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Discriminative dimensionality reduction for sensor drift compensation in electronic nose: a robust, low-rank, and sparse representation method Zhengkun Yi PII: DOI: Reference:

S0957-4174(20)30064-6 https://doi.org/10.1016/j.eswa.2020.113238 ESWA 113238

To appear in:

Expert Systems With Applications

Received date: Revised date: Accepted date:

11 March 2019 26 November 2019 22 January 2020

Please cite this article as: Zhengkun Yi , Discriminative dimensionality reduction for sensor drift compensation in electronic nose: a robust, low-rank, and sparse representation method, Expert Systems With Applications (2020), doi: https://doi.org/10.1016/j.eswa.2020.113238

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1

Highlights Propose a robust, low-rank, and sparse representation of gas sensor signals Employ the source data label information to avoid overlapping of samples Solve the formulated problem in an iterative manner. Show the performance of the proposed method on two sensor drift datasets

2

Discriminative dimensionality reduction for sensor drift compensation in electronic nose: a robust, low-rank, and sparse representation method Zhengkun Yi1,2 1

Guangdong Provincial Key Lab of Robotics and Intelligent System, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences 2 CAS Key Laboratory of Human-Machine Intelligence-Synergy Systems, Shenzhen Institutes of Advanced Technology (Corresponding author: Zhengkun Yi, E-mail: [email protected])

Abstract—Sensor drift, which is a critical issue in the field of sensor measurements, has plagued the sensor community in the past several decades. How to tackle the sensor drift problem using expert and intelligent systems has gained increasing attention. Most sensor drift compensation methods ignore the sparse and low-rank characteristics of sensor signals. In this paper, we propose a discriminative dimensionality reduction method for sensor drift compensation in the electronic nose. The proposed method consists of four major components. 1) The distribution discrepancy between source data and target data is alleviated via projecting all the data into a common subspace. 2) A sparse and low-rank reconstruction coefficient matrix is employed to preserve the global and local structures of sensor signals. 3) An error matrix is introduced to deal with outliers. 4) The source data label information is taken into consideration to avoid overlapping of samples with different labels in the common subspace. The formulated minimization problem with constraints can be solved in an iterative manner. The effectiveness of the proposed method has been verified by conducting experiments on two sensor drift datasets. The proposed method may provide new insights into the gas sensor drift compensation systems or other expert and intelligent systems. Index Terms—Sensor drift, electronic nose, dimensionality reduction, domain adaptation, transfer learning, low-rank and sparse representation.

I. INTRODUCTION

E

lectronic nose (E-nose) is a ubiquitous device used in many domains such as medical diagnosis (Damico et al., 2008; K. Yan, Zhang, Wu, Wei, & Lu, 2014), indoor and outdoor air quality monitoring (F. Hossein-Babaei & Ghafarinia, 2010; Zhang et al., 2012), fruit quality control (Pathange, Mallikarjunan, Marini, O’Keefe, & Vaughan, 2006) and detection of polluting gases from vehicles (Endres et al., 1996; Tamaekong, Liewhiran, Wisitsoraat, & Phanichphant, 2010). The key components of an electronic nose device is commonly comprised of an array of sensing elements, a condition a conditioning circuit, and an electronic system for signal processing (Fort et al., 2003). There is an increasing interest to develop various gas sensors based on different sensing principles in the past decades (X. Liu et al., 2012). As an

