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\begin{abstract}

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\keywords{Keywords1\quad Keywords2\quad Keywords3}

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\section{Introduction}

\subsection{Background}In recent years, economic vitality has become a key part to measure a region’s comprehensive competitiveness. More and more regions begin to attach importance to improving economic vitality. Many governments integrate regional resources through policy interventions and improve economic vitality through relevant policies and support, such as supporting the cultivation and introduction of talents, providing financial support for scientific and technological research and development, and reducing taxes and fees for enterprises. However, economic vitality is affected by many factors, it is difficult for government departments to find the most effective way. Therefore, in order to help solve this problem, the research on economic vitality model has become of great social significance.

\subsection{Restatement of the Problem}

In order to analyze the regional economic vitality, determine which major factors affect the economic vitality, more accurately grasp the key factors and effectively improve the regional economic vitality, The following questions are raised

1)There are huge number of factors have influence on the regional economic vitality. You have to choose a typical region to establish a suitable mathematical model which explain that how the economic vitality is affected.\par

2)According to appropriate data, taking a region to analysis the short-term and long-term effect of changes in economic policy for economic vitality using the data and conclusions from question (2).\par

3)Try to find the right indicators and use them to build models that properly measure and analyze economic vitality.\par

4)From the perspective of decision makers, provide suggestions for the region of question (2) to increase the economic vitality of the region and enhance its overall competitiveness.

\section{Problem analysis}

\subsection{How to analyse the impact of economic on demographic trends and enterprise vitality}

\qquad Take a region as an example, and pay attention to the analysis of a region, province or city rather than:

When analyzing the whole, we can see that the second question also needs to analyze a specific area

Then regional selection is more important. Then we need to establish the appropriate relationship model of the influencing factors of economic vitality,

GDP can be considered as a representative indicator of economic vitality, followed by changes in population and enterprise vitality.\par

From the perspective of trend, the independent variable is population and enterprise vitality

Which indicators can represent the vitality of an enterprise, such as the number of enterprises, the output value of enterprises and so on. The dependent variable is what you need

Analysis of regional GDP and establishment of regression models, of course, there are many kinds of regression models, linear nonlinear bias

Least square regression and so on. If the data volume is large, neural network model can be considered. Then according to the score

The results of the analysis put forward action plans to improve regional economic vitality, such as making relevant policies to attract talents and so on.

\subsection{How to analyse the impact of economic policies on economic vitality?}

\qquad From the perspective of the necessity to analyze the impact of economic policy transformation on the region (or city or province) according to the appropriate data of the survey

The short-term and long-term impact of economic vitality. Then we need to know when the economic policy of the region will be carried out

Transformation: through linear fitting of data before and after transformation, the comparison coefficient can show that transformation has a positive impact on

The influence of the region, for example, the coefficient after the transformation is much higher than that before the transformation, indicates that the policy of the region is to the land

The economic vitality of the district has a great influence.\par

Also, gray prediction, interpolation fitting, time series can also be considered

Wavelet prediction and other models are used to predict before transformation and compare the differences after transformation; for policy prediction, it can also

In order to choose Markov prediction, we mainly solve the prediction research of policy change on future economic development. Another can

Select multiple indicators to depict the economic vitality of the region, and then fit them respectively to observe which side the independent variable is to

The influence of surface is great.

\subsection{Analyze what has an impact on economic vitality?}

\qquad It is necessary to select an appropriate indicator system and establish an analysis and measurement area (or city or provincial level)

The mathematical model of economic vitality, and the ranking of cities, then this problem is a typical evaluation problem, select indicators, sort, then you need to consult the relevant data, you can consider the evaluation models are: AHP, entropy, factor analysis, fuzzy evaluation, gray correlation, etc. of course, if you want to analyze efficiency, you can also Using DEA model.\par

In addition, the neural network evaluation algorithm modified by genetic algorithm or particle swarm optimization can be selected

The neural network evaluation algorithm modified by the algorithm, which is more powerful, will increase its competitiveness.

\subsection{What can we suggest?}

\qquad It is necessary to provide development suggestions for the regions (or cities or provinces) discussed in question 2, so as to present the sustainable development with good economic vitality and stronger regional competitiveness. It can be said what kind of policies are formulated, how to improve the existing policies, how to attract talents, increase the number of enterprises, etc., and then what kind of effects will be achieved.

