Bridge performance degradation monitoring system and method based on time series cointegration analysis

Abstract

This invention relates to the field of bridge structural health monitoring technology, and discloses a bridge performance degradation monitoring system and method based on time-series cointegration analysis. It includes: acquiring non-stationary physical response time-series data of the target structure collected at multiple monitoring locations; constructing a long-term equilibrium model based on the non-stationary physical response time-series data, extracting cointegration vectors characterizing the intrinsic mechanical constraints of the target structure, and calculating an equilibrium error sequence reflecting the degree to which the target structure deviates from a healthy physical benchmark; generating a state smoothing index by performing weighted moving average processing on the equilibrium error sequence; and outputting performance degradation early warning information based on the trend drift of the state smoothing index relative to a preset control limit. This invention achieves accurate early warning of progressive structural performance degradation by using cointegration analysis to remove common-mode interference from the environment and combining statistical process control to capture minute drifts.

Description

Technical Field

[0001] This invention relates to the field of bridge structural health monitoring technology, specifically to a bridge performance degradation monitoring system and method based on time-series cointegration analysis. Background Technology

[0002] In the health monitoring of large infrastructure such as bridges, accurately identifying the long-term degradation trend of structural performance is crucial to ensuring operational safety. Existing technologies typically employ traditional correlation analysis methods, such as the Pearson correlation coefficient, to process monitoring data and assess the degree of correlation between monitoring points. However, these methods primarily measure short-term linear synchronous changes between variables and have significant limitations for structural response data strongly influenced by slowly varying environmental factors such as temperature and humidity. On the one hand, seasonal cycles and interannual variations caused by environmental factors result in non-stationary characteristics in the response data of individual monitoring points, and their short-term fluctuations can easily mask the true long-term degradation trend of the structure. On the other hand, multiple unrelated non-stationary time series may exhibit high statistical correlation due to shared environmental loads, resulting in a "spurious correlation" phenomenon. This false correlation caused by common environmental interference makes it difficult for traditional methods to effectively eliminate environmental influences and accurately characterize the inherent mechanical constraints of the structure, leading to missed or misjudged assessments of hidden, gradual structural performance degradation. Summary of the Invention

[0003] Purpose of the invention: To address the problem of low accuracy in identifying long-term structural performance degradation caused by strong environmental factors in existing technologies, this invention provides a bridge performance degradation monitoring system and method based on time-series cointegration analysis, which achieves highly sensitive early warning of slow and progressive physical performance degradation of structures.

[0004] To achieve the above objectives, the present invention adopts the following technical solution.

[0005] A bridge performance degradation monitoring method based on time-series cointegration analysis includes: acquiring non-stationary physical response time-series data of a target structure collected at multiple monitoring locations; constructing a long-term equilibrium model based on the non-stationary physical response time-series data, extracting a cointegration vector characterizing the intrinsic mechanical constraints of the target structure, and calculating an equilibrium error sequence reflecting the degree of deviation of the target structure from a healthy physical benchmark based on the cointegration vector; performing weighted moving average processing on the equilibrium error sequence to generate a state smoothing index characterizing the long-term performance state of the target structure; and outputting performance degradation early warning information of the target structure based on the trend drift of the state smoothing index relative to a preset control limit.

[0006] This invention transforms the problem of non-stationary physical monitoring affected by environmental disturbances into a statistical process control problem of stationary error sequences. It utilizes cointegration vectors to remove common-mode interference from the environment to extract a pure physical benchmark, and combines weighted moving average processing to keenly capture minute drifts that deviate from the benchmark, thereby achieving accurate monitoring of bridge performance degradation.

[0007] Furthermore, the extraction of the cointegration vector includes: integrating the non-stationary physical response time series data into a vector autoregressive model, and transforming the vector autoregressive model into a vector error correction model through difference transformation; determining the cointegration vector based on the long-term influence matrix in the vector error correction model, specifically: firstly, performing canonical correlation analysis on the long-term influence matrix to solve for the eigenvectors; then determining the cointegration rank based on the trace test or the maximum eigenvalue test; and finally calculating and determining the cointegration vector based on the cointegration rank and the eigenvectors.

[0008] This invention separates the long-term equilibrium information among multiple variables from the short-term dynamic adjustment mechanism through a vector error correction model, ensuring that the extracted cointegration vector can stably represent the inherent mechanical continuity of the structure rather than the instantaneous environmental response. It also avoids the subjectivity of manually setting the model order, thus guaranteeing the objectivity and accuracy of physical benchmark extraction.

[0009] Furthermore, the specific steps for calculating the equilibrium error sequence reflecting the degree of deviation of the target structure from the healthy physical benchmark based on the cointegration vector include: linearly combining the non-stationary physical response time-series data with the cointegration vector to generate the equilibrium error sequence reflecting the degree of deviation of the target structure from the healthy physical benchmark at the corresponding time. This process projects the high-dimensional physical response data onto the physical equilibrium subspace defined by the cointegration vector, and the generated equilibrium error sequence directly quantifies the physical deviation between the current state of the structure and the healthy benchmark.

