CN104507050A - A method for probability type fingerprint matching in WiFi (Wireless Fidelity) indoor positioning - Google Patents

Abstract

The invention provides a method for probability type fingerprint matching in WiFi (Wireless Fidelity) indoor positioning. The method is a method for probability type indoor positioning based on perception probability and kernel density estimation, and comprises five steps. Particularly, the method is a method for introducing the technologies of perception probability and non-parameter kernel density estimation into maximum likelihood probability. The method for probability type fingerprint matching in WiFi (Wireless Fidelity) indoor positioning can characterize distribution features of complex radio frequency signals more accurately through the kernel density estimation method, thus reducing a positioning error and obtaining better positioning accuracy.

Description

Probabilistic type finger print matching method in a kind of WiFi indoor positioning

Technical field

The invention provides probabilistic type finger print matching method in a kind of WiFi indoor positioning, is a kind of method perception probability and printenv Density Estimator technology are incorporated in maximum likelihood probability specifically.The method by the distribution characteristics of the complicated radiofrequency signal of Density Estimator method accurate characterization, can obtain good positioning precision, belongs to WiFi indoor positioning and wireless transmission and field of navigation technology.

Background technology

Along with modern times location and the development of airmanship, various location Based service becomes part important in Intelligent life day by day, GPS (Global Position System) (GNSS) provides high accuracy, round-the-clock positioning service for people, but because its measuring-signal can not penetrate the feature of building, effectively service cannot be positioned in highly dense groups of building district and indoor, therefore in order in the effective positioning service of indoor acquisition, indoor locating system obtains very fast development.

Indoor positioning technologies based on fingerprint matching is usually operated at two stages: off-line training step and tuning on-line stage.In off-line training step, the signal strength information from usable access point that in target area, all reference points receive forms fingerprint database.In the tuning on-line stage, the received signal strength (RSSI) of Real-time Collection is mated with the fingerprint in fingerprint database, thus obtains the positional information of positioning equipment.

Probabilistic type algorithm regards probability Estimation problem as the matching process of fingerprint in actual measurement RSSI and fingerprint base, based on the statistical property of RSSI signal, sets up the probability Distribution Model of radiofrequency signal in indoor environment, solves the uncertainty of RSSI value under complex environment.Conventional is maximum likelihood algorithm (ML) based on probabilistic type algorithm, theoretical based on Bayesian frame, posterior probability is converted into likelihood probability problem, mates maximum likelihood probability, obtain estimated position information.

The priori that Density Estimator method does not utilize relevant data to distribute, to the not additional any supposition of Data distribution8, be a kind of method from data sample data distribution characteristics itself, the distribution of sophisticated signal can be characterized more accurately, thus improve positioning precision.The wide parameter of core of kernel function reflect single sample to total body density distribution do the coverage of " contribution ".The sample changed scope of sampling within a certain period of time is limited, and each value has the possibility participating in calculating.The each sample of the larger explanation of excursion of sample value should be distributed with contribution to total body density in the larger context, and at this moment, kernel function should have larger core wide; On the contrary, the excursion of sample value is less, and the contribution Relatively centralized that each sample distributes to total body density is described, it is wide that kernel function should have less core.Therefore, when upgrading with Density Estimator algorithm, select the wide parameter of suitable core to be very important.

Based on the step that the calculating of likelihood probability in the probabilistic type indoor orientation method of fingerprint matching is crucial, the present invention, by the research to WiFi location technology and Density Estimator technology, proposes a kind of probabilistic type indoor orientation method based on perception probability and Density Estimator.

Summary of the invention

The object of the invention is to: probabilistic type finger print matching method in a kind of WiFi indoor positioning is provided, it is a kind of probabilistic type indoor orientation method based on fingerprint matching of correction, utilize parameterless Density Estimator technique computes likelihood probability, and merge mutually with the perception probability of reference point, to eliminate the discontinuity that traditional statistics with histogram method calculates, improve the positioning precision of system.

Technical scheme of the present invention:

The present invention proposes probabilistic type finger print matching method in a kind of WiFi indoor positioning.Probabilistic type indoor orientation method mainly based on probabilistic localization model, will regard a probability Estimation problem recording signal as with mating of location fingerprint database information.Posterior probability problem is converted into likelihood probability problem, utilize each location-prior RSSI statistical property information, historic state information and the environment layout information of localizing objects can also be utilized in some cases, with larger computation complexity for cost, obtain and compare based on the higher positioning precision of decision-making location technology.

