X-ray pulsar rotation parameter estimation method based on genetic algorithm

Abstract

The invention provides an X-ray pulsar rotation parameter estimation method based on a genetic algorithm. The method comprises the following implementation steps: constructing an initial population; initializing parameters; performing roulette selection on the current population; updating the selected population; updating the cross population; and obtaining an X-ray pulsar rotation parameter estimation result. According to the method, the fitness function is constructed, the simulation pulse data is combined, the key rotation parameters in the model are iteratively searched and optimized, and the robustness of observation errors and noise is enhanced. Compared with a traditional least square method or Gaussian-Newton method, the method does not depend on gradient information, can adapt to a complex search space, and improves the parameter estimation precision and stability, thereby providing more reliable time sequence basic support for high-precision position and speed calculation in a subsequent pulsar navigation system.

Description

Technical Field

[0001] The present invention belongs to the field of astrophysics and signal processing technology, and relates to a method for estimating the rotation parameters of an X-ray pulsar, specifically to a method for estimating the rotation parameters of an X-ray pulsar based on a genetic algorithm, which can be used in the fields of pulsar navigation, autonomous positioning of space targets, and the like. Background Art

[0002] X-ray pulsars are a type of neutron star that emit periodic X-ray radiation. They have extremely strong magnetic fields and high-speed rotation characteristics, and their radiation signals show obvious periodicity in time. X-ray pulsar rotation parameter estimation refers to the process of inverting and solving parameters such as the initial phase, rotation frequency and the first-order derivative and second-order derivative of its rate of change based on the received X-ray pulsar signal data using certain algorithms. It can be used in fields such as pulsar navigation and autonomous positioning of space targets.

[0003] The estimation of X-ray pulsar rotation parameters can be divided into three types: time-domain analysis, frequency-domain analysis, and model-fitting optimization algorithm. Among them, the X-ray pulsar rotation parameter estimation method based on model fitting optimization is to construct a theoretical time delay model of the pulsar, take the initial phase, rotation frequency and its derivatives as the parameters to be estimated, combine the received observation data, and use the optimization algorithm to search for the optimal parameter combination to achieve parameter inversion. For example, in the paper entitled "Research on Parameter Estimation Methods of X-ray Pulsar Timing Models" published by Professor Chen in 2022, a method for estimating the X-ray pulsar rotation parameters based on the Gauss-Newton method was proposed. This method first constructs the pulsar timing model as a nonlinear least squares problem. By linearizing the nonlinear mapping function between the pulsar phase and time, it iteratively solves the parameter solution that minimizes the residual between the observation value and the theoretical model. In each iteration, the Jacobian matrix is ​​used to approximate the Hessian matrix to update the parameter estimate and gradually converge to the local optimal solution. This method improves the accuracy of X-ray pulsar rotation parameter estimation. However, since timing models usually have strong nonlinearity, high parameter dimension, and strong sensitivity to initial values, the Gauss-Newton method relies on the accuracy of the initial parameters. In the absence of prior knowledge or poor initial values, it is easy to fall into local minima or fail to converge, affecting further improvement of estimation accuracy. In addition, this method is sensitive to noise and has poor robustness when facing measurement errors or outliers in the observation data, which limits its applicability in complex space environments. Summary of the Invention

[0004] The purpose of the present invention is to address the above-mentioned deficiencies in the prior art and to propose a method for estimating the rotation parameters of an X-ray pulsar timing model based on a genetic algorithm, so as to solve the technical problem of low estimation accuracy in the prior art.

[0005] To achieve the above object, the technical solution adopted by the present invention includes the following steps:

[0006] (1) Constructing the initial population:

[0007] Obtain N X-ray pulsar rotation parameter vectors, and take each parameter vector as an individual to construct an initial population G consisting of N individuals, where N ≥ 150 and the nth parameter vector is n∈[1,N],Φ n 、v n represent the initial phase and initial rotation frequency of the nth pulsar, Respectively represent v n The first and second derivatives of

[0008] (2) Initialization parameters:

[0009] Initialize the number of iterations of the genetic algorithm to k, the maximum number of iterations is K, K ≥ 200, and the current population is G k , the current nth individual is And let k = 1;

[0010] (3) Perform roulette selection on the current population:

[0011] The actual observed phase value Φ of the pulsar signal arrival time series T observed by the ground observation station y and the pulsar timing model for each individual The predicted phase value of Calculated Timing residuals calculate The fitness value of and through Computed The probability of selection For the current population G k Perform roulette wheel selection to obtain a selection population consisting of N individuals The nth selected individual is

[0012] (4) Update the selected population:

[0013] The population will be selected X individuals in the parent generation are used as parents, and for every two parent individuals and Perform two-point crossover, and then use the X offspring individuals obtained by crossover to Replace the parent individuals in the population to obtain a crossover population consisting of N individuals The nth crossover individual is

[0014] (5) Update the selected population:

[0015] Crossover population The Y individuals in the mutated corpus are mutated, and the Y mutated individuals obtained by mutation are Replace the individuals before mutation in the , and obtain a mutant population consisting of N individuals The nth mutant individual is

[0016] (6) Obtain the estimated results of X-ray pulsar rotation parameters:

[0017] Determine whether k=K. If so, take the offspring individual with the largest fitness value among the N offspring individuals after the crossover as the X-ray pulsar rotation parameter estimation result. Otherwise, set k=k+1. Execute step (3).