\section{Assumption}

To reduce the complexity of the problem and make it convenient for us to simulate real-life conditions,.We made some assumptions, and the model was built on the premise that they were all reasonable.

\begin{itemize}

\item There are many factors that affect security check. Our model is setting up and implemented in a relatively ideal environment.

\item Assume that there are no other factors affecting employment demand except seasonal

factors.

\item It can be predicted that the relationship and trend between economic vitality and different indicators

\item When considering that regional economic vitality is affected by policies, other factors are not taken into account

\item When using PCA model, we can think that the two main components can be a good overview of the whole

\end{itemize}

\section{Symbol Description}

\begin{table}[htbp]

\centering

\begin{tabular}{lll}

\toprule

Symbol & Definitions \

\midrule

y & the economic vitality \

$x*{1}$ & the number of corporate units \$x{2} $& the population of permanent residents \$\beta*

*{0}$,$\beta*{1}$,$\beta

*{2}$ & regression coefficient \$\epsilon{0}$ & the disturbance \lnpergdp & Logarithm of regional per capita GDP\gov & Government scale\thirdindustry & industrial structure \industry &industrialization\edu&Education\sav&Total saving rate\\bottomrule\end{tabular}\end{table}*

*P.s: Other symbols instructions will be given in the text.*

\section{Estabilishment and solution of model}

\subsection{Question1: Analysis of the impact of economic dynamism on demographic trends and enterprise vitality.}

We choose Beijing as an example. A certain quantity dependent variable that people are concerned about is affected by another quantity (independent variable). This kind of influence is usually only related (not causal), and this kind of relationship is also interfered by many random factors, so it is difficult to find out the relationship between them by mechanism analysis method. Regression analysis is the most suitable mathematical model to describe the relationship between these variables.. We represent trend of population as changes in the resident population, and replace business vitality with the number of surviving corporate units. The economic vitality of the dependent variable region is explained by GDP per capita.

\subsubsection{Data preprocessing}

Depending on the purpose of the data, check the extent to which the data explain the problem. We found that the use of double logarithm and standardized data processing can more accurately study the factors affecting dependent variables.

\subsubsection{Data description}

\begin{figure}[htbp]

\centering

\includegraphics[width=16cm]{excel1.jpg}

\caption{Data on the number of enterprises and population in Beijing in 2008-2019}

\label{fig:myphoto}

\end{figure}

We describe the data and mainly calculate the standard error, the maximum value and other statistics of the data, as shown in the following table.

\begin{table}[htbp]

\centering

\begin{tabular}{llllll}

\toprule

Variable & Obs&Mean&Std. Dev& Min& Max \

\midrule

GDP per capita(yuan)& 12 &101199.3&28632.22& 64491& 151428 \

Enterprise quantity &12&584479.2&264974.2&339808&1183000 \

Resident population(10000)&12&7.550499&.1690831&7.209858&7.683818\

time & 12&2013.5&3.605551&2008&2019 \

\bottomrule

\end{tabular}

\end{table}

\subsubsection{Establishment of multiple regression model}

Based on the above analysis, the following will establish a regression model of Beijing’s economic vitality affected by the number of enterprises and resident population.

\begin{equation}

y=\beta*{0}+\beta*{1}x*{1}+\beta*{2}x*{2}+\epsilon*{0}

\end{equation}

\subsubsection{Results and analysis of the model}

We use SPSS software to do the experiment of linear regression.

\begin{equation}

R^2=0.951

\end{equation}

\begin{equation}

F=108.63

\end{equation}

At the same time, we also find out the statistics, fitting degree and F value of the regression model.

\begin{table}[htbp]

\centering

\begin{tabular}{lllll}

\toprule

Independent variable & Unstandardized $\beta$ & Standardized $\beta$ &t & Sig \

\midrule

Constant & 4.914 & 4.914 &4.198& .002 \

Enterprise quantity & 6.035E-7 & .564 &5.887&.000 \

Resident population & .824&.492&5.128&.001 \

\bottomrule

\end{tabular}

\end{table}

According to the above data, $R^2$ represents goodness of fit, which is used to measure the fitting degree of the estimated model to the observed value. The closer it is to 1, the better the model is. The adjusted $R^2$ is more accurate than the adjusted R-square. The final adjusted $R^2$ in the figure is 0.951, indicating that the independent variable can account for 95.1\% of the change of the dependent variable.\par

