Metabolism GM and BP neural network-based satellite clock error forecasting method

Abstract

The invention discloses a satellite clock error forecasting method based on a metabolism GM and BP neural network, and belongs to the technical field of satellite navigation. The method comprises the following steps: firstly, acquiring satellite clock error historical data, constructing and using a metabolism GM (1, 1) model to carry out preliminary clock error prediction, secondly, calculating a residual error between a preliminary prediction result and a true value, constructing a prediction residual error sequence, then taking the prediction residual error sequence as a training sample, training a BP neural network model, and finally, carrying out clock error prediction. And finally, when a clock error at a future moment is predicted, a clock error value is predicted by utilizing a metabolism GM (1, 1) model, a residual error compensation value is predicted by utilizing the trained BP neural network, and the two values are superposed to obtain a final high-precision combined prediction result. According to the method, the advantages of the two models are combined, systematic errors of a single model are effectively compensated, and the accuracy and long-term stability of satellite clock error forecasting are remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of satellite navigation and positioning technology, specifically to a satellite clock error prediction method based on metabolic GM and BP neural networks. Background Technology

[0002] Global Navigation Satellite System (GNSS), with its high precision, all-weather capability, and global coverage, plays a crucial role in positioning, navigation, and timing (PNT) services. The core of the system is the onboard atomic clock, and its stability directly determines the accuracy and reliability of GNSS services. However, due to factors such as the complex space environment and the aging of the atomic clock's own physical characteristics, the operating frequency of the satellite clock will drift, resulting in a slight difference between its output time and the GNSS standard time. This difference is called satellite clock bias. Accurately predicting satellite clock bias is of key significance for real-time precise positioning, precise point positioning, and ensuring the effectiveness of navigation messages.

[0003] Currently, scholars both domestically and internationally have proposed various satellite clock error prediction models. Common prediction models mainly include: Multinomial models, such as linear polynomial (LPM) or quadratic polynomial (QPM), can fit the linear and quadratic trends in clock bias data well. They have simple algorithms, low computational cost, and can achieve good accuracy in short-term forecasts. However, satellite clock bias also contains complex nonlinear variations and random noise, which multinomial models cannot effectively characterize. As a result, the accuracy of medium- and long-term forecasts will decrease significantly with the increase of forecast duration.

[0004] Grey forecasting model: This model has good predictive effect on time series with small amount of data and insufficient information. However, the traditional GM(1,1) model is essentially an exponential fit. When the clock difference series fluctuates or undergoes nonlinear random changes, its forecast accuracy will be greatly affected, and the predictive performance of the model will gradually decrease over time.

[0005] Other models, such as the Kalman filter model, although robust, are prone to filter divergence when model parameters are set improperly or when faced with sudden environmental changes, which affects the reliability of forecasts.

[0006] In summary, existing forecasting models have limitations in describing the complex variations of satellite clock bias. They typically only capture the trend term of the clock bias sequence well, while failing to adequately fit the nonlinear, stochastic error term, making it difficult to balance short-term accuracy and long-term stability. Therefore, establishing a combined forecasting model that can simultaneously and accurately capture the main clock bias trend and compensate for stochastic error variations, thereby comprehensively improving the accuracy and stability of satellite clock bias forecasts, is a pressing technical problem to be solved in this field. Summary of the Invention

[0007] To address the problems existing in the background technology, this invention proposes a satellite clock bias combined forecasting method based on metabolic GM(1,1) and BP neural network. This method solves the problem that a single forecasting model cannot simultaneously take into account the trend characteristics and nonlinear random characteristics of the clock bias sequence, resulting in insufficient forecast accuracy and long-term stability. It achieves dynamic tracking and forecasting of the main trend of clock bias data, and then models and compensates the forecast residuals of the GM(1,1) model. This new combined forecasting model achieves step-by-step processing and accurate prediction of the deterministic trend and random error of satellite clock bias, thereby comprehensively improving forecast accuracy and stability.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: The satellite clock error prediction method based on metabolic GM and BP neural networks includes the following steps: S1. Obtain historical data of satellite clock bias as the initial clock bias sequence; S2. Construct a metabolic GM(1,1) model based on the initial clock bias sequence; S3. Based on the prediction results of the metabolism GM(1,1) model, the BP neural network model is trained to obtain the residual prediction model; S4. Combine the metabolic GM(1,1) model and the trained residual prediction model to make a combined prediction of the satellite clock bias at future times.

