Performance Analysis of Computational Intelligence Correction

Kalman ﬁlter (KF) is a widely used navigation algorithm, especially for precise positioning applications. However, the exact ﬁlter parameters must be deﬁned a priori to use standard Kalman ﬁlters for coping with low error values. But for the dynamic system model, the covariance of process noise is a priori entirely undeﬁned, which results in diﬃculties and challenges in the implementation of the conventional Kalman ﬁlter. Kalman Filter with recursive covariance estimation applied to solve those complicated functional issues, which can also be used in many other applications involving Kalaman ﬁltering technology, a modiﬁed Kalman ﬁlter called MKF-RCE. While this is a better approach, KF with SAR tuned covariance has been proposed to resolve the problem of estimation for the dynamic model. The data collected at (x: 706970.9093 m, y: 6035941.0226 m, z: 1930009.5821 m) used to illustrate the performance analysis of KF with recursive covariance and KF with computational intelligence correction by means of SAR (Search and Rescue) tuned covariance, when the covariance matrices of process and measurement noises are completely unknown in advance.

part to the advantages of recursive computation, the Kalman filter has been the subject of extensive research and implementation, especially in the field of autonomus or aided navigation [6]. Precise position estimation is the big challenge for any existing or upcoming navigation solutions [17], [18]. For several tracking and data prediction challenges [14], Kalman filter has long been used as the perfect solution. Prediction and updation are the two important steps involved in Kalman filter agorithm [2]. It is the primary module for the Kalman Filter's measurement Update and State Update [19]. Suppose the state of x k to be predicted is regulated by the following dynamics (1).
Where φ k,k−1 is the state transition matrix, H k is the observation matrix, w k is the process noise, v k is the measurement noise, γ k is the noise input matrix and y k is the measurement at t k . The state and measurement update estimates are obtained by the following equations (2). Thex k,k−1 is predicted value ofx k .
The covariances of process and measurement noise under static conditions are considered as [20,21]. The mean square eror is given by E[e k e T k ], that is equivalent to P k . This expansion carried as given below (3).
Covariance is the model parameter used to achieve low error values [12]. The next section explains how recursive covariance estimation enhanced positioning efficiency relative to the standard Kalman filter.

Modified Kalman Filter with Recursive Covariance Estimation (MKF-RCE)
In the presence of large process uncertainty and measurement noise covariance matrices, the MKF-RCE algorithm is the efficient approach to the state estimation problem [7,22]. The following section introduces the statistical study of MKF-RCE.
v k is the measurement noise at the k th step. Chosen a random variable ζ is that E[ζ k ζ T k−p ] = 0 where p ≥ k m and T is the sample period. From these samples the covariance of chosen random variable is calculated as given below (5).
Where i = 1, 2, 3, ..., k m . However, for online parameters estimating the above method of measurement is not effective since all samples from the vector cannot be obtained at any time.For real-time calculation it is also optimal to apply the recursive version of the above estimation algorithm. It can be obtained that by taking estimated covariance matrix from samples upto n and new sample n + 1, that isê n [ζ k ζ T k−i ] and with zero mean (7). Here ζ k obtained from measurements as given in (6).
From these (8) and (9) Q k ,R k values can be estimated. Those need to be substitute in standard kalman filter mathematical expressions.

Algorithm 1 -MKF-RCE
Initialization: P 0 the initial state of error covariance matrix e 0 (ζ k ζ k ) = 0,ê 0 (ζ k ζ k−1 ) = 0,Q 0 = 0,R 0 = 0 Input:measurement sequence [y k ] Output: state estimatex k , error covariance matrix P k 1 : for k = 1ton do 2 : Calculate ζ k 3 : Update estimated covariance matrix 4 : Calculate the estimatedQ k andR k 5 : Algorithm 1 can be used to deal with the state estimation problem when covariances of process and measurement noise are completely unknown. The below 2 depicts how error values reduced with MKF-RCE algorithm than standard Kalman filter.