From the perspective of F value: F value is the significance test of regression equation, which means whether the linear relationship between the explained variables and all the explained variables in the model is significant in general. Because f=108.63 > fa (k, n-k-1)= 4.102821015, the original hypothesis is rejected, that is to say, the combination of explanatory variables in the model has a significant impact on the explanatory variables.\par

In the above table, we can also see the significance test results of the independent variables. The last column is sig of t test, which is less than 0.05 in the table, indicating that the independent variables have a significant impact on the dependent variables.\par

\subsubsection{Improvement of model-based problems}

\begin{figure}[htbp]

\centering

\begin{minipage}{7cm}

\centering

\includegraphics[scale=0.3]{point1.jpg}

\caption{Scatter chart of GDP and population}

\end{minipage}

\hspace{10pt}

\begin{minipage}{7cm}

\centering

\includegraphics[scale=0.3]{point2.jpg}

\caption{Scatter chart of enterprises and GDP}

\end{minipage}

\end{figure}

Then, we found the problem from the scatter diagram. In the previous regression analysis, we all assumed that the perturbation term was spherical, but the cross-sectional data we used had the problem of heteroscedasticity. Next, we will use the white test to test the heteroscedasticity hypothesis of the model.\par

The content is that under the original assumption of the same variance:[H*{0}:E(\epsilon_i^2|X)=\sigma^2] the difference between the robust covariance matrix and the common covariance matrix converges to a zero matrix[\hat{S}-s^2S{xx}= \frac{1}{n} \sum {i=1}^n e{i}^2x{i}x{i}’-s^2\frac{1}{n} \sum{i=1}^n x{i}x{i}’=\frac{1}{n}\sum{i=1}^n(e{i}^2-s^2)x{i}x{i}’ \rightarrow{p} 0{KxK}]After applying white test, we find that heteroscedasticity does exist.Try to OLS + robust standard error. This method still carries out OLS regression, but using robust standard error, even if heteroscedasticity exists, all parameters and tests can be carried out as usual. In other words, if there is a robust standard error, the heteroscedasticity will not affect the calculation results.\par\begin{figure}[htbp]\centering\includegraphics[width=12cm]{improve2.png}\caption{Data improved by OLS + robust standard error}\label{fig:myphoto}\end{figure}From this analysis process, our experimental results are relatively ideal. We can conclude that population trends and the number of enterprises have a significant impact on economic vitality.\subsection{Question2:Analysis of the impact of economic policies on economic vitality}The common methods to estimate policy effect are: instrumental variable method, breakpoint regression, tendency score matching method, double difference method, composite control method, etc. We use the double difference method here to solve the second problem.We selected GDP data from two regions of Shanghai and Jiangsu from 2009 to 2018, and analyzed the impact of the one belt policy on the two regions. As we all know, the “one belt” policy is implemented in 2014. When the time axis was before 2014, the impact of the policy on two cities was zero. After 2014, policies began to have an impact on the city.\subsubsection{Data description}According to the original data, we generate a series of data, which is described as follows\begin{figure}[htbp]\centering\includegraphics[width=10cm]{f7.png}\caption{The image of GDP changing with time in Hebei Province from 2009 to 2018}\label{fig:myphoto}\end{figure}\begin{figure}[htbp]\centering\includegraphics[width=10cm]{f8.png}\caption{The image of GDP changing with time in Hebei Province from 2009 to 2018}\label{fig:myphoto}\end{figure}\subsubsection{Principle introduction}Grey prediction is to predict the system which contains both known and uncertain informationThen, it is to predict the gray process which changes in a certain range and is related to time.Grey prediction processes the original data to find the rule of system change, andThe data sequence with strong regularity is generated, and then the corresponding differential equation model is establishedAnd predict the future development trend of things.\parGM (1,1) uses the original discrete non negative data column to reduce the randomness through one accumulation generationThen, by establishing the differential equation model, the new discrete data series with more regularity are obtainedThe approximate estimates of the original data generated by the subtraction are solved to predict the subsequent development of the original data.
*