example, a single generic tin oxide gas sensor was developed to discriminate among complex odors for the first time (Faramarz Hossein-Babaei & Amini, 2014). Meanwhile, tremendous efforts have been devoted to improving the performance of the sensor conditioning circuits for better gas sensing. For instance, Flammini et al. (Flammini, Marioli, & Taroni, 2004) developed a low-cost and compact electronic circuit, which is capable of supporting a wide range of resistive values. This property is key to the realization of an electronic nose. Sensor drift issue is a critical issue in the field of sensors and measurement (Marco & Gutierrez-Galvez, 2012; Röck, Barsan, & Weimar, 2008). The sensor drift problem has plagued the sensor community for many years. Electronic nose as a gas sensor also suffers from the sensor drift problem. There are two types of sensor drift problems (Vergara et al., 2012). One type is called the first-order drift, which is caused by aging and poisoning, while the other type is called the second-order drift, which is caused by uncontrollable alterations of the experimental operating system such as humidity variation and temperature. However, it is not easy to classify these two types of the sensor drift problem in practical applications. Sensor drift compensation systems are intelligent systems that can automatically correct the sensor drift problems using advanced signal processing methods. How to tackle the sensor drift problem using expert and intelligent systems has gained increasing attention. Drift compensation techniques or drift correction techniques are employed to combat the sensor drift issue in the sensor drift compensation systems. The drift compensation can be implemented in many aspects such as hardware updating (Korotcenkov & Cho, 2011) and signal improvement via signal processing tools. The signal processing approaches for sensor drift compensation can be broadly divided into three categories, which include univariate methods, multivariate methods, and machine learning approaches (Q. Liu, Li, Ye, Ge, & Du, 2014). Typical univariate methods are comprised of methods such as frequency analysis, differential measurements, and baseline manipulation (Marco & Gutierrez-Galvez, 2012). The univariate methods are simple, which correct the response of each sensor independently. However, these methods are extremely sensitive to sample rate variations (Carmel, Levy,

3 Lancet, & Harel, 2003; Distante, Leo, Siciliano, & Persaud, 2002). Compared to the univariate methods, multivariate methods compensate the responses of the entire sensor array (Artursson et al., 2000; Distante, Sicilian, & Persaud, 2002). Both the univariate methods and multivariate methods explicitly compensate the sensor drift. However, it may be impossible to do the sensor drift correction in many real-world applications. Thus, machine learning methods have gained its popularity, which are employed to implicitly do the sensor drift compensation via learning from data distributions. This paper mainly focuses on developing machine learning approaches for sensor drift problems. Most sensor drift compensation methods ignore the sparse and low-rank characteristics of sensor signals. The motivation of the paper is to employ a robust, low-rank, and sparse representation of sensor signals to address the sensor drift problem. The low-rank property guarantees that each target sample can be approximately represented by its neighbors in the common subspace, while the sparse property ensures that each target sample can be represented by a few samples in the source domain. Specifically, both the source and target data are projected into a common subspace, where each target sample is assumed to be represented by a linear combination of all the source samples via a reconstruction coefficient matrix. The distribution discrepancy between source data and target data is alleviated. The reconstruction coefficient matrix is constrained to be low-rank and sparse. To deal with the outliers, an error matrix is introduced. Moreover, the proposed method takes the source data label information into consideration to avoid overlapping of samples with different labels in the common subspace. The proposed method may benefit the expert and intelligent systems by providing new insights into the gas sensor drift compensation systems. The major contributions of this paper are summarized in the following: A novel discriminative dimensionality reduction method with a robust, sparse, and low-rank representation is proposed to address the sensor drift problem in expert and intelligent systems. An alternating optimization algorithm to solve the discriminative dimensionality reduction problem with a robust, sparse, and low-rank representation. Extensive experiments to show the effective of the proposed method on the sensor drift compensation intelligent systems. The rest of the paper is organized as follows. Section II briefly introduces the related works on drift correction in

