R language uses logistic regression and afrima, ARIMA time series models to predict world population

Link to the original text :http://tecdat.cn/?p=27493 

The source of the original text is ： The official account of the tribal public

Application of this paper R Software technology , Separate use logistic Model 、ARFMA Model 、ARIMA Model 、 Time series model pair slave 2016 To 2100 The world population in . The author will 1950 Year to 2015 Years of historical data as a training set to predict 85 Years of data . The stability of the model is better after correction , Therefore, it has certain reference value .

introduction

as time goes on , The world's population is growing , In order to better grasp the progress speed and law of the world population . We use to build logistic Model and apply R Language software to analyze and predict in 2100 World population in , And compare with the predicted data , See whether the model structure is good or bad, and improve and expand the model .

Model one ：logistic Model

logistic The model is also called the growth retardation model , It is mainly used to describe when environmental resources are limited , The law of population growth . Due to some factors, the world population will eventually reach a saturation value . The blocking effect is reflected in the influence on , Make it decline with the increase of years . If it is expressed as a function of , Then it should be a subtractive function . Then there are

because bgistic Regression model is a generalized linear model based on binomial distribution family , So in R In software ,Logistic Regression analysis can be done by calling the generalized linear regression model function glm() To achieve , Its calling format is

Log< One glm(formula,family=binomial,data) among ,formula For the model to be fitted ,family=binomial It shows that the distribution is binomial ,data For selectable data frames .

By accessing relevant data on the world bank website , We will 1950 Year to 2100 Enter the population data of , And call glmnet Package to fit .

summary(lg.glm)
plot(x, y, main = " Population changes with years logistic curve ",xlab = " year ", ylab = " Population （ One hundred billion ）")

Deviance Residuals: 
      Min         1Q     Median         3Q        Max  
-0.089181  -0.028946   0.002154   0.027206   0.042212  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)
(Intercept) -23.76776   22.17527  -1.072    0.284
x             0.01046    0.01101   0.950    0.342

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 0.923810  on 82  degrees of freedom
Residual deviance: 0.082928  on 81  degrees of freedom
AIC: 13.991

Number of Fisher Scoring iterations: 6

  It can be seen from the test results that the number of people can be affected over time , And the older the year , The greater the population density ; It will eventually stay at a saturation value , And get the logistic The regression model ：

Model two ：  AFRIMA Model

Time series model can be divided into segment memory model and long memory model . The general time series analysis model has autoregression (AR) Model 、 moving average (MA) Model 、 Autoregressive moving average (ARMA) Model 、 Autoregressive integrated moving average （ARIMA） Model, etc , These models are mainly short memory models . at present , People's Empirical Research on macroeconomic variables has found , Although the long memory model has little dependence between long-distance observations, it still has research value . Fractional autoregressive moving average model (ARFMA) The model is a long memory model , It is from Granger and Joyeux (1980) as well as Hosking (1981) stay ARIMA Based on the model , It is widely used in economic and financial fields .

AFRIMA Model definition

AFRIMA The model is based on A R M A Models and ARIMA Model .

ARMA(p,q), The form of the model is :

Model implementation ：

 arfi(Diut[,2][)# establish arfima Model 
plot(Discnt[,1],Dscunt[,2])# Raw data 
points(Dicount[,1][1:66],f$fittedcol="red")# Fitting data 

 

From the result of the residual diagram ,ACF The value of is not within the dotted line , Namely, the residual error is unstable , Not white noise , So let's make a first-order difference to the data .

Model stability improvement

The first-order difference of the data makes the data more stable .

points(c(2016:2100), diffinv(pre$mean)[-1]+ Discount[66,2],col="blue")

From the result of the residual diagram ,ACF The value of and PACF The values of are all within the dotted line , That is, the residual error is stable , So the model is stable .

Model three ：  ARIMA Model  

 ARIMA Model definition

ARIMA The model is called differential autoregressive moving average model . It was written by boxy and Jenkins in 70 A famous time series prediction method proposed in the early s , Bokesi - Jenkins method . among ARIMA(p,d,q) It is called differential autoregressive moving average model ,AR It's autoregression ,p Is the autoregressive term ;MA For moving average ,q Is the moving average number of items ,d It is the difference number of times when the time series becomes stationary .

ARIMA The basic idea of the model is ： The data sequence formed by the prediction object over time is regarded as a random sequence , Use a certain mathematical model to approximately describe this sequence . Once the model is identified, it can predict the future value from the past value and current value of the time series .

ARIMA The basic idea of the model is ： The data sequence formed by the prediction object over time is regarded as a random sequence , Use a certain mathematical model to approximately describe this sequence . Once the model is identified, it can predict the future value from the past value and current value of the time series .