[0010] Furthermore, a weighted moving average is applied to the equilibrium error sequence to generate a state smoothing index characterizing the long-term performance state of the target structure. This includes: using an exponentially weighted moving average algorithm, combined with a preset smoothing coefficient and historical state values ​​from the previous time step, to recursively calculate the equilibrium error sequence and obtain the state smoothing index. By assigning higher weights to recent data and incorporating historical information, random noise interference can be effectively filtered out, while simultaneously responding quickly to small, persistent drifts in the mean of the equilibrium error, thus improving the ability to perceive gradual degradation.

[0011] Furthermore, the preset smoothing coefficient is used to adjust the weight of the current observation error on the state smoothing index; the initial value of the historical state value at the previous moment is set based on the mean of the equilibrium error sequence within the healthy baseline period. This implementation process anchors the algorithm parameters to the physical statistical characteristics of the structural health period, ensuring that the evolution starting point of the state smoothing index has a clear physical meaning and enhancing the reliability of the monitoring results.

[0012] Furthermore, the method for determining the preset control limits includes: calculating the upper and lower limits of the control chart based on the mean and standard deviation of the equilibrium error sequence within the health baseline period, combined with a preset confidence level coefficient and the smoothing coefficient corresponding to the weighted moving average processing, as the preset control limits. This invention, based on adaptive statistical boundaries of health baseline data, enables the early warning discrimination criteria to dynamically adapt to the physical characteristics of the structure itself, reducing the false alarm rate caused by fixed thresholds.

[0013] Furthermore, the step of outputting performance degradation warning information for the target structure based on the trend drift of the state smoothing index relative to the preset control limit includes: when the state smoothing index exceeds the preset control limit for a preset number of consecutive times, or exhibits a one-sided monotonic deviation trend, or the result of re-testing the stationarity of the extended equilibrium error sequence becomes non-stationary, determining that the target structure has experienced performance degradation, and outputting the performance degradation warning information.

[0014] Furthermore, this invention also provides a bridge performance degradation monitoring system based on time-series cointegration analysis, comprising: The data acquisition module is used to acquire non-stationary physical response time-series data of the target structure collected at multiple monitoring locations; The equilibrium error extraction module is used to construct a long-term equilibrium model based on the non-stationary physical response time series data, extract the cointegration vector characterizing the intrinsic mechanical constraints of the target structure, and calculate the equilibrium error sequence reflecting the degree of deviation of the target structure from the healthy physical benchmark based on the cointegration vector. The smoothing monitoring module is used to perform weighted moving average processing on the equilibrium error sequence to generate a state smoothing index that characterizes the long-term performance state of the target structure. The early warning module is used to output performance degradation warning information for the target structure based on the trend drift of the state smoothing index relative to a preset control limit. This device achieves automated execution of the above method through functional modularization, facilitating integration into existing health monitoring platforms.

[0015] Furthermore, the bridge performance degradation monitoring system of the present invention also includes multiple physical sensors deployed on the target structure, and a processor communicatively connected to the physical sensors; the physical sensors are configured to collect non-stationary physical response time-series data of the target structure at multiple monitoring locations; the processor is configured to execute the bridge performance degradation monitoring method as described above.

[0016] Beneficial Effects: Compared with existing technologies, the technical solution provided by this invention, by constructing a long-term equilibrium model to extract cointegration vectors, can transform non-stationary physical response data from multiple measurement points, which are strongly affected by environmental factors such as temperature and humidity, into a stationary equilibrium error sequence characterizing the intrinsic mechanical constraint relationship of the structure. This process essentially removes environmental common-mode interference, solving the technical problems of "pseudo-correlation" and short-term fluctuations masking long-term trends in traditional correlation analysis. Based on this, by generating a state smoothing index through weighted moving average processing of the equilibrium error sequence, and combining it with preset control limits to monitor its trend drift, the complex problem of structural degradation identification is transformed into a statistical process control problem of small mean shifts in a stationary sequence. This dual mechanism of physical benchmark extraction and statistical trend monitoring not only significantly improves the detection sensitivity of slow, progressive structural performance degradation such as concrete cracking, prestress loss, and foundation settlement, but also effectively reduces the false alarm rate caused by changes in environmental factors, providing a reliable data-driven decision-making basis for the long-term operation and maintenance of large-scale infrastructure. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of the method steps described in this invention; Figures 2(a)-2(i) are time series curves of the displacement mean of measuring points 1-9 in the embodiments of the present invention; Figure 3 This is an EWMA control chart of the cointegration relationship residuals in an embodiment of the present invention. Detailed Implementation

[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of the embodiments of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0019] As bridges age, health issues become increasingly apparent. Structural damage identification, a crucial component of bridge health monitoring, plays a decisive role in determining the bridge's health status and implementing appropriate measures. However, constantly changing environmental factors can affect monitored structural parameters, thereby impacting damage identification accuracy and increasing the likelihood of misjudgments. This invention proposes treating long-term monitoring data from multiple deflection measurement points on the structure as an interconnected "economic system," and introducing cointegration theory from econometrics to characterize and monitor its inherent long-term statistical equilibrium relationship.

