Data smoothing, point insertion and extension method and system based on Gaussian process

Abstract

The invention relates to the technical field of model parameter extraction, and provides a Gaussian process-based data smoothing, point insertion and extension method, which comprises the following steps of S1, performing normalization processing on data sets on an x axis and a y axis of a two-dimensional curve on which model parameter extraction needs to be performed; s2, a covariance function is selected, data needing to be trained are selected in the data set, the covariance between every two data points in the data needing to be trained is calculated through the covariance function, a covariance matrix is formed, and a Gaussian regression model is trained; and S3, selecting data needing to be trained, and performing data prediction operations including data smoothing, data point insertion and data extension by using the trained Gaussian regression model. According to the technical scheme, the problems of data smoothing, point insertion and extension during data preprocessing in model parameter extraction can be solved.

Description

Technical Field

[0001] The present invention relates to the technical field of model parameter extraction, and in particular to a method and system for data smoothing, interpolation and extension based on Gaussian processes. Background Art

[0002] In today's scientific research and engineering application fields, model parameterization is a crucial task. Model parameterization aims to determine the parameters in the model by analyzing and processing specific data so that the model can more accurately describe the behavior and characteristics of the actual system.

[0003] In many application scenarios, the study of the relationship between various physical quantities is often involved. For example, in the field of electronic circuits, the relationship between current and voltage is a key research object. By measuring the current value at different voltages, the curve of current changing with voltage can be obtained, such as Id-Vg (drain current and gate voltage relationship curve), Id-Vb (drain current and base voltage relationship curve), Id-Vd (drain current and drain voltage relationship curve), etc. At the same time, the relationship curves between other important parameters such as GM (transconductance), Rout (output resistance), kop trend (a certain trend parameter) and related physical quantities can also be obtained by calculation.

[0004] However, in the actual measurement process, due to the influence of various factors, the measured values ​​often fluctuate. This fluctuation makes the measured current not completely smooth with the voltage change. If the current is derived, the fluctuation of the data will be more serious. When performing automatic parameter extraction, these fluctuating scattered points will have an adverse effect on the overall fitting trend. Because the automatic parameter extraction process usually relies on analyzing and fitting the overall trend of the data to determine the parameters of the model. If there are a large number of fluctuating scattered points in the data, it will interfere with the accuracy of the fitting and make the parameter extraction results unreliable.

[0005] In order to solve this problem, data smoothing before parameter extraction is particularly important. Although general filtering algorithms, such as sliding average filtering, can reduce data fluctuations to a certain extent, they also have some limitations. The sliding average filter is too dependent on the information of surrounding points. After taking the average at the inflection point, it often cannot accurately reflect the state of the inflection point. This may cause the characteristics of the data to be blurred at the key position of the curve, affecting the accurate understanding and analysis of the actual physical phenomenon.

[0006] In addition, if Figure 1The device map shown has l (length) on the horizontal axis and w (width) on the vertical axis. In real measurements, devices are often not fully distributed. However, in order to control the trend of the fitting target with length and width, it is necessary to perform full array interpolation before the model is extracted. The traditional linear interpolation method cannot fit the trend of the existing data and is not smooth enough overall.

[0007] At the same time, some model parameter extraction curves, such as the current versus voltage curve, need to focus not only on the existing current and voltage points, but also on the current corresponding to a larger voltage, that is, the curve needs to be extended. Traditional linear or nonlinear extension methods cannot well reflect the changing trend of the original known measurement points.

[0008] In summary, in order to improve the accuracy and reliability of model parameters, it is necessary to find a more effective data smoothing method to solve the problems of insertion and extension at the same time, so as to cope with data fluctuations in various complex curves and meet the needs of controlling data trends. Summary of the invention

[0009] In view of the above problems, the purpose of the present invention is to provide a method and system for data smoothing, interpolation and extension based on Gaussian process, which can solve the problems of data smoothing, interpolation and extension during data preprocessing in model parameter extraction.

[0010] The above-mentioned object of the present invention is achieved through the following technical solutions:

[0011] A method for data smoothing, interpolation and extension based on Gaussian process, comprising the following steps:

[0012] S1: Normalize the data sets on the x-axis and y-axis of the two-dimensional curve that needs to be used for model parameter extraction;

[0013] S2: Select a covariance function, select data to be trained in the data set, and calculate the covariance between two data points in the data to be trained by using the covariance function to form a covariance matrix, and train a Gaussian regression model;

[0014] S3: Select the data to be trained and use the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation, and data extension.