In the WiFi indoor positioning that the present invention proposes, the principal character of probabilistic type finger print matching method is: be dissolved in probabilistic type indoor positioning algorithms by perception probability and Density Estimator technology, perception probability reflects the distribution character of signal to a certain extent, adopt printenv Density Estimator technique computes likelihood probability, in conjunction with perception probability, using perception likelihood probability as localization criteria to improve positioning precision.

Based on the Density Estimator method of nonparametric model, do not need to provide concrete distributed model in advance, do not need to carry out parameter Estimation yet and just can carry out modeling to system.Because traditional histogram model calculates likelihood probability value by the number of samples in statistics different grouping interval, the likelihood probability value arranged causing calculating between packet zone is discontinuous, so adopt Density Estimator to calculate likelihood function value.Due to the printenv characteristic of Density Estimator method, and the continuity of kernel function, be more suitable for the radiofrequency signal distribution under complex indoor environment.

Probabilistic type finger print matching method in a kind of WiFi indoor positioning of the present invention, comprises following step:

Step one: off-line phase, in test zone image data, sets up fingerprint database, for on-line stage matching primitives.Wherein, before image data, first test zone is surveyed, determine the position of reference point, secondly gather the sample information of some in each reference point, build fingerprint database.

Step 2: according to reference point training samples information, tries to achieve the perception probability of the relative access point of each reference point RP.Wherein, if be less than in the AP signal strength signal intensity at test position place the minimum signal strength that sample devices can perceive, indication equipment can not perceive AP signal, replaces not perceived signal strength information by a fixing signal strength signal intensity.Therefore AP signals collecting can be regarded as a Bernoulli process, to specific RP, each sampling can obtain a binary sequence B=(b ₁, b ₂... b _j..., b _n), wherein b _j∈ (0,1).Perception probability is defined as: the ratio of AP number that reference point perceives and total number of training.The perception probability of i-th RP to a jth AP is calculated as follows:

$P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)=n_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)/N\left(1\right|{\mathrm{\ω}}_i)$

Wherein, ω _ifor reference point locations, be the number of times of the jth AP that i-th RP can perceive, N (1| ω _i) be total number of training.

Step 3: mating of actual measurement RSSI and fingerprint in fingerprint base, employing Density Estimator method tries to achieve the match likelihood probability P (RSS of i-th RP to a jth AP ^j| ω _i).

First according to Bayes' theorem, maximum a posteriori probability problem is converted into maximum likelihood probability problem, Bayes' theorem expression formula is:

$P\left({\mathrm{\ω}}_i\right|\mathrm{RSS})=\frac{P\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)\·P\left({\mathrm{\ω}}_i\right)}{P\left(\mathrm{RSS}\right)}$

Wherein, P (ω _i) be the probability of corresponding reference point locations, the P (ω when not considering position history information _i) be constant, P (RSS) is constant.P (RSS| ω _i) be reference point ω _ilikelihood probability, therefore maximum a posteriori probability problem is converted into maximum likelihood probability problem, that is:

P(RSS|ω _i)＞P(RSS|ω _j)i,j＝1,2,…,m,j≠i

Be separate between each AP, therefore can obtain the expression formula of likelihood probability:

$P\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}P\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)$

Wherein, P (RSS ^j| ω _i) be the match likelihood probability of i-th RP to a jth AP.Ask the calculation procedure of match likelihood probability as follows:

1) Density Estimator equation is: ${\hat{f}}_x^j\left(x\right)=\frac{1}{\mathrm{Nh}}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}K\left(\frac{x-X_k}{h}\right)$

Wherein, likelihood probability when real-time reception signal strength signal intensity for a jth AP is x, K () is nuclear equation, and h is the wide parameter of core, X _k(1≤k≤N) is the value of the kth in tranining database.

2) kernel function K () is chosen to be gaussian kernel function

3) the wide parameter h of optimization core.