[0020] First, let me explain: assuming deflection data from two measuring points on a bridge... and If there exists a coefficient This makes the combination If it is a stationary sequence, meaning the mean and variance are constant, then... and This satisfies the cointegration relationship. When a structure suffers localized damage (such as concrete cracking or prestress loss) or systemic degradation (such as a decrease in overall stiffness or uneven foundation settlement), the long-term equilibrium relationship determined by the mechanical relationships of a healthy structure is disrupted. This disruption manifests in the loss of the previously stable equilibrium error sequence. It will become non-stationary, its mean may drift, and its variance may increase.

[0021] Example 1: This example illustrates the processing steps involved in implementing the present invention by combining actual bridge monitoring data and technical principles.

[0022] like Figure 1 As shown, this embodiment achieves accurate evaluation of the long-term performance of the target structure through higher-level logic driven by physical response data. Specifically, the method includes the following steps.

[0023] Step S100: Obtain non-stationary physical response time-series data of the target structure collected at multiple monitoring locations.

[0024] Non-stationary physical response time-series data refers to continuous observation records reflecting the mechanical behavior of a structure, collected by physical sensors deployed at key stress-bearing components of target structures such as bridges, dams, or high-rise buildings. This non-stationary physical response time-series data can also be any physical quantity capable of characterizing the mechanical state of a structure, such as strain, rotation angle, acceleration, or cable force, as long as it contains a coupling of the structure's intrinsic mechanical information and environmental disturbance information. In practical applications, simultaneous acquisition from multiple measurement points provides the necessary physical data foundation for subsequently decoupling the true mechanical behavior of the structure from complex environmental disturbances.

[0025] Step S200: Construct a long-term equilibrium model based on non-stationary physical response time series data.

[0026] Although the physical response at a single measuring point is non-stationary, the target structure in a healthy state is a holistic system with inherent mechanical continuity. The responses between measuring points exhibit inherent physical constraints determined by the structural stiffness distribution, boundary conditions, and load transfer paths during long-term evolution. This step transforms the non-stationary physical monitoring problem, which is subject to strong environmental disturbances, into a monitoring problem of a stationary error sequence, effectively isolating the influence of environmental factors.

[0027] The specific steps include: Step S201: Integrate the non-stationary physical response time-series data collected from multiple monitoring locations into a multi-dimensional vector autoregressive (VAR) model, mathematically represented as follows: .

[0028] Step S202: The VAR model is transformed into a VECM (Vector Error Correction Model) form through differential transformation.

[0029] The VAR model describes the linear dependence between the current response of each measuring point and the response at several past time points. However, VAR models built directly from raw, non-stationary data are easily affected by environmental trend terms, leading to unstable parameter estimations. Therefore, this embodiment transforms the VAR model into a VECM model form using difference transformation: .

[0030] Furthermore, in practical applications, this invention eliminates common trend terms caused by slowly varying factors such as seasonal temperature cycles and humidity changes through differential transformation, thereby extracting dynamic components that purely reflect the mechanical properties of the structure itself. The transformed VECM model contains a crucial long-term influence matrix, which essentially characterizes the inherent physical information of the bridge structure under healthy conditions, such as stiffness distribution, boundary conditions, and load transfer paths. This matrix forms the physical basis for subsequent extraction of cointegration vectors.

[0031] In the VECM model expression, the k×k matrix Π can be decomposed into Π=αβ, which in turn yields the k×r cointegration vector matrix β, where α is the k×r adjustment coefficient matrix. The specific decomposition process is implemented based on the maximum likelihood test method.

[0032] Step S203: Perform canonical correlation analysis on the long-term influence matrix to solve for the eigenvectors.

[0033] Step S204: Use trace test or maximum eigenvalue test to objectively determine the cointegration rank.

[0034] For a multi-span continuous beam bridge, there may be multiple independent mechanical equilibrium relationships between the deflections of each span. Each relationship represents an inherent constraint that prevents the structure from undergoing rigid body displacement or excessive deformation. The cointegration rank determined through statistical testing is, in effect, a quantitative assessment of the number of effective constraints currently present in the structure.

[0035] Step S205: Obtain the cointegration vector based on the cointegration rank and eigenvector.

[0036] After determining the cointegration rank, the cointegration vector can be obtained by solving the corresponding eigenvectors through canonical correlation analysis of the long-term influence matrix.

[0037] Step S300 involves performing a weighted moving average on the equilibrium error sequence to generate a state smoothing index characterizing the long-term performance state of the target structure. This includes: using an exponentially weighted moving average algorithm, combined with a preset smoothing coefficient and the historical state value from the previous time step, to recursively calculate the equilibrium error sequence and obtain the state smoothing index. The recursive calculation formula is as follows: ,middle The state smoothing index at time t. The current equilibrium error observation value. λ represents the historical state value at the previous moment, and λ is the preset smoothing coefficient. The term represents the accumulated performance state memory of the structure in the past, while The term represents the instantaneous deviation information of the current moment relative to the physical reference. Through this recursive fusion, the state smoothing index... It can effectively filter out high-frequency random disturbances caused by measurement noise or instantaneous local loads, while retaining and amplifying those persistent low-frequency trend components that reflect the evolution of the structure's intrinsic mechanical properties.

