A mobile localization algorithm based on fuzzy estimation for serious NLOS scenes

The indoor environment is intricate and the global positioning system (GPS) unable to satisfy the demand of indoor location accuracy. Therefore, the localization method based on wireless sensor network (WSN) has attached great importance and researched lately. The toughest issue to solve is the non-line of sight (NLOS) error caused by the uncertainty of the propagation environment. Hence, a location method based on hypothesis test and modified fuzzy probabilistic data association filter (HT-MFDAF) is proposed in this paper. Line-of-sight (LOS) and NLOS situations are regarded as an interactive Markov process. In the case of NLOS, we firstly identify and mitigate NLOS based on hypothesis testing theory. Then the ones which still have serious NOLS pollution is discarded by calculating similarity. Finally, the fuzzy membership degree is calculated by MFDAF, reconstructing the correlation probability to get the position estimate. The eventual location result is acquired by the Interactive Multiple Model (IMM) which weighted LOS and NLOS estimated position. Simulation and experimental results demonstrate the effectiveness of the algorithm.


Introduction
Wireless sensor network (WSN), as the hotspot of current international research, have been widely used in military [1], target tracking [2], monitoring [3], energy-efficient routing [4] and so on.Location is one of the crucial applications for WSN, which can make up for shortcomings of GPS indoor positioning accuracy [5].And it is mainly achieved by measuring the distance between the mobile node (MN) and the beacon node (BN).The main measurement methods include time of arrival [6], time difference of arrival [7], angle of arrival [8] and received signal strength [9].Moreover, LOS means that there is no barrier between BN and MN.Otherwise, it is NLOS.However, NLOS is more common in indoor environment, which is the main reason affecting the positioning accuracy [10].
On that basis, numerous algorithms and approaches have been researched.NLOS recognition is a research focus [10][11][12][13][14][15][16][17].In [11,12], the identified NLOS measurements are discarded.A chi square test is proposed in [11] to identify NLOS situation.Then only the measured values under LOS situation is used for positioning.[12] identifies and discards NLOS measurements by a threshold determined by NP theorem.This kind of approach availably removes impacts of NLOS measurements, but the location errors will sharply rise if the quantity of LOS BNs is not enough.To deal with this problem, creating virtual base stations (VBS) in uncharted region is presented in [13].And VBS are generated by mirroring the NLOS BS on the transmitting wall, which can consider as LOS BS.Therefore, there will be enough BSs to ensure accurate positioning.Another approach is to use appropriately weighted LOS and NLOS measurements [10,14,15].A method based on gaussian mixture model (GIMM) and extended Kalman filter (EKF) is proposed in [10], where the GIMM is used to identify LOS and NLOS.Moreover, feature matching also can reduce NLOS error, which is based on the pre-created databases, such as offline training [14,15].But if the weight of NLOS is large, the positioning accuracy will be dramatically reduced.Due to the occlusion of obstacles, the NLOS measurements always have a positive deviation.Thus, it can be used as constraints to mitigate NLOS errors and improve localization accuracy [16,17].A cooperative localization algorithm is proposed in [16].The geometric relationship of distance deviation is used as a constraint in semi definite programming.And the constrained Cramer Rao bound is derived to prove that can effectively reduce NLOS errors.A through-the-wall (TTW) model is proposed in [17], which focuses on reducing NLOS errors caused by walls.It describes the relationship between LOS and NLOS through the relative dielectric constant and thickness of the wall.Simulation reflects better positioning accuracy of the model.
