method. Owing to the challenges, such as high computational cost and complexity, the inaccurate properties of initial conditions, boundary conditions, geometrical conditions, and physical property parameters, it is hard for the former to achieve combustion diagnosis tasks in real-world applications. With the development of modern measurement technologies, the latter has attracted more and more attentions.

Conventional flame monitoring systems are only used to judge whether there is a flame in a combustion space, and it is difficult to realize the quantitative measurement of the combustion process parameters. A lot of technologies based on the radiation image and radiation energy signal processing had been developed for 3D temperature distribution measurements [1–6].

Optical CCD methods are based on the radiation transfer characteristics of the flue gas in a combustion system, which focus on the mathematical relationship between the radiation accumulation image and 3D radiation energy distributions, and thus the temperature distribution profiles are reconstructed via an appropriate algorithm [7–11]. Though this kind of method has been accepted for industrial measurements, the accuracy is relatively low, the measurement systems are complex and costly, and the final result is sensitive to exterior conditions. Particularly, the increase of the spatial scale of the measurement area will lead to the rapid degeneration of the precision.

The acoustic tomography (AT) method employs an appropriate algorithm to realize the reconstruction of the temperature distribution profiles from the given acoustic wave time-of-flight (TOF) data and has gained extensive acceptances in a variety of areas [12–14]. The effectiveness of the inversion method has a great influence on practical applications of the AT technology. Unfortunately, owing to the ill-posed nature in the AT inverse problem, seeking for an efficient inversion method with a good numerical stabilizing to ensure a high-precision reconstruction remains a formidable challenge. Particularly, with the augment of the spatial scale of the measurement domain, the above challenges become more serious.

Are there other measurement strategies for 3D temperature distributions? The development of modern measurement instruments makes it possible to measure scattered temperature values. Is it possible to predict 3D temperature distribution from the finite number of the temperature measurement data? If the answer is positive, a natural problem will appear, that is, how to achieve it? With such insight in mind, in this work, we put forward a new methodology to predict 3D temperature distributions from the finite local temperature measurement values, and the main highlights of the study can be summarized as follows.

(1) A new methodology is proposed to predict 3D temperature distribution from the limited number of the scattered temperature measurement data. The proposed method includes two key phases. In the first phase, traditional measurement technologies are employed to measure the local temperature values in a large-scale 3D measurement area. In the second phase, the GPR method is developed to predict 3D temperature distribution details. Numerical simulation results indicate that the execution the GPR method is easy and can accurately predict 3D temperature distributions under a relatively low SA. Furthermore, the GPR method holds satisfactory robustness and good adaptability of coping with complicated, nonlinear, and high-dimensional problems.

(2) A 3D combustion experiment system is constructed, and a series of combustion experiments are conducted. The effectiveness and robustness of the GPR method are further validated by experimental results. As a result, a practicable and efficient method is introduced for 3D temperature distribution measurements.

In accordance with the research target, the rest of this paper is arranged as follows. In Section 2, we propose a new methodology for 3D temperature distribution predictions, and the appealing superiorities are summarized. Section 3 introduces the principle of the GPR method. In Section 4, we specify the prediction procedure, and attractive superiorities are summarized. Section 5 performs numerical simulations and discusses numerical results. In Section 6, a 3D experiment system is constructed; a series of the combustion experiments are carried out to validate the practicability and the effectiveness of the proposed prediction method. Finally, we draw conclusions of the study in Section 7.

#### 2. Methodology

The development of modern measurement instruments makes it possible to measure scattered temperature values. In this study, the 3D temperature distribution measurement problem will predict unknown temperature distributions in terms of the provided finite observations.

For convenience, 3D temperature distribution predictions from the limited number of the measurement data can be formulized to be a tensor completion (TC) problem. Mathematically speaking, a 3D measurement area can be represented as a tensor by means of the discretization process, in which some entries are missing. The TC task tries to recovery the tensor with missing data in terms of a given sampling set , where is a subset of the complete set. For the convenience of calculation, the solution of the TC problem can be written as the following optimization problem with the introduction of the low rank constraint [15]:where the elements of and in the set are given while the remnant elements are missing.