[0038] As one implementation method, a preset smoothing coefficient is used to adjust the weight of the current observation error on the state smoothing index; the initial value of the historical state value at the previous moment is set based on the mean of the equilibrium error sequence within the health baseline period.

[0039] Although the equilibrium error sequence theoretically eliminates environmental trend terms, in actual engineering monitoring, it may still contain short-term random fluctuations caused by measurement noise, instantaneous local loads, or abnormal data transmission. Directly judging based on the original equilibrium error can easily lead to false alarms. Therefore, this step uses weighted moving average processing as a physical state memory and fusion mechanism. By recursively weighting historical error information and current observations, high-frequency random disturbances are filtered out, thereby extracting a state smoothing index that can truly reflect the long-term performance evolution trend of the structure.

[0040] Furthermore, it is pointed out that the smoothing coefficient λ in this embodiment has a clear physical adjustment meaning, and its value directly determines the sensitivity characteristics of the monitoring system to different types of structural damage. When λ is small, it means that historical states are given higher weight, and the system exhibits "long memory" characteristics. This configuration is more sensitive to the extremely slow-developing gradual degradation such as concrete creep, prestressed tendon corrosion and fracture, or uneven foundation settlement, and can accumulate significant degradation trends from long-term small drifts. Conversely, when λ is large, the system exhibits "short memory" characteristics, responds more quickly to current observations, and is suitable for capturing sudden damage events such as vehicle collisions and sudden component fractures. In practical engineering applications, considering that the performance degradation of large infrastructure is usually a long-term cumulative process, and that monitoring data inevitably contains environmental noise, it is recommended to set the preferred range of λ to 0.1 to 0.3. Within this range, the system maintains sufficient noise suppression capability while being able to keenly capture the slow decay of structural stiffness on a timescale of several days to several weeks. For example, if λ is 0.1, the system effectively averages historical information from approximately 20 sampling periods, making it highly suitable for monitoring long-term stiffness degradation under seasonal temperature cycling. If λ is 0.3, the system's response speed to recent changes is improved by about three times, making it more suitable for bridge structures operating in complex environments with frequent load variations. It should be understood that although this embodiment provides a preferred range of 0.1 to 0.3, in other embodiments, the specific value of λ can be adaptively adjusted according to the type of the target structure, its service life, data acquisition frequency, and the expected damage mode, as long as it can achieve a physical fusion of historical memory and current observations.

[0041] Initial values ​​for historical state values Its design is also subject to strict physical constraints. In this embodiment, It must be set as the mean of the equilibrium error sequence within the healthy baseline period. Since the equilibrium error sequence in a healthy state should theoretically exhibit a stationary process with zero mean, therefore... It is typically initialized to 0. The physical significance of this setting lies in precisely anchoring the starting point of the state smoothing index evolution to a healthy physical benchmark of the structure. Only when the starting point represents an "absolutely healthy" reference state can subsequent... Only any systematic drift relative to the zero line has clear physical interpretability, that is, it directly quantifies the degree to which the structure deviates from a healthy baseline. If If the value is arbitrarily set to the mean of an unhealthy period or other arbitrary constant, the generated state smoothing index will contain an initial bias that cannot be eliminated. This will cause subsequent warning and control limits to lose their physical reference, leading to a large number of false alarms or missed alarms. Therefore, [the following text is incomplete and requires further context: "will"] Tying the approach to statistical characteristics of the health baseline period is a key prerequisite for ensuring the physical legitimacy of the entire smooth monitoring logic.

[0042] Through the above-mentioned physical anchoring mechanism for parameters, this embodiment transforms the abstract exponentially weighted moving average algorithm into a physical state tracker specifically tailored for structural performance degradation characteristics, effectively distinguishing it from general data processing methods.

[0043] Step S400: Based on the trend drift of the state smoothing index relative to the preset control limit, output performance degradation warning information of the target structure, including: when the state smoothing index exceeds the preset control limit for a preset number of consecutive times, or shows a one-sided monotonic deviation trend, or the result of re-testing the stationarity of the expanded equilibrium error sequence becomes non-stationary, determine that the target structure has experienced performance degradation, and output performance degradation warning information.

[0044] This preset control limit is a statistical boundary that can dynamically adapt to the physical characteristics of the structure and the parameters of the monitoring algorithm. In the calculation formula, the standard deviation of the equilibrium error sequence within the healthy baseline period is introduced. In addition to characterizing the inherent physical fluctuations of a structure in a healthy state, a smoothing coefficient λ corresponding to the weighted moving average processing is specifically incorporated. This is because the statistical variance of the state smoothing index, after exponential weighted moving average processing, scales with changes in the value of λ. If this factor is ignored and the standard deviation of the original data is directly applied, the control limits will be too wide or too narrow, thus losing the accuracy of the warning. By including the smoothing coefficient in the calculation method, the control limits can accurately match the statistical distribution characteristics of the state smoothing index, achieving adaptive service of the statistical boundary to the needs of physical diagnosis. For example, when a smaller λ value is selected to capture slow degradation, the volatility of the state smoothing index is compressed, and the calculated control limits will also be narrowed accordingly, thus ensuring that the detection sensitivity for small drifts is not diluted.

