Acceleration-Displacement Relationship Identification Method and Device Based on Bayesian Inference

Abstract

The present invention relates to a method for identifying the acceleration-displacement relationship based on Bayesian inference. Using the basic theoretical framework of Bayesian inference and aiming at the basic functional relationship between acceleration and displacement, starting from the displacement signal, a method that can achieve displacement optimization through multiple iterations is proposed. The acceleration-displacement analytical relationship obtained by a single-degree-of-freedom system is used to conduct a comparative analysis under different working conditions such as different noises and artificial baselines through the displacement time history obtained based on Bayesian inference, which proves the applicability and feasibility of this method. Finally, large-scale shaking table model test data are selected, and by comparing high-quality and poor acceleration sensing signals, the displacement data and its uncertainty are inversely obtained respectively. The results show that this method has certain advantages in characterizing the acceleration-displacement relationship and can achieve displacement solution without relying on the processing of acceleration signals.

Description

Technical Field

[0001] The present invention belongs to the field of dynamic displacement testing, and particularly relates to a method and device for identifying the acceleration-displacement relationship based on Bayesian inference. Background Art

[0002] Dynamic displacement is an important physical quantity in the fields of earthquake engineering, military weapon design, and structural health monitoring. However, in the actual testing process, usually only the vibration acceleration signal can be directly measured. Due to the influence of uncertain testing conditions such as the environment, the acceleration signal inevitably contains low-frequency and high-frequency noises, resulting in obvious drift phenomena in the velocity and displacement time histories during the acceleration integration process. Therefore, it is of great scientific research and practical engineering application significance to reasonably and scientifically obtain the acceleration-displacement relationship. Summary of the Invention

[0003] The present invention adopts the basic theoretical framework of Bayesian inference. For the basic functional relationship between acceleration and displacement, starting from the displacement signal, a method for realizing displacement optimization through multiple iterations is proposed. The implementable method for identifying the acceleration-displacement relationship based on Bayesian inference proposed by the present invention includes the step of solving the posterior distribution of displacement d according to formula (3):

[0004]

[0005] where J(d) represents the posterior distribution of displacement d; A is a matrix, d is a displacement vector, a is an acceleration vector, Δt is the acceleration sampling time interval, and λ is a penalty parameter.

[0006] The beneficial effects of the present invention are as follows: Based on the Bayesian theory framework, a Bayesian learning and identification method for dynamic displacement is constructed. The displacement response is inversely obtained for different noise conditions (white noise, artificial noise), and the identified dynamic displacement is basically consistent with the analytical displacement; and using the large-scale shaking table test data, the displacements inversely obtained from acceleration sensing signals with different performances are compared, and their uncertainties are analyzed. The results show that: The Bayesian learning and identification method for dynamic displacement has certain advantages in characterizing the acceleration-displacement relationship, and can realize displacement solution without relying on the processing of acceleration signals, thus avoiding the displacement integration distortion caused by noise accumulation errors. Description of the Drawings

[0007] Figure 1 Input of single-degree-of-freedom system model and time history curves of top acceleration and displacement;

[0008] Figure 2 Comparison of displacement time history based on Bayesian inference and analytical displacement time history curve (no noise);

[0009] Figure 3 Parameter iteration process;

[0010] Figure 4 Baseline, noisy data;

[0011] Figure 5 Comparison of displacement time history and analytical displacement time history curves based on Bayesian inference (two - stage noise);

[0012] Figure 6 Shaking table model test design of high - rise structures;

[0013] Figure 7(a) Acceleration and displacement time history of the same floor test - Acceleration time history of measuring point A;

[0014] Figure 7(b) Acceleration and displacement time history of the same floor test - Acceleration time history of measuring point B;

[0015] Figure 7(c) Acceleration and displacement time history of the same floor test - Displacement time history of measuring points A and B;

[0016] Figure 8(a) Identification of displacement response at floor measuring points A and B and comparison with test time history - Identification of displacement response at floor measuring point A;

