This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
An Indoor Positioning Algorithm with Kernel Direct Discriminant Analysis Yubin Xu, Member, IEEE, Zhian Deng, and Weixiao Meng, Member, IEEE Communication Research Center, Harbin Institute of Technology, 150001, China
[email protected],
[email protected],
[email protected] Abstract—Location estimation based on received signal strength (RSS) in WLAN environment is an attractive method for indoor positioning system. Unfortunately, due to the explicit nonlinearity and uncertainty of RSS signal, the traditional approaches always fail to deliver good location accuracy. This paper presents a novel positioning algorithm with kernel direct discriminant analysis (KDDA). We deploy the KDDA to map the original RSS vectors into a kernel feature space for feature extraction. The experimental results show that the proposed algorithm leads to higher location accuracy over the traditional algorithms including weighted k-nearest neighbor, maximum likelihood and kernel method. The performance improvement can be attributed to that the nonlinear discriminative location information can be efficiently extracted, while the redundant location information is considered as noise and discarded adaptively. Keywords-received signal strength (RSS); location estimation; wireless local area network (WLAN); kernel direct discriminant analysis
I.
INTRODUCTION
Recently, the location based services (LBS) in indoor environments have been receiving increasing interests for the wide range of personal and commercial applications, such as personalized information delivery, context awareness and healthcare [1],[2]. The received signal strength (RSS) based indoor positioning system [3],[4] has been regarded as a key complement to the existing global positioning system (GPS) which cannot work well in the complex indoor environments. The RSS based indoor positioning system can be achieved by pervasive available wireless networks and modern personal computing devices (e.g. PDA, notebooks) with wireless network interface cards installed. It not only requires no extra device, but also performs well in the non-line-of-sight (NLOS) circumstances. The key challenging problem [5] in RSS based indoor positioning is how to map the received signal strength (RSS) vector from p access points (APs) to a 2D Cartesian coordinate location, which can be described as the mapping rule h(⋅) : R p → R 2 . The accurate mapping depends on the fundamental assumption that RSS signal of different physical locations show complex but distinguishable patterns. If the discriminative feature between these patterns could be sufficiently extracted, the similarity measure between the feature representations can be used for pattern recognition. RSS of multiple APs at some selected physical calibration points are recorded to create the training samples set, which is always
referred as radiomap. In the radiomap, the physical calibration points with corresponding feature representation samples are called fingerprints. The location estimation can be considered as the pattern matching between a new RSS observation and the fingerprints stored in the radiomap. However, in real indoor environments, the distribution of RSS always shows explicit nonlinear and multimodal property for the well-known reasons of severe multipath, NLOS propagation and the effect of user body shadow [6],[7]. Traditional approaches such as weighted k-nearest neighbor (WKNN) algorithm [8], maximum likelihood (ML) algorithm [9] always fail to deliver good performance since the original RSS vectors are directly used as input features. For kernel method (KM) [10], though it becomes easier to detect the nonlinear relationship between RSS and physical location, there is still no effective method to extract discriminative features automatically. In this paper, a novel indoor positioning system with kernel direct discriminant analysis (KDDA) is proposed. The original RSS vectors are mapped into a kernel feature space by KDDA for feature extraction, which is referred as KDDA transformation. The location estimation can be obtained by the Euclidean distance measure between kernel features, which we call KDDA distance. Experimental results have shown that the proposed approach can obtain higher location accuracy than traditional approaches. The improvement can be attributed to that the nonlinear discriminative features between different fingerprints can be efficiently extracted by KDDA transformation, while the redundant information is regarded as noise and discarded adaptively. The remainder of the paper is organized as follows. Section 2 first reviews the kernel direct discriminant analysis, and then the proposed positioning algorithm using KDDA is described. In section 3, the experimental setup and the analysis of experimental results is presented. The last section gives some conclusions. II.