4) S _kfor the real-time signal strength of the kth AP that equipment receives.Can try to achieve match likelihood probability is $P\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i){\hat{f}}_{S_k}\left(S_k\right|1,{\mathrm{\ω}}_i)$

Wherein, in Density Estimator method, because the flatness of value on the Density Estimator curve based on training sample of the wide parameter h of core has larger impact, h is less, Density Estimator curve may be rough, although can reflect the information that each training sample comprises better, density Estimation is partial to probability density to distribute be confined to very much near observation data, causes density function estimation to have the peak value of a lot of mistake; On the contrary, if h is larger, Density Estimator curve is more smooth, but density Estimation is just loose for probability density contribution get Tai Kai, also lost by the part key character causing comprising in sample data, so need to be in optimized selection h, it is as follows to the Optimization Steps realizing h that employing minimizes mean square error:

1) suppose that true nuclear equation is f, then Density Estimator equation and the integrated square error between true nuclear equation is: $\mathrm{MISE}\left(\hat{f}\right)=E\{\∫[\hat{f}\left(x\right)-f\left(x\right){]}^2\mathrm{dx}\}$

2) due to integrand non-negative, MISE equation is rewritten as:

$\begin{array}{c}\mathrm{MISE}\left(\hat{f}\right)=\∫E[\hat{f}\left(x\right)-f\left(x\right){]}^2\\ =\∫[E\hat{f}\left(x\right)-f\left(x\right){]}^2\mathrm{dx}+\∫\mathrm{var}\hat{f}\left(x\right)\mathrm{dx}\end{array}$

3) suppose that nuclear equation K (u) is continuous, true cuclear density Equation f bounded, and second derivative is continuous.Define two constant α and β, wherein according to Taylor expansion, MISE can expand into following equation:

$\mathrm{MISE}\left(\hat{f}\right)\frac{1}{4}{h}^4{\mathrm{\α}}^2{\left\{{\▿}^{2}f\left(x\right)\right\}}^2\mathrm{dx}+N^{-1}{h}^{-n}\mathrm{\β}$

4) therefore, minimize mean square error MISE, the optimization solution that can obtain the wide parameter of core is:

$h_{\mathrm{opt}}^{n+4}=\mathrm{d\β}{\mathrm{\α}}^{-2}{\left\{{\left({\▿}^{2}f\right)}^2\right\}}^{-1}{N}^{-1}$

5) when kernel function is Gauss equation, the optimization solution of the wide parameter of core is:

$h_{\mathrm{opt}}={\left(\frac{4N}{d+2}\right)}^{1/(n+4)}\widehat{\mathrm{\σ}}$

Wherein, n is AP number, for the mean variance of all AP.

Step 4: according to perception probability and match likelihood probability, calculates perception likelihood probability P'(RSS| ω _i).

First, replace not perceived signal strength signal intensity with a particular value C, try to achieve the perception likelihood probability P'(RSS of i-th RP to a jth AP ^j| ω _i).Accounting equation is as follows:

$P^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)=\{P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i){\hat{f}}_{S_k}\left(S_k\right|1,{\mathrm{\ω}}_i){\}}^{b_j}\×\{[1-P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)]{\hat{f}}_{S_k}\left(C\right|1,{\mathrm{\ω}}_i){\}}^{(1-b_j)}$

Secondly, can try to achieve perception likelihood probability is: $P^{\′}\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}{P}^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)$

Step 5: by maximum likelihood probability (ML) method, try to achieve the coordinate of measuring position.

The calculation procedure of the coordinate of measuring position is as follows:

1) by ML method, the sequence number of trying to achieve the maximum reference point of perception likelihood probability is:

$t=\underset{i\∈[1,m]}{\arg \max }\left[\underset{1\≤j\≤n}{\mathrm{\Π}}{P}^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)\right]$

2) set reference point sequence number as the coordinate of t be (x _t, y _t), the estimated coordinates can trying to achieve measuring position is:

$(\hat{x},\hat{y})=(x_t,y_t)$

The invention has the advantages that:

One, perception probability and Density Estimator method are dissolved in maximum likelihood probability method, are a kind of probabilistic type indoor orientation method of correction, can indoor position accuracy be improved.

Two, employing Density Estimator method overcomes the discontinuity that statistics with histogram method calculates likelihood probability.

Three, Density Estimator method does not utilize the priori that relevant data distributes, and to the not additional any supposition of Data distribution8, is a kind of method from data sample data distribution characteristics itself, can characterizes the distribution of sophisticated signal more accurately.

Accompanying drawing explanation

Fig. 1 WiFi indoor positioning scene graph.