[0017] Figure 8(b) Identification of displacement response at floor measuring points A and B and comparison with test time history - Comparison of displacement response and test time history at floor measuring points A and B

[0018] Figure 9(a) Displacement comparison at point A under different acceleration processing methods - Comparison of displacement response at point A with Bayesian and test time history under different baseline corrections;

[0019] Figure 9(b) Displacement comparison at point A under different acceleration processing methods - Comparison of displacement time history at point A under different filtering ranges;

[0020] Figure 10 Analysis of displacement uncertainty identification by multi - segment offset of acceleration;

[0021] Figure 11 Probability density distribution of points A - F on the displacement time history curve. Specific implementation manner

[0022] The testing and characterization of physical quantities during vibration are common scientific problems in multiple industries such as earthquake engineering, military weapon design, structural health monitoring, aircraft, and medicine. Many research results point to the key conversion index of the acceleration-displacement relationship. During the actual testing process, vibration acceleration signals are usually directly measured and then displacement data are obtained through integration. The integration methods are generally divided into time-domain methods and frequency-domain methods. The time-domain integration method directly performs first and second integrations on the measured acceleration signals to obtain velocity responses and displacement responses; the frequency-domain integration method converts time-domain signals into frequency-domain signals through Fourier transform, then performs integration calculations in the frequency domain, and finally reconstructs time-domain signals through inverse Fourier transform. Since acceleration testing is often affected by noise interference, drift phenomena will occur in the integrated velocity and displacement. Many scholars have carried out research work in this field and achieved some research results. Research by Pintelon et al. shows that it is difficult to obtain relatively accurate velocity or displacement using time-domain integration represented by the Newton-Cotes integration formula, and using high-order integration algorithms such as the fourth-order Runge-Kutta method, interference noise will cause greater errors. Boore et al. pointed out that ground motion records inevitably contain low-frequency and high-frequency noise. High-frequency noise needs to be filtered by a low-pass filter, and low-frequency noise requires baseline correction to restore the permanent displacement caused by the earthquake, but there is no standard and effective method for determining the segmentation parameters. Dai Zhijun et al. proposed a new method that is optimized and can automatically identify baseline segmentation for the baseline correction problem using the L1 norm method. Scholars such as Zheng Shuiming and Chen Weizhen corrected the baseline of the acceleration and integrated it to give displacement results with significantly improved accuracy; Lee et al. started their research from the perspective of designing more adaptable signal filters and proposed the FDM-FIR filter; scholars such as Hong et al. improved the FDM-FIR filter and proposed the FEM-FIR filter, determined the regularization coefficients of the control equation in the frequency domain, and combined with the finite element idea, regarded acceleration as the bending moment of the beam, velocity as the rotation angle, and displacement as the deflection, and used the shape function to calculate the vibration response of the structure. These methods target the noise terms in the acceleration signal, achieve noise reduction through various signal processing means and make the integrated displacement more reasonable, but it is extremely difficult to achieve the integration condition of zero noise. Therefore, there are more or less deviations in the results of integrating to obtain dynamic displacement.

[0023] In addition, machine learning based on probability statistics has received great attention from the industrial and academic communities in recent years and has achieved many important successful applications in fields such as vision, speech, natural language, biology, and structural health monitoring. Among them, the Bayesian method, as an important branch, has been widely applied, from single-variable classification and regression to multi-variable structured output prediction, from supervised learning to unsupervised and semi-supervised learning.

[0024] Integral relationship of the acceleration-displacement relationship

[0025] The measured acceleration signal a = [a1, a2, … a N T , and the vector representation of the corresponding displacement signal is d = [d1, d2, … d N T , given the initial conditions, the displacement can be solved by the method of numerical integration of acceleration values (such as the Newmark-β method, Wilson-θ method, etc.). The numerical method can solve the integral relationship between acceleration and displacement to a certain extent, but the low-frequency drift problem will cause the displacement data obtained by acceleration integration to be too large (the low-frequency lower limit is too low) or too small (the low-frequency lower limit is too high).