[0015] Furthermore, in step S1, the x-axis and y-axis data of the two-dimensional curve for which the model parameter extraction is required are normalized, specifically:

[0016] Use the standard deviation normalized z-score method;

[0017] For the x-axis data, calculate the mean μ of the x-axis data x, that is, sum all x-axis data values ​​and divide them by the total number of data points to calculate the standard deviation σ of the x-axis data x , that is, first calculate the score of the difference between each data point and the mean, sum the total number of post-processing data points, and then take the square root to normalize each x-axis data point. The calculation formula is:

[0018]

[0019] For the y-axis data, calculate the mean μ of the y-axis data y , that is, sum all the y-axis data values ​​and divide them by the total number of data points to calculate the standard deviation σ of the y-axis data y , that is, first calculate the score of the difference between each data point and the mean, sum the total number of post-processing data points, and then take the square root to normalize each y-axis data point. The calculation formula is:

[0020]

[0021] Visualize the normalized x-axis and y-axis data to see whether the data distribution is concentrated within a reasonable range. If problems are found in the normalized data, including extreme values ​​or unreasonable distribution, adjust the normalization method or process the outliers separately and then normalize again.

[0022] Further, in step S2, the covariance function is selected, specifically:

[0023] The Gaussian kernel function is selected as the covariance function, and the basic form is as follows:

[0024]

[0025] Among them, K(x i , x j ) represents the data point x i and x j The covariance between them, σ is a hyperparameter, which is related to the variance of the Gaussian process and determines the amplitude of the function. l is a hyperparameter, which is related to the length of the input space and determines the smoothness of the function. When l is small, the function changes more dramatically, and when l is large, the function is smoother. exp represents the natural exponential function, ||x i -x j || represents the data point x i and x j The Euclidean distance between them.

[0026] Further, in step S2, data to be trained are selected from the data set, and the covariance between two data points in the data to be trained is calculated by the covariance function to form a covariance matrix, and a Gaussian regression model is trained, specifically:

[0027] For the data {x1, x2, ..., x n}, the covariance between two data points is calculated by the covariance function to form an n×n covariance matrix K nn ;

[0028] Considering the influence of noise, assuming ∈ i The mean is 0 and the variance is The normal distribution of The final covariance matrix is where I is the identity matrix, is the noise variance;

[0029] σ is determined by maximizing the log-marginal likelihood function 2 , l 2 and The value of , the log marginal likelihood function is:

[0030]

[0031] Among them, p(y|X) is the probability density function, y=[y1,y1,…,y n ] T is the target value vector of the training data, X = [x1, x2, ..., x n ] T is the input vector of training data, is a multivariate normal distribution;

[0032] The constructed covariance matrix is ​​used as the Gaussian regression model to perform data prediction operations of the Gaussian process including data smoothing, data interpolation, and data extension.

[0033] Further, in step S3, the data to be trained is selected and the data is smoothed using the trained Gaussian regression model, specifically:

[0034] For any prediction point x * , calculate and train each point x i The covariance vector k of * =[K(x * ,x1),K(x * ,x2),…,K(x * ,x n )] T , where K(x * ,x i ) is the predicted point x * and training point x iThe covariance between them calculated by the covariance function;

[0035] Calculate the predicted point x * Its own covariance k ** =K(x * ,x * );

[0036] Calculate the predicted point x * The mean value u * and variance

[0037] Among them, the mean u * for:

[0038]

[0039] is the covariance vector k * The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data;

[0040] variance for:

[0041]

[0042] k ** is the covariance of the prediction point itself, k * is the covariance vector between the prediction point and the training point, is the covariance matrix The inverse matrix of is the noise variance;

[0043] For the data line segment where the selected data to be trained is located, the data is discretized into multiple prediction points x * , for each prediction point x * Calculate the mean u * , with the mean value u * Connecting them together gives us the predicted curve, which is smoother than the original curve.

[0044] Further, in step S3, the data to be trained is selected and the data interpolation is performed using the trained Gaussian regression model, specifically:

[0045] Predict the insertion point position in the selected area. For any insertion point position x interp , calculate and train each point x i The covariance vector of

[0046] k interp =[K(xinterp ,x1),K(x interp ,x2),…,K(x interp ,x n )] T , where K(x interp ,x i ) is the insertion point position x interp and training point x i The covariance between them calculated by the covariance function;

[0047] Calculate the insertion point position x interp Its own covariance k interp,interp =K(x interp ,x interp );

[0048] Calculate the insertion point position x interp The mean value u interp and variance

[0049] Among them, the mean u interp for:

[0050]

[0051] is the covariance vector k interp The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data;

[0052] variance for:

[0053]

[0054] k interp,interp is the insertion point position x interp Its own covariance, k interp is the insertion point position x interp The covariance vector with the training points, is the covariance matrix The inverse matrix of is the noise variance;

[0055] With the insertion point position x interp The mean value u interp As the insertion point position x interp The values ​​are plotted and highlighted in the graph.