Fig. 2 the method for the invention flow chart.

The flow chart of the wide parameter optimization process of Fig. 3 Density Estimator method center.

Fig. 4 is based on position error figure of the present invention.

Fig. 5 is based on the position error figure of the wide parameter optimization of core of the present invention.

In figure, symbol, code name are described as follows:

AP Access Point WAP (wireless access point)

KDE kernel density estimation Density Estimator

ML maximum likelihood maximum likelihood

MISE mean integrated square error integrated square error

CDF cumulative distribution function cumulative distribution function

Embodiment

Below in conjunction with accompanying drawing 1,2,3,4, introduce the probabilistic type indoor orientation method based on perception probability and Density Estimator proposed by the invention in detail for a concrete indoor test environment.

See Fig. 1, be a concrete indoor test area planar figure, in this test zone, have 5 access point AP, and room area is divided into the grid of 1.1m*1.1m; Corridor area is divided into the grid of 1.2m*1.2m.Whole test zone is divided into 195 grid altogether, and the center of each grid is the position of reference point RP.

See Fig. 2, it is the method for the invention flow chart.For a concrete indoor test environment, probabilistic type finger print matching method in a kind of WiFi indoor positioning of the present invention, its step is as follows:

Step one: gather training samples information at test zone to 195 RP, each reference point sampling number is 80 times, and using signal strength signal intensity average in the training samples information of trying to achieve as fingerprint, set up fingerprint database, for on-line stage matching primitives.

Step 2: according to reference point training samples information, tries to achieve the perception probability of 195 relative 5 access point AP of reference point RP concrete formula is as follows:

$P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)=n_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)/N\left(1\right|{\mathrm{\ω}}_i)$

Step 3: test position (10.95,1.65), the actual measurement RSSI value mean vector obtained is $\stackrel{\→}{S}=(-59.714,-50.114,-63.829,-89.2,-84.971),$ Actual measurement RSSI vector with mating of fingerprint in fingerprint base, Density Estimator method is adopted to try to achieve the match likelihood probability P (RSS of i-th RP to a jth AP successively ^j| ω _i).

Step 4: according to perception probability and match likelihood probability, calculates the perception likelihood probability of i-th RP to a jth AP, thus obtains perception likelihood probability P'(RSS| ω _i).Computing formula is as follows:

$P^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)=\{P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i){\hat{f}}_{S_k}\left(S_k\right|1,{\mathrm{\ω}}_i){\}}^{b_j}\×\{[1-P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)]{\hat{f}}_{S_k}\left(C\right|1,{\mathrm{\ω}}_i){\}}^{(1-b_j)}---\left(1\right)$

$P^{\′}\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}{P}^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)---\left(2\right)$

Wherein, calculate the perception likelihood probability of i-th RP to a jth AP according to (1) formula, in formula, constant C is set to-100dbm, calculate perception likelihood probability according to (2) formula.

Step 5: by maximum likelihood probability method, tries to achieve the coordinate of measuring position.

The calculation procedure of the coordinate of measuring position is as follows:

By ML method, the sequence number of trying to achieve the maximum reference point of perception likelihood probability is t=37, and computing formula is as follows:

$t=\underset{i\∈[1,m]}{\arg \max }\left[\underset{1\≤j\≤n}{\mathrm{\Π}}{P}^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)\right]$

Wherein, reference point sequence number be 37 coordinate be (10.95,1.65), in the hope of the estimated coordinates of measuring position be (10.58,1.65).

One-time positioning is completed by above five steps, be (10.95 at coordinate figure, 1.65) point is measured, by obtaining probabilistic type likelihood probability method based on perception probability and Density Estimator, the estimated coordinates value obtained is (10.58,, and can to obtain position error be 0.37m 1.65).

See Fig. 3, the optimization procedure of the wide parameter of center density estimation method center of the present invention, the wide parameter of core is the biggest factor affecting Density Estimator, calculate the integrated square error between Density Estimator equation and true nuclear equation, and integration type is carried out abbreviation by Taylor expansion, and minimizing integrated square error, the optimal solution obtaining the wide parameter of core is h=6.87.

As shown in Figure 4, the error CDF figure between positioning result and physical location is obtained according to method of the present invention actual experiment in the scene shown in Fig. 1, in figure, dotted line represents the positioning result adopting statistics with histogram to obtain, what solid line represented is the positioning result adopting the inventive method to obtain, can find out that the location mean error obtained according to the present invention is 2.0941 meters from the statistics of lower list one, improve 34.2% relative to statistics with histogram method positioning precision.