electronic noses and relevant machine learning approaches. Subsequently, Section III presents the proposed discriminative dimensionality reduction methods with robust, low-rank, and sparse representation. In section IV, the results of our method and the competing methods on two public gas sensor datasets are compared. Finally, Section V concludes the work and discusses the future work. II. RELATED WORK In this section, the machine learning methods for anti-drift in electronic noses are firstly reviewed in Section II-A. Subsequently, Section II-B provides a brief review of subspace learning methods. In Section II-C we present another quite related machine learning topic, i.e., domain adaptation, which is widely used in many applications in addition to sensor drift correction. A. Sensor drift compensation in electronic nose with machine learning approaches Sensor drift compensation using machine learning methods has gained increasing attention in the past decades. Vergara et al. (Vergara et al., 2012) proposed to use classifier ensembles for the gas recognition problem on an extensive gas sensor drift dataset. The dataset is comprised of six gases in total, which is collected using an array of 16 metal-oxide gas sensors over a period of three year. The concentration of the gas ranges from 10 to 1000 ppmv. Zhang et al. (Lei Zhang & Zhang, 2015) developed two extreme learning machine-based methods for sensor drift problem. The proposed methods are called domain adaptation extreme learning machines, which leverage a limited number of labeled samples from the target domain for gas recognition. Compared to the classic extreme learning machines, the proposed methods have comparable computational efficiency. Ziyatdinov et al. (Ziyatdinov et al., 2010) addressed the sensor drift problem with common principal component analysis (CPCA). This approach sought to search for components that are common for all gases. Zhang et al. (Zhang et al., 2017) solved the sensor drift issue in an unsupervised manner. The unsupervised subspace learning approach aimed to combat the distribution difference by minimizing the mean distribution discrepancy. The subspace search could be efficiently solved by the aid of eigenvalue decomposition. Table I shows the comparison of related approaches.

Table I. Comparison of related approaches. Method PCA

Reference (Maćkiewicz & Ratajczak, 1993)

Advantages Simple and easy to be used in real systems

LPP

(He & Niyogi, 2004)

Exploit the local structure of the data

Disadvantages The distribution difference minimization between the source and target domain is missing Difficult to specify the parameter in the similarity function; The distribution difference minimization between the source and target domain is missing

4 LDA

CPCA DS

(Mika, Ratsch, Weston, Scholkopf, & Mullers, 1999) (Ziyatdinov et al., 2010) (Fonollosa, Fernández, Gutiérrez-Gálvez, Huerta, & Marco, 2016)

Exploit the label information for subspace projection Utilize common components for sensor drift compensation Explicitly calibrate the data in the target domain

B. Dimensionality reduction methods via subspace projection Subspace learning aims to search a low-dimensional subspace by optimizing certain objective functions. Principal component analysis (PCA) is an unsupervised subspace learning method (Maćkiewicz & Ratajczak, 1993). PCA is an orthogonal linear transformation that preserves most variance of the data in the subspace. The possibly correlated variables are transformed into uncorrelated ones. As another unsupervised subspace method, locality preserving projection (LPP) (He & Niyogi, 2004) optimally preserves the neighborhood structure of the data set. LPP can discover the nonlinear structure of the data manifold. Unlike both PCA and LPP, Linear discriminant analysis (LDA) (Mika, Ratsch, Weston, Scholkopf, & Mullers, 1999) is a supervised subspace method. LDA finds the subspace through maximizing the between-class variance and at the same time minimizing the within-class variance. Marginal Fisher Analysis (MFA) (S. Yan et al., 2007) is a graph embedding method for dimensionality reduction. MFA alleviates the limitations of LDA using graph embedding tricks, where two graphs are designed to characterize the within-class compactness and between-class separability. C. Machine learning for domain adaptation There is a major assumption of classic machine learning approaches, i.e., the distribution of training data is consistent with that of test data. However, various real-world applications have the distribution discrepancy issue, for instance, sentiment analysis (Pan, Ni, Sun, Yang, & Chen, 2010), cross-system recommendation (Zhao et al., 2013), text classification (Zhuang et al., 2012), and indoor WiFi localization (Pan, Shen, Yang, & Kwok, 2008). Thus, transfer learning as a machine learning technique is proposed to address the distribution discrepancy issue (Pan & Yang, 2010). Domain adaptation (Boqing Gong, Yuan Shi, Fei Sha, & Grauman, 2012; Gopalan, Ruonan Li, & Chellappa, 2011) as a branch of transfer learning aims to improve the algorithm performance in the target domain. Both the information from the source domain and target domain are utilized in domain adaptation. A domain adaptation method called transfer component analysis (TCA) was proposed by Pan et al. (Pan, Tsang, Kwok, & Yang, 2011). TCA aimed to subspace through minimizing the Maximum Mean Discrepancy (MMD) (Gretton, Borgwardt, Rasch, Schoelkopf, & and Smola, 2007). Transfer components spanned a subspace where distributions of the source data and target data are close to each other. An extension of TCA was formulated in a semi-supervised learning setting (Pan et al., 2011). Jiang et al. (Jiang, Huang,