PROPOSED POSITIONING ALGORITHM USING KDDA
A brief overview of the proposed KDDA based positioning system is shown in Fig.1. During the offline phase, the RSS values from all APs are recorded at selected physical calibration points. The RSS vector samples labeled with corresponding 2-dimensional coordinates are stored in the radiomap. We deploy KDDA transformation on original radiomap to produce a new transformed radiomap with kernel
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feature representation. During the online phase, after feature extraction of the observation RSS vectors by KDDA transformation, KDDA distance between the fingerprints and observation point with kernel feature representation is calculated. The location is estimated by averaging the coordinates of the k-nearest fingerprints whose weight are the inverse KDDA distances. Data Collection
Radiomap (original data)
Offline phase
Transformed Radiomap
KDDA Transformation
KDDA Distance Calculation
Measured RSS
Online phase
eigenvectors P=[ p1
SWTH be the between- and within-class scatter matrices in the space F respectively: 1 C S BTW = ∑ Ci (φi − φ )(φi − φ )T (1) L i =1 Ci
1 ∑∑ (φ (z ij ) − φi )(φ (z ij ) − φi )T L i =1 j =1
(2)
Where φi = 1 Ci ∑ j =i 1φ (z ij ) is the mean of class Zi ; C
φ = 1 L ∑ i =1 ∑ j =1φ (z ij ) is the average of the ensemble; Ci is Ci
the element number in Zi ; L = ∑ i =1 Ci is the element number C
for the ensemble. Assuming ψ = [ψ 1 ,…ψ m ] is the set of optimal discriminant basis vectors. The D-LDA algorithm seeks it by solving the eigenvalue problem: ψ * = arg max( F (ψ )) (3) ψ
F (ψ ) =
p m ] of UT SWTH U (with size M × M)
with the m smallest eigenvalues [ λ1′ ,
input space to a high-dimensional feature space F , S BTW and
C
λM ] . The eigenvectors with the smaller eigenvalues
nonzero space of SBTW can be obtained by the m ( ≤ M)
Estimated Location
A. Kernel Direct Discriminant Analysis The KDDA [11] algorithm generalizes the D-LDA [12] algorithm into its kernel version. It seeks a set of optimal discriminant basis vectors which maximize the ratio of between- and within-class scatter in the kernel feature space. Let φ : z ∈ R p → φ (z ) ∈ F be a nonlinear mapping from the
SWTH =
[ λ1
contain little discriminative information and will be discarded as noise. Supposing U = VΛ b−1/ 2 , Λ b = diag [ λ1 λM ] , a M × M diagonal matrix, then: UT S BTW U = Ι (5) The intersection space of the null space of SWTH and the
KDDA Transformation
Figure. 1 Architecture of the proposed KDDA based positioning system.
C
The detailed solution of ψ to maximize F (⋅) can be found in Ref. [11]. It can be sought by the intersection space of the null space of SWTH and the nonzero space of SBTW. The nonzero space of SBTW is obtained by preserving the eigenvectors V = [ v1 v M ] with the corresponding M biggest eigenvalues
tr(ψ T S BTW ψ ) tr(ψT SWTH ψ )
(4)
Let z ∈ R p be RSS vectors of p APs, class Z i represent the fingerprint with Ci corresponding RSS vector samples, φ (z ij ) be the kernel representation of the RSS vector. The kernel feature representation of z ij can be obtained by the projection of φ (z ij ) onto the optimal discriminant basis vectors ψ .
λm′ ]. We discard the
eigenvectors with the larger eigenvalues, since these eigenvectors lead to the larger feature variation. Supposing a matrix Q=UP, then we can derive that： QT SWTH Q = Λ w
where Λ w = diag [ λ1′ ,
(6)
λm′ ], a m×m diagonal matrix. Finally,
the m optimal discriminant basis vectors can be given as: ψ = QΛ −w1/ 2 . The basis vectors form a low-dimensional
subspace in F, which we call the kernel feature space. Using the kernel techniques, KDDA transformation of the input vector r can be represented as: f (r ) = Θ ⋅ γ (φ ( r ) )
(7)
where k ( ⋅, ⋅) is the kernel function, Θ is a m×L matrix.