[0056] Further, in step S3, the data to be trained is selected and the data extension is performed using the trained Gaussian regression model, specifically:

[0057] For the data points in the selected extension area, prediction is performed. For any extension position x extend , calculate and train each point x i The covariance vector of

[0058] k extend =[K(x extend ,x1),K(x extend ,x2),…,K(x extend ,x n )] T , where K(x extend ,x i ) is the extended position x extend and training point x i The covariance between them calculated by the covariance function;

[0059] Calculate the extended position x extend Its own covariance k extend,extend =K(x extend ,x extend );

[0060] Calculate the extended position x extend The mean value u extend and variance

[0061] Among them, the mean u extend for:

[0062]

[0063] is the covariance vector k extend The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data;

[0064] variance for:

[0065]

[0066] k extend,extend is the extended position x extend Its own covariance, k extend is the extended position x extend The covariance vector with the training points, is the covariance matrix The inverse matrix of is the noise variance;

[0067] To extend the position x extend The mean value u extendAs the extended position x extend The value of the extended position x is calculated based on extend The predicted value is plotted as the curve after the data is extended.

[0068] A system for performing the above-mentioned method for data smoothing, interpolation and extension based on Gaussian process, comprising:

[0069] A normalization processing module is used to normalize the data sets on the x-axis and y-axis of the two-dimensional curve that needs to be used for model parameter extraction;

[0070] A covariance matrix construction module is used to select a covariance function, select data to be trained in the data set, and calculate the covariance between two data points in the data to be trained by the covariance function to form a covariance matrix and train a Gaussian regression model;

[0071] The parameter data processing module is used to select the data that needs to be trained and use the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation, and data extension.

[0072] A computer device comprises a memory and one or more processors, wherein the memory stores computer codes, and when the computer codes are executed by the one or more processors, the one or more processors execute the above method.

[0073] A computer-readable storage medium stores computer codes. When the computer codes are executed, the above method is executed.

[0074] Compared with the prior art, the present invention has at least one of the following beneficial effects:

[0075] (1) A method for data smoothing, interpolation and extension based on Gaussian process is provided, characterized by comprising: S1: normalizing the data sets on the x-axis and y-axis of the two-dimensional curve for model parameter extraction; S2: selecting a covariance function, selecting the data to be trained in the data set, and calculating the covariance between the two data points in the data to be trained by the covariance function to form a covariance matrix, and training a Gaussian regression model; S3: selecting the data to be trained and using the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation and data extension. The above technical solution simultaneously solves the problems of data smoothing, interpolation and extension when the model is extracted, can fully cope with data fluctuations in various complex curves and meet the needs of controlling data trends, ensures that the model extraction process in the field of scientific research and engineering applications is more accurate and reliable, and helps to more accurately describe and analyze the actual system behavior and characteristics.

[0076] (2) The solution of the present invention can effectively deal with the problem of fluctuations in measured values. In actual measurements, due to the influence of various factors, the measured data often fluctuates, especially when performing operations such as derivatives on current, the fluctuations will be more serious. The data smoothing method of the present invention can reduce these fluctuations and avoid the adverse effects of the scatter points of the fluctuations on the overall fitting trend in the process of automated parameter extraction, thereby improving the accuracy of model parameter extraction. Compared with general filtering algorithms (such as sliding average filtering), the solution of the present invention overcomes its limitations. The averaging operation of the sliding average filter at the inflection point will blur the data features, while the solution of the present invention can more accurately retain the data features at the key positions of the curve, which helps to more accurately understand and analyze actual physical phenomena.

[0077] (3) When processing device map and other data, the traditional linear interpolation method cannot fit the trend of existing data, resulting in an overall lack of smoothness. However, the solution of the present invention can perform full array interpolation before model parameter extraction, and the interpolation result can better fit the trend of existing data, ensuring that when processing device data with non-full array distribution, the trend of the fitting target with length and width can be accurately controlled, thereby improving the reliability of model parameter extraction.