The distribution difference minimization between the source and target domain is missing The label information is ignored in the projection Explicit calibration is impossible in real systems

Huang, & Yen, 2017) proposed an algorithm called integration of global and local metrics for domain adaptation learning (IGLDA). Compared to TCA, IGLDA utilized the source data label information to achieve a better classification performance. Long et al. proposed a unified framework to learn the classifier and feature-invariant subspace simultaneously. The framework achieved both distribution adaptation and label propagation, whose name is Adaptation Regularization based Transfer Learning (ARTL) (Long, Wang, Ding, Pan, & Yu, 2014). III. PROPOSED METHOD Section III-A provides the notation employed throughout this paper. Subsequently, Section III-B presents the proposed discriminative subspace method for domain adaptation in sensor drift compensation using a low-rank and sparse representation. Section III-C presents a solution to the proposed approach. A. Notations [ ] The source data is denoted as [ ] and the target data is denoted as . and are the numbers of samples in source domain and target domain, respectively. and are the dimension of original space and the dimension of the subspace, respectively. The projection matrix and reconstruction matrix are respectively represented as and . [ ] denotes the label matrix. is the number of classes. The k-th entry of is 1 if has a label of ( ) and the other entries of is 0. B. Discriminative domain adaptation for sensor drift compensation via a robust, low-rank, and sparse representation In this paper, it is assumed that the target data can be linearly represented by the source data in multiple subspaces (Shao, Kit, & Fu, 2014). Therefore, the problem can be formulated as ( ) . (1) The problem (1) is non-convex and NP-hard (G. Liu et al., 2013; Wright, Ganesh, Rao, Peng, & Ma, 2009). The nuclear norm is employed to take place of the non-convex objective function. Thus, problem (1) is converted to the following problem ‖ ‖ . (2) The sparse constraint is introduced in the objective function as well. Therefore, each target sample can be represented by

5 sparse samples in the source domain. In addition, a noise term is introduced to robustify the proposed method by requiring the noise to be sparse. The problem can be reformulated as follows ‖ ‖ ‖ ‖ ‖ ‖ (3) where and are trade-off parameters. To exploit the label information in the source domain, a discriminative term ( ) is added in the objective function. We follow the formulation in (Shiming Xiang, Feiping Nie, Gaofeng Meng, Chunhong Pan, & Changshui Zhang, 2012), where a non-negative label relaxation matrix is introduced to make the samples with different labels far away from each other. The formulation with the discriminative term is given by ‖ ‖ ‖ ‖ ‖ ‖ ( ) where

(

as

{

)

‖

( and

)‖ .

The formulation of the discriminative term means that the dimensionality of the common subspace is fixed to . We leave the exploration of the variable dimensionality of the subspace to future work. C. Solution to the optimization problem The problem (4) is non-convex and it can be solved in an iterative manner by update each variable and simultaneously fix the other variables. The problem (4) is reformulated by introducing two auxiliary variables and in the following. ‖ ‖ ‖ ‖ ‖ ‖ (

)‖

‖ ‖ ‖ ‖ ) (6) where is a penalty parameter and , are Lagrange multipliers. The main steps for solving problem (6) are in the following. Update : The projection matrix can be updated by solving the optimization problem given by ‖ 〈 ( )‖ 〉

‖

‖

‖

‖

‖

(

)‖

‖

‖ ‖

‖

where

‖

‖

,

, and

.