γ (φ ( r ) ) = 1
L ⋅ ⎡⎣ k ( r1 (1) , r ) k ( r1 ( 2 ) , r )
k ( rC ( n ) , r ) ⎤⎦
T
,
The exact expression of transformation matrix Θ can be seen in Ref. [11]. The ultimate kernel feature representation matrix Γ can be obtained by the projection of the kernel matrix onto the set of optimal discriminant basis vectors: Γ = ΘK .
B. KDDA Distance Calculation for Location Estimation As stated above, the kernel feature representation of the radiomap can be obtained from the original fingerprint data by KDDA transformation. The radial basis function is chosen as the kernel function: 2 − r − r′ k (r, r ′) = (2π ) − p / 2 σ − p exp( ) (8) 2σ 2 where r, r ′ are both the RSS vector samples in the radiomap, p is the AP number used in the algorithm, k (⋅, ⋅) is the kernel function, σ is the kernel width. The parameter σ controls the width of the kernel and has a great effect on the shape of the kernel feature space. Its optimal value can be estimated by the total average variance. The proposed KDDA distance measure can be considered as the Euclidean distance in the kernel
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feature space. First, the original kernel representation K ′ of observation RSS vector can be obtained:
α (t ) = k (r (t ), r ′), K ′ = (α (1),
, α ( L ))T Then ultimate kernel feature representation Γ′ is: Γ′ = ΘK ′
(9) (10)
where Γ′ is a m ×1 vector. Let Γi be the average kernel feature representation at fingerprint i . The KDDA distance between the observation point and the fingerprint i can be represented as: di = Γ′ − Γ i 2 (11) And finally results in 2D Cartesian location estimation: k
p′ = ∑ p i ⋅ wi
(12)
i =1
wi =
1 di
k
1
∑d j =1
(13)
j
with: p′… coordinate estimation at observation point, pi … coordinate at fingerprint i , d j … KDDA distance between observation point and
the fingerprint j , k … number of the nearest fingerprints. Each fingerprint with corresponding kernel feature representations in the kernel feature space is treated as one class. Then the numerator and the denominator of F (⋅) in (4)
discriminative location information. The location estimation can be achieved by matching the k-nearest fingerprints by KDDA distance measure directly. III.
ANALYSIS OF THE EXPERIMENTAL RESULT
A. Experimental Setup RSS data were recorded in a realistic WLAN indoor environment, as shown in Fig.2. The dimension of test environment is about 24 × 20 m 2 . Every location in the environment was covered by four to nine IEEE 802.11b/g APs (another five APs are in the other place not shown). An ASUS laptop with Intel PRO/3945ABG IEEE Wireless Card was installed to gather RSS values from nearby APs. RSS measurements were achieved by publicly available network sniffer software, NetStumbler.
100 RSS samples for 78 physical points separated by 1 m along the corridor and the test rooms were recorded at a rate of 2 samples/sec. 40 locations were chosen as fingerprints and the other 38 locations as observation points. The grid distance was 2 m except for few fingerprints in the corner of the corridor with 1 m spacing. 4 new RSS samples were averaged to obtain a new RSS observation for location estimation. Gaussian distribution was used in the ML algorithm. In WKNN algorithm, the constant k was set to the found optimal value. The KM algorithm was implemented as described in Ref. [10].
represent the between-class scatter S BTW and within-class scatter SWTH of the fingerprints in the kernel feature space, respectively. S BTW measures the separation ability among different fingerprints (between-class) and thus the bigger it is, the more the system tolerates the location feature variation and noise. In contrast, SWTH measures the location feature variation on the average of fingerprint (within-class). The RSS location feature is more robust when it becomes smaller. As a result, the optimal discriminant basis vectors satisfying (3) lead to the best separation between different fingerprints in the kernel feature space, because the distance between different fingerprints is maximized while retains the same within-class scatter. Moreover, the original RSS vectors are mapped into high dimensional kernel space, in which different patterns of nonlinear relationship can be detected. Therefore, the nonlinear discriminative location information can be sufficiently extracted and enhanced by the KDDA transformation. Different from the kernel method (KM), the KDDA transformation not just maps the original RSS vectors into high dimensional kernel space, but also complies with Fisher criterion which can maximize the ratio of between- and within-class scatter. After creation of the kernel feature space which contains sufficient and enhanced nonlinear discriminative location information, the pattern recognition methods can be used for location estimation. The KDDA distance calculation is applied to extract the nonlinear
Figure. 2 Part of the 12-th floor of Communication Research Center, Harbin Institute of Technology, where we perform the experiment. The dash line represents the path of data collection along the corridor, the stars show the test rooms, and the black dots indicate the locations of the APs.
