CN113571200A - Prediction method of infectious disease spread based on graph filter-vector autoregressive model - Google Patents

Prediction method of infectious disease spread based on graph filter-vector autoregressive model Download PDF

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CN113571200A
CN113571200A CN202110868810.9A CN202110868810A CN113571200A CN 113571200 A CN113571200 A CN 113571200A CN 202110868810 A CN202110868810 A CN 202110868810A CN 113571200 A CN113571200 A CN 113571200A
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蒋俊正
李文娟
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Abstract

本发明公开了基于图滤波器‑向量自回归模型的传染病传播预测方法，其特征在于，包括如下步骤：1）构建图模型；2）预测模型设计；3）图滤波器参数优化设计；4）利用优化后的VAR模型对时变图信号进行预测。这种方法实用性好，能提高传染病预测的准确率，进而为预防提高决策依据。

Figure 202110868810

The invention discloses a method for predicting the spread of infectious diseases based on a graph filter-vector autoregressive model, which is characterized by comprising the following steps: 1) constructing a graph model; 2) designing a prediction model; 3) optimizing the design of graph filter parameters; 4) ) using the optimized VAR model to predict the time-varying graph signal. This method has good practicability and can improve the accuracy of infectious disease prediction, thereby improving the decision-making basis for prevention.

Figure 202110868810

Description

Infectious disease propagation prediction method based on graph filter-vector autoregressive model
Technical Field
The invention relates to the field of infectious disease prediction, in particular to an infectious disease propagation prediction method based on a graph filter-vector autoregressive model.
Background
Infectious diseases can be rapidly spread in a large range, so that great influence and harm are brought to human beings, and the infectious diseases further become one of the main factors endangering human health. Infectious atypical pneumonia in 2003, influenza a H1N1 in 2009, and novel coronavirus pneumonia outbreaks in 2019 are all examples of very typical infectious disease transmission. The spreading process is predicted according to the monitored infectious disease data and early warning is carried out in time, so that countermeasures are taken, and the harm of infectious diseases is greatly reduced. Infectious diseases are global public health problems threatening human society, so that the research related to prediction of transmission of infectious diseases is of great significance to prevention and control of infectious diseases.
At present, the models and methods for predicting infectious diseases are widely applied mainly to a propagation dynamics model, a markov model, a time series method, a multiple regression model, and the like. The classical susceptibility-infection (SI) model proposed by kermanck and Mckendrick can reasonably predict the transmission of infectious diseases from person to person; gomez et al propose a discrete markov chain method for contagious disease transmission based on contact, solving the problem of the number of people in contact during the transmission process that depends on one node; siregar et al have studied the onset of hemorrhagic fever of dengue fever under the influence of climate by adopting a time series regression method, provide the basis for prevention and control of dengue fever; vector Autoregressive (VAR) models, which are a type of multivariate regression model, are also often used in time series prediction problems. The infectious disease transmission process is mainly affected by human-to-human interaction in time and space, and the crowd flow between regions is a very critical influencing factor. However, in the conventional prediction method, much attention is paid to the time correlation between data as a time series, and the intrinsic topology of the data, that is, the correlation between regions is ignored. Therefore, how to better depict the relationship between data is also an important issue, and if infectious disease data is modeled as a graph signal and a region is modeled as a node on the graph, the time correlation between time series data and the correlation between regions can be well described.
Nowadays, Graph Signal Processing (GSP) is receiving attention from researchers as an emerging field. GSP models data on irregular discrete domains as graphs, and uses edges on the graph models to depict the relationship between data. In many practical applications, for example, data of a wireless sensor network may change with time, have a time-varying characteristic, and can be modeled as a time-varying graph signal. Tay et al denoise the time-varying image signal using an efficient image filter; isufi et al predict the time-varying graph signals by using an approximate time-vertex stationarity hypothesis and an extended VAR model, and solve the dimension problem existing in multivariate time sequence evolution prediction, but the method only considers the correlation between the prediction time and the previous time signals in the parameter estimation problem, and does not consider the correlation between each time before the prediction time and the previous time signals.
Disclosure of Invention
The invention aims to provide an infectious disease propagation prediction method based on a graph filter-vector autoregressive model, aiming at the defects of the prior art. The method has good practicability, can improve the accuracy of infectious disease prediction, and further improves decision basis for prevention.