[0018] Compared with the prior art, the present invention has the following advantages:

[0019] The present invention calculates the fitness value of each individual based on the timing residual calculated by the pulsar timing model for the predicted phase value of each individual, and performs roulette wheel selection, crossover and mutation on the current population according to the fitness value of each individual. This avoids the problems of the least squares method or gradient optimization algorithm based on the existing technology that are prone to falling into local optimality and being sensitive to initial values ​​during the estimation process, effectively improves the accuracy and stability of the rotation parameter estimation, and enhances the adaptability of the pulsar timing model under high-noise observation data. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 It is a flow chart for implementing the present invention.

[0021] Figure 2 This is a simulation comparison diagram of the estimation results of the rotation parameter estimation method of the present invention and the prior art. DETAILED DESCRIPTION

[0022] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0023] Reference Figure 1 , the present invention comprises the following steps:

[0024] (1) Constructing the initial population:

[0025] Obtain N X-ray pulsar rotation parameter vectors, and take each parameter vector as an individual to construct an initial population G consisting of N individuals, where N ≥ 150 and the nth parameter vector is n∈[1,N],Φ n 、v nrepresent the initial phase and initial rotation frequency of the nth pulsar, Respectively represent v n The first and second order derivatives of .

[0026] (2) Initialization parameters:

[0027] Initialize the number of iterations of the genetic algorithm to k, the maximum number of iterations is K, K ≥ 200, and the current population is G k , the current nth individual is And let k = 1;

[0028] (3) Perform roulette selection on the current population:

[0029] The actual observed phase value Φ of the pulsar signal arrival time series observed by ground-based observation stations y and the pulsar timing model for each individual The predicted phase value of Calculated Timing residuals calculate The fitness value of

[0030]

[0031] Among them, Φ n (t) represents the signal phase at time t calculated by the timing model, t0 represents the epoch reference time of the pulsar signal, ∑ represents the summation operation, and |·| represents the absolute value operation.

[0032] pass Computed The probability of selection For the current population G k Perform roulette wheel selection to obtain a selection population consisting of N individuals The nth selected individual is

[0033]

[0034] Where T represents the contemporary population G k The sum of all individual fitnesses.

[0035] Through each individual The probability of choosing roulette wheel calculate The cumulative probability distribution Q of roulette n , and judge Q n Does the pre-set random threshold r satisfy Q n-1 <r≤Q n If so, individuals can be selected repeatedly Enter the selected population Select enough N individuals to obtain a selected population consisting of N individuals

[0036]

[0037] r∈[0,1].

[0038] (4) Update the selected population:

[0039] The population will be selected X individuals in the parent generation are used as parents, and for every two parent individuals and Perform two-point crossover, and then use the X offspring individuals obtained by crossover to Replace the parent individuals in the population to obtain a crossover population consisting of N individuals The nth crossover individual is Parent individual and The individual performs two-point crossover, and the generated and The corresponding offspring individuals and The calculation formulas are:

[0040]

[0041] Among them, α is the weighting coefficient.

[0042] (5) Update the cross population:

[0043] Crossover population The Y individuals in the mutated corpus are mutated, and the Y mutated individuals obtained by mutation are Replace the individuals before mutation in the , and obtain a mutant population consisting of N individuals The nth mutant individual is For the y-th crossover individual The variation formula is:

[0044]

[0045] Among them, β is the variation range control factor; δ is the random disturbance value that obeys the standard normal distribution.

[0046] (6) Obtain the estimated results of X-ray pulsar rotation parameters:

[0047] Determine whether k=K. If so, take the offspring individual with the largest fitness value among the N offspring individuals after the crossover as the X-ray pulsar rotation parameter estimation result. Otherwise, set k=k+1. Execute step (3).

[0048] The following is a description of the technical effects of the present invention in conjunction with simulation experiments:

[0049] 1. Simulation conditions and content:

[0050] The simulation hardware is a microcomputer with the following parameters: CPU: Intel(R) Core(TM) i5-8265U @ 1.60GHz to 1.80GHz; RAM: 8.00GB; operating system: Windows 10. The simulation software is MATALB2022a. The simulation parameters for the pulsar PSR_B0531+21 are shown in Table 1.