[0045] This embodiment constructs a triple early warning triggering mechanism to address different types of structural damage modes. The first triggering condition is that the state smoothing index exceeds a preset control limit for a predetermined number of consecutive times. This mechanism is mainly used to eliminate random anomalies caused by instantaneous environmental interference or sensor noise. Only when the deviation is persistent is it confirmed as a change in physical state, effectively reducing the false alarm rate. The second triggering condition is that the state smoothing index exhibits a unilateral monotonic deviation trend. Even if its value has not yet exceeded the control limit, if it shows a continuous upward or downward unidirectional evolution characteristic over a long observation window, it often indicates the occurrence of progressive degradation such as increased concrete creep, accumulated prestress loss, or support aging. This mechanism compensates for the insensitivity of simple threshold discrimination to early weak trends. The third triggering condition is that the result of re-testing the stationarity of the expanded equilibrium error sequence becomes non-stationary. In traditional methods, only the mean shift is usually considered; however, in this invention, stationarity itself is regarded as a direct representation of the stability of the structural mechanical relationship. When newly added monitoring data causes the previously established stationary equilibrium error sequence during the healthy baseline period to transform into a non-stationary sequence, this physically signifies that the inherent mechanical constraints maintaining the long-term equilibrium relationship between the responses of each measuring point have been fundamentally disrupted or reconstructed. It is no longer merely a quantitative change in degree, but a qualitative change in the system state. Therefore, the failure of the stationarity test is the most conclusive evidence of structural damage. Even if the state smoothing index may not yet significantly exceed the limit due to hysteresis, a high-level warning should be triggered immediately.

[0046] like Figure 3 The EWMA control chart showing the cointegration relationship residuals visually illustrates examples of the three triggering scenarios: the blue scatter plot on the left represents the healthy baseline period, where data fluctuates smoothly within control limits; the red scatter plot on the right, after entering the testing period, shows that the state smoothing index (shown by the solid green line) initially exhibits a clear unilateral upward trend, followed by multiple consecutive sampling points exceeding the upper control limit (UCL) indicated by the dashed red line. Real-time verification by the backend confirms that the extended equilibrium error sequence within this period has lost its stationarity. All three factors collectively confirm the occurrence of structural performance degradation. It should be understood that the three triggering conditions are logically related by "OR," meaning that an alert is output when any condition is met. However, in practical applications, a combined triggering mode can be configured according to operational needs, or other auxiliary criteria can be added to adapt to the monitoring requirements of structures with different security levels.

[0047] Example 2: In this example, n deflection measurement points are selected on the bridge to acquire data at T time points, forming an n×T time series dataset. Based on this time series dataset (non-stationary physical response time series data is selected as deflection time series data), the implementation process of this invention includes: (1) Data preprocessing and stationarity test The raw data is preprocessed to remove outliers, and a unit root test is performed on the deflection time series data for each measuring point. Specifically, the ADF test (Augmented Dickey-Fowler test) is used: model parameters are estimated using the least squares method, the ADF statistic is calculated, and the ADF statistic is compared with critical values ​​(usually at 1%, 5%, or 10% significance levels). If the ADF statistic > the critical value, the deflection time series data is considered non-stationary. In this application, during the healthy operation phase of the bridge, due to the seasonal cycles and interannual variations in ambient temperature, individual deflection sequences typically exhibit non-stationary physical response time series data (containing trends and seasonal cycles). This process is a prerequisite for cointegration analysis.

[0048] (2) Cointegration test and vector error correction model establishment

[0049] During the healthy baseline period, the first step in conducting cointegration tests and establishing a vector error correction model (VECM) is to integrate the high-dimensional time series data of each measurement point into a vector autoregressive (VAR) model.

[0050] use Let represent the observed values ​​of all deflection measurement points at time t. Let be a k-dimensional time series vector, where k is the number of deflection measurement points. First, determine the optimal hysteresis order p based on the information criterion, and establish the VAR(p) model, mathematically represented as: .in, Both are k×k coefficient matrices. The variable that measures the value at a lag of one period (i.e., time t-1) relative to the current time. The linear effect; similarly, To measure the impact of the lag two-period error, and so on. Measuring lag The effects of time. Each element in these matrices represents the marginal effect of the lagged value of one variable on the current value of another variable, which are parameters that the vector autoregressive model needs to estimate. It is also a k×k random error term vector, usually assumed to be white noise, that is, with a mean of zero and a covariance matrix that is constant (and uncorrelated with past values), representing random shocks or measurement noise that cannot be explained by the vector autoregressive model.

