总策略为中文书写，GPT/DeepL翻译，英文语料作为prompt微调表述。

abstract

综合模型概述

  • In order to solve the problem of the impact of the US election results on the Sino-US economy, this paper uses Lagrange interpolation and principal component analysis to complement and reduce the historical data, build a predictive model based on BP neural network, and use the VAR model to analyze the future economy development status, and establish an analytic hierarchy model based on multiple linear regression to evaluate and analyze the results of the general election.(B2020230040158) 包括数据处理，模型建立及其对应问题。

每问解答概述

先重复问题

  • Question 1 requires a quantitative analysis……(B2020230040158)
  • Question 2 is to analyze the impact on……(B2020230040158)
  • Question 3 requires combining the mathematical models of Question 1 and Question 2 to propose……(B2020230040158)

每个模型的动机/基础/前提

  • Due to the large number of policy indicators, it is considered to conduct principal component analysis on policy indicators, and to establish a prediction model based on BP neural network, using historical data of principal component indicators to predict various principal component indicators in the next 4 years.(B2020230040158)
  • Finally, the VAR model is used to ……(B2020230040158)
  • Assuming that……(B2020230040158)
  • By constructing an analytic hierarchy model……the index weights are obtained according to……(B2020230040158)

求解答案/解释分析答案

创新点阐述

  • The innovation of this paper is to adjust the prediction results of the BP neural network based on the impact of the epidemic to make it more in line with the actual situation, and use the VAR model to analyze and determine the future economic development status. In the analytic hierarchy process, the multiple linear regression method is used to construct the judgment matrix, which avoids the shortcoming ofthe subjectiveness ofthe expert score.(B2020230040158)

Problem background and restatement

background

问题重述

(1)(2)(3)分点

Problem analysis

每问分析

  • Secondly, different candidates adopting different policy propositions will result in different economic development conditions. In order to characterize this impact, select corresponding related policy indicators for each policy, including infrastructure investment, tax income, average CO2 emissions, average domestic general government health expenditure, unemployment rate, total import and export, trade deficit, stock price general 500 index, total international migrants and percentage of education expenditure. On the other hand, in order to quantitatively judge the economic development status, relevant economic indicators such as Gross Domestic Product 、Producer Price Index 、average Gross National Income are selected. Collect historical data for these indicators.(B2020230040158) 动机-目的-方法

  • Different from problem 1, only two candidates' China policies will affect China's economy in problem 2. Therefore, in problem 2, indicators related to China's policies are selected to characterize the impact of different candidates on China's economy, including China's foreign trade cargo throughput, US imports and exports to China, US dollar to RMB exchange rate and US federal fund interest rate. On the other hand, in order to quantitatively judge the development of China's economy, China's total import and export and customs duties are selected as indicators to measure China's economy. As the first one is similar to other principal component analysis, it is not necessary to extract principal component analysis.(B2020230040158) 叙述每问之间的逻辑性

Problem hypothesis

it is assumed that

Symbol description

基本模型叙述

插值叙述

  • After checking the acquired data, it is found that some indicators only have partial data values. The main purpose of data preprocessing is to make data interpolation. Commonly used interpolation methods include mean/median/mode interpolation, fixed value, nearest neighbor interpolation, regression method and interpolation method, considering that the results of interpolation method are more accurate and only need to use the information of known points, therefore, using Lagrange interpolation method to impute missing values. Data interpolation can not only improve data quality, but also improve the accuracy of fitting and prediction in the model. The basic steps of Lagrange interpolation are as follows: step1/2/3(B2020230040158)

spearman相关系数

  • Among them, the Spearman correlation coefficient is a method to study the convergence between two groups of variables. It does not require high sample size, and does not need to assume the normality of the population, and good results can be obtained. Therefore, here we use the Spearman rank correlation coefficient to test.(apmcm2112772/C)

回归之后的T检验和F检验(apmcm2112772/C)

  • F test：We can see that the p-value of the F test is 0, rejecting the null hypothesis: the coefficients of all independent variables are 0, and the F test is passed. Moreover, the value of the determination coefficient is as high as 0.9826, indicating that the fitting effect of the model is still relatively good.

  • T test：From the above table, we can see that 3 variables of the model: ,the p-value of the t-test is all equal to 0, and the values of the remaining variables and constant terms are also less than 0.10. The more excellent ones have passed the t-test, indicating The Linear Relationship between the independent variable and the dependent variable of the model is significant, and the model fitting effect is better.

  • 与原数据比较可视化

数据降维阐述

  • After obtaining the data of 10 indicators, we found that if all these indicators are used to evaluate the ecological environment of Saihanba, the complexity of the model will be slightly larger. In order to simplify the model, we can extract several of the most representative characteristic indicators from these 10 indicators as evaluation criteria. Among them, the entropy method is a method that can objectively assign weights, which can reflect the effective information amount of the indicator. The larger the entropy value, the smaller the effective information amount of the indicator, and the smaller the weight of the indicator. Compared with the subjective weighting method, the entropy weight method is more explanatory for indicators. Therefore, we can use the entropy method to more accurately extract the most representative feature indicators among the above 10 indicators.

