Orthogonal mode empirical likelihood inference method of longitudinal part nonlinear model

Abstract

The invention relates to the technical field of statistics and data analysis, and discloses an orthogonal mode empirical likelihood inference method for a longitudinal part nonlinear model, which comprises the following steps of: constructing a longitudinal part nonlinear model which comprises a linear parameter component and an unknown non-parameter component; approaching a non-parametric component by using a B-spline method, and converting a non-parametric function into a finite dimension parameter problem; carrying out QR decomposition on the model covariable matrix, and realizing the decomposition of the model through orthogonal projection; defining an auxiliary random vector based on modal regression, and constructing an empirical log-likelihood ratio; estimation values of linear parameters and non-parameter components are obtained by optimizing the empirical log-likelihood ratio. Through combination of modal regression, an orthogonal projection technology and an empirical likelihood method, robustness, high efficiency and applicability are significantly improved. By means of modal regression, dependence of a traditional method on normal distribution is overcome, and the method is excellent in performance under error distribution of skewness, thick tail and the like.

Description

Technical Field

[0001] The invention relates to the technical field of statistics and data analysis, in particular to an orthogonal modal empirical likelihood inference method for a longitudinal partial nonlinear model. Background Art

[0002] Longitudinal data are widely used in fields such as biology, economics, and environmental science to study the dynamic characteristics of the same object at multiple time points. In order to analyze such data, some nonlinear models have become commonly used modeling tools. The model flexibly describes the complex relationship between variables by combining linear and nonparametric components. In actual inference, traditional methods such as least squares regression (OLS) and maximum likelihood estimation (MLE) rely on the normality assumption of error distribution and are suitable for situations with simple data distribution and large sample size. However, these methods face significant limitations in complex data environments and are difficult to ensure robustness and estimation efficiency.

[0003] The existing technologies have the following main problems when processing some nonlinear models of longitudinal data: First, they rely heavily on the normality of the error distribution. When faced with skewed, fat-tailed or outlier error distributions, the estimation results often lose their validity; second, they do not fully consider the intra-group correlation of longitudinal data, resulting in low inference accuracy of linear parameters and non-parametric components; third, some methods lack perfect theoretical support, making it difficult to prove the consistency and asymptotic normality of the estimates, which limits their application in complex practical scenarios such as biomedicine and economics. Therefore, there is an urgent need for an inference method that is robust, has a perfect theoretical basis, and is adapted to the characteristics of longitudinal data to make up for these shortcomings. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides an orthogonal modal empirical likelihood inference method for a longitudinal partially nonlinear model, which solves the problem in the prior art that it is difficult to effectively handle complex error distribution and intra-group correlation in longitudinal data.

[0005] To achieve the above purpose, the present invention is implemented by the following technical scheme: an orthogonal modal empirical likelihood inference method for a longitudinal partial nonlinear model comprises the following steps:

[0006] A longitudinal partial nonlinear model is constructed, wherein the model includes unknown linear parameter components and unknown nonparametric components;

[0007] The nonparametric components are approximated by the B-spline method, and the nonparametric function is transformed into a finite-dimensional parametric problem;

[0008] Perform QR decomposition on the model covariate matrix and achieve model decomposition through orthogonal projection;

[0009] Define auxiliary random vectors based on modal regression and construct empirical log-likelihood ratios;

[0010] By optimizing the empirical log-likelihood ratio, estimates of linear parameters and nonparametric components are obtained;

[0011] Based on the asymptotic properties of the log-likelihood ratio, confidence intervals for the linear parametric and nonparametric components are constructed.

[0012] Preferably, the longitudinal partial nonlinear model is in the form of:

[0013] Y ij =f(X ij ,β)+g(Z ij )+ε ij ,i=1,…,n;j=1,…,m i .

[0014] Y ij is the response variable of the i-th individual at the j-th observation;

[0015] X ij is the p-dimensional covariate vector;

[0016] β is the unknown parameter vector;

[0017] g(Z ij ) is an unknown non-parametric function;

[0018] ε ij is the error term.