On the other hand, [6,[18][19][20] do not require NLOS identification for location under mixed LOS-NLOS conditions.Interacting multiple mode (IMM) [18] designs two parallel filters for LOS and NLOS models predicted at the same time.The final output values of the two models are obtained by weighted fusion of likelihood function and transition probability, which efficiently mitigate the NLOS effects.Therefore, numerous localization algorithms combined with IMM model have been proposed, such as base on Kalman filter [18] and unscented Kalman filter (UKF) [19].Moreover, another beneficial method is weighted least squares weighted least square (WSL) [6,20].A location method under unknown model is proposed in [20], which derivates an equilibrium parameter and additional path loss.The unknown parameters and positions are solved by robust weighted least square (RWLS) iteration.In [6], the variance of the estimated range is obtained by M estimation or MM estimation to calculate weights.Then the location is estimated by WSL and extrapolated single propagation UKF.It is worth noting that these methods use all the measured values, especially the IMM model.It will require a lot of calculation when there is a quantity of BSs.
Recently, fuzzy theory is widely used in target tracking and location.Fuzzy inference [21,22] and fuzzy clustering [24][25][26] are the two most commonly used techniques.Ho [21] proposes a selective adaptive method for adjusting noise covariance using fuzzy logic.Then the author extends the algorithm [22].In this algorithm, EKF is added on the original basis.The fuzzy-tuned EKF bank with NLOS bias is used for state estimation, which significantly improves the positioning accuracy.In [23], RSS is used to calculate the distance between MN and BN. which is nonlinear and obtained by interpolating several linear equations by fuzzy technology.Additionally, fuzzy clustering is mainly used for data association [24][25][26] based on the similarity characteristics of fuzzy membership degree and correlation probability.Such as fuzzy c-means clustering [24] and maximum entropy fuzzy clustering [25].The concept of rough set is introduced in [26], which is combined with fuzzy set to calculate the incidence matrix.These methods greatly reduce the complexity of probabilistic data association (PDA), but the improvement in accuracy is limited [24].
In this paper, we propose a modified fuzzy PDA filter and make a NLOS recognition and mitigation of the initial distance measurements by hypothesis testing.Then the location estimation that still has NLOS errors are rejected by calculating the similarity.Finally, the modified fuzzy PDA filter is used to calculate corresponding fuzzy memberships to get final estimated location under NLOS model.The contributions of our algorithm are as follows: 1.A NLOS recognition and mitigation method is proposed, which is based on the different distribution of measurement noise and NLOS noise.It has a distinct mitigation performance on the measurement distance contaminated by NLOS. 2. The distance is processed in advance and the serious NLOS pollution points are discarded.The algorithm has been proved to have better performance through simulations and experiments, especially in the serious NLOS scenes.3. The algorithm can be easily expanded to multi-target tracking.And compared with the traditional data association filter, the modified fuzzy PDA filter has lower computational cost and complexity.
The structure of the paper is organized as below.Section 2 introduces the signal model.The proposed algorithm is explained in detail in Section 3. Simulations and experiments are presented in Section 4, and Section 5 draws the conclusion.