[0078] (4) For situations where the curve needs to be extended, such as the current versus voltage curve needs to focus on the current corresponding to a larger voltage, the traditional linear or nonlinear extension method cannot well reflect the change trend of the original known measurement point. The solution of the present invention can effectively solve this problem. When extending the curve, it can better extend it according to the trend of the original measurement point, ensuring that the extended curve can accurately reflect the relationship between the actual physical quantities, thereby improving the accuracy and reliability of the model parameter extraction. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1It is a schematic diagram of the length and width dot matrix of the device map of the present invention;

[0080] Figure 2 This is a flow chart of Gaussian process training and prediction for a two-dimensional curve of the present invention;

[0081] Figure 3 It is an overall flow chart of the method for data smoothing, interpolation and extension based on Gaussian process of the present invention;

[0082] Figure 4 This is a schematic diagram of data smoothing of the present invention;

[0083] Figure 5 This is a schematic diagram of data insertion points of the present invention;

[0084] Figure 6 It is a schematic diagram of data extension of the present invention;

[0085] Figure 7 It is the overall structural diagram of the system of data smoothing, interpolation and extension based on Gaussian process of the present invention. DETAILED DESCRIPTION

[0086] In order to make the purpose, technical solution and advantages of the embodiments of the present application clearer, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0087] Those skilled in the art will appreciate that, unless otherwise stated, the singular forms "a", "an", "said" and "the" used herein may also include plural forms. It should be further understood that the term "comprising" used in the specification of the present invention refers to the presence of the features, integers, steps, operations, elements and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components and / or groups thereof.

[0088] The data smoothing, interpolation and extension based on Gaussian process of the present invention is an effective data analysis method.

[0089] 1. Data Smoothing

[0090] principle:

[0091] A Gaussian process is a random process whose characteristic is that any finite number of random variables obeys a joint Gaussian distribution. In data smoothing, the Gaussian process is used to model the data and smooth the noise in the data by estimating the potential function of the data.

[0092] Specifically, for a given data set, it is assumed that there is a potential functional relationship between the data points, which can be described by a Gaussian process. By estimating the parameters of the Gaussian process, a smoothed function can be obtained, which can better reflect the true trend of the data while reducing the impact of noise.

[0093] step:

[0094] Data preparation: Collect the data set that needs to be smoothed.

[0095] Select kernel function: The kernel function of the Gaussian process determines the similarity measure between data points. Common kernel functions include square exponential kernel, linear kernel, etc. Select the appropriate kernel function according to the characteristics of the data.

[0096] Parameter estimation: Use maximum likelihood estimation or Bayesian methods to estimate the parameters of the Gaussian process. These parameters include the parameters of the kernel function and the noise variance.

[0097] Data smoothing: Use the estimated Gaussian process model to smooth the data. For each data point, the smoothed value is obtained by calculating its conditional expectation under the Gaussian process.

[0098] 2. Data insertion point

[0099] principle:

[0100] In data interpolation, the predictive power of Gaussian processes is used to estimate the values ​​of missing data points. Since Gaussian processes are able to model the underlying function of the data, the values ​​of unknown data points can be predicted from known data points.

[0101] Specifically, for a given data set and a location where a missing data point is located, the Gaussian process model is used to predict the function value at that location. The prediction result is a probability distribution that reflects the range of possible values ​​for that location.

[0102] step:

[0103] Data preparation: Identify the dataset that needs to be interpolated and determine the location of the missing data points.

[0104] Build Gaussian process model: Use the same method as data smoothing to build a Gaussian process model.

[0105] Interpolation prediction: Use the established Gaussian process model to predict missing data points. The interpolation point value can be obtained by calculating the predicted mean or taking the median of the probability distribution.

[0106] 3. Data Extension

[0107] principle:

[0108] Data extension refers to the use of Gaussian process models to expand the range of data. By modeling known data, the values ​​of data points beyond the known data range can be predicted, thereby achieving data extension.

[0109] Specifically, assuming that the range of known data is limited, the data within this range is modeled through the Gaussian process model, and then the model is used to predict data points outside this range. The prediction results can provide an estimate of the unknown data, thereby achieving data extension.

[0110] step:

[0111] Data preparation: Determine the data set that needs to be extended, and determine the direction and scope of the extension.

[0112] Build Gaussian process model: Use the same method as data smoothing to build a Gaussian process model.

[0113] Data extension: Use the established Gaussian process model to predict the data points within the extension range. The prediction range can be gradually expanded as needed until the required extension degree is reached.

[0114] In summary, data smoothing, data interpolation, and data extension based on Gaussian processes is a powerful data analysis method that can effectively deal with problems such as noise, missing values, and range limitations in the data, providing a better data foundation for data analysis and modeling.