[0019] Preferably, the non-parametric component g(Z ij ) is expressed by B-spline basis function approximation:

[0020]

[0021] B k (Z ij ) is the B-spline basis function;

[0022] γ k is the basis function coefficient;

[0023] K is the number of basis functions.

[0024] Preferably, the covariate matrix W i Through QR decomposition, we get:

[0025]

[0026] in:

[0027]

[0028] W ij =(B1(Z ij ),…,B K(Z q )) T ;

[0029] Q i =(Q i1 ,Q i2 ) is an orthogonal matrix;

[0030] R i is an upper triangular matrix.

[0031] Preferably, the model is transformed by orthographic projection into:

[0032]

[0033] in:

[0034]

[0035] Preferably, the modal regression auxiliary random vector is defined as:

[0036]

[0037] in:

[0038] is the Gaussian kernel function;

[0039] h is the bandwidth parameter.

[0040] Preferably, the empirical log-likelihood ratio of the parameter component β is defined as:

[0041] constraint:

[0042] Preferably, the Lagrangian multiplication λ is solved by optimizing the Lagrangian function:

[0043]

[0044] Where λ is the Lagrange multiplier.

[0045] Preferably, the empirical log-likelihood ratio of the non-parametric component is defined as:

[0046] constraint:

[0047] Preferably,

[0048]

[0049] The consistency of the nonparametric estimation is verified based on the asymptotic properties of the log-likelihood ratio.

[0050] The present invention provides an orthogonal modal empirical likelihood inference method for a longitudinal partial nonlinear model, which has the following beneficial effects:

[0051] 1. The present invention adopts modal regression technology and maximizes the modal position of the conditional density, thus avoiding the reliance of the traditional least squares method on the assumption of normality of the error distribution, especially when the error distribution is skewed, fat-tailed, or contains outliers, it can still maintain a high estimation efficiency. At the same time, the kernel function and bandwidth selection mechanism are used to achieve adaptive adjustment of different distribution characteristics to ensure the robustness of the method under various error environments. Comparative experiments show that whether it is normal, skewed or fat-tailed distribution, the mean square error of the parameter estimation of the present invention is significantly lower than that of the traditional method, and the confidence interval coverage is closer to the theoretical value.

[0052] 2. The present invention effectively decomposes the correlation between linear parameters and non-parametric components in longitudinal data through orthogonal projection technology, and solves the impact of intra-group correlation in longitudinal data on model estimation. The model after orthogonal decomposition reduces the dimensional complexity of parameter estimation, making inference more efficient. At the same time, the non-parametric components are accurately modeled using B-spline approximation technology, and the asymptotic chi-square distribution characteristics of the empirical likelihood method are combined to construct a more reliable confidence interval. Experimental data verify that the present invention can simultaneously achieve efficient and accurate inference of linear parameters and non-parametric components under the intra-group correlation environment of longitudinal data, which is significantly better than existing methods.

[0053] 3. The present invention ensures the consistency and asymptotic normality of the estimation through the strict design of the assumptions; combined with tools such as Taylor expansion, central limit theorem and Slutsky theorem, the asymptotic properties of parameter estimation and non-parametric fitting are theoretically proved. In addition, the present method has wide applicability in longitudinal data analysis in different fields, such as patient indicator modeling in biomedicine, long-term trend analysis in economics, and climate data research in environmental science. Regardless of the sample size and error distribution characteristics, the present invention can provide robust and reliable inference results. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a stereogram of the present invention. DETAILED DESCRIPTION

[0055] The following will be combined with the drawings in the specification of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0056] Please see attached Figure 1The embodiment of the present invention provides an orthogonal modal empirical likelihood inference method for a longitudinal partial nonlinear model. The present invention decomposes the model through orthogonal projection technology, uses modal regression to enhance the robustness to the non-normality of error distribution, and combines empirical likelihood inference to construct estimates and confidence intervals with a clear theoretical basis, which is applicable to a variety of complex data scenarios.