[0026] Acceleration-Displacement Relationship Based on Bayesian Inference

[0027] In the embodiments of the present invention, an algorithm can be designed based on Bayesian inference from the relationship between acceleration and displacement (Formula 1):

[0028] Ad = aΔt 2 (1)

[0029] Among them, matrix A is a difference matrix, d is a displacement vector, a is an acceleration vector, Δt is the acceleration sampling time interval, and the central difference matrix is shown in Formula (2).

[0030]

[0031] Among them, represents a real matrix of dimension N×(N + 2); N represents the data length.

[0032] It should be noted that due to the existence of noise signals in the acceleration test process, problems such as underdetermination and ill-conditioning will occur in the solution process of Formula (1).

[0033] In some embodiments, the regularization constraint method is used to rewrite Formula (1) into Formula (3): In some specific embodiments, the acceleration-displacement relationship identification method based on Bayesian inference includes the step of solving the posterior distribution of displacement d according to Formula (3):

[0034]

[0035] Among them, J(d) represents the posterior distribution of displacement d; A is a matrix, d is a displacement vector, a is an acceleration vector, Δt is the acceleration sampling time interval, and λ is a penalty parameter.

[0036] In some specific embodiments, the matrix A is a central difference matrix, and the central difference matrix is shown in Formula (2):

[0037] ​​

[0038] Among them, a real matrix with dimensions N×(N + 2) is shown; N represents the data length.

[0039] In some specific embodiments, the posterior distribution of the displacement d is estimated using the maximum a posteriori probability represented by formulas (4) and (5):

[0040]

[0041]

[0042] Wherein: represents the update / posterior probability density function of the displacement d, d is the displacement vector, σ 2 and τ 2 represent hyperparameters, is represented as aΔt 2 , is the likelihood function, p(d|τ 2 ) is the prior probability density function of the displacement vector d, p(σ 2 ) and p(τ 2 ) are the conjugate prior probability density functions of the hyperparameters σ 2 and τ 2 .

[0043] In some specific embodiments, the prior probability density function p(d|τ 2 ) of the displacement vector d is characterized by the Gaussian distributions of formulas (6) and (7) respectively:

[0044]

[0045]

[0046] Wherein, σ N exp() and τ N exp() represent the conjugate prior distributions of the time series, and exp() represents the exponential function with base e.

[0047] In some specific embodiments, the conjugate prior probability density function p(σ 2 ) is represented by the inverse gamma distribution of formula (8):

[0048]

[0049] Wherein, α0 and β0 are the hyperparameters of the conjugate prior distribution.

[0050] In some specific embodiments, the conjugate prior probability density function p(τ 2 ) is represented by the inverse gamma distribution of formula (9):

[0051]

[0052] Among them, α1 and β1 are the hyperparameters of the conjugate prior distribution.

[0053] In some specific embodiments, substituting formulas (6), (7), (8), and (9) into formula (4) gives formula (10):

[0054]

[0055] Taking the logarithm of formula (10) and respectively taking the derivatives of d, σ 2 , τ 2 and setting them to 0, an iterative relationship between the three target quantities can be obtained, and the optimal solution is obtained by optimizing and solving according to formulas (11), (12), and (13):

[0056]

[0057]

[0058]

[0059] In some specific embodiments, the optimal solution of the displacement d is obtained through multiple iterations, and the parameters converge after multiple iterations. The calculation convergence condition is shown in formula (14), where ε is taken as 1×10 -6 ;

[0060]

[0061] 1. Method verification

[0062] To verify the feasibility of the method of the embodiments of the present invention, a single-degree-of-freedom system (as shown in Figure 1 ) is designed. The Newmark-β method is used to calculate the top acceleration (ü) and displacement (u) of the single-degree-of-freedom system. The two have an analytical relationship. The Kobe earthquake record is selected as the base input (ü g ), and the applicability and feasibility of the method are verified. The input ground motion and the obtained acceleration and displacement time history curves at the top are shown in Figure 1 .