[0115] like Figure 2 As shown, data smoothing, data interpolation, and data extension all go through the process from Step 1 to Step 4. The difference is that in Step 4, the data for data smoothing prediction is the same as that in Step 3; the data interpolation prediction area needs to add the location to be inserted in Step 3; the data extension prediction area needs to extend a part based on the Step 3 data. Example:

[0116] In Step 3, x=

[2345] , y=[0.10.30.50.70.8]. If the data is smooth, then x'=x, and y' is predicted. If a point is inserted at x=2.5, then x'=[22.5345], and y' is predicted. If the data is extended, then x'=[23455.56], and y' is predicted.

[0117] (1) Normalization of x-axis and y-axis data

[0118] Normalize the data to the same data scale to eliminate the impact of dimension and improve the stability of calculation.

[0119] (2) Selecting the covariance function

[0120] Select a suitable covariance function to make the predicted data smoother, so that the model fits the data better and has stronger generalization ability.

[0121] (3) Select the data to be trained

[0122] The Gaussian regression model is trained using these data. The data that can be selected include curves (current-voltage relationship, capacitance-voltage relationship, etc.) and KOP (the relationship between various key indicators such as Idlin, Idsat, Vtlin, Vtsat, etc. and size), both of which can be optimized using this method.

[0123] (4) Select the area to be predicted

[0124] By predicting the data in these areas using a Gaussian regression model, we can eliminate point jumps while retaining the characteristics of the original data, making the curve smoother.

[0125] (4) Perform Gaussian process calculation to obtain the above y'

[0126] The following is an explanation through specific implementation:

[0127] First embodiment

[0128] like Figure 3 As shown, this embodiment provides a method for data smoothing, interpolation and extension based on Gaussian process, including the following steps:

[0129] S1: Normalize the data sets on the x-axis and y-axis of the two-dimensional curve that needs to be used for model parameter extraction.

[0130] In this embodiment, step S1 is specifically as follows:

[0131] Use the standard deviation normalized z-score method;

[0132] For the x-axis data, calculate the mean μ of the x-axis data x , that is, sum all x-axis data values ​​and divide them by the total number of data points to calculate the standard deviation σ of the x-axis data x , that is, first calculate the score of the difference between each data point and the mean, sum the total number of post-processing data points, and then take the square root to normalize each x-axis data point. The calculation formula is:

[0133]

[0134] For the y-axis data, calculate the mean μ of the y-axis data y , that is, sum all the y-axis data values ​​and divide them by the total number of data points to calculate the standard deviation σ of the y-axis data y, that is, first calculate the score of the difference between each data point and the mean, sum the total number of post-processing data points, and then take the square root to normalize each y-axis data point. The calculation formula is:

[0135]

[0136] Visualize the normalized x-axis and y-axis data to see whether the data distribution is concentrated within a reasonable range. If problems are found in the normalized data, including extreme values ​​or unreasonable distribution, adjust the normalization method or process the outliers separately and then normalize again.

[0137] S2: Select a covariance function, select data to be trained in the data set, and calculate the covariance between any two data points in the data to be trained using the covariance function to form a covariance matrix, and train a Gaussian regression model.

[0138] The covariance function is also called the kernel function. The kernel function is the core of a Gaussian process and determines the properties of a Gaussian process. The kernel function generates a covariance matrix (correlation coefficient matrix) in the Gaussian process to measure the "distance" between any two points. Different kernel functions have different measurement methods, and the properties of the resulting Gaussian process are also different.

[0139] In this embodiment, step S2 is specifically as follows:

[0140] The Gaussian kernel function is selected as the covariance function, and the basic form is as follows:

[0141]

[0142] Among them, K(x i , x j ) represents the data point x i and x j The covariance between them, σ is a hyperparameter, which is related to the variance of the Gaussian process and determines the amplitude of the function. l is a hyperparameter, which is related to the length of the input space and determines the smoothness of the function. When l is small, the function changes more dramatically, and when l is large, the function is smoother. exp represents the natural exponential function, ||x i -x j || represents the data point x i and x j The Euclidean distance between them. In addition, the hyperfunctions σ and l can be set directly, such as σ is set to 1 and l is set to 2. It is also possible to make predictions by maximizing the log-marginal likelihood function, which will be described below.

[0143] For the data {x1, x2, ..., x n}, the covariance between two data points is calculated by the covariance function to form an n×n covariance matrix K nn ;

[0144] Considering the influence of noise, assuming ∈ i The mean is 0 and the variance is The normal distribution of The final covariance matrix is where I is the identity matrix, is the noise variance;

[0145] σ is determined by maximizing the log-marginal likelihood function 2 , l 2 and The value of , the log marginal likelihood function is:

[0146]

[0147] Where p(y|X) is the probability density function, y=[y1,y1,...,y n ] T is the target value vector of the training data, X = [x1, x2, ..., x n ] T is the input vector of training data, is a multivariate normal distribution;

[0148] The constructed covariance matrix is ​​used as the Gaussian regression model to perform data prediction operations of the Gaussian process including data smoothing, data interpolation, and data extension.