[0051] Table 1

[0052]

[0053] In this embodiment, the errors of various parameters set in the simulation are shown in Table 2:

[0054] Table 2

[0055]

[0056] The simulation results of the timing residuals after fitting of the present invention and the existing X-ray pulsar timing model parameter estimation method are compared. Figure 2 and as shown in Table 3.

[0057] 2. Analysis of simulation results:

[0058] Reference Figure 2 , Figure 2 (a) is the timing residual after iteration of the existing technology; the timing residual after fitting of the existing technology is on the order of 10-10s; Figure 2 (b) is the timing residual after iteration of the present invention. The timing residual after fitting of the present invention is on the order of 10-11s, which is significantly better than the previous two, that is, there is a significant improvement in accuracy.

[0059] Table 3

[0060]

[0061] Table 3 shows that when the observation period is 30 days and the sample length is 1000, the estimation error of the present invention is significantly lower than that of the prior art, indicating that the present invention has significantly improved accuracy compared with the prior art.

[0062] The above description is only a specific example of the present invention and does not constitute any limitation to the present invention. Obviously, for professionals in this field, after understanding the content and principles of the present invention, it is possible to make various modifications and changes in any form and details without departing from the principles and structure of the present invention. However, these modifications and changes based on the ideas of the present invention are still within the scope of protection of the claims of the present invention.

Claims

1. A method for estimating X-ray pulsar rotation parameters based on genetic algorithm, characterized in that: The steps include: (1) Constructing the initial population: Obtain N X-ray pulsar rotation parameter vectors, and take each parameter vector as an individual to construct an initial population G consisting of N individuals, where N ≥ 150 and the nth parameter vector is n∈[1,N],Φ n 、v n represent the initial phase and initial rotation frequency of the nth pulsar, Respectively represent v n The first and second derivatives of (2) Initialization parameters: Initialize the number of iterations of the genetic algorithm to k, the maximum number of iterations is K, K ≥ 200, and the current population is G k , the current nth individual is And let k = 1; (3) Perform roulette selection on the current population: The actual observed phase value Φ of the pulsar signal arrival time series observed by ground-based observation stations y and the pulsar timing model for each individual The predicted phase value of Calculated Timing residuals calculate The fitness value of and through Computed The probability of selection For the current population G k Perform roulette wheel selection to obtain a selection population consisting of N individuals The nth selected individual is (4) Update the selected population: The population will be selected X individuals in the parent generation are used as parents, and for every two parent individuals and Perform two-point crossover, and then use the X offspring individuals obtained by crossover to Replace the parent individuals in the population to obtain a crossover population consisting of N individuals The nth crossover individual is (5) Update the cross population: Crossover population The Y individuals in the mutated corpus are mutated, and the Y mutated individuals obtained by mutation are Replace the individuals before mutation in the , and obtain a mutant population consisting of N individuals The nth mutant individual is (6) Obtain the estimated results of X-ray pulsar rotation parameters: Determine whether k=K. If so, take the offspring individual with the largest fitness value among the N offspring individuals after the crossover as the X-ray pulsar rotation parameter estimation result. Otherwise, set k=k+1. Execute step (3).

2. The method according to claim 1, characterized in that The timing model described in step (3) is calculated as follows: Among them, Φ n (t) represents the signal phase at time t calculated by the timing model, and t0 represents the epoch reference time of the pulsar signal.

3. The method according to claim 1, characterized in that Described in step (3) The fitness value of The calculation formula is: Here, ∑ represents the sum operation, and |·| represents the absolute value operation.

4. The method according to claim 1, wherein Described in step (3) The probability of selection The calculation formula is: Where T represents the contemporary population G k The sum of all individual fitnesses.

5. The method according to claim 1, characterized in that The current population G described in step (3) k To make a roulette selection, the steps are: Through each individual The probability of choosing roulette wheel calculate The cumulative probability distribution Q of roulette n , and judge Q n Does the pre-set random threshold r satisfy Q n-1 <r≤Q n If so, individuals can be selected repeatedly Enter the selected population Select enough N individuals to obtain a selected population consisting of N individuals in: r∈[0,1]。 6. The method according to claim 1, characterized in that For every two parent individuals described in step (4) and The individual performs two-point crossover, and the generated and The corresponding offspring individuals and The calculation formulas are: Among them, α is the weighting coefficient.

7. The method according to claim 1, characterized in that The population after the two points crossover described in step (5) The Y crossover individuals in the mutate, where the y-th crossover individual The variation formula is: Among them, β is the variation range control factor; δ is the random disturbance value that obeys the standard normal distribution.

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