The technical scheme for realizing the purpose of the invention is as follows:
the infectious disease propagation prediction method based on the graph filter-vector autoregressive model comprises the following steps:
1) constructing a graph model: infectious disease transmission time series data
Figure BDA0003188306260000021
Each row in the system represents one area, N areas are in total, each column represents the number of cases at a certain moment in all areas, and T is in totalpThe data of each moment takes the region as the nodes on the graph, and the nodes are connected by edges according to the characteristics of different data to construct the topological structure of the graph and transmit the infectious disease data
Figure BDA0003188306260000022
Modelling as a time-varying graph signal
Figure BDA0003188306260000023
Namely, the data at each moment is a graph signal;
2) designing a prediction model: when a Graph Filter-Vector Autoregressive (GF-VAR) model predicts infection propagation, a time-varying Graph signal X is predicted by the VAR model, then coefficients of the VAR model are designed by the aid of concepts of a Graph Laplace matrix and a Graph Filter, and the signal value at the time t can be representedShown as front TpFunction of the signal at 1 instant, TpAnd 2, modeling the time-varying graph signal by adopting a VAR model, wherein the expression is shown as formula (1):
Figure BDA0003188306260000024
wherein epsilontThe error vector is a random vector with mean value of 0 and positive definite covariance matrix, and the graph filter
Figure BDA0003188306260000025
As a coefficient matrix, aq,pIs the coefficient of the Laplace matrix of order p, and Q is the highest order of the Laplace matrix;
3) and (3) optimally designing parameters of the graph filter: analyzing the prediction model according to step 2), considering the correlation between each moment of the time-varying graph signal X and the previous moment, and using the parameter a of the GF-VAR model graph filterq,pIs expressed as an optimization problem as shown in equation (2):
Figure BDA0003188306260000031
equation (2) is summarized as an unconstrained optimization problem as shown in equation (3):
Figure BDA0003188306260000032
the inputs in the problem shown in equation (3) are the signal x and the Laplace matrix LGCan be solved to Q × (T)p-1) parameters aq,p；
4) Predicting a time-varying graph signal by using the optimized VAR model, and quantizing by using a normalized minimum mean square error root (rNMSE) formula to estimate the performance of the model, wherein the error formula is shown as a formula (4):
Figure BDA0003188306260000033
wherein xtThe true number of cases for the node at time t,
Figure BDA0003188306260000034
is the predicted number of cases.
The unconstrained optimization problem described in step 3) as shown in equation (3) can be transformed into a vector of coefficients for a graph filter
Figure BDA0003188306260000035
The least squares problem of (2) is shown in equation (5):
Figure BDA0003188306260000036
wherein b is,
Figure BDA0003188306260000037
C is defined as shown in formula (6):
Figure BDA0003188306260000038
the parameters involved in equation (6) are shown in equation (7):
Figure BDA0003188306260000041
the theoretical solution to the least squares problem in equation (5) is as shown in equation (8):
Figure BDA0003188306260000042
thereby obtaining Qx (T)p-1) map filter parameters aq,pAnd finally obtaining the graph filter.
The unconstrained optimization problem shown in the formula (3) in the step 3) is solved by adopting a convex optimization toolkit cvx, and then the convex optimization toolkit cvx is solved to obtain the graph filter.
Compared with the existing infectious disease prediction model and method, the technical scheme is that the graph filter is used as a coefficient matrix of the VAR model, the graph topology is embedded into the model, and the optimization problem of the parameters of the graph filter is proposed based on the correlation between each moment before the prediction moment and the previous moment signals.
Drawings
FIG. 1 is a graph comparing 6-step prediction errors of susceptible-infected SI data in examples;
FIG. 2 is a comparison of 6-step prediction error for susceptibility-infection but not disease-infection-recovery-susceptibility SEIRS data in the examples;
FIG. 3 is a graph comparing 10-step prediction errors of the data of the German novel coronavirus pneumonia COVID-19 in the example.
Detailed Description
The invention will be further elucidated with reference to the drawings and examples, without however being limited thereto.
Example (b):
the infectious disease propagation prediction method based on the graph filter-vector autoregressive model comprises the following steps:
1) constructing a graph model: infectious disease transmission time series data
Figure BDA0003188306260000043
Each row in the system represents one area, N areas are in total, each column represents the number of cases at a certain moment in all areas, and T is in totalpThe data of each moment takes the region as the nodes on the graph, and the nodes are connected by edges according to the characteristics of different data to construct the topological structure of the graph and transmit the infectious disease data
Figure BDA0003188306260000051
Modelling as a time-varying graph signal
Figure BDA0003188306260000052
Namely, the data at each moment is a graph signal;