[0009] Specifically, step S2 includes: S21. Based on the initial clock difference sequence, establish a grey prediction model GM(1,1); S22. Calculate the predicted clock difference data for the next moment using the grey prediction model GM(1,1); S23. A new satellite clock bias data sequence is obtained through the sequence metabolism update method, and it is used as the initial clock bias sequence for the next prediction. S24. Repeat steps S21 to S23 until the clock difference data prediction for the predetermined forecast duration is completed.

[0010] Specifically, step S21 includes: (1) For the initial clock difference sequence Performing one accumulation generation operation yields the accumulation generation sequence. The formula for calculating each accumulated data point is: ; in, ; (2) Based on a single accumulation sequence Calculate the next nearest mean generation sequence , No. k The nearest data points to the mean The calculation formula is: ; (3) Establish the first-order differential equation: ; Among them, parameters For development coefficients, parameters The amount of gray action; (4) Integrate both sides of the differential equation to transform it into the form of a difference equation: ; Then convert it into matrix form: ; Recorded as ; (5) Solve the matrix equation obtained in step (4) using the least squares method.

[0011] Specifically, the least squares solution obtained in step (5) is: ; Then the parameters are obtained. and The estimated value and Based on the estimated parameters and The prediction results of the cumulative generated sequence are as follows: ; The grey prediction model for the satellite clock error sequence obtained through cumulative subtraction and restoration is as follows: .

[0012] Specifically, the sequence metabolism update method in step S23 is as follows: the predicted clock bias data obtained in step S22 is added to the end of the currently used satellite clock bias data sequence, and at the same time, the oldest historical clock bias data is removed from the beginning of the sequence, thereby generating a new satellite clock bias data sequence.

[0013] Specifically, the training data for the BP neural network in step S3 is: the predicted residual sequence obtained by comparing the clock error prediction sequence predicted by the metabolic GM(1,1) model with the true value of the original clock error sequence and subtracting them.

[0014] Specifically, the implementation process of the BP neural network algorithm in step S3 includes: S31. Assign initial values ​​to each weight and bias of the BP neural network, and determine the activation function; S32. Input the original data and calculate the output error of the output layer. The formula is as follows: ; in, This is the actual value. This is a predicted value; S33. The error is backpropagated from the output layer through the hidden layers to the input layer, calculated layer by layer, and propagated based on the error of the previous layer and the current weights. The calculation formula is as follows: ; in, For the error of the later layer, For connection weights, The derivative of the activation function; S34. Update and adjust the weights using gradient descent: ; in, Input value; The learning rate; S35. Repeat steps S32 to S34 to iterate continuously. Each time forward and backward propagation is performed, the weights are continuously updated to gradually reduce the error until the prediction error is lower than the predetermined value.

[0015] Specifically, step S4 includes: S41. Predict the clock bias values ​​for future moments using the GM(1,1) metabolic model. S42. Use the trained residual prediction model to predict the residual compensation value at future time points; S43. The clock error prediction value and the residual compensation value are superimposed to obtain the final satellite clock error combination prediction result.

[0016] In summary, the beneficial technical effects of the present invention are as follows: 1. Significantly improved forecast accuracy: This invention decomposes the satellite clock error forecasting problem into two stages: trend term forecasting and residual term compensation. The metabolic GM(1,1) model is responsible for capturing and forecasting the main, deterministic trend changes in the clock error data, completing the preliminary forecast. The BP neural network is specifically designed to model the forecast residuals of the GM(1,1) model, accurately learning and forecasting the complex nonlinear random errors that are difficult for traditional models to characterize. By superimposing the forecast results of the two, effective compensation for the residuals is achieved, making the final combined forecast result closer to the true value than any single model, and the forecast accuracy is greatly improved.

[0017] 2. Enhanced long-term stability of forecast results: In long-term forecasts, the forecast error of a single model can accumulate rapidly over time, leading to model failure. This invention effectively suppresses the cumulative effect of error by specifically modeling and predicting the forecast residuals. Even in a longer forecast time domain, the combined model can still provide stable residual compensation, keeping the forecast error within a small range, thereby significantly enhancing the long-term stability and reliability of the forecast results.