〈

〉

〈

〉 ‖ ‖ )

(‖ ‖

‖

‖

‖

‖

‖

‖ ‖

‖

‖

‖

‖

‖

‖

‖

(9)

‖

Table II. Gas sensor drift dataset from UCSD with period of collection and sample count of different types of gases. Batch ID 1 2 3 4 5 6 7 8 9 10

Month 1, 2 3, 4, 8~10 11~13 14, 15 16 17~20 21 22, 23 24, 30 36

Ethanol 90 164 365 64 28 514 649 30 61 600

Ethylene 98 334 490 43 40 574 662 30 55 600

Ammonia 83 100 216 12 20 110 360 40 100 600

Acetaldehyde 30 109 240 30 46 29 744 33 75 600

Acetone 70 532 275 12 63 606 630 143 78 600

Toluene 74 5 0 0 0 467 568 18 101 600

Total 445 1244 1586 161 197 2300 3613 294 470 3600

Table III. Recognition accuracy (%) of the sensor drift dataset from UCSD. Bold values represent the best results. Batch ID PCASVM LDASVM CC-PCA SVM-rbf SVM-gfk

Batch 2 82.40 47.27 67.00 74.36 72.75

3 84.80 57.76 48.50 61.03 70.08

4 80.12 50.93 41.00 50.93 60.75

5 75.13 62.44 35.50 18.27 75.08

(7)

The solution of problem (7) has a closed form which is given by ( ) ( ) (8) where is a small positive constant to guarantee the solution numerically more stable. Update : The reconstruction matrix can be updated by solving the optimization problem given by 〈 〉

)‖

‖

〉

(‖

. (5) The augmented Lagrange multiplier (ALM) of problem (5) is given by (

〉 〈

‖

,

‖

〈 〉

(4) is defined

is Hadamad product operator.

‖

‖ ‖ 〈

6 73.57 41.48 55.00 28.26 73.82

7 56.16 37.42 31.00 28.81 54.53

8 48.64 68.37 56.50 20.07 55.44

9 67.45 52.34 46.50 34.26 69.62

10 49.14 31.17 30.50 34.47 41.78

Average 68.60 49.91 45.72 38.94 63.76

6 SVM-comgfk ML-rbf ML-comgfk ELM-rbf OSC GLSW DS Proposed Method

where

74.47 42.25 80.25 70.63 88.10 78.38 69.37 90.35

70.15 73.69 74.99 66.44 66.71 69.36 46.28 85.56

,

59.78 75.53 78.79 66.83 54.66 80.75 41.61 62.73

75.09 66.75 67.41 63.45 53.81 74.62 58.88 61.93

, and

. The solution of problem (9) also has a closed form which is given by ( ) ( ) (10) Update : can be updated by solving the optimization problem given by ‖ ‖ 〈 〉 ‖ ‖ ‖

‖

‖

‖

.

(11) The problem (11) has a closed form solution, which is given by ⁄

(

)

73.99 77.51 77.82 69.73 65.13 69.43 48.83 71.57

54.59 54.43 71.68 51.23 63.71 44.28 32.83 56.08

55.88 33.50 49.96 49.76 36.05 48.64 23.47 46.26

70.23 23.57 50.79 49.83 40.21 67.87 72.55 52.55

41.85 34.92 53.79 33.50 40.08 46.58 29.03 67.44

64.00 53.57 67.28 57.93 56.5 64.43 46.98 66.05

Update : The multipliers and the ): can be updated by step-size of the iteration ( ( ) ( ) (19) ( ) ( ) { IV. EXPERIMENTS AND RESULTS In this section, experiments are performed to demonstrate the effectiveness of the proposed approach on two public sensor drift datasets. The datasets include one from UCSD and the other from CQU. The proposed discriminative dimensionality reduction method is compared with the competing methods. A. Experiment on sensor drift dataset from UCSD

(12) ( ) ( ) where is the thresholding operator. ( ) ( ) (| | ) . The singular value decomposition of is . Update : can be updated by solving the optimization problem given by ‖

‖

‖

‖

〈

〉 ‖

‖

‖

‖

.