[0063] Using the present method, the comparison between the displacement time history calculated with the simulated top acceleration data as the input and the analytical displacement is shown in Figure 2 , Figure 3 which is the iterative process of the parameters. After adding 5%, 10%, and 20% white noise to the acceleration signal respectively, the relationship between the identified displacement parameters and the analytical displacement parameters is shown in Table 1. From Figure 2 , Figure 3, as can be seen from Table 1: By using the Bayesian inference method, the calculation parameters can be iterated automatically, and the displacement time history data can also be consistent with the analytical solution after multiple iterations. Even when random noise is added to the acceleration signal, the displacement can still be effectively identified.

[0064] Relationship between displacement parameters identified from acceleration signal with added noise and analytical displacement parameters in Table 1

[0065]

[0066] In earthquake engineering applications, due to various interferences in strong ground motion data, such as zero drift caused by the change of the amplifier with the ambient temperature, instability of the low-frequency performance outside the sensor frequency range, and noise and vibration interference in the surrounding environment, the velocity and displacement obtained by integrating and transforming the ground motion signal deviating from the baseline may be completely distorted. To simulate such engineering application problems, the present invention designs a Kobe earthquake record with an additional typical two-stage artificial noise model. The original earthquake record, baseline, and ground motion after adding artificial noise are as Figure 4 shown. Using the Figure 4 ground motion noise-added record as the calculation input, the comparison between the displacement time history given by the Bayesian inference method and the analytical displacement time history is as Figure 5 shown. As can be seen from the figure: The displacement identification given by the Bayesian inference can better process the ground motion data with baseline drift and give a reasonable displacement time history. After adding the two-stage artificial noise, the strong nonlinearity will affect the structural displacement response. As the baseline amplitude decreases, the calculation accuracy will gradually increase and approach the analytical solution, indicating that this algorithm has strong robustness. In the processing of ground motion data, the displacement time history can be accurately identified and can adapt to different noise conditions.

[0067] 2 Applications in shaking table model tests of major structural engineering

[0068] 2.1 Shaking table model test of high-rise structure

[0069] Applying the method proposed by the present invention to the large-scale shaking table model test of the structure, a six-story single-tube steel structure house is selected. The structural form and sensor layout are as Figure 6 shown. The data of this test are selected because the acceleration sensor X6E (measurement point A) on the same floor has good working performance, while the acceleration sensor X6W (measurement point B) has an abnormal state, and the displacements on both sides are measured, which provides good data for the verification of this method in practical applications. The accelerations and displacements on the same floor are shown in Figures 7(a) and 7(b). As can be seen from Figure 7(c): The displacement responses on the same floor are relatively basically consistent, and the acceleration data at measurement point B is abnormal (the sensor is tilted or loose).

[0070] 2.2 Data analysis

[0071] The displacement time history and frequency domain response obtained by identifying the acceleration data using the algorithm proposed in the embodiments of the present invention are shown in Figure 8 (measurement point A) (measurement point B). For measurement point A, the identified displacement data is in good agreement with the measured displacement both in the time domain and the frequency domain. When there are significant multiple and multi-segment offsets in the acceleration signal (measurement point B), from the perspective of the displacement time history waveform, it can be found that it is relatively consistent with the test waveform, and the peak value is slightly smaller than the test displacement, and the frequency domain responses of the two are basically the same. For measurement point B, an uncertainty analysis method is used to measure the uncertainty of the identified displacement time history. For Gaussian random variables, the covariance matrix can be solved using the Hessian Matrix (the two are inverse relationships), and the diagonal and non-diagonal elements are shown in formulas (15) and (16). Since the displacement objective function is implicit, the difference method is used to solve its matrix elements, and the perturbation amounts (Δd l and Δd l' ) are both taken as 1×10 -3 of the peak value of the identified displacement.