[0051] By performing a difference transformation, the vector autoregressive model is equivalently transformed into a vector error correction model (VECM) form. express: ,in:

[0052]

[0053] In the above formula, Denotes the first - order difference of a variable, i.e., the short - term change from t - 1 to t. Is any one of the coefficient matrices in the coefficient matrix of the above - mentioned optimal lag order p. Is a k×k matrix, called the "long - term impact matrix" or "compression matrix", which contains all long - term equilibrium information. When there is a cointegration relationship between variables, the rank of the matrix Will be reduced (0 < rank(Π) < k, at this time Π can be decomposed into Π = αβ. β is a k×r matrix of cointegration vectors, and each column represents an independent long - term equilibrium relationship, that is, the linear combination β′ Should be zero in the equilibrium state. Usually, β′ Is called the "equilibrium error". α is a k×r matrix of adjustment coefficients, which measures the speed and direction of each variable's callback to the equilibrium state in the short - term when the previous period deviates from the equilibrium. Is the equilibrium error sequence, that is, the error term.

[0054] To determine the cointegration rank r, the maximum likelihood test method is usually adopted. In specific operations, first estimate the above vector error correction model, and obtain the eigenvalues By solving a generalized eigenvalue problem. Then use two likelihood ratio statistics for hypothesis testing: the trace test and the maximum eigenvalue test.

[0055] The calculation formula of the trace statistic is: . Where: T is the sample size (number of observation periods). k is the total number of variables (corresponding to the number of deflection measurement points). Use To represent the th eigenvalue estimated from the data, sorted from largest to smallest: . These eigenvalues come from the canonical correlation analysis of the Π matrix and reflect the explanatory strength of each potential cointegration combination for the long - term relationship of the system. Is the cointegration rank in the null hypothesis.

[0056] The null hypothesis of the test is "cointegration rank ≤ r", and the alternative hypothesis is "cointegration rank > r". The statistic is constructed by accumulating the logarithmic transformation of the last k - r eigenvalues. If these smaller eigenvalues are significantly non - zero, the statistic value is larger, thus rejecting the null hypothesis.

[0057] Is used to test the null hypothesis "there are at most r cointegration vectors" and the alternative hypothesis "there are more than r cointegration vectors". The maximum eigenvalue statistic is . Represents the estimated value of the (r + 1)th eigenvalue. When the null hypothesis is "cointegration rank equals r" (that is, there are r cointegration relationships), theoretically the (r + 1)th eigenvalue should be zero. This value reflects the strength of the (r + 1)th potential cointegration combination. If A value significantly greater than zero implies the existence of at least r+1 cointegration relationships, thus the null hypothesis should be rejected.

[0058] It is used to test the null hypothesis "there are r cointegrating vectors" and the alternative hypothesis "there are r+1 cointegrating vectors". In practice, it is usually tested sequentially starting from r=0 until the first r value that cannot reject the null hypothesis is reached. This value is the cointegration rank.

[0059] Once the cointegration rank r is determined, maximum likelihood estimation can be performed on the VECM model under the constraint rank(Π)=r, yielding estimates of the cointegration vector matrix β and the adjustment coefficient matrix α. The estimation process is based on the two-step method proposed by Johansen: first from... and Remove short-term dynamic terms (i.e.) from the middle. The residuals are obtained by considering the influence of the residuals, and then the eigenvectors are solved by canonical correlation analysis to obtain an estimate of β. Finally, α and β are obtained using the least squares method or generalized least squares method. The estimation was performed. Finally, the equilibrium error sequence for the healthy baseline period was constructed: It represents the degree of deviation of each cointegrating combination from the long-term equilibrium at time t-1. Under healthy conditions, this sequence should exhibit a stationary process with a mean of zero, and its statistical characteristics will become the core basis for subsequent online monitoring.

[0060] (3) After establishing the cointegration model for the healthy baseline period, the real-time collected monitoring data is substituted into the constructed statistical framework. The structural degradation is determined by monitoring the changes in the statistical characteristics of the equilibrium error sequence. The specific process is as follows: (31) Calculate the equilibrium error at the current time. When new monitoring data Upon arrival (also a k-dimensional column vector containing deflection or strain readings at all measuring points at the current moment), it is first substituted into the cointegration vector matrix β estimated during the healthy baseline period to calculate the degree of deviation from the long-term equilibrium relationship at the current moment. The calculation formula is as follows:

[0061] The symbols here have the same meaning as before: β is a k×r cointegration vector matrix, where each column represents an independent long-term linear combination; β′ is its transpose, with dimensions r×k; It is the observation vector from the previous time step; the product result It is an r×1 vector representing r equilibrium error values ​​calculated based on the state at time t-1, at time t. Under healthy conditions, these error values ​​should fluctuate randomly around zero; once structural damage causes a change in the long-term mechanical relationship between the measuring points, these error values ​​may exhibit a systematic shift.

[0062] (32) Statistical process monitoring of the equilibrium error sequence

[0063] After calculating the degree of deviation from the equilibrium relationship at the current moment, it is not possible to make a judgment based solely on a single point value, because occasional random fluctuations always exist. An exponentially weighted moving average (EWMA) control chart is used to monitor its small but persistent drift.