Signal model
MN is assumed to move in a plane area with N beacon nodes (BNs).The location of the BNs is known.The transmission channels between the MN and the BN changes between LOS and NLOS conditions, which is denoted as a two-state Markov process.As shown in Fig. 1.
The state vector of the MN is indicated as X(s) = [x(s), y(s), ̇x(s), ̇y(s)] T , where (x(s), y(s)) and ( ̇x(s), ̇y(s)) denote the position and speed of the MN respectively and s indicates a time step.
Then the state equation is modeled as where, where, T s is the sampling period, F denotes the state tran- sition matrix of the MN.And C is described the random acceleration caused by the process driving noise v(s) that be assumed follow the Gaussian distribution v(s) ∼ N(0, Q(s)).
In this paper, the distance between the MN and BN is obtained based on TOA at time step s, which can be defined as where, d(x(s)) is the real distance between BN and MN.And the noise n given by where, n L and b NLOS represent range and NLOS measurement noise respectively, which can be modeled as the Gaussian distribution n L ∼ N(0, 2 L ) and b NLOS ∼ N( n , 2 n ) [29].More- over, the probability density function (PDF) of b NLOS can be derived as d(x(s)) can be expressed as where, (x BS n , y BS n ) represents the coordinate of n-th BN. (1)

The main procedures of the algorithm
Firstly, we make a NLOS recognition and mitigation of the distance measurements by hypothesis testing.In this step, the mean m 1 and standard deviation (STD) 1 of the measured distance sequence before the current moment are calculated firstly (assuming that the sequence length is M).The mean and variance of LOS and NLOS are different, so that we can compare the difference value with threshold that calculated by hypothesis testing to judge the current state.But once threshold range in the initial recognition is too small, which easily lead to wrong assumptions.Therefore, it is necessary to verify the assumptions of the first step.The mean m 2 and STD 2 of the measured distance sequence after the current moment (including the current moment) are calculated and compared with m 1 and 1 .We also make a comparison between thresholds and results to verify the hypothesis.Specific details are described in part C of this section.When the current state is verified NLOS, the effect of the NLOS can be greatly eliminated by subtracting estimated bias from measurements.Then C 3 N subgroups can be obtained when there are N BNs.Next, the same number of estimated positions can be obtained by the least-squares (LS).Once more than one measurement of the subgroup is NLOS propagation, the position estimation will have deviation errors and outliers are generated.Since the more the coordinate points deviate, the smaller the corresponding similarity is.Therefore, outliers are found and discarded by calculating the similarity.Finally, the fuzzy membership degree of the remaining position estimation is calculated by MFDAF, reconstructing the correlation probability to get the position estimate.The eventual location result is acquired by the IMM which weighted LOS and NLOS estimated position.

Interaction and LOS update
1. Interaction: Firstly, the output of the LOS and NLOS model from the previous time is interacted, which as the initial state in the current loop.We assumed that the state and covariance matrix interaction value of model j separately express by X0j (s − 1|s − 1) and P 0j (s − 1|s − 1) .The interaction process is as follows where, Xi (s − 1|s − 1) and P i (s − 1|s − 1) denote the state output value of mode i at time s-1 and ( 7)

Model matched:
The results of interaction between the two models obtained in the above process are matched.
Then the position and covariance of MN can be predicted as 3. LOS Update: Under the LOS model, the update process is achieved by EKF.Firstly, the innovation and innovation covariance matrix at time s are calculated where, R * 1 (s) is covariance matrix of the measurement error vector.And Jacobian matrix H 1 (s) in Eq. ( 15) is expressed as Then, the likelihood function under LOS model can be obtained

Fig. 2 Flow chart of proposed algorithm
And gain K 1 (s) is calculated by Finally, the state variable estimation and error covariance matrix are modified based on Kalman gain matrix, which given by

NLOS identification and mitigation
As shown in Fig. 3, in the case of NLOS (j = 2) we carry out NLOS recognition and mitigation for the initial measured distance.More detail be described in the following section.

Initial NLOS Identification:
The initial identification uses the sequence of past distance measurements.Specifically, suppose that the current time instant is s.Then the mean and STD of the past sequence of M original distance measurements are computed by where, M is the length of the sequence.Specifically, M should be less than the samples per second of MN.For instance, if the sampling frequency is 5HZ, M can be set to about 3. When the speed of MN is 1 m/s, the change between the sequences is no more than 0.6 m.It is guaranteed that there is only a minor change between sequences, especially compared to the noise.Hence, the approach is uncomplicated and credible.
Suppose the distance at the current moment is denoted by d s and compared with m 1 .Here, we mainly compare whether the difference between the measured distance at the current moment and at previous moments is within a normal range.On the one hand, if the difference is within a predetermined range, which is The current measurement is assumed to be under NLOS state.But if the hypothesis is changed to LOS. t1 , t1 and t1 are threshold, that are discussed in Sect.3. Then the mean and STD in Eqs.(21) and (22) are updated by d s , which are used for the next moment measurement and identification.
Next, a new measurement is taken and the identification procedure continues.On the other hand, if Eq. ( 24) do not holds and the current propagation hypothesis is NLOS.
Moreover, while Eq. ( 23) is not satisfied, and the LOS is supposed.There might be wrong decisions when t1 and t2 are small.Therefore, verifying the hypothesis is important, which can greatly enhance the correctness of decision-making.

Hypothesis Verification:
To verify the decision that exceeds the threshold range, the mean and STD of the measured sequence after current moment are obtained by ( 21)

