Simplified unscented particle filter for nonlinear/non ... - IEEE Xplore

Journal of Systems Engineering and Electronics Vol. 24, No. 3, June 2013, pp.537–544

Simplified unscented particle filter for nonlinear/non-Gaussian Bayesian estimation Junyi Zuo1,* , Yingna Jia2 , and Quanxue Gao3 1. School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China; 2. Aviation Equipment Research Institute, Qing’an Group Corporation Limited, Xi’an 710077, China; 3. State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, China

Abstract: Particle filters have been widely used in nonlinear/nonGaussian Bayesian state estimation problems. However, efficient distribution of the limited number of particles in state space remains a critical issue in designing a particle filter. A simplified unscented particle filter (SUPF) is presented, where particles are drawn partly from the transition prior density (TPD) and partly from the Gaussian approximate posterior density (GAPD) obtained by a unscented Kalman filter. The ratio of the number of particles drawn from TPD to the number of particles drawn from GAPD is adaptively determined by the maximum likelihood ratio (MLR). The MLR is defined to measure how well the particles, drawn from the TPD, match the likelihood model. It is shown that the particle set generated by this sampling strategy is more close to the significant region in state space and tends to yield more accurate results. Simulation results demonstrate that the versatility and estimation accuracy of SUPF exceed that of standard particle filter, extended Kalman particle filter and unscented particle filter.

Keywords: nonlinear filtering, particle filter, unscented Kalman filter, importance density function.

DOI: 10.1109/JSEE.2013.00062

1. Introduction Nonlinear filtering problems arise in many fields including economics, statistics and engineering (such as communications, target tracking [1], fault diagnosis [2], satellite navigation and flight vehicle attitude estimation [3]). In recent years, importance sampling-based filters have been used to address these problems, where a distribution is represented by a weighted set of samples (or particles), which are propagated through the dynamic system using importance sampling to sequentially update the posterior distribution. This is called a sequential importance sampling (SIS) filter [4]. Gordon et al. introduced the Manuscript received March 28, 2012. *Corresponding author. This work was supported by the National Natural Science Foundation of China (61271296).

resampling step in SIS to tackle the degeneracy problem, and, as a result, obtained the sampling importance resampling (SIR) algorithm (also called standard particle filter, SPF) [5]. However, the resampling step introduces the new problem of particle impoverishment [4]. A common way to overcome the particle impoverishment problem is to incorporate the current measurement information in sampling process. Following this idea, Pitt et al. [6] proposed the auxiliary particle filter (APF), whose advantage lies in that it naturally generates points from the samples at previous time index, which, conditioned on the current measurement, are more likely to be close to the true state. However, if the process noise is large, the use of APF tends to degrade the performance. In [7], a particle filter using correlation of observation was proposed, where the weights of the particles are proportional to the correlation coefficient of the observations. When the likelihood model has a bimodal nature, the algorithm can achieve more accurate estimates than SPF and APF. By selecting a good importance density function (IDF), which contains the current measurement information, the particle impoverishment problem can also be alleviated. Following this idea, Doucet et al. proposed the extended Kalman particle filter (EPF) [8], which uses the extended Kalman filter (EKF) to generate IDF. The unscented particle filter (UPF) [9] utilizes the unscented Kalman filter (UKF) to generate IDF. The quadrature Kalman particle filter [10] uses the quadrature Kalman filter [11] to generate IDF. The common idea of these filters and their variants [12,13] is running a suboptimal filter for each particle to obtain the required IDF. There are two main problems in such filters despite the fact that they perform well in some situations. (i) Their extremely high computational costs make them impractical for real time applications; (ii) They do not always outperform SPF and sometimes yield much poorer results than SPF. This is the case, for example, for systems with

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Journal of Systems Engineering and Electronics Vol. 24, No. 3, June 2013

large measurement noise variance. To address these problems, a simplified unscented particle filter (SUPF) is proposed in this paper. In the importance sampling step, we firstly run the UKF algorithm, which plays a role of pre-filter, to obtain the Gaussian approximate posterior density (GAPD), and then draw the particles partly from transition prior density (TPD) and partly from the GAPD. The ratio of the number of particles drawn from TPD, to the number of particles drawn from GAPD is adjusted by the maximum likelihood ratio (MLR), which makes SUPF perform well at different measurement noise levels. Furthermore, SUPF implements UKF only once at each time step, therefore, its computational cost is significantly reduced in comparison with UPF and EPF. The remainder of this paper is organized as follows. A brief review of Bayesian estimation and SPF are presented in Section 2. In Section 3, we firstly illustrate the importance of efficient distribution of particles in state space, and then explain the inefficiency of the sampling strategy of SPF in cases such as small measurement noise variance; SUPF is presented at the end of this section. In Section 4, the improved performance of SUPF is demonstrated by numerical simulations. Finally, conclusions are drawn in Section 5.