[0064] The equilibrium error sequence for the j-th cointegration relation (j=1,…,r) Its recursive formula is:

[0065] in, It is the actual observed value of the j-th equilibrium error at the current time (i.e. (the j-th component). It is the smoothing coefficient (usually taken as 0.1≤). ≤0.3), used to control the length of historical data memory. The larger the value, the more sensitive it is to recent data. This represents the exponentially weighted moving average statistic at the current moment, which is a weighted average of all historical observations. It is the exponentially weighted moving average statistic of the previous time step, with its initial value. It is usually set as the mean of the error series during the healthy baseline period (generally 0).

[0066] The core idea of ​​this recursive formula is to give higher weight to recent data, thereby enabling the rapid capture of small shifts in the mean of the error sequence.

[0067] (33) Set control limits and make early warning judgments.

[0068] In order to determine Whether the changes are statistically significant requires calculating the upper and lower limits of the control chart based on the baseline health data. The formula for calculating the control limits is:

[0069] in: It is the mean of the equilibrium error during the healthy baseline period (usually assumed to be 0). The standard deviation of the equilibrium error during the health baseline period is estimated from the baseline period data. The width factor of the control limit (usually 2 or 3, corresponding to a confidence level of about 95% or 99.7%) determines the sensitivity of the alarm. It has the same meaning as the smoothing coefficient mentioned above.

[0070] By calculating each cointegration relationship Value. When any Multiple consecutive (more than 3 points) deviations from the UCL or LCL control limits, or the manifestation of a clear trend deviation (such as a one-sided monotonically increasing error that fails to regress over a long period), can be used to determine long-term performance degradation. Furthermore, the stationarity of the extended equilibrium error series, including new data, can be periodically retested. If the test result changes from "stationary" to "non-stationary," it provides another strong piece of evidence of damage. Through these steps, cointegration analysis transforms the structural health monitoring problem into a statistical process control problem of a set of stationary error series, enabling early detection and quantitative warning of slow structural degradation.

[0071] Example 3: Data from a certain bridge over the past six months was selected for correlation analysis (9 vertical displacement measurement points).

[0072] 1) Data Acquisition: All monitoring data were acquired and converted to daily averages to remove intraday noise and focus on long-term degradation trends. The data was divided into a healthy baseline period (first 70%) and a testing period (last 30%). The time series curves of the displacement mean values ​​at measuring points 1-9, as shown in Figures 2(a)-2(i), visually demonstrate this non-stationary physical response characteristic modulated by the environment. This step, through simultaneous acquisition at multiple measuring points, provides the necessary physical data foundation for subsequently decoupling the true mechanical behavior of the structure from complex environmental disturbances.

[0073] 2) Stationarity test: The ADF test was used to perform a unit root test on the healthy period sequence of each measurement point to confirm that it was a non-stationary I(1) process. The sequence after first difference was tested again to verify that it was a stationary process (cointegration premise).

[0074] 3) Johansen cointegration test: Determine the optimal lag order p using the AIC criterion; in this bridge data, p=1. Call jcitest to perform trace and maximum eigenvalue tests to determine the cointegration rank r; in this bridge data, r=8. This indicates that there are seven independent "constraints" that bind the long-term motions of these 14 variables together.

[0075] Since the main objective of this scheme is to monitor the long-term performance degradation of the overall structure, usually only one comprehensive cointegration relation is needed. Therefore, the first cointegration relation (i.e., r=1) can be used to establish the relevant cointegration vector.

[0076] Extract the cointegration vector matrix β and the adjustment coefficient matrix α from the maximum likelihood estimation MLEs.

[0077] 4) Establish VECM and calculate equalization error: Calculate the equilibrium error sequence during the healthy period based on β. Based on the bridge data, the relevant equations are established as follows: =1.289251×Y1-9.777194×Y2+10.546430×Y3+0.841103×Y4+0.024842×Y5+6.439657×Y6-4.901364×Y7-6.067510×Y8-0.180130×Y9 (Long-term equilibrium relationship) The equilibrium error is then subjected to an ADF test to verify its stationarity (mean close to 0). 5) Online monitoring and EWMA control chart: For the test period data, calculate the equilibrium error at each time point in sequence.

[0078] Applying the EWMA recurrence formula Calculate the time-varying control limits and compare them with the steady-state control limits. If the limits are exceeded, an early warning is triggered.

[0079] 6) Results Visualization: Time history curves for each measurement point, scatter plots of equilibrium errors in cointegration relationships, EWMA statistics and control limits charts, etc. Figure 3 As shown in the figure. Monitoring results indicate that the equilibrium error exceeded the control limit on day 25 of the testing period, which may indicate structural performance degradation.