2) designing a prediction model: when the GF-VAR model predicts the spread of infectious diseases, the VAR model is firstly adopted to predict a time-varying graph signal X, then the concept of a graph Laplace matrix and a graph filter is utilized to design the coefficient of the VAR model, and the signal value at the moment T can be represented as the previous TpFunction of the signal at 1 instant, TpAnd 2, modeling the time-varying graph signal by adopting a VAR model, wherein the expression is shown as formula (1):
Figure BDA0003188306260000053
wherein epsilontThe error vector is a random vector with mean value of 0 and positive definite covariance matrix, and the graph filter
Figure BDA0003188306260000054
As a coefficient matrix, the time correlation between time series data and the correlation between regions can be well described, graph topology is embedded into a model, aq,pThe coefficient of a p-order Laplace matrix is adopted, Q is the highest order of the Laplace matrix, the formula (1) is used as a one-step prediction result expression, and n-step prediction results can be obtained by performing iterative computation on the one-step prediction result expression for n times;
3) and (3) optimally designing parameters of the graph filter: analyzing the prediction model according to step 2), considering the correlation between each moment of the time-varying graph signal X and the previous moment, and using the parameter a of the GF-VAR model graph filterq,pIs expressed as an optimization problem as shown in equation (2):
Figure BDA0003188306260000055
equation (2) is summarized as an unconstrained optimization problem as shown in equation (3):
Figure BDA0003188306260000056
the inputs in the problem shown in equation (3) are the signal x and the Laplace matrix LGCan be solved to Q × (T)p-1) parameters aq,p；
4) Predicting a time-varying graph signal by using the optimized VAR model, and quantizing by using a normalized minimum mean square error root (rNMSE) formula to estimate the performance of the model, wherein the error formula is shown as a formula (4):
Figure BDA0003188306260000061
wherein xtThe true number of cases for the node at time t,
Figure BDA0003188306260000062
is the predicted number of cases.
The unconstrained optimization problem described in step 3) as shown in equation (3) can be transformed into a vector of coefficients for a graph filter
Figure BDA0003188306260000063
The least squares problem of (2) is shown in equation (5):
Figure BDA0003188306260000064
wherein b is,
Figure BDA0003188306260000065
C is defined as shown in formula (6):
Figure BDA0003188306260000066
the parameters involved in equation (6) are shown in equation (7):
Figure BDA0003188306260000067
the theoretical solution to the least squares problem in equation (5) is as shown in equation (8):
Figure BDA0003188306260000068
thereby obtaining Qx (T)p-1) map filter parameters aq,pAnd finally obtaining the graph filter.
The unconstrained optimization problem shown in the formula (3) in the step 3) is solved by adopting a convex optimization toolkit cvx, and then the convex optimization toolkit cvx is solved to obtain the graph filter.
Simulation experiment: simulations the performance of the proposed model was tested using mock-generated disease data (data generated by the susceptibility-infection (SI) model, data generated by the susceptibility-infection but not the pathogenesis-infection-recovery-susceptibility (SEIRS) model) and german new coronavirus pneumonia (COVID-19) data for simulated disease data based on flight network propagation at 487 days at 125 international airports: constructing graphs with the node number N being 125 and constructing a weight matrix W based on flight information by using SI data and SEIRS dataGAnd using the data from which the mean value within the sample has been subtracted as a signal, constructing a 6-neighbor graph in which the number of nodes is N100 for the german covi-19 data including 100 infected persons in 507 days of the region, and setting a parameter TpConstructing a weighted adjacency matrix W with 3 and Q with 5GNormalized Laplace matrix LnorSelecting successive TpData of dayxt：
Figure BDA0003188306260000071
As initial data, then, the optimization problem equation (3) is solved by the CVX toolkit to obtain Qx (T)p-1) parameters aq,pAnd finally, obtaining the parameter aq,pAnd data
Figure BDA0003188306260000072
Substituting the prediction model to obtain the predicted value
Figure BDA0003188306260000073
And will be
Figure BDA0003188306260000074
And one-step prediction of results
Figure BDA0003188306260000075
As new continuous TpData of day xtThen, the next time is predicted.
Comparing the method with a G-VARMA model and a GP-VAR model, and quantifying by a normalized minimum mean square error root rNMSE formula to evaluate the performance of the model, wherein the error is shown as a formula (4):
Figure BDA0003188306260000076
wherein xtThe true number of cases for the node at time t,
Figure BDA0003188306260000077
is the predicted number of cases.
In the simulation experiment process, the method of the embodiment is compared with a G-VARMA model and a GP-VAR model proposed by Isufi and the like, SI data and SEIRS data are subjected to 6-step prediction, German COVID-19 data are subjected to 10-step prediction, the prediction result of each data is visualized to obtain the prediction effect shown in figures 1, 2 and 3, different data are observed, and compared with the method, the GF-VAR model of the method of the embodiment has better stability, and the good prediction effect can be kept on different types of data.