2. Particle filter Consider the following nonlinear state space model xt = f t (xt−1 ) + v t−1

(1)

z t = ht (xt ) + w t

(2)

where t is the time index, {xt , t ∈ N} denotes the state sequence, {z t , t ∈ N} denotes the measurement sequence, {v t−1 , t ∈ N} denotes an independent identically distributed (IID) process noise sequence with known probability density functions (PDF), {wt , t ∈ N} denotes an IID measurement noise sequence, also with known PDF, and f t (·) and ht (·) are some known functions. The objective of filtering is to recursively estimate xt based on the set of all available measurements z 1:t = {z 1 , z 2 , . . . , z t } up to time t. The filtering problem can be solved by calculating the posterior PDF p(xt |z 1:t ). In principle, once the initial PDF p(x0 |z 0 ) ≡ p(x0 ) is available, p(xt |z 1:t ) can be obtained, recursively, in two stages: prediction and update. Suppose that the required PDF p(xt−1 |z 1:t−1 ) is available. The prediction stage involves using the system model (1) to obtain the prior PDF of the state at time t via the following equation p(xt |z 1:t−1 ) = p(xt |xt−1 )p(xt−1 |z 1:t−1 )dxt−1 . (3)

The probabilistic model of the state evolution p(xt |xt−1 ) is defined by the system equation (1). The update stage involves using the measurement z t to update the prior via the following Bayes’s rule p(xt |z 1:t ) =

p(z t |xt )p(xt |z 1:t−1 ) p(z t |z 1:t−1 )

(4)

where p(z t |z 1:t−1 ) is the normalizing constant. The likelihood model p(z t |xt ) is defined by the measurement equation (2). The principle of a particle filter is to implement the recursive Bayesian filter described above by the Monte Carlo method. The posterior PDF p(xt |z 1:t ) is represented by a set of N random particles as p(xt |z 1:t ) ≈

N

πti δ(xt − xit )

(5)

i=1

where δ(·) denotes the Dirac delta function, and {xit }i=1,...,N represents the N particles at time t, the weight of the ith particle is denoted as πti , satisfying N πti = 1. If the particles are drawn from the postei=1

rior density p(xt |z 1:t ), the question of filtering will be simplified. Unfortunately, p(xt |z 1:t ) is unknown in general. Alternatively, particles can be drawn from an IDF q(xt |xit−1 , z t ) and the corresponding weight πti can be calculated recursively as follows [4]: i πti ∝ πt−1

p(z t |xit )p(xit |xit−1 ) . q(xit |xit−1 , z t )

(6)

In SPF, the TPD is selected as the IDF for simplicity, i.e., q(xt |xit−1 , z t ) = p(xt |xit−1 ), and then (6) becomes i p(z t |xit ). Furthermore, the resampling step is πti ∝ πt−1 introduced in SPF at each time step to overcome the dei generacy problem, which results in πt−1 = 1/N, ∀i ∈ {1, 2, . . . , N }. Thus, (6) becomes πti ∝ p(z t |xit ).

(7)

Finally, the optimal state estimation in the minimummean-square-error (MMSE) sense can be expressed as

xt =

N

xit πti .

(8)

i=1

3. SUPF The superiority of particle filter lies in its strong ability to represent the propagation of a complex PDF in a nonlinear state space. However, in practice, the number of particles is always finite and cannot be very large for computational

Junyi Zuo et al.: Simplified unscented particle filter for nonlinear/non-Gaussian Bayesian estimation

reasons. As illustrated in Fig. 1, the measurement update of particle filter is strictly restricted by the distribution of particles in state space. The particles drawn from TPD are represented as the hollow circles in Fig. 1. When the current measurement becomes available and the particles gain their weights, no matter how the weights are assigned, the probable location of the state estimate certainly lies between the leftmost and rightmost hollow circles as the weights of particles are nonnegative and sum to 1.