[0080] To highlight the technical advantages of this invention, this embodiment also introduces traditional Pearson correlation analysis as a comparative example. Analysis of data from the same testing period revealed that, because all measuring points are driven by the same seasonal temperature field, the Pearson correlation coefficient between any two measuring points is as high as 0.9 or more, exhibiting a very strong "pseudo-correlation" phenomenon. This false high correlation caused by environmental common-mode interference completely masks the changes in local mechanical relationships caused by support detachment, making it impossible for traditional methods to identify any anomalies. In contrast, the equilibrium error sequence extracted by this invention through cointegration analysis physically eliminates the common trend term caused by temperature, retaining only the deviation information reflecting the mechanical state of the structure itself. This allows it to keenly capture hidden progressive degradation signals under strong environmental interference. This comparative result fully demonstrates the significant progress of this invention in solving the problems of "short-term fluctuations masking long-term trends" and "pseudo-correlation" described in the background art, and also provides a reliable data-driven tool for the long-term performance evaluation of large infrastructure in complex service environments.

[0081] It should be understood that the above-mentioned test cases are only illustrative and are not intended to limit the scope of protection of this invention. In other application scenarios, the selection of cointegration rank, the setting of smoothing coefficient, and the configuration of warning threshold can be adaptively adjusted according to different structural types and data characteristics.

Claims

1. A bridge performance degradation monitoring method based on time series cointegration analysis, characterized in that, The implementation process of this method includes: Acquire non-stationary physical response time-series data of the target structure collected at multiple monitoring locations; A long-term equilibrium model is constructed based on the non-stationary physical response time series data. A cointegration vector characterizing the intrinsic mechanical constraints of the target structure is extracted, and an equilibrium error sequence reflecting the degree of deviation of the target structure from the healthy physical benchmark is calculated based on the cointegration vector. The equilibrium error sequence is subjected to a weighted moving average to generate a state smoothing index that characterizes the long-term performance state of the target structure. Based on the trend drift of the state smoothing index relative to the preset control limit, the performance degradation warning information of the target structure is output.

2. The bridge performance degradation monitoring method according to claim 1, characterized in that, The extraction of the cointegration vector specifically includes: The non-stationary physical response time series data are integrated into a vector autoregressive model, and the vector autoregressive model is transformed into a vector error correction model through difference transformation. Based on the long-term influence matrix in the vector error correction model, the cointegration vector is determined by: firstly, performing canonical correlation analysis on the long-term influence matrix to solve for the eigenvectors; then determining the cointegration rank based on the trace test or the maximum eigenvalue test; and finally obtaining the cointegration vector based on the cointegration rank and the eigenvector matrix.

3. The bridge performance degradation monitoring method according to claim 1, characterized in that, The equilibrium error sequence is obtained by linearly combining the non-stationary physical response time series data with the cointegration vector, and then calculating it based on the degree to which the target structure deviates from the healthy physical baseline at the corresponding time.

4. The bridge performance degradation monitoring method according to claim 1, characterized in that, The state smoothing index is obtained by using an exponentially weighted moving average algorithm, combined with a preset smoothing coefficient and the historical state value of the previous time step, and then recursively calculating the equilibrium error sequence.

5. The bridge performance degradation monitoring method according to claim 4, characterized in that: The preset smoothing coefficient is used to adjust the weight of the influence of the current observation error on the state smoothing index; The initial value of the historical state value at the previous moment is set based on the mean of the equilibrium error sequence within the healthy baseline period.

6. The bridge performance degradation monitoring method according to claim 1, characterized in that, The method for determining the preset control limits includes calculating the upper and lower limits of the control chart based on the mean and standard deviation of the equilibrium error sequence within the health baseline period, combined with the preset confidence level coefficient and the smoothing coefficient corresponding to the weighted moving average processing, thereby serving as the preset control limits.

7. The bridge performance degradation monitoring method according to claim 6, characterized in that, This method utilizes the trend drift of the state smoothing index relative to a preset control limit to output performance degradation warning information for the target structure, specifically including: When the state smoothing index exceeds the preset control limit for a preset number of consecutive times, or exhibits a one-sided monotonic deviation trend, or the result of re-testing the stationarity of the expanded equilibrium error sequence becomes non-stationary, the target structure is determined to have experienced performance degradation, and the performance degradation warning information is output.

8. A bridge performance degradation monitoring system based on time-series cointegration analysis, characterized in that, include: The data acquisition module is used to acquire non-stationary physical response time-series data of the target structure collected at multiple monitoring locations; The equilibrium error extraction module is used to construct a long-term equilibrium model based on the non-stationary physical response time series data, extract the cointegration vector characterizing the intrinsic mechanical constraints of the target structure, and calculate the equilibrium error sequence reflecting the degree of deviation of the target structure from the healthy physical benchmark based on the cointegration vector. The smoothing monitoring module is used to perform weighted moving average processing on the equilibrium error sequence to generate a state smoothing index that characterizes the long-term performance state of the target structure. The early warning module is used to output performance degradation early warning information of the target structure based on the trend drift of the state smoothing index relative to the preset control limit.

9. The bridge performance degradation monitoring system according to claim 8, characterized in that, The system includes multiple physical sensors deployed on the target structure, and a processor that is communicatively connected to the physical sensors; The physical sensor is configured to collect non-stationary physical response time-series data of the target structure at multiple monitoring locations; The processor is configured to perform the bridge performance degradation monitoring method as described in any one of claims 1 to 7.

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