Claims (3)

1.基于图滤波器-向量自回归模型的传染病传播预测方法，其特征在于，包括如下步骤：1. the method for predicting the spread of infectious diseases based on the graph filter-vector autoregressive model, is characterized in that, comprises the steps: 1)构建图模型：传染病传播时间序列数据
Figure FDA0003188306250000011
中的每一行代表一个地区，共有N个地区，每一列代表所有地区的某个时刻的病例人数，共Tp个时刻的数据，将地区作为图上节点，并根据不同数据的特性将节点用边相连，构建图的拓扑结构，将传染病传播数据
Figure FDA0003188306250000012
建模为时变图信号
Figure FDA0003188306250000013
即每一时刻的数据均为一个图信号；1) Build a graph model: infectious disease spread time series data
Figure FDA0003188306250000011
Each row represents a region, there are N regions in total, each column represents the number of cases in all regions at a certain time, and the data at T p moments in total. Regions are used as nodes on the graph, and nodes are used according to the characteristics of different data. Edges are connected to construct the topology of the graph, and the infectious disease spread data
Figure FDA0003188306250000012
Modeled as a time-varying graph signal
Figure FDA0003188306250000013
That is, the data at each moment is a graph signal; 2)预测模型设计：首先采用VAR模型对时变图信号X进行预测，然后利用图拉普拉斯矩阵和图滤波器的概念对VAR模型的系数进行设计,假设t时刻的信号值表示为之前Tp-1个时刻的信号的函数，Tp≥2,采用VAR模型对时变图信号进行建模，表达式如公式(1)所示：2) Prediction model design: First use the VAR model to predict the time-varying graph signal X, and then use the concept of graph Laplacian matrix and graph filter to design the coefficients of the VAR model, assuming that the signal value at time t is expressed as before The function of the signal at T p -1 time, T p ≥ 2, the VAR model is used to model the time-varying graph signal, and the expression is shown in formula (1):
Figure FDA0003188306250000014
Figure FDA0003188306250000014
 其中，εt为误差向量，是一个均值为0、协方差矩阵正定的随机向量，图滤波器
Figure FDA0003188306250000015
作为系数矩阵，aq,p为p阶拉普拉斯矩阵的系数，Q为拉普拉斯矩阵的最高阶数；Among them, ε t is the error vector, which is a random vector with a mean value of 0 and a positive definite covariance matrix. The graph filter
Figure FDA0003188306250000015
As a coefficient matrix, a q, p are the coefficients of the p-order Laplace matrix, and Q is the highest order of the Laplace matrix; 3)图滤波器参数优化设计：将GF-VAR模型图滤波器的参数aq,p的估计问题表述为如公式(2)所示的优化问题：3) Graph filter parameter optimization design: The estimation problem of the parameters a q, p of the GF-VAR model graph filter is formulated as the optimization problem shown in formula (2):
Figure FDA0003188306250000016
Figure FDA0003188306250000016
 公式(2)归结为如公式(3)所示的无约束优化问题：Equation (2) boils down to an unconstrained optimization problem as shown in Equation (3):