[0057] 1. Model construction

[0058] The longitudinal part nonlinear model is as follows:

[0059] Y ij =f(X ij ,β)+g(Z ij )+ε ij ,i=1,…,n;j=1,…,m i .

[0060] in:

[0061] Y ij is the response variable of the i-th individual at the j-th observation;

[0062] X ij is the p-dimensional covariate vector;

[0063] β is the unknown parameter vector;

[0064] g(Z ij ): unknown non-parametric smoothing function;

[0065] ∈ ij : Random error, which may be skewed or fat-tailed distribution;

[0066] Z ij : One-dimensional covariate.

[0067] Assumptions: The data within the groups are correlated but the data between the groups are independent;

[0068] The nonparametric function g(·) is continuous and smooth on the interval [a,b];

[0069] Covariate X ij and Z ij Independent of each other.

[0070] 2. B-spline approximation of nonparametric components

[0071] 2.1B-spline approximation principle

[0072] Nonparametric function g(Z ij ) is approximated by B-spline basis functions on the interval [a,b]:

[0073]

[0074] in:

[0075] B k (Z ij ) is the K-order B-spline basis function;

[0076] γ k are the basis function coefficients;

[0077] K is the number of basis functions, which is determined by the number of internal knots and the spline order.

[0078] 2.2 Construction of spline basis functions

[0079] Select a node t in the interval [a,b] k :

[0080] Define the 0th order basis function:

[0081]

[0082] Recursive definition of k-order spline basis function:

[0083]

[0084] 2.3 Nonparametric Partial Matrix Expression

[0085] Construct a spline basis matrix B(Z) whose jth row is [B1(Z ij ),…,B K (Z ij )]. The non-parametric part is expressed as:

[0086] g(Z ij )≈B(Z ij )γ,

[0087] Convert the model to:

[0088] Y ij =f(X ij ),β)+B(Z ij )γ+ε ij .

[0089] 3. Model decomposition and orthogonal projection

[0090] 3.1 Matrix Representation

[0091] The model is expressed in matrix form as:

[0092] Y=f(X,β)+Bγ+ε,

[0093] in:

[0094] Y is the nm×1 response variable vector;

[0095] X is the nm×p covariate matrix;

[0096] B is the nm×K spline basis matrix.

[0097] 3.2 QR decomposition covariate matrix

[0098] To W i Perform QR decomposition:

[0099]

[0100] in:

[0101] W i =(W i1 ,…,W imi ) T ;

[0102]

[0103] Q i is an orthogonal matrix;

[0104] R i is an upper triangular matrix.

[0105] 3.3 Projected Model

[0106] Left multiplication Get the model after orthogonal projection:

[0107]

[0108] in:

[0109]

[0110]

[0111] At this point, the model is divided into a parametric part and a non-parametric part.

[0112] 4. Construction of random vectors assisted by modal regression

[0113] Construct auxiliary random vectors by modal regression:

[0114]

[0115] in:

[0116] Kernel Function Using Gaussian kernel function

[0117] The introduction of auxiliary random vectors enhances the robustness to the deviation of error distribution from normality.

[0118] 5. Empirical Log-Likelihood Inference of Parametric and Nonparametric Components

[0119] 5.1 Parameter Components

[0120] Construct the empirical log-likelihood ratio:

[0121] constraint:

[0122] Use the Lagrangian method to solve the sample weights:

[0123]

[0124] Maximize the log-likelihood ratio, obtain parameter estimates, and construct confidence intervals.

[0125] 5.2 Nonparametric Components

[0126] Construct a nonparametric auxiliary random vector:

[0127]

[0128] Define the empirical log-likelihood ratio for the nonparametric components:

[0129] constraint:

[0130] Maximize the likelihood ratio and estimate g(Z ij ), and verify its consistency.

[0131] 6. Numerical simulation and practical application

[0132] The performance of the method was verified by numerical simulation, and different sample sizes (n=200, 400) and error distributions (normal, skewed, and fat-tailed distributions) were set to evaluate parameter estimation bias, nonparametric component fitting effect, and confidence interval coverage. The experimental results show that the present invention is robust and efficient under complex data conditions.