(13) The problem (13) has a closed form solution (Wright et al., 2009), which is given by (

)

(14)

( ) (| | ). where Update : can be updated by solving the optimization problem given by ‖ ‖ 〈 〉 ‖

‖ ‖ ‖

‖

‖

(15)

Similar to problem (13), the solution of problem (15) is ( )

(16)

Update : can be updated by solving the optimization problem given by ‖ ( )‖ . (17) The solution to problem (17) (Shiming Xiang et al., 2012) is ) (( ) . (18)

Fig 1. Samples in batch 1~10 are projected to 2D subspace using principal component analysis. The time-varying sensor drift can be easily observed.

The UCSD sensor drift dataset was collected by Vergara et al (Vergara et al., 2012). There are 13910 samples in total collected using an electronic nose with 16 gas sensors. The collection period lasted for 36 months starting from January 2018. Six types of gases with different concentrations were detected. The gases were comprised of Acetaldehyde, Acetone, Ammonia, Ethanol, Ethylene, and Toluene. According to the sample acquisition time, the sample set was split into ten batches. Table II provides the detailed information such as the acquisition time and the sample count of each batch. Eight Features were extracted from sensor signals for each sensor. Thus, the feature vector was 128-dimensional for each sample. The experimental protocol in (Q. Liu et al., 2014) was adopted in this paper. Specifically, batch 1 is adopted as the samples in the source domain. The label information in the source domain is available. Other batches are adopted as the samples in the target domains whose labels need to be predicted. Fig 1 shows the 2D projection of the samples in batch 1~10. It is easy to observe that the sensor signals are time-varying, i.e., the

7 distribution difference between the source domain and target domain is time-dependent. The proposed method is compared with the competing methods in Table III. Both LDA and PCA are baseline subspace approaches. Component correction based principal component analysis (CC-PCA) is a multivariate method (Artursson et al., 2000). Orthogonal signal correction (OSC) (Wold, Antti, Lindgren, & Öhman, 1998) is another multivariate method similar to CC-PCA. Multi-class support vector machine (SVM) with RBF kernel (SVM-rbf), the geodesic flow kernel (SVM-gfk), and the combination kernel (SVM-comgfk) are methods presented in (Q. Liu et al., 2014). ML-rbf and ML-comfgk are semi-supervised methods with manifold regularization (Q. Liu et al., 2014). Both generalized least squares weighting (GLSW) (Fernandez, Guney, Gutierrez-Galvez, & Marco, 2016) and direct standardization (DS) (Fonollosa, Fernández, Gutiérrez-Gálvez, Huerta, & Marco, 2016) are calibration transfer methods. The proposed approach can achieve comparable performance as the best competing methods in terms of average classification accuracy. Moreover, the proposed method performs best in three drift correction tasks, i.e., batch 1 batch 2, batch 1 batch 3, and batch 1 batch 10. B. Experiment on sensor drift dataset from CQU The CQU sensor drift dataset was collected by Zhang et al (Zhang et al., 2017). There are 1604 samples in total collected using multiple E-nose devices of the same model. Thus, the difference of the device is one reason causing the sensor drift issue. There are three batches in total in the dataset including batch master collected five years earlier than the batches slave 1 and slave 2. Six types of gases with different concentrations were detected. The gases were comprised of Ammonia, Benzene, Carbon monoxide, Formaldehyde, Nitrogen dioxide, and Toluene. Table IV summarizes the detailed information such as the sample count of each batch. Feature extraction were performed for each sensor resulting in a 6-dimensional feature vector for each sample. The experimental protocol (Zhang et al., 2017) was adopted in this paper. Specifically, master is adopted as the samples in the source domain. The label information in the source domain is available. Other batches (i.e., slave 1 and slave 2) are adopted as the samples in the target domains whose labels need to be predicted. Fig 2 shows the 2D projection of the samples in each batch. It is easy to observe that the sensor signals are time-varying, i.e., the distribution difference between the source domain and target domain is time-dependent.