Calculate feature weights and establish entropy values

  • 详见apmcm2112772/C

TOPSIS method to evaluate XXX impact

  • 详见apmcm2112772/C

主成分分析叙述

(B2020230040158)

Presidential candidates Trump and Biden have expressed different policy propositions in response to COVID-19 measures, international trade, and economic development. The election of each candidate would have varied impacts on economic development. We selected relevant indicators for each policy to reflect its impact, serving as a foundation for analyzing the effects of the policies proposed by each candidate. Each statistical indicator reflects a segment of the economic situation under the respective policies, but given the correlations among indicators, dimensionality reduction and simplification are necessary to extract potential comprehensive indicators that describe economic conditions. This study uses principal component analysis (PCA) to process economic indicators, aiming to use a few comprehensive indicators to encapsulate the information each original indicator carries.

  1. Principle of Principal Component Analysis

The goal of PCA is to retain the most significant components in the original data during dimensionality reduction, thereby maximizing the variance of the original data. This approach replaces multiple single indicators with fewer comprehensive indicators, essentially substituting ( p ) indicators ( X_p ) with ( k ) principal components ( Y_k ). In mathematical terms, the ( p ) indicators ( X_p ) are linearly combined to form ( Y_k ).

The information in a random variable can be quantified by its dispersion (variance). Therefore, in PCA, the combination with the largest variance (( (Y_1) )) is used as the first principal component. If ( Y_1 ) alone cannot represent all ( p ) indicators, the next comprehensive indicator ( Y_2 ) is extracted. To ensure effective information extraction, ( (Y_1, Y_2) = 0 ) is set, meaning ( Y_1 ) and ( Y_2 ) are uncorrelated, allowing ( Y_2 ) to independently capture additional information.

  1. Standardization of Indicator Variables

Before conducting PCA, each indicator variable is usually standardized to avoid unreasonable weighting due to unit selection. This prevents significant variance differences that could skew PCA results. Standardization is generally done by using ( X = ), where ( E(X_i) ) is the mean, and ( S_i ) is the standard deviation of the indicator.

  1. Steps of Principal Component Analysis
  1. Standardize the Raw Index Data:
    • Transform the raw data into a standardized form to build a matrix of standardized values.
  2. Calculate the Correlation Coefficient Matrix:
    • Calculate the correlation between each variable and construct a correlation coefficient matrix ( R ) among variables.
  3. Solve for Eigenvalues and Eigenvectors:
    • Compute eigenvalues and eigenvectors from the correlation matrix ( R ).
  4. Calculate Contribution Rates:
    • Compute the contribution rate ( p_i = ) and the cumulative contribution rate to determine the number of principal components according to the 0.85 principle.
  5. Formulate Principal Component Expressions:
    • Define each principal component ( Y_i = l_{i1}X_1 + l_{i2}X_2 + + l_{ip}X_p ). The load matrix of principal components is rotated using the maximum variance method to simplify interpretation.
  6. Calculate Component Scores:
    • Return to the original data to compute the scores of the selected principal components.
  1. Results of Principal Component Analysis

The Kaiser-Meyer-Olkin (KMO) statistic for 10 policy indicators is 0.754 (greater than 0.6), and Bartlett’s Test of Sphericity yields a significance value of 0.000 (less than 0.05), indicating that PCA is appropriate for this dataset.

Through PCA, we derive the principal component contribution rates for the 10 indicators. As seen in Table 6 and Figure 1, the cumulative contribution rate of the first three principal components exceeds 85%. Additionally, Figure 2's scree plot shows that the eigenvalues for the first three components are all greater than 1. Thus, we select the first three principal components to represent the original 10 indicators.

BP神经网络叙述

(B2020230040158)

BP Neural Network can approximate any non-linear mapping with arbitrary precision and has learning and self-adaptability. The connection value of the network can be modified to respond to changes in the system. Additionally, the multi-input multi-output network constructed by it has good fault tolerance and can generate a system with strong robustness. Due to these characteristics, BP Neural Network is widely used and can better adapt to the system characteristics of the multi-index variables in this problem, making it possible to predict the policy indicators of the two presidential candidates in the next term.

The BP Neural Network mainly includes three layers: input layer, hidden layer, and output layer. Each layer is composed of many parallel computing neurons. The principle is as follows: the working signal obtains the network error through forward propagation, and the error signal uses backpropagation to adjust the network. During forward propagation, the input layer's working signal is transformed by the hidden layer and transmitted to the output layer, where the output signal is obtained. If the output signal does not meet the given standard, an error backpropagation process is performed. The difference between the actual output value from the forward operation and the expected output value is used as the error signal, which is transmitted from the output layer to the input layer. This error signal is continuously fed back to adjust the network weights, obtaining the ideal output value by continuously correcting the weights.