Fig. 2

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The cases that the sampling strategy of SPF is efficient

Suppose that at time t − 1, the particles {xit−1 }i=1,...,N i and their normalized weights {πt−1 }i=1,...,N are available. The filtered estimate and its covariance can be expressed as N i xt−1 = πt−1 xit−1 (9) i=1

and

P t−1 =

N

i πt−1 (xit−1 − xt−1 )(xit−1 − xt−1 )T . (10)

i=1

Fig. 1 The cases that the sampling strategy of SPF is inefficient

This example reflects the fact that the key step in designing a particle filter is to select a proper IDF, particles drawn from which should cover the region where the true state lies. In other words, the support of IDF should include the support of the posterior PDF, which is the necessary condition for applying the importance sampling theory. However, because the IDF of SPF is independent of the current measurement and the state space is explored without any knowledge of the measurement; the necessary condition may not be met, which makes SPF inefficient. As can be seen from Fig. 1, when the likelihood model is aiguilleslike (which means that there is rich information on the state xt in the measurement z t and the true state is in the high-likelihood-region with high probability) or locates at the tail of TPD, the majority or even all of the particles (hollow circles in Fig. 1) of SPF are far away from the significant region, resulting in degraded estimation accuracy. However, when the variance of measurement noise is large and the support of TPD covers the significant region well, the sampling strategy of SPF will be efficient (see Fig. 2). From the above discussion, it can be seen that, to improve the estimation accuracy, it is crucial to ensure that there are enough particles in high significant region. To achieve this purpose, we only draw N1 (N1 < N ) particles from TPD, while the remaining N2 (N2 = N − N1 ) particles are drawn from GAPD, obtained by running a suboptimal filter (such as UKF) in advance.

According to pre-selected sampling strategy [14], sigma points {χj,t−1 }j=1,...,2n+1 with weights {Wmeanj }j=1,...,2n+1 and {Wcovariancej }j=1,...,2n+1 can be drawn from the Gaussian distribution N (xt−1 ; xt−1 , P t−1 ), where Wmeanj and Wcovariancej are the weights of the jth sigma points for calculation the mean and covariance respectively. n is the dimension of the state vector. We then implement the following UKF algorithm [15] χj,t|t−1 = f t (χj,t−1 ) xUKF t|t−1 = Σ xx t|t−1 =

2n j=0

2n

Wmeanj χj,t|t−1

(11) (12)

j=0

Wcovariancej [χj,t|t−1 − xUKF t|t−1 ]·

T [χj,t|t−1 − xUKF t|t−1 ] + Q.

(13)

Calculate the new sigma points {ξj,t|t−1 }j=1,...,2n+1 according to the known Gaussian density N (xt ; xUKF t|t−1 , ) and the pre-selected sampling strategy, then Σ xx t|t−1 γ j,t|t−1 = ht (ξ j,t|t−1 ) z UKF t|t−1 = Σ zz t|t−1

=

2n j=0

2n

Wmeanj γ j,t|t−1

(14) (15)

j=0

Wcovariancej [γ j,t|t−1 − z UKF t|t−1 ]·

T [γ j,t|t−1 − z UKF t|t−1 ] + R

(16)

540


Σ xz t|t−1 =

2n j=0

Wcovariancej [ξj,t|t−1 − xUKF t|t−1 ]·

T [γ j,t|t−1 − z UKF t|t−1 ]

(17)

zz −1 K t = Σ xz t|t−1 (Σ t|t−1 )

(18)

UKF = xUKF xUKF t|t t|t−1 + K t (z t − z t|t−1 )

(19)

zz T Σ t|t = Σ xx t|t−1 − K t Σ t|t−1 K t

(20)

where Q and R are the covariance matrices of system noise and measurement noise, respectively. With the known xUKF and Σ t|t , the GAPD can be ext|t pressed as NUKF (xt ; xUKF t|t , Σ t|t ). From Fig. 1, it can be seen that when particles are partly drawn from NUKF (xt ; xUKF t|t , Σ t|t ) (represented as gray circles), the estimation accuracy will be improved. The reason for this is that these particles, in general, are close to the high significant region and contribute greatly to the estimate due to their large weights. In comparison, for the situation illustrated in Fig. 2, it is unnecessary to draw many particles from GAPD because the TPD covers the significant region well and high-quality particles can be easily obtained by using TPD as IDF. In this case, if we still draw a large number of particles from GAPD, the sampling process may be misled by the suboptimal filter, which is harmful to the estimation accuracy. Therefore, it is reasonable that N1 should be adjusted according to the quality of particles drawn from p(xt |xit−1 ). For this purpose, we define a new parameter (21) α(t) N1 /N and then use the following MLR λ(t) to measure the quality of particles max{p(z t |xit )}i=1,...,N λ(t) . max{p(z t |xt )}xt ∈Ω x