The modeling process mainly includes ：

First step ： Autoregressive process

Make Yt Express t In the period of GDP. If we put Yt The model of is written as

(Y_t-δ)=α_1 (Y_(t-1)-δ)+u_t

among δ yes Y The average of , and ut It has zero mean and constant variance σ^2 Uncorrelated random error term of ( namely ut It's white noise ), Then Cheng Yt Follow a first-order autoregressive or AR(1) random process .

P The form of order autoregressive function is written as ：

(Y_t-δ)=α_1 (Y_(t-1)-δ)+α_2 (Y_(t-2)-δ)+α_3 (Y_(t-3)-δ)+⋯+α_p2 (Y_(t-p)-δ)+u_t

There are only Y This variable , There are no other variables . Can be interpreted as “ Let the data speak for themselves ”.

The second step ： Moving average process

Above AR The process is not to produce Y The only possible mechanism . If Y The model of is described as

Y_t=μ+β_0 u_t+β_1 u_(t-1)

among μ Is constant ,u White noise ( Zero mean 、 Constant variance 、 Non autocorrelation ) Random error term .t In the period of Y Equal to a constant plus a moving average of the present and past error terms . said Y Follow a first-order moving average or MA(1) The process .

q The order moving average can be written as ：

Y_t=μ+β_0 u_t+β_1 u_(t-1)+β_2 u_(t-2)+⋯+β_q u_(t-q)

Self regressive moving average process

If Y Both AR and MA Characteristics of , It is ARMA The process .Y It can be written.

Y_t=θ+α_1 Y_(t-1)+β_0 u_t+β_1 u_(t-1)

There is an autoregressive term and a moving average term , So he's a ARMA(1,1) The process .Θ It's a constant term .

ARMA(p,q) In the process p An autoregressive sum q Moving average .

The third step ： Autoregressive quadrature moving average process

All the above is based on the fact that the data is stable , But many times the time data is non-stationary , That is, single integer ( Simple product ) Of , Generally, nonstationary data can be obtained by difference . So if we talk about a time series difference d Time , Become stable , And then use AEMA(p,q) Model , Then we say that the original time series is AEIMA(p,d,q), Autoregressive quadrature moving average time series .AEIMA(p,0,q)=AEMA(p,q).

Usually ,ARIMA The modeling steps are 4 Stages : Sequence stationarity test , Preliminary identification of the model , Model parameter estimation and model diagnosis analysis . 

Model implementation

Step one ： distinguish . Find the right p、d、 and q value . It can be solved by correlation diagram and partial correlation diagram .

Step two ： It is estimated that . Estimate the parameters of autoregressive and moving average terms included in the model week . Sometimes the least square method can be used , Sometimes it is necessary to use nonlinear estimation methods .( The software can automatically complete )

Step three ： The diagnosis ( test ). See if the calculated residual is white noise , yes , Then accept the fitting ; No , Then do it again .

Step four ： forecast . More reliable in the short term .

say concretely ：

First , See if there is any obvious trend, Do you need differencing Then model .

from ACF and PACF As a result , The sequence did not fall within the dotted line quickly , therefore , The sequence is unstable . The sequence is differentiated .

     Draw ACF and PACF, Decide which model to use by looking at the picture （ARMA（p,q）,ARIMA And so on. ）.

From the data results after the difference ,ACF stay 8 After the step, it begins to fall into the dotted line ,PACF stay 2 The step will soon fall into the dotted line , therefore p=8,q=2,d=1.

xz1=automa(Dist[1:66,2],ic=c('bic'),trace=T)# Automatically find the best arima Model 

 ARIMA(2,2,2)                    : -1058.701
 ARIMA(0,2,0)                    : -1026.504
 ARIMA(1,2,0)                    : -1049.834
 ARIMA(0,2,1)                    : -1048.83
 ARIMA(1,2,2)                    : -1062.532
 ARIMA(1,2,1)                    : -1050.673
 ARIMA(1,2,3)                    : -1058.723
 ARIMA(2,2,3)                    : Inf
 ARIMA(0,2,2)                    : -1060.99

 Best model: ARIMA(1,2,2)        

 

 

       From the residual ACF As a result , The sequence quickly and stably falls into the dotted line range , The model is stable .

plot(Discnt[,1],Disnt[,2])
points(c(2016:2100) ,Diunt[66,2]+diffinv(xzrcast$mean)[-1],col="red")# Predictive value 2015 To 2100 year 

 

From the residual ACF As a result , The sequence quickly and stably falls into the dotted line range , The model is stable .

In order to test that the prediction error is a normal distribution with zero mean , We can draw the histogram of prediction error , And the average value is zero 、 Plot of normal distribution of standard variance to prediction error .

points(mst$mids, myst$density, type="l", col="blue", lwd=2)
}
plotForecrrors(xzrrecast $residuals)

reference

【1】 Statistical simulation and its R Realization , Xiao Zhihong , Wuhan university press

【2】Logistic Application of model in population prediction , Yan Huizhen , Journal of Dalian University of Technology , The first 27 Volume No 4 period  

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