[0072]

[0073]

[0074] Figure 9 shows the comparison of the displacement time history of measurement point A under the traditional methods (baseline correction, filtering processing) with the Bayesian method and the test results. It can be seen from the figure that when different baselines and filtering ranges [0.5 Hz, 30 Hz] are selected for processing, it will have a significant impact on the calculation accuracy of the displacement time history, especially the selection of the lower limit of the filtering range will have a relatively large impact on the calculation accuracy. The Bayesian method uses the information update algorithm to obtain the optimal parameters while ensuring the calculation accuracy. Figure 10 The comparison between the positive and negative one standard deviations of the identified displacement and the measured results is given. It can be seen from the figure that the differences between the identified displacement and the measurement are all within one standard deviation; using the identified displacement amounts and covariance of each sampling point for Monte Carlo simulation (1000 times), the probability distribution of each sampling point of the identified displacement can be given. Figure 11 The probability distributions of the identified displacement points at six moments A, B, C, D, E, and F are given. There are similar distribution characteristics at any moment, and finally the identified displacement with probability meaning can be given.

[0075] 3 Conclusions

[0076] Based on the basic framework of Bayesian theory, the present invention constructs a Bayesian learning and identification method for dynamic displacement, and at least has the following technical effects:

[0077] (1) For the practical problem of obtaining displacement using acceleration, the theoretical derivation of displacement identification is first given, and the acceleration and displacement analytical data given by a single-degree-of-freedom system simulation are used to simulate different noise (white noise, artificial noise) conditions. The results show that the identified dynamic displacement is basically consistent with the analytical displacement.

[0078] (2) Using the large shaking table test data of high-rise structures, by comparing the acceleration sensing signals with good and poor performance, the dynamic displacement data of the floors are inversely obtained and the uncertainty of the data is analyzed, and the applicability and feasibility of the method are verified. The results show that the acceleration-displacement identification method proposed in this paper can give the displacement time history more accurately, and has good performance in both the time domain and the frequency domain. This method has certain advantages in characterizing the acceleration-displacement relationship, and can solve the displacement without relying on the processing of the acceleration signal, thus avoiding the distortion of the integral solution of the displacement caused by the noise accumulation error.

[0079] (3) From the perspective of Bayesian learning, this invention constructs the acceleration-displacement relationship and gives the optimal solution of displacement with the meaning of probability distribution, which is different from the traditional signal processing method (only giving the integral displacement amount and unable to give the probability of the solution). This invention focuses on constructing the theoretical framework of the method and verifying it. In practical applications, the selection of high-precision difference matrices and penalty functions is a key issue. For large deformation problems (such as near-field earthquake displacements, structural collapses, etc.), high-precision difference formulas will have a certain corrective significance for the algorithm stability.

[0080] The acceleration-displacement analytical relationship obtained by using a single-degree-of-freedom system is used to compare and analyze the displacement time history obtained by Bayesian inference under different noise and artificial baseline conditions, etc., to prove the applicability and feasibility of the method. Finally, the large shaking table model test data are selected, and by comparing the good and poor acceleration sensing signals, the displacement data and its uncertainty are inversely obtained respectively. The results show that this method has certain advantages in characterizing the acceleration-displacement relationship and can solve the displacement without relying on the processing of the acceleration signal.

[0081] The embodiments and functional operations of the subject matter described in this specification can be implemented in: digital electronic circuits, tangibly implemented computer software or firmware, computer hardware, including the structures disclosed in this specification and their structural equivalents, or a combination of one or more of the above. The embodiments of the subject matter described in this specification can be implemented as one or more computer programs, that is, one or more modules of computer program instructions encoded on one or more tangible non-transitory program carriers for being executed by a data processing device or controlling the operation of a data processing device.

[0082] As an alternative or in addition, program instructions can be encoded on an artificially generated propagated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, which is generated as encoded information to be transmitted to a suitable receiver device that executes with a data processing apparatus. A computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of the foregoing devices.