Table 1

	Density Estimator method	Statistics with histogram method
			Mean error distance (m)	2.0941	3.1846

As shown in Figure 5, the position error CDF that obtains after the wide parametric optimal solution of the core obtained according to method of the present invention figure, the position error that what in figure, dotted line represented is h value when being 2, position error during h gets optimal solution 6.87 that what solid line represented is.Therefore, the optimization of the wide parameter of the core in the present invention can obtain good positioning precision.

In sum, probabilistic type finger print matching method in a kind of WiFi indoor positioning provided by the present invention, is based on WiFi fingerprint matching indoor positioning technologies, and adopts Density Estimator technique computes likelihood probability.Feature of the present invention is the method for Density Estimator from data sample data distribution characteristics itself, does not utilize the priori that relevant data distributes, to the additional any supposition of Data distribution8, and can the distribution of signal in complex indoor environment more accurately.In addition, due to the Density Estimator continuity of a function of the present invention, overcome the discreteness that traditional statistics with histogram method calculates likelihood probability, and in conjunction with perception probability, adopt perception likelihood probability as measure coefficient, improve positioning precision.

Claims (1)

1. a probabilistic type finger print matching method in WiFi indoor positioning, is characterized in that: it comprises the following steps:

Step one: off-line phase, in test zone image data, sets up fingerprint database, for on-line stage matching primitives; Wherein, before image data, first test zone is surveyed, determine the position of reference point, secondly gather the sample information of some in each reference point, build fingerprint database;

Step 2: according to reference point training samples information, tries to achieve the perception probability of the relative access point of each reference point RP; Wherein, if be less than in the AP signal strength signal intensity at test position place the minimum signal strength that sample devices perceives, indication equipment can not perceive AP signal, not perceived signal strength information is replaced by a fixing signal strength signal intensity, therefore AP signals collecting is regarded as a Bernoulli process, to specific RP, each sampling acquisition binary sequence B=(b ₁, b ₂... b _j..., b _n), wherein b _j∈ (0,1); Perception probability is defined as: the ratio of AP number that reference point perceives and total number of training; The perception probability of i-th RP to a jth AP is calculated as follows:

$P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)=n_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)/N\left(1\right|{\mathrm{\ω}}_i)$

Wherein, ω _ifor reference point locations, be the number of times of the jth AP that i-th RP can perceive, N (1| ω _i) be total number of training;

Step 3: mating of actual measurement RSSI and fingerprint in fingerprint base, employing Density Estimator method tries to achieve the match likelihood probability P (RSS of i-th RP to a jth AP ^j| ω _i);

First according to Bayes' theorem, maximum a posteriori probability problem is converted into maximum likelihood probability problem, Bayes' theorem expression formula is:

$P\left({\mathrm{\ω}}_i\right|\mathrm{RSS})=\frac{P\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)\·P\left({\mathrm{\ω}}_i\right)}{P\left(\mathrm{RSS}\right)}$

Wherein, P (ω _i) be the probability of corresponding reference point locations, the P (ω when not considering position history information _i) be constant, P (RSS) is constant; P (RSS| ω _i) be reference point ω _ilikelihood probability, therefore maximum a posteriori probability problem is converted into maximum likelihood probability problem, that is:

P(RSS|ω _i)＞P(RSS|ω _j) i,j＝1,2,…,m,j≠i

Be separate between each AP, therefore obtain the expression formula of likelihood probability:

$P\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}P\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)$

Wherein, P (RSS ^j| ω _i) be the match likelihood probability of i-th RP to a jth AP, ask the calculation procedure of match likelihood probability as follows:

1) Density Estimator equation is: ${\hat{f}}_x^j\left(x\right)=\frac{1}{\mathrm{Nh}}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}K\left(\frac{x-X_k}{h}\right)$

Wherein, likelihood probability when real-time reception signal strength signal intensity for a jth AP is x, K () is nuclear equation, and h is the wide parameter of core, X _k(1≤k≤N) is the value of the kth in tranining database;

2) (# is chosen to be gaussian kernel function to kernel function K

3) the wide parameter h of optimization core;