[0018] 3. The model has stronger adaptability and generalization ability: The metabolic GM(1,1) model adopted in this invention can dynamically adapt to the latest operating status of the satellite clock through the metabolic data update mechanism, and is more sensitive to clock bias changes. At the same time, the BP neural network can learn the unique residual patterns of satellite clocks of different types and under different states through training. Therefore, the combined model of this invention is not limited to a specific type of satellite clock, and has good adaptability and generalization ability for clock bias sequences with different linear and nonlinear characteristics. Attached Figure Description

[0019] Figure 1 This is a flowchart of the combined model clock error prediction process of the present invention; Figure 2 This is a flowchart of satellite clock bias prediction for the metabolic GM(1,1) model in this invention; Figure 3 This is a graph showing the variation of 2-hour and 6-hour satellite clock error predictions for PRN03, PRN05, PRN10, and PRN11 satellites in an embodiment of the present invention. Figure 4 This is a graph showing the variation of 2-hour and 6-hour satellite clock error predictions for PRN15, PRN16, PRN22, and PRN27 satellites in an embodiment of the present invention. Figure 5 This is a graph showing the variation of 2-hour forecast and 18-hour satellite clock error forecast for PRN03, PRN05, PRN10, and PRN11 satellites in an embodiment of the present invention. Figure 6 This is a graph showing the variation of 2-hour forecast and 18-hour satellite clock error forecast for PRN15, PRN16, PRN22, and PRN27 satellites in an embodiment of the present invention. Figure 7 This is a graph showing the variation of 4-hour and 18-hour satellite clock error predictions for PRN03, PRN05, PRN10, and PRN11 satellites in an embodiment of the present invention. Figure 8 This is a graph showing the variation of 4-hour and 18-hour satellite clock error forecasts for PRN15, PRN16, PRN22, and PRN27 satellites in an embodiment of the present invention. Figure 9This is a graph showing the changes in the average forecast accuracy, stability, and improvement rate of the three forecasting methods in this embodiment of the invention. Detailed Implementation

[0020] To make the technical means, creative features, objectives and effects of this invention clearer and easier to understand, the invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0021] Example This invention uses high-precision GPS clock bias data from the first day of week 2377 released by the GNSS Analysis Center of Wuhan University for model verification. The experimental data covers more than 30 satellites in orbit. Eight representative sets of rubidium clock data from the GPS system were randomly selected for the experiment, specifically from PRN03, PRN05, PRN10, PRN11, PRN15, PRN16, PRN22, and PRN27 satellites. Their specific information is shown in the table below: Table 1 shows the relevant information of the selected satellites.

[0022] According to the observed data, PRN03, PRN05, PRN10 and PRN27 show a decreasing trend, while PRN11, PRN15, PRN16 and PRN22 show an increasing trend. Furthermore, the clock error curves of PRN05 and PRN22 exhibit significant nonlinear characteristics.

[0023] This embodiment constructs three forecasting models based on observational data: LPM, GM(1,1), and the combined forecasting model of this invention. It also conducts comparative experiments on three forecasting methods: forecasting six hours based on two-hour data, forecasting eighteen hours based on two-hour data, and forecasting eighteen hours based on four-hour data.

[0024] When using this invention to make predictions based on observational data, based on, for example Figure 1 The combined model shown executes the satellite clock prediction method based on metabolic GM and BP neural networks of the present invention, specifically performing the following steps: S1. Obtain historical data of satellite clock bias as the initial clock bias sequence; S2, Constructing based on the initial clock difference sequence as follows Figure 2 The steps for the GM(1,1) metabolic model shown are as follows: S21. Based on the initial clock difference sequence, establish a grey prediction model GM(1,1): (1) For the initial clock difference sequence Performing one accumulation generation operation yields the accumulation generation sequence. The formula for calculating each accumulated data point is: ; in, ; (2) Based on a single accumulation sequence Calculate the next nearest mean generation sequence , No. k The nearest data points to the mean The calculation formula is: ; (3) Establish the first-order differential equation: ; Among them, parameters For development coefficients, parameters The amount of gray action; (4) Integrate both sides of the differential equation to transform it into the form of a difference equation: ; Then convert it into matrix form: ; Recorded as ; (5) Solve the matrix equation obtained in step (4) using the least squares method.

[0025] Specifically, the least squares solution obtained in step (5) is: ; Then the parameters are obtained. and The estimated value and Based on the estimated parameters and The prediction results of the cumulative generated sequence are as follows: ; The grey prediction model for the satellite clock error sequence obtained through cumulative subtraction and restoration is as follows: .

[0026] S22. Calculate the predicted clock difference data for the next moment using the grey prediction model GM(1,1); S23. A new satellite clock bias data sequence is obtained through the sequence metabolism update method, and it is used as the initial clock bias sequence for the next prediction. The sequence metabolism update method is as follows: the predicted clock bias data obtained in step S22 is added to the end of the currently used satellite clock bias data sequence, and at the same time, the oldest historical clock bias data is removed from the beginning of the sequence, thereby generating a new satellite clock bias data sequence.

[0027] S24. Repeat steps S21 to S23 until the clock difference data prediction for the predetermined forecast duration is completed.