[0083] The term "data processing apparatus" encompasses all kinds of devices, apparatuses, and machines for processing data, including, by way of example, programmable processors, computers, or multiprocessors or multi-computers. The apparatus can include dedicated logic circuitry, e.g., FPGAs (Field Programmable Gate Arrays) or ASICs (Application Specific Integrated Circuits). The apparatus can also include code that creates an execution environment for relevant computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them, in addition to including hardware.

[0084] A computer program (which may also be referred to as or described as a program, software, a software application, a module, a software module, a script, or code) can be written in any form of programming language, including a compiled language or an interpreted language or a declarative or procedural language, and the computer program can be deployed in any form, including as a stand-alone program or as a module, a component, a subroutine, or other unit suitable for use in a computing environment. A computer program may or may not correspond to a file in a file system. The program can be stored in a part of a file that holds other programs or data, e.g., in one or more scripts such as in a markup language document; in a single file dedicated to the relevant program; or in multiple cooperating files, e.g., files that hold one or more modules, subroutines, or portions of code. The computer program can be deployed to execute on one computer or multiple computers, which are located at one place, or distributed to multiple locations and interconnected by a communication network.

[0085] The processing and logical flows described in this specification can be performed by one or more programmable computers that execute one or more computer programs by operating on input data and generating output to perform a function. The processing and logical flows can also be performed by dedicated logic circuitry, e.g., FPGAs (Field Programmable Gate Arrays) or ASICs (Application Specific Integrated Circuits), and the apparatus can also be implemented as dedicated logic circuitry.

[0086] A computer suitable for implementing a computer program includes and, by way of example, may be based on a general-purpose microprocessor or a special-purpose microprocessor or both of the foregoing processors, or any other kind of central processing unit. Generally, the central processing unit will receive instructions and data from a read-only memory or a random access memory or both. The main elements of a computer are a central processing unit for running or executing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include or be operatively coupled to receive data from or transfer data to one or more mass storage devices for storing data, or both receive and transfer, such as magnetic disks, magneto-optical disks, or optical disks. However, a computer need not have such devices. In addition, a computer may be embedded in another device, such as a mobile phone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a global positioning system (GPS) receiver, or a removable storage device, such as a universal serial bus (USB) flash drive, etc.

[0087] Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media, and memory devices, by way of example, including: semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks. Processors and memories may be supplemented by or incorporated into dedicated logic circuitry.

[0088] To send interactions with a user, embodiments of the subject matter described in this specification may be implemented on a computer having: a display device, such as a CRT (cathode ray tube) or LCD (liquid crystal display) monitor, for displaying information to the user; and a keyboard and a pointing device such as a mouse or a trackball by which the user may send input to the computer. Other kinds of devices may also be used to send interactions with the user; for example, the feedback provided to the user may be any form of sensory feedback, such as visual feedback, auditory feedback, or tactile feedback; and input from the user may be received in any form, including acoustic input, voice input, or tactile input. Additionally, a computer may interact with a user by sending a document to and receiving a document from a device used by the user; for example, by sending a web page to a web browser on a client device of the user in response to a request received from the web browser.

[0089] Embodiments of the subject matter described in this specification can be implemented in a computing system that includes backend components such as, for example, a data server, or includes middleware components such as, for example, an application server, or includes frontend components such as, for example, a client computer having a graphical user interface or a web browser through which a user can interact with an implementation of the subject matter described in this specification, or the computer system can include any combination of one or more of such backend, middleware, or frontend components. The components in the system can be interconnected to each other by any form or medium of digital data communication such as, for example, a communication network. Examples of communication networks include local area networks ("LAN") and wide area networks ("WAN"), such as, for example, the Internet. A computing system can include clients and servers. Clients and servers are typically located remotely from each other and typically interact through a communication network. The relationship between a client and a server is created by computer programs that run on respective computers and have a client-server relationship with each other.