[0133] Practical applications can be extended to longitudinal data analysis in biomedicine, economics, and environmental sciences.

[0134] Example 1: Longitudinal partial nonlinear model inference based on normal error distribution

[0135] Steps and process parameters:

[0136] Model settings:

[0137] Model form:

[0138] Y ij =f(X ij ,β)+g(Z ij )+εij ,ε ij ~N(0,σ 2 ),

[0139] Where: f(X ij ,β)=exp{X 1,ij β1+X 2,ij β2}.

[0140] X 1,ij and X 2,ij is a covariate generated from a standard normal distribution N(0,1);

[0141] Z ij Generated from uniform distribution U(0,1);

[0142] g(Z ij )=sin(2πZ ij ).

[0143] Nonparametric component-wise approximation:

[0144] B-spline order: 3rd order;

[0145] Number of spline nodes: 6.

[0146] Bandwidth and kernel function selection:

[0147] Kernel function: Gaussian kernel

[0148] Bandwidth: h = 0.05.

[0149] Estimation method:

[0150] Orthogonal modal empirical likelihood estimation is performed for the linear parameter β and the non-parametric component g(Z) respectively;

[0151] Number of simulation repetitions: 500 times.

[0152] Comparative Example 1: Ordinary Least Squares Method Based on Normal Error Distribution

[0153] Steps and process parameters:

[0154] The model setting and generation method are the same as in Example 1.

[0155] Estimation method:

[0156] The ordinary least squares (OLS) method was used to estimate the parameters of the model;

[0157] The nonparametric part uses polynomial fitting instead of spline approximation.

[0158] Example 2: Longitudinal partial nonlinear model inference based on skewed error distribution

[0159] Steps and process parameters:

[0160] Model settings:

[0161] Model form:

[0162] Y ij =f(X ij ,β)+g(Z ij )+ε ij ,∈ ij ~Skewed distribution (γ=2)

[0163] in:

[0164] The skewed distribution was generated by using the standard normal distribution with one-sided clipping;

[0165] The other settings are the same as those in Example 1.

[0166] Nonparametric component-wise approximation:

[0167] B-spline order: 4;

[0168] Number of spline nodes: 8.

[0169] Bandwidth and kernel function selection:

[0170] Kernel function: triangular kernel

[0171] Bandwidth h = 0.1.

[0172] Estimation method:

[0173] Modal empirical likelihood estimation, with 500 simulations repeated.

[0174] Comparative Example 2: Maximum Likelihood Estimation Based on Skewed Error Distribution

[0175] Steps and process parameters:

[0176] The model setting and generation method are the same as in Example 2.

[0177] Estimation method:

[0178] The parametric and nonparametric parts were estimated using maximum likelihood estimation (MLE).

[0179] Example 3: Longitudinal Partial Nonlinear Model Inference Based on Thick-Tailed Error Distribution

[0180] Steps and process parameters:

[0181] Model settings:

[0182] Model form:

[0183] Y ij=f(X ij ,β)+g(Z ij )+ε ij ,∈ ij ~t(3)

[0184] in:

[0185] t(3) is the t distribution with 3 degrees of freedom;

[0186] The other settings are the same as those in Example 1.

[0187] Nonparametric component-wise approximation:

[0188] B-spline order: 5;

[0189] Number of spline nodes: 10.

[0190] Bandwidth and kernel function selection:

[0191] Kernel function: Epanechnikov kernel

[0192] Bandwidth: h = 0.15.

[0193] Estimation method:

[0194] Modal empirical likelihood estimation, the simulation is repeated 1000 times.

[0195] Comparative Example 3: Kernel Density Estimation Based on Thick-Tailed Error Distribution

[0196] Steps and process parameters:

[0197] The model setting and generation method are the same as in Example 3.

[0198] Estimation method:

[0199] The fit was adjusted for model error using kernel density estimation.