It is noticeable that the TC problem has attracted the increasing attention over the past several years and has been successfully applied to signal and image processing area. A variety of numerical methods have been developed for solving the problem, and the interested readers are referred to [16, 17] for more details. But, it is essential to stress that seeking an efficient method to solve the TC problem is a formidable task. Besides, numerous applications indicate that the TC method requires a high SA to ensure a satisfactory solution. Obviously, in practical applications, it is hard to satisfy such requirement due to the limitation of measurement conditions and costs.

We find that if the mapping of the measurement point positions and the corresponding temperature values are abstracted, the temperature distribution at other positions in a 3D measurement area can be predicted. In a mathematical notation, the mapping between the measurement point positions and the corresponding temperature values can be formulized bywhere means the spatial coordinates of the measurement positions; is the temperature value corresponding to the measurement position; is a mapping describing the correlation between the spatial coordinates and the temperature values. In (2), if the mapping is obtained, we can estimate the temperature distribution at other positions in a 3D measurement area.

According to the analysis presented above, in this work, we put forward an alternative methodology to overcome the drawbacks of the TC technique to predict 3D temperature distributions, which can be divided into the following five steps.

*Step 1. *In accordance with practical demands, a 3D measurement area is appropriately determined.

*Step 2. *Acquire finite temperature measurement data using one of conventional measurement technologies.

*Step 3. *The raw measurement data is refined by an appropriate method to ameliorate the data quality.

*Step 4. *Abstract the mapping between the temperature value and the measurement position according to the finite measured temperature values.

*Step 5. *Use the mapping abstracted in Step to predict 3D temperature distributions at other locations in a measurement area.

#### 3. Gaussian Process Regression Method

The above discussions result in a fact that, in order to successfully predict 3D temperature distributions, we must seek an effective method to abstract the mapping between the temperature information and the position information in terms of the finite temperature measurement values. Naturally, in the rest of the paper, we will answer the above problems to achieve the goal of predicting 3D temperature distributions.

A variety of methods are available for the estimation of the mapping function, , including the multivariate linear regression (MLR) method, the multivariate nonlinear regression (MNR) method, the robust estimation (RE) method [18], the regularized MLR (RMLR) method, the artificial neural network (ANN) technique [19, 20], and the GPR method [21–30]. The least squares based MLR (LSMLR) method is the simplest and holds a closed solution, but the estimation results are far from satisfactory. Considering the inaccurate attributes of the measurement data, the RE method is introduced to improve the robustness of the estimation. The M-estimator is one kind of the RE methods, which has been successfully applied to different fields. Popular M-estimation functions include the Huber function, the Cauchy function, and the Fair function. In practical applications, a key task for the M-estimator is to determine the form of the M-estimation function. The RMLR technique is developed to improve the numerical stability of the LSMLR method. For the complicated regression tasks, the results estimated by the MNR often exceed the LSMLR method, but the determination of the form of the regression function is formidable in real-world applications. Owing to the superior fitness capacity, the ANN technique has found wide applications in different areas. In real-world applications, how to select a suitable network structure is full of challenges. Due to the prominent superiorities, for example, satisfactory generalization ability, high robustness, and low computational complexity, the GPR method is successfully introduced to a variety of areas, including image and signal processing, and wide speed prediction, to cope with hard prediction tasks. Inspired by successes of the GPR method, in this study, the method is introduced to abstract the mapping from the position information to the temperature values.

The GPR method is a kernel based learning machine and holds a good adaptability to deal with complicated, nonlinear, and high-dimensional problems. The execution of the GPR method includes two key phases, the training process and the prediction process, and more details can be found in [21–30].

##### 3.1. Prediction Process

Given a training set , where means the spatial coordinate of the measurement positions in a measurement area; denotes the temperature corresponding to the measurement positions. In the finite set of the given , as a set of random variables, , obeys the joint Gaussian distribution. The statistical characters of the GP can be clearly represented via the mean function and the covariance function :where and mean random variables.