The competing methods contain SVM, PCA, LDA, and calibration transfer methods (GLSW and DS). The recognition accuracy of the sensor drift dataset from CQU is shown in Table V. The proposed method outperforms competing ones in terms of average classification accuracy. The highest classification accuracy of 54.87 is achieved. Moreover, the proposed approach performs best in each task, i.e., master slave 1, and master slave 2.

Fig 2. Samples in batch master, slave 1 and slave 2 are projected to 2D subspace using principal component analysis respectively. The time-varying sensor drift can be easily observed.

C. Parameter sensitivity analysis There are two parameters to be tuned in the proposed approach. Specifically, the parameters contain two trade-off parameters and . The parameters are tuned from the set * + for both datasets. Fig 3 shows the classification accuracy of the proposed method on the UCSD dataset by tuning two trade-off parameters and . Fig 4 shows the classification accuracy of the proposed method on the CQU dataset by tuning two trade-off parameters and . In summary, based on the experimental results on the two public gas dataset, we find that the proposed method has improved the system performance using a robust, sparse, and low-rank representation. This indicates that a sparse and low-rank representation in the common subspace is beneficial for the solving of the gas sensor drift problem. The proposed method is a general framework for drift compensation problem. It can be extended to other expert and intelligent systems, in particular, other sensor drift systems.

Table IV. Complex E-nose dataset from CQU with dimension of features (DoF) and sample count of different types of gases. Master Slave 1 Slave 2

DoF 6 6 6

Ammonia 60 81 84

Benzene 72 108 87

Carbon monoxide 58 98 95

Formaldehyde 126 108 108

Nitrogen dioxide 38 107 108

Toluene 66 106 94

Total 420 608 576

Table V. Recognition accuracy of the sensor drift dataset from CQU. Bold values represent the best results. The proposed method outperforms the competing ones in both tasks. Task master slave 1

SVM 45.89

PCA 46.22

LDA 42.11

GLSW 41.45

DS 40.30

Proposed Method 49.84

8 master slave 2 Average

31.08 38.49

41.84 44.03

41.32 41.72

48.09 44.77

39.76 40.03

59.90 54.87

settings such as the semi-supervised transfer learning are also interesting. We will investigate a new method in the case that some samples in the target domain are available. ACKNOWLEDGMENT The authors would like to thank Dr. Vergara’s group from USCD and Dr. Zhang’s group for providing the sensor drift datasets. Zhengkun Yi contributes to all the aspects of the paper.

The credit of this paper goes to Zhengkun Yi.

The authors have declared that no competing interests. Fig 3. Classification accuracy of the proposed method on the UCSD dataset by tuning two trade-off parameters and .

REFERENCES

Fig 4. Classification accuracy of the proposed method on the CQU dataset by tuning two trade-off parameters and .

V. CONCLUSION In this paper, a discriminative dimensionality reduction method has been proposed for gas sensor drift compensation systems. The proposed method has improved the system performance using a robust, sparse, and low-rank representation. Experimental results have demonstrated the effectiveness of the proposed method on two public gas sensor datasets. The proposed method is a general framework for drift compensation problem. It can be extended to other expert and intelligent systems, in particular, other sensor drift systems. The proposed method has some limitations. One limitation is that the dimension of the common subspace has to be fixed, which results in a less flexible algorithm. Another limitation is the grid search of the optimal values of the trade-off parameters. A more feasible approach is needed for the optimization of trade-off parameters. In the future, the research can be extended in several aspects. (1) We will relax the constraint of the fixed subspace dimension and develop a more flexible subspace projection method. (2) We will investigate a method to optimize the trade-off parameters instead of a grid search method. (3) Gas sensor signals are complicated and nonlinear in real systems. Therefore, we will study a nonlinear version of the current method. A potential method is to use the kernel trick. (4) Other

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