[0028] S3. Compare the clock error prediction sequence predicted by the metabolic GM(1,1) model with the actual values ​​of the original clock error sequence, subtract the predicted residual sequence to obtain the prediction residual sequence, and use the prediction residual sequence to train the BP neural network model to obtain the residual prediction model; the specific implementation process of the BP neural network algorithm includes: S31. Assign initial values ​​to each weight and bias of the BP neural network, and determine the activation function; S32. Input the original data and calculate the output error of the output layer. The formula is as follows: ; in, This is the actual value. This is a predicted value; S33. The error is backpropagated from the output layer through the hidden layers to the input layer, calculated layer by layer, and propagated based on the error of the previous layer and the current weights. The calculation formula is as follows: ; in, For the error of the later layer, For connection weights, The derivative of the activation function; S34. Update and adjust the weights using gradient descent: ; in, Input value; The learning rate; S35. Repeat steps S32 to S34 to iterate continuously. Each time forward and backward propagation is performed, the weights are continuously updated to gradually reduce the error until the prediction error is lower than the predetermined value.

[0029] S4. Combining the metabolic GM(1,1) model and the trained residual prediction model, perform a combined prediction of satellite clock bias at future times: S41. Predict the clock bias values ​​for future moments using the GM(1,1) metabolic model. S42. Use the trained residual prediction model to predict the residual compensation value at future time points; S43. The clock error prediction value and the residual compensation value are superimposed to obtain the final satellite clock error combination prediction result.

[0030] Finally, this embodiment calculates the difference between the model output and the concurrent precise clock difference data released by the GNSS Analysis Center of Wuhan University to obtain the prediction residual sequence of each model. The root mean square error (RMS) and range (i.e., the maximum error value minus the minimum error value) are used to evaluate and compare the prediction accuracy and stability of each model. The formulas for calculating the root mean square error and range are as follows: ; ; like Figures 3 to 8 The table below shows the forecast error variations for each model and forecast method. The statistical results of the 2-hour forecast to 6-hour satellite clock error forecast are as follows: Table 2. Statistical results of satellite clock error prediction from 2 h to 6 h (unit: ns)

[0031] The statistical results of the 2-hour forecast to 18-hour satellite clock error prediction are shown in the table below: Table 3. Statistical results of satellite clock error prediction for 2-hour forecast to 18-hour forecast (unit: ns)

[0032] The statistical results of the 4-hour forecast to 18-hour satellite clock error prediction are shown in the table below: Table 4. Statistical results of satellite clock error prediction for 4-hour forecast and 18-hour forecast (unit: ns)

[0033] like Figure 9 The figure shows the changes in average forecast accuracy, stability, and improvement rate of the three forecasting methods. The average forecast accuracy and improvement rate of each model for the three forecasting methods are shown in the table below: Table 5. Average forecast accuracy and improvement rate of each model using the three forecasting methods

[0034] As shown in Tables 2 to 4, among the three forecasting methods, the combined model consistently exhibits the highest forecast accuracy and stability. In the method of forecasting six hours from two-hour data, the average forecast accuracy of the first-order polynomial model is 0.72 ns, and the average forecast stability is 1.37 ns; the average forecast accuracy of the grey model is 0.70 ns, and the average forecast stability is 1.33 ns; the average forecast accuracy of the combined model of the present invention is 0.60 ns, and the average forecast stability is 1.14 ns. In the method of forecasting eighteen hours from two-hour data, the average forecast accuracy of the first-order polynomial model is 2.12 ns, and the average forecast stability is 3.82 ns; the average forecast accuracy of the grey model is 2.00 ns, and the average forecast stability is 3.58 ns; the average forecast accuracy of the combined model of the present invention is 1.67 ns, and the average forecast stability is 2.94 ns. In the method of forecasting 18 hours from 4-hour data, the average forecast accuracy of the first-order polynomial model is 1.39 ns, and the average forecast stability is 2.70 ns; the average forecast accuracy of the grey model is 1.23 ns, and the average forecast stability is 2.41 ns; the average forecast accuracy of the combined model of the present invention is 0.95 ns, and the average forecast stability is 1.93 ns.

[0035] As shown in Table 5, the average forecast accuracy and stability of the three forecasting methods are 1.41 ns and 2.63 ns for the first-order polynomial model, 1.31 ns and 2.44 ns for the grey model, and 1.07 ns and 2.00 ns for the combined model of the present invention. The average forecast accuracy and stability improvement rates of the combined model compared with the first-order polynomial model and the grey model are 24.11% and 18.32%, and 23.95% and 18.03%, respectively.