[0090] Although this specification contains many specific implementation details, these should not be construed as limitations on the scope of any invention or on the scope of what can be claimed, but rather as illustrations of features that may be specific to particular embodiments of a particular invention. The specific features described in the context of separate embodiments in this specification can also be implemented in combination in a single embodiment. Conversely, the various features described in the context of a single embodiment can also be implemented independently in multiple embodiments or in any suitable sub-combination. Additionally, although the features may be described above as acting in combination and even initially claimed as such, one or more features from the claimed combination may in some cases be excluded from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0091] Similarly, although operations are depicted in the drawings in a particular order, it should not be understood that such operations are required to be performed in the particular order shown or in sequential order to achieve a desired result, or that all illustrated operations are to be performed. In certain circumstances, multitasking and parallel processing may be advantageous. Additionally, the separation of various system modules and components in the above embodiments should not be understood as required in all embodiments, and it should be understood that program components and systems can generally be integrated in a single software product or packaged into multiple software products.

[0092] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the acts recited in the claims can be performed in a different order and still achieve the desired result. As one example, in order to achieve the desired result, the processes described in the figures do not necessarily require the particular order or sequence shown. In certain implementations, multitasking and parallel processing may be advantageous.

Claims

1. A method for identifying the acceleration-displacement relationship based on Bayesian inference, characterized in that: Including the step of solving the posterior distribution of the target displacement d according to formula (3): Wherein, J(d) represents the posterior distribution of the displacement d; A is a matrix, d is a displacement vector, a is an acceleration vector, Δt is the acceleration sampling time interval, and λ is a penalty parameter; The matrix A is a central difference matrix, and the central difference matrix is shown in formula (2): Among them, a real matrix with dimensions of N×(N + 2); N represents the data length; The maximum a posteriori probability represented by formula (4) and formula (5) is used to estimate the posterior distribution of the displacement d: Wherein: represents the updated / posterior probability density function of the displacement d, where d is the displacement vector, and σ 2 and τ 2 represent hyperparameters, is expressed as aΔt 2 , is the likelihood function, p(d|τ 2 ) is the prior probability density function of the displacement vector d, and p(σ 2 ) and p(τ 2 ) are the conjugate prior probability density functions of the hyperparameters σ 2 and τ 2 .

2. The method according to claim 1, characterized in that: The prior probability density function p(d|τ 2 ) of the displacement vector d is characterized by Gaussian distributions of formula (6) and formula (7) respectively: Among them, σ N exp() and τ N exp() represents the conjugate prior distribution of the time series, and exp() represents the exponential function with base e.

3. The method according to claim 2, characterized in that: The conjugate prior probability density function p(σ 2 ) is represented by the inverse gamma distribution of formula (8): Wherein, α0 and β0 are hyperparameters of the conjugate prior distribution.

4. The method according to claim 3, characterized in that: The conjugate prior probability density function p(τ 2 ) is represented by the inverse gamma distribution of formula (9): Wherein, α1 and β1 are hyperparameters of the conjugate prior distribution.

5. The method according to claim 4, characterized in that: Substituting formula (6), formula (7), formula (8) and formula (9) into formula (4) gives formula (10): Take the logarithm of formula (10) and take the derivatives of the target variables d, σ 2 , τ 2 and set them to 0, we can obtain the iterative relationship between the three target variables, and optimize and solve them by formulas (11), (12), and (13): The optimal solution of the displacement d is obtained through multiple iterations, and the parameters converge after multiple iterations. The convergence condition is calculated as shown in formula (14), where ε is taken as 1×10 -6 ; 6. A sensor, characterized in that: The sensor includes at least one processor; and a memory that stores instructions, and when the instructions are executed by the at least one processor, the steps of the method according to any one of claims 1-5 are implemented.

7. A computer-readable storage medium, on which a computer program / instructions are stored, characterized in that, When the computer program / instructions are executed by a processor, the steps of the method according to any one of claims 1-5 are implemented.

8. A computer program product, comprising a computer program / instructions, characterized in that, When the computer program / instructions are executed by a processor, the steps of the method according to any one of claims 1-5 are implemented.

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