B. Analysis of the Experimental Results Three classical positioning algorithms, including Kernel, ML, and WKNN, are compared with the KDDA based positioning algorithm. We adopt the error distance as the performance metric, which is a Euclidean distance between the estimated location coordinate and the true coordinate. The location accuracy indicates the cumulative error distance distribution. Fig. 3 shows the location accuracy comparison of various algorithms. At error distance of 2 m, location accuracy of KDDA, Kernel, ML, and WKNN algorithm are 70.3%, 62.5%, 62.7%, 58.4% respectively. At error distance of 3 m, location accuracy for these algorithms are 85.7%, 77.7%, 78.7%, 74.9% respectively. It means that KDDA performs
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much better than other algorithms in terms of location accuracy. Table 1 shows that KDDA leads to 21.8% mean location error improvement than traditional algorithms. We can see that the kernel method has a comparable performance with the ML method, while WKNN performs the worst. That is because the statistical information is not sufficiently used in the WKNN algorithm.
Cumulative error distribution
The discriminative information quantity is indicated by the cumulative percentage of obtained eigenvalues versus the number of remain eigenvectors M. Fig. 4(a) shows that the mean location error is the lowest when M is 7. That is, 97.6% useful nonlinear discriminative information is preserved and the other 2.4% is discarded as noise, as presented in Fig. 4(b). Fig. 4(b) shows that the mean location error is decreasing with bigger M until M=7, while with more than 7 remain eigenvectors, the mean location error increases. The eigenvectors corresponding to smaller eigenvalues can be regarded as noise and discarded for little discriminative location information contained. The parameter m is set to the optimal value m=5 for M=7. In contrast to parameter M, parameter m indicates the number of remain eigenvectors of UT SWTH U with the m smallest eigenvalues. The eigenvectors with the larger eigenvalues are discarded since they lead to the higher feature variation and thus degrade the location accuracy. The proposed algorithm can adaptively extract nonlinear discriminative location information by tuning the parameter M and m, while the redundant location information is regarded as noise and discarded. Therefore, only useful discriminative location information is deployed and the location accuracy is improved by KDDA transformation. 100% 80% 60% KDDA
40%
Kernel WKNN
20% 0%
ML
0
1
2
3 4 5 Error distance(m)
6
7
PERFORMANCE COMPARISON OF VARIOUS ALGORITHMS
Within 2 m
Within 3 m
KDDA
1.61
70.3
85.7
Kernel
2.10
62.5
77.7
ML
2.06
62.7
78.7
WKNN
2.37
58.4
74.9
CONCLUSIONS
This paper proposes a novel KDDA based indoor positioning algorithm. The original RSS vectors are mapped into a kernel feature space for feature extraction by KDDA transformation. Each fingerprint in the kernel feature space is best separated from each other and so that the nonlinear discriminative location information between different fingerprints can be efficiently extracted and enhanced by KDDA distance calculation, while the redundant information is discarded as noise adaptively by tuning the remaining eigenvector number. As experimental result shows, the KDDA based positioning algorithm leads to 21.8% mean location error improvement than traditional algorithms. ACKNOWLEDGMENT The authors would like to thank the reviewers for useful comments and corrections. This work was supported by the National High-Tech Research & Development Program of China under Grant 2008AA12Z305. REFERENCES
Location Accuracy (%)
Mean Location Error (m)
Algorithm
IV.
8
Figure. 3 Location accuracy comparison of various algorithms. TABLE I.
Figure. 4 Effect of different number of remain eigenvectors M on the mean location error. (a) Mean location error versus remain eigenvector number M . (b) Cumulative percentage of remain eigenvalues versus remain eigenvector number M .
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