If the observation value, , is perturbed by noises, the model of the GPR problem can be expressed aswhere stands for the independent random variables, which obeys the Gaussian distribution:where stands for the variance of the noise.

Similarly, the prior distribution of can be specified as

Eventually, we can express the joint prior distribution of and the prediction as follows:where represents a symmetric and positive definite covariance matrix with the dimensionality ; stands for a covariance matrix between the training set and the test point , and its dimensionality ; defines the covariance of .

Meanwhile, we can write the joint posterior distribution of the prediction as follows:where and define the prediction mean and covariance corresponding to , respectively.

It is necessary to mention that the covariance matrix is positive definite for the limited data set, which is consistent with the property of the kernel in Mercer’s theorem, and thus the covariance function and kernel function are equivalent. Consequentially, we can rewrite (9) as follows:where .

##### 3.2. Training Process

The selection of various covariance functions is crucial for practical applications of the GPR method. Currently, a variety of covariance functions are available. The squared exponential function has been successfully applied to different areas, and it has following form:where means the set of the hyperparameters and defines the variance of the kernel function.

The optimal hyperparameter can be solved by the maximum likelihood estimate. The negative log marginal likelihood function can be expressed as

In order to obtain the optimal solution of (13), the partial derivatives with respect to can be specified as follows:where and .

Finally, we can compute the prediction mean and variance corresponding to a new input provided that the optimal hyperparameter is determined.

The execution details of the GPR technique can be divided into the following three steps.

*Step 1. *According to the temperature measurement values and the measurement position coordinates, a set of training samples, (), in which represents the coordinates of the measurement positions in a 3D measurement area and stands for the temperature values, is determined.

*Step 2. *Solve (13) using a suitable method to get the optimal hyperparameters.

*Step 3. *The mean and variance of other positions can be computed via solving (9) and (10).

#### 4. Prediction Procedure of 3D Temperature Distributions

Owing to the conspicuous superiorities, for example, satisfactory generalization ability, easy numerical execution, low computational complexity and cost, and high robustness, in this study, the GPR method is employed to predict 3D temperature distributions. The prediction details can be divided into the following five steps.

*Step 1. *A 3D measurement area is determined according to the requirements of practical measurements.

*Step 2. *Measure the finite temperature data via traditional measurement technologies according to practical measurement conditions.

*Step 3. *Refine the raw measurement data to improve the data quality.

*Step 4. *The GPR model is used to abstract the mapping between the temperature values and the position information.

*Step 5. *Use the trained GPR model to predict 3D temperature distributions of other positions at a predetermined measurement area.

#### 5. Numerical Simulations

In this section, we use the temperature distribution function to generate the temperature distribution of the whole measurement area firstly. The known temperature data is obtained by random sampling with different sampling ratio, and other temperature data is considered unknown. Then, we perform numerical simulations to make a fair assessment for the GPR method, and the results are compared with the MNR method, the generalized regression neural network (GRNN) method, and the TC methods [17]. For computational convenience, all algorithms are implemented by the MATLAB software.

The mean relative error (MRE) is used to make a quantitative evaluation for the prediction accuracy of the GPR method, which is defined bywhere and denote the true values and the estimated values, respectively.

In order to simulate a real-world measurement environment, the simulation data is perturbed with the normal distribution random number with different noise variances (NVs), which can be formulated aswhere ; represents the standard deviation and means a normal distribution random number with the mean of 0 and the standard deviation of 1; the original and noisy measurement data are defined as and , respectively.

We use the SA to make a quantitative evaluation for the competing methods, which is shown in (17), that is,where the number of the samples and the number of the unknown variables are defined as SN and VN, respectively.

##### 5.1. Case 1

In order to fairly assess the effectiveness and robustness, the following temperature distribution is reconstructed:where , , and . The temperature distribution is shown in Figure 1.