Fig. 3 Identification flowchart
The two sequences are shown in Fig. 4. In the following, we call the measurement of the first sequence S 1 and the second S 2 .
When the Eq. ( 25) is true in the initial identification and where, m1 is the threshold for the mean difference and t2 is another STD threshold.Then the current measurement is decided as NLOS.However, once a condition in Eq. ( 29) is not true, the assumption that time s is under NLOS propagation is rejected.
Similarly, if Eq. ( 26) holds in the initial identification and where, m2 is another threshold for the mean difference.The current measurement will be decided as LOS.But the original LOS decision is also negated if one of the conditions in Eq. ( 30) is not true.Moreover, if both Eqs. ( 29) and (30) are not satisfied, we return to check Eq. (24).And if Eq. ( 24) holds, the hypothesis is changed from NLOS to LOS.
Note that in the hypothesis testing, it is inevitable to produce incorrect decisions.However, since the identification is based on a sliding window of measurements, no serious error propagation phenomenon would occur.Also, the online distance measurement variance estimation would usually strengthen the correct decision and reject the incorrect decision.Therefore, the presented identification algorithm would be robust in the presence of incorrect decisions.3. Threshold calculation: In the above recognition and verification process, numerous thresholds are needed.They are calculated based on the hypothesis testing theory and described in detail in this section.In particular, the PDF of NLOS and LOS are different.Therefore, the threshold can be determined accordingly when the probability of false alarm (PFA) is given.Specifically, it is assumed that the distributions of the distance measurement noise and bias error are known a priori such as based on process- ing field measurements and modeling the noise and bias error.In this case, the threshold is calculated in advance, so it doesn't take time.But the noise distribution in real scene may be unknown, which can be obtained by offline measurement and online estimation (e.g.HMM [27] and GMM [28]).When these methods are not used, we can also estimate a smaller value based on experience.The method is still effective.However, investigation of such an adaptive technique is beyond the scope of this paper.In the remainder of this section, the details of selecting the thresholds in the presence of known distance measurement noise and bias error statistics are described.
Due to the occlusion of obstacles, the propagation time of NLOS will be longer, which is always the positive errors.And it is usually assumed as a Gaussian or exponential distribution when comparing the accuracy of localization algorithm [29,30].Thus, the measurement errors and NLOS errors are assumed as Gaussian distribution N(0, 2 L ) and N( n , 2 n ) respectively in this paper.Firstly, if S 1 sequence is considered as LOS propagation.The mean and STD are zero and Gaussian distribution which depends on whether the state of current moment is NLOS or LOS.If the current measurement is under LOS, the PDF of f can be denoted as where, LL = L √ 1 + 1∕ M .On the other hand, when it is NLOS, the PDF is where, LN = √ 2 L M + 2 n .Figure 5 shows an example of the two PDFs in this case.We define the probability of detection (POD) as the probability of determining NLOS as NLOS, while PFA is the probability of determining LOS as NLOS.Explicitly, the two PDFs are overlap.Therefore, if a larger POD is required, the PFA will not be small enough, and vice versa.Therefore, a trade-off should be made.And POD and PFA are respectively derived as where, Q( • ) is the standard Q-function.When a PFA or POD value is given, the threshold is calculated.
It is worth mentioning that the bias mean is subtracted when the current measured distance is verified as NLOS.Hence, it is essential to keep PFA in a smaller range, which is essential.
However, as we mentioned above, the initial decision may be wrong when the value of t1 and t2 are small.And the ideal PFA and POD values cannot be obtained simultaneously.For example, when required value is determined, we set the PFA at 95%, but the POD of this time is decreased to 66%.Therefore, it is indispensable to verify the hypothesis for resolving such a contradiction.
In the analysis of hypothesis verification, the threshold is calculated based on the mean difference m 2,1 between S 1 and S 2 .Firstly, the case that S 1 is the LOS propagation state is analyzed.When S 2 is also in LOS propagation state, the PDF of m 2,1 is expressed as where, σLL =  L √ 2∕ M. Then when S 2 is under NLOS propagation, m 2,1 has a PDF given by where, σLN = 7 shows the two PDFs in this case.
It is easy to observe that the two PDFs hardly overlap, even when the M is small.The STD of m 2,1 is significantly reduced because of the averaging operation.Therefore, by selecting the appropriate threshold, we can almost always get the right hypothesis.PFA and POD correspond as follows (38) On the other hand, if the assumption of S 1 is in the NLOS propagation state, also the PDFs can be expressed as Eqs.( 43) or (44) respectively with S 2 in the LOS or NLOS.
It can be seen that the overlap of the two density functions is small, so as long as the appropriate threshold is selected, the decision in the first stage can be verified correctly.The PFA and POD at this time are As mentioned above, all thresholds are calculated by the Q function which guided by the value of PFA and POD.