[0149] S3: Select the data to be trained and use the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation, and data extension.

[0150] (1) Data smoothing

[0151] For any prediction point x * , calculate and train each point x i The covariance vector k of * =[K(x * ,x1),K(x * ,x2),…,K(x * ,x n )] T , where K(x * ,x i ) is the predicted point x * and training point x i The covariance between them calculated by the covariance function;

[0152] Calculate the predicted point x * Its own covariance k ** =K(x * ,x * );

[0153] Calculate the predicted point x * The mean value u * and variance

[0154] Among them, the mean u * for:

[0155]

[0156] is the covariance vector k * The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data;

[0157] variance for:

[0158]

[0159] k ** is the covariance of the prediction point itself, k * is the covariance vector between the prediction point and the training point, is the covariance matrix The inverse matrix of is the noise variance;

[0160] For the data line segment where the selected data to be trained is located, the data is discretized into multiple prediction points x * , for each prediction point x * Calculate the mean u * , with the mean value u * Connecting them together gives us the predicted curve, which is smoother than the original curve.

[0161] For example, Figure 4 As shown in the figure, the x-axis and y-axis data have been normalized, where the circled points are the actual measurement points, and the dotted line is the curve connecting the actual measurement points. The data before 0.42 do not need to be trained and predicted, so the circled points are not drawn.

[0162] The smoothing process is Figure 2 In the flowchart, the obtained y' is a smooth curve.

[0163] When smoothing the data, Step 3 in the process selects the line segment to be smoothed. In this figure, it is from x=0.42 to x=1. This line segment is used for training. When predicting in Step 4, the data line segment selected in the prediction area is the same as Step 3, and the prediction curve is obtained, which is the solid line curve. Vgs is the voltage, and GM is the derivative of the current with respect to the voltage.

[0164] (2) Data insertion point

[0165] Predict the insertion point position in the selected area. For any insertion point position x interp , calculate and train each point x i The covariance vector of

[0166] k interp =[K(x interp ,x1),K(x interp ,x2),…,K(x interp ,x n )] T , where K(x interp ,x i ) is the insertion point position x interp and training point x i The covariance between them calculated by the covariance function;

[0167] Calculate the insertion point position x interp Its own covariance k interp,interp =K(x interp ,x interp );

[0168] Calculate the insertion point position x interp The mean value u interp and variance

[0169] Among them, the mean u interp for:

[0170]

[0171] is the covariance vector k interp The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data;

[0172] variance for:

[0173]

[0174] k interp,interp is the insertion point position x interp Its own covariance, kinterp is the insertion point position x interp The covariance vector with the training points, is the covariance matrix The inverse matrix of is the noise variance;

[0175] With the insertion point position x interp The mean value u interp As the insertion point position x interp The values ​​are plotted and highlighted in the graph.

[0176] For example, Figure 5 As shown in the figure, Step 1, Step 2, and Step 3 are the same as data smoothing. The difference is that Step 4 includes not only the x value of Step 3 but also the location of the interpolation point when selecting the prediction area. The interpolation point location is required to be inside Dtep3, that is, x must be greater than 0.42 and less than 1.

[0177] Figure 5 The calculation includes both the data smoothing at the same position as Step 2 and the prediction of the interpolation points at x = 0.95, 0.85, and 0.75, as indicated by the black arrows in the figure. Figure 2 In the above equation, the obtained y' is the interpolation curve.

[0178] (3) Data extension

[0179] For the data points in the selected extension area, prediction is performed. For any extension position x extend , calculate and train each point x i The covariance vector of

[0180] k extend =[K(x extend ,x1),K(x extend ,x2),…,K(x extend ,x n )] T , where K(x extend ,x i ) is the extended position x extend and training point x i The covariance between them calculated by the covariance function;

[0181] Calculate the extended position x extend Its own covariance k extend,extend =K(x extend ,x extend );

[0182] Calculate the extended position x extend The mean value u extend and variance

[0183] Among them, the mean u extend for:

[0184]

[0185] is the covariance vector k extend The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data;

[0186] variance for:

[0187]

[0188] k extend,extend is the extended position x extend Its own covariance, k extend is the extended position x extend The covariance vector with the training points, is the covariance matrix The inverse matrix of is the noise variance;

[0189] To extend the position x extend The mean value u extend As the extended position x extend The value of the extended position x is calculated based on extend The predicted value is plotted as the curve after the data is extended.