(22)

Once z t is available, z t will be fixed and p(z t |xt ) can be regarded as a function of xt . max{p(z t |xit )}i=1,...,N is the maximum likelihood of N particles drawn from TPD, and max{p(z t |xt )} is the maximum likelihood for xt ∈ Ωx (Ωx denotes the whole state space), which can be evaluated analytically in general. ∀i ∈ {1, . . . , N }, we have xit ∈ Ωx and consequentially 0 < λ(t) 1. With the known λ(t), α(t) can be chosen as α(t) = α0 + (1 − α0 )λ(t)

(23)

where α0 ∈ [0, 1] is a predetermined constant, selected as 0.4 in this paper. According to the range of λ(t), we have α0 < α(t) 1. The parameter α0 makes the new filter free of being misled by UKF, because, indicated by (23), at least [α0 N ] particles are drawn from TPD, no matter

how well the particles match the likelihood model ([·] denotes the rounding operation). Clearly, the larger the λ(t) is, the more the particles will be drawn from TPD. When λ(t) approaches to 1 and consequentially α(t) → 1, the behavior of SUPF becomes similar to that of SPF. The SUPF resamples at every time step, the weights of all particles are updated by (7). Although (7) is not the best choice to update the weights of particles drawn from GAPD, it is motivated by two considerations to do so. Firstly, the filtering algorithm will be greatly simplified [4]. Secondly, the loss in estimation accuracy caused by the errors of the weights of a small portion of particles is often much smaller than the gain in estimation accuracy brought by the more proper distribution of particles in the state space. The iteration of SUPF can be described as follows: Algorithm 1 SUPF 1: Initialization step: t = 0 • Draw N particles {xi0 }i=1,...,N from p(x0 ) and set weights π0i = 1/N for i = 1, . . . , N ; 2: For t = 1, 2, . . . • Select N = [α0 N ] particles randomly and draw them from p(xt |xit−1 )(i = 1, . . . , N ); • Calculate α(t) according to (22) and (23); and Σ t|t , and then construct • Run the UKF to obtain xUKF t|t , Σ the GAPD NUKF (xt ; xUKF t|t ); t|t • Draw N − [α(t)N ] particles from NUKF (xt ; xUKF t|t , Σ t|t ); • Draw the remaining [α(t)N ] − [α0 N ] particles from TPD; • Calculate weights according to (7) and normalize them; • Resample; N X t = xit /N ; • Output: x End for

i=1

Remark 1 There are some guidelines for selecting the parameter N . On the one hand, too large N will increase the computational burden because more particles need to be drawn and more probabilities p(z t |xit ) need to be evaluated and, on the other hand, may cause the algorithm failure because, suppose N → ∞, the particles will, in theory, take all possible values of the state space, resulting in λ(t) ≡ 1. Therefore, N should not be larger than N . On the contrary, if N is too small, the particle set, containing N particles, is not representative, and λ(t) cannot reflect whether the particle set, containing N particles, match the likelihood function well. A rule of thumb would be [α0 N ] N N (α0 = 0.4). In Algorithm 1, we choose N = [α0 N ] to save the computational resources.

4. Simulation results In this section, the proposed filter is compared with EKF, UKF, SPF, EPF and UPF by two examples. All the filters are implemented on a PC with 1.7 GHz CPU. The root


mean square error (RMSE) is used to measure the estimation accuracy. The smaller the RMSE means, the higher the estimation accuracy. RMSE can be defined as two forms: ⎛ ⎞ M (xt,j − x t,j )2 /M ⎠ (24) RMSE(t) = ⎝

541

more likely to be close to the true value, and, consequently, can help to improve the estimation accuracy.

j=1

T RMSE(j) = (x

t,j

2

− x t,j ) /T .