*\subsubsection{Establishment and analysis of the model}1)Establishment\parAssume that $x^{(0)}=\left(x^{(0)}(1), x^{(0)}(2), \cdots, x^{(0)}(n)\right)$ is the nonnegative original,We did Accumulating Generation Operator once and generated a new sequence $x^{(1)}=\left(x^{(1)}(1), x^{(1)}(2), \cdots, x^{(1)}(n)\right)$\parWe take equation $x^{(0)}(k)+a z^{(1)}(k)=b$ as the basic form of GM (1,1) model (k = 2,3,… N).(X we define the change of GDP, Z we define the change of time series)Where b is the amount of ash used, and -a is the development coefficient.\begin{figure}[htbp]\centering\includegraphics[width=12cm]{f1.png}\caption{The image of GDP changing with time in Hebei Province from 2009 to 2018}\label{fig:myphoto}\end{figure}2)check of quasi exponential law\parIn order to ensure the preciseness of the model, we need to test the quasi exponential law of the model before building it.\parAfter the test, we will get an image about the series ratio. The vertical coordinate of the image represents the smoothness of the data inflection point, as long as the value below the critical value is considered.Here are the test results.\begin{figure}[htbp]\centering\includegraphics[width=10cm]{f3.png}\caption{We can see from the figure that the smoothness of the first two years is not ideal, so we do not consider the impact of the government’s macro-control policies on the GDP of Hebei Province from 2014 to 2015}\label{fig:myphoto}\end{figure}3)Selected model\parThere are three types of model selection:\begin{itemize}\item Traditional GM (1,1) prediction\item New information GM (1,1) prediction\item Metabolism GM(1,1) prediction\end{itemize}\begin{figure}[htbp]\centering\includegraphics[width=10cm]{f2.png}\caption{We can see that the three models work well, in which the traditional information and the new information models overlap}\label{fig:myphoto}
*

*\subsubsection{Data evaluation and forecast}1)Data evaluation(1) residual inspectionIt can be divided into three types: absolute residual, relative residual and average relative residual\parIf the residual is less than 20\%, it is considered that GM (1,1) meets the general requirements for the fitting of the original data.\parIf the residuals are less than 10\%, GM (1,1) is considered to have a good fitting effect on the original data.(2) grade ratio deviation inspectionFirst, the order ratio $\sigma_{k}$ of the original data is calculated from $x^{(0)}(k-1)$ and $x^{(0)}(k-1)$:According to the predicted development coefficient (a), the corresponding stage ratio deviation and average stage ratio deviation are calculated\parIf the deviation is less than 0.2, it is considered that GM (1,1) can fit the original data to the general requirements.\parIf the deviation is less than 0.1, GM (1,1) is considered to be a good fit for the original data.\begin{figure}[htbp]\centering\includegraphics[width=10cm]{f5.png}\caption{The figure above is the residual test, and the figure below is the level ratio deviation test}\label{fig:myphoto}\end{figure}After the above two methods are tested, we come to the conclusion that GM (1,1) fits the original data very well.2)Forecast future trendsDue to the requirements of the subject, we should divide it into short-term and long-term ones. For short-term ones, we set the time as two years and for long-term ones as 50 years.\begin{figure}[htbp]\centering\includegraphics[width=10cm]{f4.png}\caption{Forecast the change of GDP in Hebei Province by 2020}\label{fig:myphoto}\end{figure}\begin{figure}[htbp]\centering\begin{minipage}{7cm}\centering\includegraphics[scale=0.4]{f4.png}\caption{Forecast the change of GDP in Hebei Province by 2020}\end{minipage}\hspace{10pt}\begin{minipage}{7cm}\centering\includegraphics[scale=0.4]{f6.png}\caption{Forecast the change of GDP in Hebei Province by 2068}\end{minipage}\end{figure}
*

*\subsection{Question3:A mathematical model for analyzing the vitality of regional economy}When using statistical analysis method to study multi variable subjects, too many variables will increase the complexity of the subject. People naturally want to get more information with fewer variables. In many cases, there is a certain correlation between variables. When there is a certain correlation between two variables, it can be explained that the two variables reflect the information of this subject has a certain overlap. Principal component analysis is to delete redundant variables (closely related variables) and establish as few new variables as possible, so that these new variables are irrelevant, and these new variables keep the original information as much as possible in reflecting the information of the subject.\par\subsubsection{Data description}
*

