国际数学建模大赛一等奖论文 - 图文

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various eras and forecasting error of linear combination.if Assumed conditions of AR(p) has not been satisfied，this shape could be considered to use. The shape of model is always satisfied with Stationary condition，because its value of parameter less intense temporal series than parameter P in AR model. So comparatively large random variation will not change the direction of temporal series Identification condition

If k?q，autocorrelation coefficient rk?0，or autocorrelation coefficient rk obey N(0,1/n(1?2?r)211/2) and the number of (rk?2/n(1??r)2)≤4.5%,

21211that is, the autocorrelation coefficient of stationary Time Series rk is step-q truncation, and partial correlation coefficient ?kprogressively attenuate but not truncation, then the sequence is MA(q) model.

Actually，in the process of MA，the functions PACF presents unilateral descending or the Damped Oscillation，so the function ACF is used to differentiate.(all the autocorrelation coefficient equals to 0 from step-q) Reversible condition First order：?1?1

Second order：?2?1,?1??2?1

If Reversible condition has been satisfied，the model MA(q) could transform into model AR(p)

Autoregressive Moving Average model ARIMA(p, d, q)

yt??1yt?1??2yt?2????pyt?p??t??1?t?2??2?t?3????p?t?p

Symbols of formula：p and q are estimated Autoregressive order and Moving Average ；?and ?are not-zero undermined coefficient;

?t is an independent error termytis Stationary, normal and aero mean Time Series

Implications of model

The ratio of using two multinomial is similar to a longer AR multinomial，that is，

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the p?q thereinto is less than the step-p of the model AR(p). The first two models are particular cases of model ARIMA(p, d, q)

An ARMA process may be the superimposing among the process of AR and MA，the process of several AR，the process of AR and ARMA Identification condition

Both partial correlation of the Stationary Time Series ?kand Autocorrelation coefficient rk are not truncation，but comparatively fast converge to 0，so the time series may model ARIMA(p, d, q). In practice，this model is always used. So the main job of modeling is to solve the value ofp,qand ?,? examine values of ?t and yt Order of model

AIC rule: which is rule of smallest unit of information，and it shows the best prediction of order and parameter of model ARMA，and the problem of less sample data included is fitted. The aim is to judge which random process is close to the developing process of prediction target. When there is a concrete application，array the order of models by hierarchy，calculate AIC value separately, finally be sure to let the smallest order of model be the suitable order of model If model parameter Maximum Likelihood Estimation，

AIC?(n?d)log?2?2(p?q?2)

If model parameter Least Square Estimation，AIC?nlog?2?(p?q?)logn In the formula：n stands for sample number，?2 stands for Fitting Residual sum of squares ，d,p,q are parameters

Among them，the range limit of p,q as followed，if n is on the smaller side，take the ratio of n，if n is comparatively large，take multiple of lognIn practice，p,q is not more than 2.

Autoregressive Integrated Moving Average model ARIMA(p, d, q) Both partial correlation of the Stationary Time Series ?kand Autocorrelation

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coefficient rk are not truncation，then slowly converge，so the time series may model ARIMA(p, d, q) Implications of model

The shape of model is similar to model ARMA（p ,q）but data must experience special processing. Especially when linear session is non-stationary，model ARMA（p, q）can not be used directly，but taking use of Finite order difference makes non-Stationary Time Series tranquilization.，in practice，d is always not more than 2. If cyclical swings exist in Time Series，then differences according to time period，aims to change Time Series influenced for a long time by Random error into Time Series just temporarily influenced.

That is，new series is accord with model ARMA(p, q) after difference，and original is accord with model ARIMA(p ,d, q).

(3) Solving the Cases of the Yorkshire Ripper

After looking up all the venues related to Peter Sutcliffe[4]，use the software of Google Earth to label them. Omit those that do not have definite address or are far away. Finally 18 valid data have been got according to the time order, as shown in illustration.

Picture 1

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To verify time series possibility of predicted next crime. Firstly, use the software Google Earth to get the distance between the continuous two localities where crimes are committed

Tag number Distance 1-2 2-3 3-4 4-5 5-6 16.10 21.52 29.25 0.88 3.92 Tag number 6-7 7-9 9-10 10-13 13-14 Distance 0.75 5.43 13.23 1.46 17.01 Tag number 14-16 16-18 18-19 19-20 20-21 Distance 7.23 15.32 10.07 23.15 23.01 Table 1 continuous 2 localities distance

Take use of insured data among 11groups and predict the data of Group 12 and 13，then compare with the actual data Time Series analysis of SPSS data tested

Make a column diagram：examine the normality and zero mean

Picture 2

It can be seen in this diagram that standard deviation is 9.47，data exchange is needed. And use index conversion to transform data

3)Make a related diagram：examine stationary and periodicity

Use Autocorrelation and partial correlation to analyze，because time series sample data n>50 and hysteresis cycle are required，so enactment Maximum Number of Lags