(25)

t=1

Both of them are based on M simulation runs with T time steps. xt,j denotes the true value of one component of the state at time t of the jth simulation run and x t,j is the estimate of xt,j . The averaging operation of RMSE(t) is carried out over M simulation runs for each time step; whereas the averaging operation of RMSE(j) is over T time steps for each simulation run. In this section, the RMSE curves are calculated by (24), and the average M RMSE is defined by RMSE(j)/M .

Fig. 3

Filtering result comparison of the six filters in simulation run

j=1

Example 1 The state space model is taken from [9, 10, 16] as follows: xt = ft (xt−1 ) + vt−1

(26)

zt = ht (xt ) + wt

(27)

where ft (x t−1 ) 2 = 1 + sin(0.04π(t − 1)) + 0.5xt−1 , 0.2xt , t 30 ht (xt ) = , vt−1 is a Ga(3, 2) ran0.5xt − 2, t > 30 dom variable modeling the process noise. wt is a zero mean Gaussian random variable with variance R modeling the measurement noise. Given the noisy measurements zt , six filters are used to estimate the state sequence xt for t = 1, . . . , 50. Fig. 3 shows the true value and the estimates of the six filters in a simulation run with N = 130 and R = 0.000 1. For clarity, only a fragment (from time index 8 to 31) of the curve is shown. It can be seen that SUPF is more accurate than other filters especially at time 10. The estimate of SUPF (8.819) at time index 10 is very close to the true value (8.941), whereas the calculation result of SPF (6.133) deviates from the true state significantly. This can be explained by Fig. 4. Fig. 4 (a) corresponds to SPF, where the left figure represents the prior PDF obtained by using p(xt |xit−1 ) as IDF. It is observed that all the particles are not only far away from the true value, but also lie on the same side of it. In this situation, the estimation accuracy is poor no matter how the weights of particles are assigned. Fig. 4 (b) corresponds to SUPF, where some, but not all, particles are drawn from GAPD. These particles are

Fig. 4

Sampling result comparison of SPF and SUPF

Fig. 5 plots the RMSE curves obtained by 200 simulation runs with N = 130 and R = 0.000 1. It can be seen that SUPF performs much better than UPF, while UPF performs better than EKF, UKF, EPF and SPF. In Fig. 6, the average RMSE are shown for R = 0.000 1, 0.001, 0.01, 0.1 and 1 without changes in other parameters (Notice the common logarithmic scale of the horizontal axis). From Fig. 6, it is observed that, when R = 0.000 1, the average RMSE of SUPF is the smallest

542


among the six filters. Due to ineffective sampling strategy, SPF performs only better than EKF and EPF, but worse than UKF, UPF and SUPF. As R increases from 0.000 1 to 1, the performances of the six filters change quite different. For SPF, the accuracy is, firstly, increased, and then decreased, while the accuracies of UPF and SUPF are decreased monotonously. Among them, the increment of RMSE of SPF is the smallest, which makes it become the best filter for R = 1, whereas the increment of RMSE of UPF is the largest, which makes it become the worst filter. Different from UPF, SUPF always performs well at different measurement noise levels. When R = 1, its performance is only slightly poorer than that of SPF, but much better than other filters. In fact, in this case, most of the particles in SUPF are drawn from TPD and the UKF module in SUPF almost does not take effect. The RMSE curves of the six filters for R = 1 are shown in Fig. 7.

Fig. 7

RMSE curves of the six filters for R = 1

Fig. 8 shows the average run-time per simulation run of SPF, EPF, UPF and SUPF at different measurement noise levels. As expected, the run-time of SUPF is almost the same as that of SPF; both run-time of them is much smaller than that of EPF and UPF. Note that the run-time of SUPF is slightly smaller than that of SPF in the case of small measurement noise variance. This may be due to the fact that drawing samples from Gaussian distribution is much easier than drawing them from Gamma distribution, and the saved run-time for this reason is much than the increased run-time caused by the additional UKF algorithm in SUPF.