[0190] For example, Figure 6 As shown in Figure 1, similar to data interpolation, Step 1, Step 2, and Step 3 are the same as data smoothing. The difference is that in Step 4, when selecting the prediction area, not only the x value of Step 3 is included, but also the prediction area where x is greater than 1. Figure 6 The predicted extension curve includes three more points x=1.02, x=1.04, and x=1.06.

[0191] Second embodiment

[0192] like Figure 7 As shown, this embodiment provides a system for performing the method for data smoothing, interpolation and extension based on Gaussian process as in the first embodiment, including:

[0193] Normalization processing module 1, used for normalizing the data sets on the x-axis and y-axis of the two-dimensional curve that needs to be modeled;

[0194] A covariance matrix construction module 2 is used to select a covariance function, select data to be trained in the data set, and calculate the covariance between two data points in the data to be trained by the covariance function to form a covariance matrix and train a Gaussian regression model;

[0195] The parameter data processing module 3 is used to select the data that needs to be trained and use the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation, and data extension.

[0196] A computer-readable storage medium stores a computer code. When the computer code is executed, the above method is executed. A person skilled in the art can understand that all or part of the steps in the various methods of the above embodiments can be completed by instructing related hardware through a program. The program can be stored in a computer-readable storage medium. The storage medium can include: a read-only memory (ROM), a random access memory (RAM), a disk or an optical disk, etc.

[0197] The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.

[0198] The technical features of the above-described embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above-described embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0199] It should be noted that the above embodiments can be freely combined as needed. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered as the protection scope of the present invention.

Claims

1. A method for data smoothing, interpolation and extension based on Gaussian processes, characterized in that: The following steps are involved: S1: Normalize the data sets on the x-axis and y-axis of the two-dimensional curve that needs to be used for model parameter extraction; S2: Select a covariance function, select data to be trained in the data set, and calculate the covariance between two data points in the data to be trained by using the covariance function to form a covariance matrix, and train a Gaussian regression model; S3: Select the data to be trained and use the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation, and data extension.

2. The method for data smoothing, interpolation and extension based on Gaussian process according to claim 1, characterized in that: In step S1, the x-axis and y-axis data of the two-dimensional curve for which the model parameter extraction is required are normalized, specifically: Use the standard deviation normalized z-score method; For the x-axis data, calculate the mean μ of the x-axis data x , that is, sum all x-axis data values ​​and divide them by the total number of data points to calculate the standard deviation σ of the x-axis data x , that is, first calculate the score of the difference between each data point and the mean, sum the total number of post-processing data points, and then take the square root to normalize each x-axis data point. The calculation formula is: For the y-axis data, calculate the mean μ of the y-axis data y , that is, sum all the y-axis data values ​​and divide them by the total number of data points to calculate the standard deviation σ of the y-axis data y , that is, first calculate the score of the difference between each data point and the mean, sum the total number of post-processing data points, and then take the square root to normalize each y-axis data point. The calculation formula is: Visualize the normalized x-axis and y-axis data to see whether the data distribution is concentrated within a reasonable range. If problems are found in the normalized data, including extreme values ​​or unreasonable distribution, adjust the normalization method or process the outliers separately and then normalize again.

3. The method for data smoothing, interpolation and extension based on Gaussian process according to claim 1, characterized in that: In step S2, the covariance function is selected, specifically: The Gaussian kernel function is selected as the covariance function, and the basic form is as follows: Among them, K(x i , x j ) represents the data point x i and x j The covariance between them, σ is a hyperparameter related to the variance of the Gaussian process and determines the amplitude of the function. l is a hyperparameter related to the length of the input space and determines the smoothness of the function. When l is small, the function changes more dramatically, and when l is large, the function is smoother. exp represents the natural exponential function, ‖x i -x j ‖ represents the data point x i and x j The Euclidean distance between them.

4. The method for data smoothing, interpolation and extension based on Gaussian process according to claim 3, characterized in that: In step S2, data to be trained are selected from the data set, and the covariance between two data points in the data to be trained is calculated by the covariance function to form a covariance matrix, and a Gaussian regression model is trained, specifically: For the data {x1, x2, ..., x n }, the covariance between two data points is calculated by the covariance function to form an n×n covariance matrix K nn ; Considering the influence of noise, assuming ∈ i The mean is 0 and the variance is The normal distribution of The final covariance matrix is where I is the identity matrix, is the noise variance; σ is determined by maximizing the log-marginal likelihood function 2 , l 2 and The value of , the log marginal likelihood function is: Among them, p(y|X) is the probability density function, y=[y1,y1,…,y n ] T is the target value vector of the training data, X = [x1, x2, ..., x n ] T is the input vector of training data, is a multivariate normal distribution; The constructed covariance matrix is ​​used as the Gaussian regression model to perform data prediction operations of the Gaussian process including data smoothing, data interpolation, and data extension.