MFDAF
1. Outlier Detection: At least 3 measured distances are needed when the location of MN is calculated by LS.Therefore, R = C 3 N location estimates are obtained with N BNs at most, which denoted as (x i (s), ŷi (s)), i = 1, 2, ..., R (42) Although the measured distance has been processed above, the effect is not perfect.Consequently, there may be estimated coordinates with deviation errors, which are called outlier and shown in Fig. 9.
Firstly, the similarity S ij between two coordinate points is calculated as where, D ij is the distance between two coordinate points and have given by where, 0 < S ij < 1.When D ij is small, S ij will be close to 1. Conversely, S ij will tend to zero if the D ij gets larger.In this way, S ij will have greater curvature.That makes it applicable to all data sets, not just the set of distances in a particular case.Mathematically, is defined as where, D is the mean distance among the pairs of coor- dinate points.When S ij >  , these two coordinate points are considered largely relevant.is the threshold of similarity which takes a value in the range 0 ~ 1. 0.7 is quite robust as shown by our experiments.
(47)  By traversing R coordinate points, R i is regarded as the number point whose similarity with the i-th point is satisfied S ij >  .Then the following hypothesis can be obtained.
where, R s is the control coefficient of quantity with the range [0, R].And it is better to take 20% of the total quantity.If H 0 is true, coordinate points are regarded as under LOS propagation.Or else coordinate points are outlier and removed.

A brief introduction to PDAF: A detailed derivation
of PDAF can be found in [31] while it is just briefly described here.The measurement model is the same as above.Suppose the set of validate measurements at time s is Z r (s) , r is the number of validated measurements at time s.The cumulative set of validated measurements up to time k is denoted as Z S .At each scan, a validation gate, centered around the predicted measurement of the target, is setup to select the measurements associated probabilistically with the target.The validation region is where, S(s) is the covariance of the innovation corresponding to the true measurement.If the EKF is used to estimate the state of a target, the prediction and update process of PDAF is the same as Eq. ( 12) to Eq. ( 18).The differences are the estimation of the state variable and error covariance matrix.
The state update equation of PDAF is where, the combined innovation is The overall covariance associated with the updated state is i (s) denotes the probability that the i-th measurement comes from the target at time s, 0 (s) denotes the prob- ability that none of the measurements is originated from the target.The derivation of the probability i (s) is briefly described here, and the details can be found in [31].
To evaluate the association probabilities, the conditioning is broken down into the past data Z S−1 and the latest data Z(s) .A probabilistic inference can be made on both the number of measurements in the validation region (from the clutter density, if know) and on their locations, expressed as where c s is a normalization constant, j (s) denotes the event that the measurement z j (s) is originated from the target, 0 (s) denotes the event that none of the measure- ments is originated from the target.
The association probabilities are where where, V s (s) is the volume of the ellipsoidal validation region, P D is the probability of detection, P G is the prob- ability that the target measurement falls in the m-dimensional validation region, is the spatial density of false measurements.
At the same time, the weights i (s) ) fulfill the con- straint 3. MFDAF for NLOS update: MFDAF is used for data association and update in NLOS case, which is based on fuzzy c-means clustering (FCM) and probabilistic (57) data association (PDA).FCM is a process to partition a given set of data points into subgroups according to a certain similarity.After partition, each point in the data space is categorized into clusters via membership.The membership value reflects the closeness of the data point to clusters.
The idea developed in this paper is the possibility of incorporating the membership grades supplied by the fuzzy algorithm as a direct counterpart of the association probability in PDAF while keeping the main structure of PDAF.Indeed, the justification of such substitution can be carried out from different viewpoints: (i) The interpretation of the association probability quantities as degrees of agreement between measurement and targets means that both joint probabilities and membership grades have the same interpretative setting.(ii) The general constraints governing the construction of the joint probabilities are kept preserved in the fuzzy setting.(iii) PDAF needs to calculate the validation region every time, and there is a situation that the target cannot be matched.PDAF when multiple are associated to the same measurement, PDAF needs to calculate all possible correlation scenarios, with an explosive increase in calculation [32].FCM only needs to iterate membership based on distance.
Therefore, the ij can be replaced by u ij .Firstly, supposed there is a set of data Z i , and let c and C j (1 ≤ j ≤ c) respectively denote number of classes and center.Consider the dynamic structure of the system and its connection to previous measurements, the objective function based on FCM is modeled by subject to where, indicates the degree of ambiguity.It is used to modify the shape of membership functions under different categories.(Z j ) old is relevant knowledge of the last time of j-th class, which is a part of X(s − 1|s − 1) .F p j is the part of the matrix F which represents the position component asso- ciated with the target j.Thus F p j (Z j ) old represents the pre- dicted prototype of the j-th class.The last constant K is a parameter ranging from 0 to 1, depending on whether we (64) prefer the current observation or the known state.Generally, the value of K should be greater than 0.5, which denotes that current observations are more trusted.Then C j and distance matrix d ij are be calculated as the u ij can be updated to The objective function is minimized by iterating the u ij and C j .
We know that PDA can be used for multi target tracking, while FCM can also have multiple clustering centers (i.e., multiple targets).Therefore, this MFDAF can be extended to applications of multi target tracking.In this paper, we just consider the case of one MN.Therefore, the Z i can be expressed as the coordinate estimation Z r (s) satisfied Eq. (50).Then the innova- tion v 2,r (s) between prediction coordinate and Z r (s) is given by where, Ẑ(s|s − 1) is the position estimation prediction, which given by where, G = 1 0 0 0 0 1 0 0 is the observation matrix.Then let Finally, according to the probability calculated by Eq. ( 68), the state and variance vector can be updated.The innovation covariance matrix S 2 (s) and gain matrix K 2 (s) can be got by Then the state and covariance are updated as (65) where, Ps with the weighted innovation E(s) corrects the meas- urement uncertainty.And Λ 2 (s) indicates likelihood function.