BP Neural Network Algorithm Steps

  1. Clarify the Input and Output Variables
    Define the input and output variables and corresponding parameters:
    [ X_k = [x_{k1}, x_{k2}, , x_{kM}] (k = 1, 2, , N) ] where ( N ) is the number of samples (32 for Biden, 47 for Trump), and ( M ) is the number of main modeling variables (10).

  2. Initialize the Network
    Construct the initial network by randomly assigning smaller values to the network weights. The weight matrix for the input to the hidden layer and the hidden to the output layer at iteration ( t ) is represented as: [ W_{IJ}(t) =

    \[\begin{bmatrix} w_{I1}(t) & w_{I2}(t) & \dots & w_{IJ}(t) \\ \vdots & \vdots & \ddots & \vdots \\ w_{I1}(t) & w_{I2}(t) & \dots & w_{IJ}(t) \end{bmatrix}\]

    ] The input and output data are standardized and transformed for direct calculation. This paper uses linear normalization as follows: [ x^* = + y_{} ]

  3. Sample Selection
    Randomly select a sample ( X_k ) from the training samples, along with the corresponding expected output ( D_k ).

  4. Forward Propagation
    Using the selected input sample ( X_k ), conduct a forward operation through each layer's neurons in the BP Neural Network, resulting in the output signal ( O_n(t) = v_{Kk}(t) ). The Sigmoid function is used as the activation function: [ f(x) = ]

  5. Calculate Error
    Determine the network error ( E(t) ) as the difference between the network output ( O_n(t) ) and the expected output ( D_n(t) ). If the error meets the accuracy requirements, proceed to Step 8. Otherwise, go to Step 6.

  6. Check Iteration Condition
    Determine if the next iteration meets the termination condition. If it does, go to Step 8. If not, calculate the local gradient of each neuron in each layer through the error backpropagation process.

  7. Weight Update
    Use the learning rate to update the weights according to the following equations: [ w_{ij}(t+1) = w_{ij}(t) + j(t) x{ni}(t) ] [ w_{jk}(t+1) = w_{jk}(t) + k(t) O{nj}(t) ] Return to Step 4 for the next forward propagation.

  8. Completion Check
    If all training samples are completed, the calculation ends. Otherwise, return to Step 3.

This algorithm enables the BP Neural Network to learn and adapt to the data, providing robust predictions for policy indicators.

Performance optimization of BP Neural Network

不同神经元和层数的影响，列表，可视化

Neural network testing and verification

可视化测试集上的结果

结合自身问题

基于问题的神经网络建模可视化

apmcm2112772/C

神经网络动机介绍

Therefore, we need to find out the mapping relationship between these two pairs of indicators, namely: (19) This is a mapping relationship from 5 dimensions to 3 dimensions. We cannot simply use multiple regression to establish a suitable mathematical model. In the field of artificial intelligence, deep learning solves this problem well. Therefore, we decided to use BP Neural Network to achieve this mapping relationship.(apmcm2112772/C)

结合题目的输入隐藏输出层的构建

apmcm2112772/C

VAR

(B2020230040158)

This section aims to analyze the relationship between principal component indicators and economic indicators, and then based on the values of the principal component indicators predicted by the BP Neural Network above, the predicted values ofvarious economic indicators in the United States in the next stage can be obtained, and then the economic development trends of the United States when different candidates are elected can be quantitatively analyzed. Vector Autoregressive Model (VAR) is widely used in economics. It mainly uses actual economic data to determine the dynamic structure of the economic system. It is often used to analyze the relationship between interconnected indicators and the impact of random disturbances on the economic system. In view of the built-in VAR model in the economic software EViews10, this paper uses EViews10 software to establish a VAR-based prediction model of US economic indicators. The specific steps are as follows:

Step1: Test the stationarity of the data series, which is the prerequisite for establishing the VAR model.

Step2: Determining the optimal lag order is an important problem that the VAR model needs to solve.

Step3: Establish a VAR model with a lag order of 2, and get the relationship between GDP and Y1, Y2, and Y3 as shown in the equation (the coefficients retain two decimal places).

AHP

The construction of judgment matrix

详见亚太杯B2020230040158

模型建立与分析

分析

  • It can be seen from the above figure that the overall gap between the predicted value and the actual value of China policy indicators is small, and the fluctuation direction of the curve tends to be consistent, which indicates that the accuracy of the model is high, the prediction effect is good, and the predicted value data credibility is high.(B2020230040158)

模型评价与改进

评价

  • In terms of data sources, this paper selects data from multiple economic databases to ensure reliable and abundant data sources.(数据获取方面)(B2020230040158)
  • Secondly, through the comparative analysis of BP neural network errors of different parameters, the optimal network parameters are determined.Based on this, the BP neural network which can accurately predict the main component values is established and adjusted according to……(神经网络方面)(B2020230040158)

Promotion

  • In this paper, BP Neural Network adjustment indicators value fitting and forecasting method based on epidemic situation, the economic prediction model based on VAR and the analytic hierarchy process model based on multiple linear regression can be applied to predict the impact of any two presidents on the economy of China and the United States in the future election.(B2020230040158)