[0200] Comparative Experimental Design and Results

[0201] Experimental design

[0202] Experimental objective: To verify the superiority of the proposed method for parameter estimation and non-parametric component approximation under different error distributions.

[0203] Experimental indicators:

[0204] Mean square error (MSE) of linear parameter estimates;

[0205] Mean square error of the nonparametric component fit;

[0206] Confidence interval coverage (the proportion of true value covered). Experimental settings:

[0207] Simulation repetition times: 500 times;

[0208] Sample size: n = 200,400;

[0209] Comparative method: the present invention and the comparative example method.

[0210] Experimental results:

[0211]

[0212]

[0213] Through examples and comparative experiments, it is verified that the orthogonal modal empirical likelihood inference method of the present invention is robust, efficient and has a high confidence interval coverage under different error distribution conditions, which is significantly better than the existing methods.

[0214] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An orthogonal modal empirical likelihood inference method for longitudinal partial nonlinear models, characterized in that: The following steps are involved: A longitudinal partial nonlinear model is constructed, wherein the model includes unknown parametric components and unknown nonparametric components; The nonparametric components are approximated by the B-spline method, and the nonparametric function is transformed into a finite-dimensional parametric problem; Perform QR decomposition on the model covariate matrix and achieve model decomposition through orthogonal projection; Define auxiliary random vectors based on modal regression and construct empirical log-likelihood ratios; By optimizing the empirical log-likelihood ratio, the estimated values ​​of the parametric and non-parametric components are obtained; Based on the asymptotic properties of the log-likelihood ratio, confidence intervals for the parametric and nonparametric components are constructed.

2. The orthogonal modal empirical likelihood inference method for the longitudinal partial nonlinear model according to claim 1, characterized in that: The longitudinal part nonlinear model is in the form of: Y ij =f(X ij ,β)+g(Z ij )+ε ij ,i=1,…,n;j=1,…,m i . in: Y ij is the response variable of the i-th individual at the j-th observation; X ij is the p-dimensional covariate vector; β is the unknown parameter vector; g(Z ij ) is an unknown non-parametric function; ε ij is the error term.

3. The orthogonal modal empirical likelihood inference method for the longitudinal partial nonlinear model according to claim 1, characterized in that: The nonparametric component g(Z ij ) is expressed by B-spline basis function approximation: B k (Z ij ) is the B-spline basis function; γ k is the basis function coefficient; K is the number of basis functions.

4. The orthogonal modal empirical likelihood inference method for the longitudinal partial nonlinear model according to claim 1, characterized in that: The covariate matrix W i Through QR decomposition, we get: in: IN ij =(B1(Z ij ),…,B K (WITH ij )) T ; Q i =(Q i1 ,Q i2 ) is an orthogonal matrix; R i is an upper triangular matrix.

5. The orthogonal modal empirical likelihood inference method for the longitudinal partial nonlinear model according to claim 1, characterized in that: Transform the model via orthographic projection: in:

6. The orthogonal modal empirical likelihood inference method for longitudinal partial nonlinear model according to claim 1, characterized in that: The modal regression auxiliary random vector is defined as: in: is the Gaussian kernel function; h is the bandwidth parameter.

7. The orthogonal modal empirical likelihood inference method for a longitudinal partial nonlinear model according to claim 1, characterized in that: The empirical log-likelihood ratio of the parameter component is defined as:

8. The orthogonal modal empirical likelihood inference method for the longitudinal partial nonlinear model according to claim 7, characterized in that: Solve the Lagrangian multiplication λ by optimizing the Lagrangian function: Where λ is the Lagrange multiplier.

9. The orthogonal modal empirical likelihood inference method for a longitudinal partial nonlinear model according to claim 1, characterized in that: The empirical log-likelihood ratio of the nonparametric component is defined as:

10. The orthogonal modal empirical likelihood inference method for longitudinal partial nonlinear model according to claim 9, characterized in that: The consistency of the nonparametric estimation is verified based on the asymptotic properties of the log-likelihood ratio.

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