IMM model probability update and combination
1. Model probability update: The probability of two models j (s) can be updated where,c denotes normalization factor, obtained by 2. Combination: The final state vector estimation X(s|s) and error covariance matrix P(s|s) are 4 Simulation and experimental results

Simulation results
In this section, we designed simulations and experiments to verify the advantages of HT-MFDAF algorithm and it is compared with the existing localization algorithms.The MN is assumed to move along a fixed route in an area of 100m × 100m that randomly distributed 6 BNs.The sample length is 100, and the sample interval is equal to 0.5.The initial state sector and covariance matrix of MN are X(0) = [0m, 20m, 1m∕s, 0.5m∕s] T and P(0|0) = diag(1 2 , 1 2 , 1 2 , 1 2 ) , the Markov transition matrix initial value was p = 0.995 0.005 0.005 0.995 , and the error covariance is R = I N , where N denotes the number of base nodes.Moreover, the NLOS distance errors are randomly produced according to the probability P NLOS , which obeys Gaussian distribution.Particularly, in the NLOS recognition and verification phase, we let L = 3.The threshold t1 , t2 , m1 and m2 are calculated when the PFA is 80%, and we make t1 and t2 respectively equal to the STD of S 1 and S 2 in LOS propagation state.In order to get more accurate results, the simulation results are all obtained by 1000 Monte Carlo runs.We mainly compared PIMM (IMM-MPDA) [30], RIMM (IMM-REKF) [33] EIMM (IMM-EKF) [18] and PFNN (based on the particle filter and the nearest neighbor association algorithm) [34] with the same parameter.Two indicators are used to evaluate the performance of the algorithm, which are cumulative distribution function (CDF) of the mean error distance (MED) and root mean square error (RMSE) respectively.
where, S indicates the total number of time steps.
The NLOS error and measurement noise obey Gaussian distribution N(0, 2 L ) and N( n , 2 n ) respectively.The default parameters in this simulation are shown in Table 1.Figures 10,11, and 12 illustrate that with the change of the P NLOS , n and 2 n , the RMSE comparison charts of five algorithms are displayed.It is worth noting that other parameters are the same.
It can be observed distinctly from Fig. 10, as the P NLOS increases, the positioning errors of all algorithms show an upward trend.This is because the error of measuring distance becomes larger due to the increase of P NLOS .When P NLOS is below 0.3, the performance of PIMM and HT-MFDAF is very close.However, as the P NLOS con- tinues to rise, it is made evident that the precision of HT-MFDAF is superior to other algorithms, while the positioning error of PIMM increases at a faster speed.Based on the analysis above, it is not difficult to see that HT-MFDAF has strong robustness, especially in severe NLOS environments.
(81)    Then CDF of MED is obtained by 1000 Monte Carlo simulations, which is shown in Fig. 13.The blue curve represents the CDF of the location error of HT-MFDAF, which has always been the fastest convergence.Furthermore, the 90% localization error of HT-MFDA, PIMM, RIMM EIMM and PF-NN is less than 3.4965 m, 5.1279 m, 6.2198 m 6.4165 m and 7.36 m, respectively.From the data and figure, it is made evident that HT-MFDAF is significantly superior to other algorithms.