5. The method for data smoothing, interpolation and extension based on Gaussian process according to claim 4, characterized in that: In step S3, the data to be trained is selected and the trained Gaussian regression model is used to smooth the data, specifically: For any prediction point x * , calculate and train each point x i The covariance vector k of * =[K(x * ,x1),K(x * ,x2),...,K(x * ,x n )] T , where K(x * ,x i ) is the predicted point x * and training point x i The covariance between them calculated by the covariance function; Calculate the predicted point x * Its own covariance k ** =K(x * ,x * ); Calculate the predicted point x * The mean value u * and variance Among them, the mean u * for: is the covariance vector k * The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data; variance for: k ** is the covariance of the prediction point itself, k * is the covariance vector between the prediction point and the training point, is the covariance matrix The inverse matrix of is the noise variance; For the data line segment where the selected data to be trained is located, the data is discretized into multiple prediction points x * , for each prediction point x * Calculate the mean u * , with the mean value u * Connecting them together gives us the predicted curve, which is smoother than the original curve.

6. The method for data smoothing, interpolation and extension based on Gaussian process according to claim 4, characterized in that: In step S3, the data to be trained is selected and the data interpolation is performed using the trained Gaussian regression model, specifically: Predict the insertion point position in the selected area. For any insertion point position x interp , calculate and train each point x i The covariance vector of k interp =[K(x interp ,x1),K(x interp ,x2),...,K(x interp ,x n )] T , where K(x interp ,x i ) is the insertion point position x interp and training point x i The covariance between them calculated by the covariance function; Calculate the insertion point position x interp Its own covariance k interp,interp =K(x interp ,x interp ); Calculate the insertion point position x interp The mean value u interp and variance Among them, the mean u interp for: is the covariance vector k interp The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data; variance for: k interp,interp is the insertion point position x interp Its own covariance, k interp is the insertion point position x interp The covariance vector with the training points, is the covariance matrix The inverse matrix, σ n 2 is the noise variance; With the insertion point position x interp The mean value u interp As the insertion point position x interp The values ​​are plotted and highlighted in the graph.

7. The method for data smoothing, interpolation and extension based on Gaussian process according to claim 4, characterized in that: In step S3, the data to be trained is selected and the data is extended using the trained Gaussian regression model, specifically: For the data points in the selected extension area, prediction is performed. For any extension position x extend , calculate and train each point x i The covariance vector of k extend =[K(x extend ,x1),K(x extend ,x2),…,K(x extend ,x n )] T , where K(x extend ,x i ) is the extended position x extend and training point x i The covariance between them calculated by the covariance function; Calculate the extended position x extend Its own covariance k extend,extend =K(x extend ,x extend ); Calculate the extended position x extend The mean value u extend and variance Among them, the mean u extend for: is the covariance vector k extend The transpose of is the covariance matrix The inverse matrix of , y is the target value vector of the training data; variance for: k extend,extend is the extended position x extend Its own covariance, k extend is the extended position x extend The covariance vector with the training points, is the covariance matrix The inverse matrix of is the noise variance; To extend the position x extend The mean value u extend As the extended position x extend The value of the extended position x is calculated based on extend The predicted value is plotted as the curve after the data is extended.

8. A system for performing the method for data smoothing, interpolation and extension based on Gaussian process according to any one of claims 1 to 7, characterized in that: include: A normalization processing module is used to normalize the data sets on the x-axis and y-axis of the two-dimensional curve that needs to be used for model parameter extraction; A covariance matrix construction module is used to select a covariance function, select data to be trained in the data set, and calculate the covariance between two data points in the data to be trained by the covariance function to form a covariance matrix and train a Gaussian regression model; The parameter data processing module is used to select the data that needs to be trained and use the trained Gaussian regression model to perform data prediction operations including data smoothing, data interpolation, and data extension.

9. A computer device comprising a memory and one or more processors, wherein the memory stores computer codes, and when the computer codes are executed by the one or more processors, the one or more processors execute the method according to any one of claims 1 to 7. 10 . A computer-readable storage medium storing a computer code. When the computer code is executed, the method according to claim 1 is executed.

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