Fig. 5

RMSE curves of the six filters for R = 0.000 1

Fig. 8 Run-time comparison of SPF, EPF, UPF and SUPF at different measurement noises

Example 2 Consider a target tracking problem where the state space model can be described as Fig. 6 Average RMSE of the six filters at different measurement noise variances

xt = F xt−1 + Gv t−1

(28)

z t = h(xt ) + w t

(29)


where

⎡

1 ⎢ 0 F =⎣ 0 0

1 1 0 0

0 0 1 0

⎤

0 0 ⎥, 1 ⎦ 1

⎡

⎤

0.5 0 ⎢ 1 0 ⎥, G=⎣ 0 0.5 ⎦ 0 1

xt = [xt , x˙ t , yt , y˙ t ]T , h(xt ) = [ x2t + yt2 arctan (yt /xt )]T . Process noise v t−1 = [vx vy ]T is assumed to be a zero-mean Gaussian random vector with covariance matrix Q = diag [(10 m/s2 )2 (10 m/s2 )2 ]. xt , x˙ t , yt and y˙ t denote the target positions and velocities in x and y directions respectively. z t denotes the measurement at time t, which is composed of two components: range and azimuth. wt is the glint measurement noise modeled as a mixture of two zero-mean Gaussian distributions p(w) = (1 − ε)N1 (w; 0, R1 ) + εN2 (w; 0, R2 ) where R1 = diag [(10 m)2 (1 mrad)2 ], R2 = diag [(10 m)2 (20 mrad)2 ]. ε= 0.05 is the glint probability. Initial state is x0 = [4 000 m −20 m/s 4 000 m 0 m/s]T , and P 0 = diag [(10 m)2 (5 m/s)2 (10 m)2 (5 m/s)2 ]. Fig. 9 shows the RMSE plots of the positions in x and y directions for 150 simulation runs.

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The number of particles used for SPF, EPF, UPF and SUPF is N = 1 000. Table 1 lists the average run-time and the average RMSE of each filter. It can be seen that the average RMSE of SUPF is the smallest among the six filters. As expected, the run-time of SPF and SUPF is almost the same, and both run-time of them is far less than that of EPF and UPF. Table 1

Average run-time and average RMSEs of the six filters

when tracking a target in x-y plane Filter EKF UKF SPF EPF UPF SUPF

RMSE y/m 13.21 13.19 9.87 15.00 15.80 7.04

RMSE x/m 13.67 13.66 9.59 15.52 16.30 6.63

Run-time/s — — 9.71 23.06 36.51 9.61

5. Conclusions A new particle filter, called SUPF, is proposed to estimate the state of a nonlinear/non-Gaussian system, where particles are drawn partly from TPD and partly from GAPD, obtained by UKF. Thus, large deviation of the particle set from high significant region can be effectively avoided and the sampling efficiency can be greatly improved. Through numerical simulation, the following conclusions can be drawn: (i) MLR, as a simple measurement of the quality of particles, is reliable. It enables SUPF to realize adaptive sampling from TPD and GAPD. (ii) Although SUPF is computationally more expensive than EKF and UKF, its run-time is much less than that of EPF and UPF, and almost the same as that of SPF. (iii) Compared with SPF, EPF and UPF, SUPF is versatile and has satisfactory estimation accuracy at different measurement noise levels.

References

Fig. 9

RMSE for positions in x and y directions of the six filters

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Biographies Junyi Zuo was born in 1975. He received the B.S. and M.S. degrees from the School of Aeronautics, Northwestern Polytechnical University, in 2000 and 2003 respectively, and the Ph.D. degree from the Institute of Information and Control, School of Automation, Northwestern Polytechnical University in 2008. Currently, he is a lecturer at the School of Aeronautics, Northwestern Polytechnical University. His research interests include nonlinear filtering, information fusion, target tracking, guidance, navigation and control of flight vehicle, and aircraft system identification. E-mail: [email protected]

Yingna Jia was born in 1978. She received the B.S. degree from Agriculture University of Hebei, in 2001, and the M.S. degree from the Institute of Information and Control, School of Automation, Northwestern Polytechnical University, in 2004. From 2004 to 2009, she was an engineer of the 610 Institute in Aviation Industry Corporation of China. Currently, she is now an engineer of Aviation Equipment Research Institute in Qing’an Group Corporation Limited. Her research interests include complex system modeling, optimal estimation, control theory and its applications. E-mail: [email protected]

Quanxue Gao was born in 1975. He received the B.E. degree in 1998 from University of Xi’an Highway, the M.S. degree in 2001 from GanSu University of Science and Technology, and the Ph.D. degree from Northwestern Polytechnical University, in 2005. In 2006 and 2007, he was an associate research assistant in the Hong Kong Polytechnic University. He is currently an associate professor at State Key Laboratory of Integrated Service Networks in Xidian University. His research interests include manifold learning, biometric recognition, machine learning, dimensionality reduction, and statistical pattern recognition. E-mail: [email protected]