Figure FDA0003188306250000017
Figure FDA0003188306250000017
 公式(3)所示问题中的输入为信号x与拉普拉斯矩阵LG,解得Q×(Tp-1)个参数aq,p；The input in the problem shown in formula (3) is the signal x and the Laplacian matrix L G , and the Q×(T p -1) parameters a q,p are obtained from the solution; 4)利用优化后的VAR模型对时变图信号进行预测，采用归一化最小均方误差根rNMSE公式进行量化，评估模型性能，误差公式如公式(4)所示：4) Use the optimized VAR model to predict the time-varying graph signal, use the normalized minimum mean square error root rNMSE formula to quantify, and evaluate the performance of the model. The error formula is shown in formula (4):
Figure FDA0003188306250000018
Figure FDA0003188306250000018
 其中xt为t时刻的节点的真实病例数，
Figure FDA0003188306250000019
为预测的病例人数。where x t is the true number of cases at the node at time t,
Figure FDA0003188306250000019
is the predicted number of cases. 2.根据权利要求1所述的基于图滤波器-向量自回归模型的传染病传播预测方法，其特征在于，步骤3)中所述的如公式(3)所示无约束优化问题转化为关于图滤波器系数向量
Figure FDA0003188306250000021
的最小二乘问题如公式(5)所示：2. the method for predicting the spread of infectious diseases based on graph filter-vector autoregressive model according to claim 1, is characterized in that, described in step 3), the unconstrained optimization problem as shown in formula (3) is converted into about. Graph filter coefficient vector
Figure FDA0003188306250000021
The least squares problem of , is shown in formula (5):
Figure FDA0003188306250000022
Figure FDA0003188306250000022
 其中b、
Figure FDA0003188306250000023
C定义如公式(6)所示：where b,
Figure FDA0003188306250000023
C is defined as formula (6):
Figure FDA0003188306250000024
Figure FDA0003188306250000024
 公式(6)中涉及的参数如公式(7)所示：The parameters involved in formula (6) are shown in formula (7):
Figure FDA0003188306250000025
Figure FDA0003188306250000025
 则公式(5)中最小二乘问题的理论解为如公式(8)所示：Then the theoretical solution of the least squares problem in formula (5) is as shown in formula (8):
Figure FDA0003188306250000026
Figure FDA0003188306250000026
 从而得到Q×(Tp-1)个图滤波器参数aq,p，最后得到图滤波器。Thus, Q×(T p -1) graph filter parameters a q,p are obtained, and finally a graph filter is obtained. 3.根据权利要求1所述的基于图滤波器-向量自回归模型的传染病传播预测方法，其特征在于，步骤3)中所述的如公式(3)所示无约束优化问题采用凸优化工具包cvx求解得到，进而求解得图滤波器。3. the infectious disease propagation prediction method based on graph filter-vector autoregressive model according to claim 1, is characterized in that, described in step 3) as shown in formula (3) unconstrained optimization problem adopts convex optimization The toolkit cvx solves it, and then solves the graph filter.
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