Search and Rescue Optimization (SAR)
SAR is a metaheuristic algorithm inspired by explorations behavior during search and rescue operations. The performance of SAR is proved better than classical optimization techniques [13]. Real world engineering problems are difficult to resolve using existing optimization techniques, there gradient information is required [8]. Hence, a prominent optimization, which does not depend on gradient data is required to find the solution for constrained engineering problems. SAR is a such technique which does not need gradient information. Many creatures use different strategies while searching their goals [9]. rescue operation is one type of group explorations and search is a systematic operation. Hold clue and abandoned clue are two important types of clues in search and rescue operations by human beings. Every optimization consists a fitness function to achieve optimal solutions [3,10,11]. In SAR mathematical model, human's position is the solution and clue is the fitness. While finding main clues, some clues are left. Those left clues are stored in a memory matrix (M) and position matrix consists of human's positions, both the matrices dimensions are equal. The found clues are stored in clue matrix (C). The clue matrix is given as in (10).
The main two control parameters of SAR algorithm are Social Effect (SE) and Maximum Unsuccessful Search Number (MU). In social phase, one random clue has to be assumed and need to check for clues around the positions. In individual phase of SAR, humans search for clues around their current positions. In boundary control state, the solutions obtained in both social and individual phases has to be located in solution space, if they are out of limits, they should be changed. Like this in each iteration the group members have to check their positions with previous positions. If the position is greater than previous position, the previous position is stored in memory matrix, oterwise it is not updated.
Where, M k is the position of k th stored clue, and X i is the previous position.

Kalman Filter with SAR Tuned Covariance
Kalman filter is a navigation solution used to estimate the anonymous state of a dynamic system and to suppress noise existing in aviation control systems [16]. However, the effect is more susceptible to parameters of KF, whose choice purely based on previous experience of an operator. In this paper the parameter tuning using SAR is presented. And condition number of observation matrix is also considered as performance metric by checking condition number is reaching unity or not.  The fitness function (12) for SAR optimization to achieve optimal covariances of process and measurement noise is Mean radial spherical error (MRSE) = (σ 2 x ) + (σ 2 y ) + (σ 2 z ) Circular error probability (CEP) = 0.56σx + 0.62σy Spherica error probability (SEP) = 0.51(σx + σy + σz) For the error analysis of state estimation using SAR tuned covariance the following expressions can be used. Infinity norm of the forward error Relative forward error of state estimate Relative backward error of state estimate Where z is the measurement residual. Error magnification factor

Condition number of observation matrix (H) is
The proposed Kalman Filter with SAR tuned covariance improves the positioning performance by updating the covariances of process and measurement noises using SAR optimization and by setting objectives as condition number and error maginification factor criteria. The presented work is depicted in 1. The performance analysis is based on 2D position accuracy measures and 3D accuracy measures [1], which are listed in 1.

Results
To illustrate the performance analysis, data collected by GPS receiver located at Andhra University, Visakhapatnam is used. The position estimation is done over a period of 23hr 56 mins(2640 epochs) with initial reference location (x:1208443.5605, y:5966808.3895, z:1897080.4657m).The collected data is sampled at an interval of 30 sec. The receiver located in coastal area of Andhra Pradesh is estimated using conventional Kalman

Conclusion
The standard KF, MKF-RCE and proposed SAR tuned KF performances are assessed interms of precision and accuracy. The DGPS receiver located at Andhra University, Andhra Pradesh,(Lat/lon: 17.72 0 N/83.32 0 E) data used in the analysis. In the proposed technique, the parameters of Kalman filter are optimized by search and rescue optimization technique. The suggested SAR tuned KF was used in position estimation and its performance is evaluated with various SAM. Results depicting that SAR-KF converges in less time and has an accuracy difference of 1.19 meters circular error probability when compared to MKF-RCE technique. It is observed that SAR-KF has high accuracy and faster convergence rate than conventional Kalman filter over southern region of Indian subcontinent.