[0036] Therefore, the satellite clock bias prediction method based on metabolic GM and BP neural networks provided by this invention achieves a closed-loop prediction process of "preliminary prediction - fine residual compensation - high-precision combined output" by using a metabolic GM(1,1) model to dynamically track and predict the main trend of clock bias data, utilizing the powerful nonlinear fitting ability of BP neural networks to model and compensate the prediction residuals of the former, and finally constructing a combined prediction model. This solves the pain point of traditional models being unable to simultaneously take into account deterministic trends and complex nonlinear random components, resulting in insufficient prediction accuracy and long-term stability. It improves the accuracy of satellite clock bias prediction, the stability of the model, and the generalization ability to different clock bias characteristics, significantly suppresses the cumulative error in long-term prediction, and enhances the model's ability to capture nonlinear changes in clock bias. It provides an efficient, accurate, and structurally clear solution for medium- and long-term prediction of time series data with similar trends and random characteristics in GNSS and other fields.

[0037] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. The above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A satellite clock bias prediction method based on metabolic GM and BP neural networks, characterized in that, Includes the following steps: S1. Obtain historical data of satellite clock bias as the initial clock bias sequence; S2. Construct a metabolic GM(1,1) model based on the initial clock bias sequence; S3. Based on the prediction results of the metabolism GM(1,1) model, the BP neural network model is trained to obtain the residual prediction model; S4. Combine the metabolic GM(1,1) model and the trained residual prediction model to make a combined prediction of the satellite clock bias at future times.

2. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 1, characterized in that, Step S2 specifically includes: S21. Based on the initial clock difference sequence, establish a grey prediction model GM(1,1); S22. Calculate the predicted clock difference data for the next moment using the grey prediction model GM(1,1); S23. A new satellite clock bias data sequence is obtained through the sequence metabolism update method, and it is used as the initial clock bias sequence for the next prediction. S24. Repeat steps S21 to S23 until the clock difference data prediction for the predetermined forecast duration is completed.

3. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 2, characterized in that, Step S21 specifically includes: (1) For the initial clock difference sequence Performing one accumulation generation operation yields the accumulation generation sequence. The formula for calculating each accumulated data point is: ； in, ; (2) Based on a single accumulation sequence Calculate the next nearest mean generation sequence , No. k The nearest data points to the mean The calculation formula is: ； (3) Establish the first-order differential equation: ； Among them, parameters For development coefficients, parameters The amount of gray action; (4) Integrate both sides of the differential equation to transform it into the form of a difference equation: ； Then convert it into matrix form: ； Recorded as ; (5) Solve the matrix equation obtained in step (4) using the least squares method.

4. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 3, characterized in that, The least squares solution obtained in step (5) is: ； Then the parameters are obtained. and The estimated value and Based on the estimated parameters and The prediction results of the cumulative generated sequence are as follows: ； The grey prediction model for the satellite clock error sequence obtained through cumulative subtraction and restoration is as follows: 。 5. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 2, characterized in that, The sequence metabolism update method in step S23 is as follows: the predicted clock bias data obtained in step S22 is added to the end of the currently used satellite clock bias data sequence, and at the same time, the oldest historical clock bias data is removed from the beginning of the sequence, thereby generating a new satellite clock bias data sequence.

6. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 1, characterized in that, The specific training data for the BP neural network in step S3 is as follows: the predicted clock error sequence predicted by the metabolic GM(1,1) model is compared with the true value of the original clock error sequence, and the predicted residual sequence is calculated by subtracting the two.

7. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 6, characterized in that, The specific implementation process of the BP neural network algorithm in step S3 includes: S31. Assign initial values ​​to each weight and bias of the BP neural network, and determine the activation function; S32. Input the original data and calculate the output error of the output layer. The formula is as follows: ； in, This is the actual value. This is a predicted value; S33. The error is backpropagated from the output layer through the hidden layers to the input layer, calculated layer by layer, and propagated based on the error of the previous layer and the current weights. The calculation formula is as follows: ； in, For the error of the later layer, For connection weights, The derivative of the activation function; S34. Update and adjust the weights using gradient descent: ； in, Input value; The learning rate; S35. Repeat steps S32 to S34 to iterate continuously. Each time forward and backward propagation is performed, the weights are continuously updated to gradually reduce the error until the prediction error is lower than the predetermined value.

8. The satellite clock bias prediction method based on metabolic GM and BP neural networks according to claim 1, characterized in that, Step S4 specifically includes: S41. Predict the clock bias values ​​for future moments using the GM(1,1) metabolic model. S42. Use the trained residual prediction model to predict the residual compensation value at future time points; S43. The clock error prediction value and the residual compensation value are superimposed to obtain the final satellite clock error combination prediction result.

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