Experimental results
To prove the effectiveness of HT-MFDAF, the experiment was set up in real indoor scenarios.To better evaluate the efficiency of HT-MFDAF, we conducted experiments under 4 BNs and 5 BNs respectively.It is worth to notice that the 4 BNs in the first experiment were all indoors.Therefore, when the MN moves in the corridor, all the BNs are in the NLOS state.Next the BN in the corridor was added.Figures 17 and 18 separately illustrate the positioning error of each sampling point and CDF of the positioning error.As can be observed from Fig. 14, the positioning accuracy of HT-MFDAF is the highest at almost every sampling point.In more detail, the average positioning accuracy of HT-MFDAF is about 12.29%, 16.01%, 16.7% and 16.97% higher than that of the RIMM, EIMM, PF-NN and PIMM.From Fig. 18, HT-MFDAF still converges fastest.And the 90% localization error of HT-MFDA, RIMM, EIMM, PF-NN and PIMM is less than 2.462 m, 2.627 m, 2.979 m, 3.064 m and 3.102 m respectively.In summary, we verify that HT-MFDAF has better positioning performance through experiments.

Computation Time Comparison:
In order to more intuitively reflect the time complexity of the algorithm, we used the running time of the algorithm for illustration.Table 2 shows the comparison of the running time of each algorithm.The five algorithms are coded using MATLAB R2022a and tested on a Windows 10 Professional workstation with Intel(R) Core (TM) i7-8700 @3.20 GHz and 8.00 GB RAM.It can be seen that HT-MFDAF has a higher time complexity compared to EIMM and RIMM.Because it cascades filters.However, its time complexity is significantly lower than PIMM.Meanwhile the time for a single processing is still far less than the interval between single sample (the sampling frequency is 1 Hz).Therefore, the algorithm can be applied for online tracking.

Conclusions and future works
A localization method based on hypothesis test for NLOS recognition and modified fuzzy probabilistic data association filtering (HT-MFDAF) is proposed in this paper.Firstly, under the LOS model, EKF is used to update.Meanwhile, the initial distance measurements are processed by hypothesis test under the NLOS model.Then a series of position estimates are obtained by grouping and LS estimation.Next the outliers are judged and abandoned by calculating the similarity.Finally, the fuzzy membership degree of the remaining position estimation is calculated by MFDAF, reconstructing the correlation probability to get the position estimate.Simulations and experimental results, compared with PIMM, RIMM and EIMM.The results demonstrate the effectiveness of HT-MFDAF algorithm, especially in severe NLOS situations.When implementing the algorithm, prior knowledge of noise should be required for the hypothesis testing of the first stage.Or else, the information of noise has to compute in advance.Furthermore, HT-MFDAF can be more robustly applied to the tracking of different targets in more areas by building measurement models in future work.And it can also be extended to the scene of multitarget tracking.

Figure 2
Figure 2 illustrates the algorithm flow chart of this paper.

Fig. 5 Fig. 6
Fig. 5 An example of the two PDFs when NLOS is assumed

Fig. 8
Fig. 8 An example of two PDFs for NLOS hypothesis verification

Figure 11 distinctly
Figure 11 distinctly reflects the change in RMSE when the n increases from 2 to 9 m.It is obvious that HT-MFDAF increases at a relatively slow speed and the accuracy is

1 .
Fig. Deployment of the environment

Fig. 15
Fig.15 Positioning error of each sampling point

Fig. 17
Fig.17 Positioning error of each sampling point

Table 1
The default parameters in simulation