*Here are 19 cities’ specific data on nine different indicators. We will build a model based on these nine data and select the main components to summarize the original data*

\subsubsection{Calculation steps of principal component analysis(PCA)}

First, suppose there are n samples and P indexes, then we get the sample matrix of n * p

x=\left[\begin{array}{cccc}{x*{11}} & {x*{12}} & {\cdots} & {x*{1 p}} \ {x*{21}} & {x*{22}} & {\cdots} & {x*{2 p}} \ {\vdots} & {\vdots} & {\ddots} & {\vdots} \ {x*{n 1}} & {x*{n 2}} & {\cdots} & {x*{n p}}\end{array}\right]=\left(x*{1}, x*{2}, \cdots, x*{p}\right)

1.We standardized it:

According to the calculated mean \overline{x*{j}}=\frac{1}{n} \sum*{i=1}^{n} x*{i j}*

,and standard deviationS{j}=\sqrt{\frac{\sum*{i=1}^{n}\left(x*{i j}-\overline{x*{j}}\right)}{n-1}}*,

Standardized data of calculationX{i j}=\frac{x*{i j}-\bar{x}*{j}}{S*{j}}*,

The original sample is standardized into this matrix

X=\left[\begin{array}{llll}{X{11}} & {X*{12}} & {\dots} & {X*{1 p}} \ {X*{21}} & {X*{22}} & {\dots} & {X*{2 p}} \ {\vdots} & {\vdots} & {\ddots} & {\vdots} \ {X*{n 1}} & {X*{n 2}} & {\cdots} &{X*{np}}\end{array}\right]=\left(X*{1}, X*{2}, \cdots, X*{p}\right)*

2.Then we calculate the covariance matrix of the standardized samples of the model

R=\frac{\sum{k=1}^{n}\left(x*{k i}-\bar{x}*{i}\right)\left(x*{k j}-\bar{x}*{j}\right)}{\sqrt{\sum*{k=1}^{n}\left(x*{k i}-\bar{x}*{i}\right)^{2} \sum*{k=1}^{n}\left(x*{k j}-\bar{x}*{j}\right)^{2}}}

3.Calculating eigenvalues and eigenvectors of R.\par

eigenvalues:[\lambda*{0}\geq\ \lambda*{1}\geq\ \dots \geq\ \lambda*{p}\geq\ 0](R is a positive semidefinite matrix, and $\operatorname{tr}(R)=\sum{k=1}^{p} \lambda {k}=p$)\pareigenvectors:a{1}=\left[\begin{array}{c}{a{11}} \ {a{21}} \ {\vdots} \ {a{p 1}}\end{array}\right], a{2}=\left[\begin{array}{c}{a{12}} \ {a{22}} \ {\vdots} \ {a{p 2}}\end{array}\right], \cdots, a{p}=\left[\begin{array}{c}{a{1 p}} \ {a{2 p}} \ {\vdots} \ {a{p p}}\end{array}\right]\par4.Calculation of principal component contribution rate and cumulative contribution rate.\parContribution rate=$\frac{\lambda{i}}{\sum{k=1}^{p} \lambda{k}}(i=1,2, \cdots, p)$\parCumulative contribution rate=$\frac{\sum{k=1}^{i} \lambda{\mathrm{k}}}{\sum{k=1}^{p} \lambda{k}}(i=1,2, \cdots, p)$\subsubsection{Solution and analysis of the PCA model}In principal component analysis, we should first ensure the cumulative contribution of the first several principal components extractedThe contribution rate reaches a high level, and then all the extracted principal components must be able to be givenAn explanation that conforms to the actual background and significance.The meaning of the explanation of principal component is a little fuzzy, unlike that of the original variableSo it is clear and exact that this is the price that variables have to pay in the process of dimensionality reduction.Therefore, extractionThe number of principal components m of should generally be significantly smaller than the number of original variables p (unless P itself is smaller), noThe “advantage” of dimension reduction may not offset the “disadvantage” of principal component meaning which is not as clear as the original variable.Based on the introduction of the above principles and steps, we can get a correlation coefficient matrix:\par
*

* \begin{figure}[htbp]\centering\includegraphics[width=15cm]{relation.png}\caption{correlation coefficient matrix(As the color gets darker becomes more relevant)}\label{fig:myphoto}\end{figure}\par
*