4) S _kfor the real-time signal strength of the kth AP that equipment receives, trying to achieve match likelihood probability is $P\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)={\hat{f}}_{S_k}\left(S_k\right|1,{\mathrm{\ω}}_i)$

Wherein, in Density Estimator method, because the flatness of value on the Density Estimator curve based on training sample of the wide parameter h of core has larger impact, h is less, Density Estimator curve is rough, although can reflect the information that each training sample comprises better, density Estimation is partial to probability density to distribute be confined to very much near observation data, causes density function estimation to have the peak value of a lot of mistake; On the contrary, if h is larger, Density Estimator curve is more smooth, but density Estimation is just loose for probability density contribution get Tai Kai, also lost by the part key character causing comprising in sample data, so need to be in optimized selection h, it is as follows to the Optimization Steps realizing h that employing minimizes mean square error:

1) suppose that true nuclear equation is f, then Density Estimator equation and the integrated square error between true nuclear equation is: $\mathrm{MISE}\left(\hat{f}\right)=E\{\∫[\hat{f}\left(x\right)-f\left(x\right){]}^2\mathrm{dx}\}$

2) due to integrand non-negative, MISE equation is rewritten as:

$\begin{array}{c}\mathrm{MISE}\left(\hat{f}\right)=\∫E[\hat{f}\left(x\right)-f\left(x\right){]}^2\mathrm{dx}\\ =\∫[E\hat{f}\left(x\right)-f\left(x\right){]}^2\mathrm{dx}+\∫\mathrm{var}\hat{f}\left(x\right)\mathrm{dx}\end{array}$

3) suppose that nuclear equation K (u) is continuous, true cuclear density Equation f bounded, and second derivative is continuous, defines two constant α and β, wherein β=∫ { K (t) } ²dt, according to Taylor expansion, MISE expands into following equation:

$\mathrm{MISE}\left(\hat{f}\right)\≈\frac{1}{4}{h}^4{\mathrm{\α}}^2\∫\{{\▿}^{2}d\left(x\right){\}}^2\mathrm{dx}+N^{-1}{h}^{-n}\mathrm{\β}$

4) therefore, minimize mean square error MISE, the optimization solution obtaining the wide parameter of core is:

$h_{\mathrm{opt}}^{n+4}=d{\mathrm{\β\α}}^{-2}\{{\left({\▿}^{2}f\right)}^2{\}}^{-1}{N}^{-1}$

5) when kernel function is Gauss equation, the optimization solution of the wide parameter of core is:

$h_{\mathrm{opt}}={\left(\frac{4N}{d+2}\right)}^{1/(n+4)}\widehat{\mathrm{\σ}}$

Wherein, n is AP number, for the mean variance of all AP;

Step 4: according to perception probability and match likelihood probability, calculates perception likelihood probability P'(RSS| ω _i);

First, replace not perceived signal strength signal intensity with a particular value C, try to achieve the perception likelihood probability P'(RSS of i-th RP to a jth AP ^j| ω _i), accounting equation is as follows:

$P^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)=\{P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i){\hat{f}}_{S_k}{\left(S_k\right|1,{\mathrm{\ω}}_i)\}}^{b_j}\×\{[1-P_{b_j|\mathrm{\ω}}\left(1\right|{\mathrm{\ω}}_i)]{\hat{f}}_{S_k}{\left(C\right|1,{\mathrm{\ω}}_i)\}}^{(1-b_j)}$

Secondly, trying to achieve perception likelihood probability is: $P^{\′}\left(\mathrm{RSS}\right|{\mathrm{\ω}}_i)=\underset{j=1}{\overset{n}{\mathrm{\Π}}}{P}^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)$

Step 5: by maximum likelihood probability (ML) method, try to achieve the coordinate of measuring position;

The calculation procedure of the coordinate of measuring position is as follows:

1) by ML method, the sequence number of trying to achieve the maximum reference point of perception likelihood probability is:

$t=\underset{i\∈[1,m]}{\arg \max }\left[\underset{1\≤j\≤n}{\mathrm{\Π}}{P}^{\′}\left({\mathrm{RSS}}^j\right|{\mathrm{\ω}}_i)\right]$

2) set reference point sequence number as the coordinate of t be (x _t, y _t), the estimated coordinates can trying to achieve measuring position is:

$(\hat{x},\hat{y})=(x_t,y_t).$