科学计算科学计算 (33).pdf

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1、5.5 Curve Fitting5.5 Curve FittingCase of population forecastPrinciples of curve fittingImplementation of curve fittingPractical applicationYear179018001810182018301840185018601870188018901900Population（million）3.95.37.29.612.917.123.231.438.650.263.076.0Year19101920193019401950196019701980199020002

2、010Population（million）92.0105.7122.8131.7150.7179.3203.2226.5248.7281.4308.71.1.Case of population Case of population forecastforecastPopulation growth is a popular concern in the world today.Forecasting population growthtrends is a common practice in every country.The data for a countrys censuses b

3、etween 1790and 2010 is shown in the table below.Please forecast population of the country in 2020.Solving ideas:Find a function to approximate these data,and then calculate the value of theprediction point based on the function found.x=1790:10:2010;y=3.9,5.3,7.2,9.6,12.9,17.1,23.2,31.4,38.6,50.2,63.

4、0,76.0,92.0,105.7,122.8,131.7,150.7,179.3,203.2,226.5,248.7,281.4,308.7;plot(x,y,*);p=polyfit(x,y,3);polyval(p,2020)plot(x,y,*,x,polyval(p,x);ans=339.8Two questions：(1)What kind of function is adopted as approximating function?polynomial function(2)How to calculate the minimum error?least square met

5、hodxx1x2xkxnyy1y2ykyny=f(x)y=g(x)Create a function g(x)to approximate f(x),and minimize the error under some condition.i=g(xi)-f(xi)(i=1,2,3,n)2.Principles of curve fitting2.Principles of curve fittingSimilar to data interpolation,curve fitting is also a method of function approximation.The least sq

6、uare method is a mathematical optimization technique.By minimizing thesum of squares of errors,it can find the best function matching for the data.Assume p(x)is a polynomial functionp(x)=amxm+am-1xm-1+a1x+a0If the value ofis minimal,then p(x)is the approximation function of the original function.()2

7、1p xyiiin=3.Implementation of curve fitting3.Implementation of curve fittingIn MATLAB,the function used for polynomial fitting is the polyfit function.The aim of thisfunction is to figure out the coefficients of fitting polynomial with the least squared errors.Itssyntax is:(1)P=polyfit(X,Y,m)(2)P,S=

8、polyfit(X,Y,m)(3)P,S,mu=polyfit(X,Y,m):According to the sample data X and Y,an mth polynomial P and itserror data S at the sampling point are generated.mu is a binary vector,mu(1)is mean(X),mu(2)isstd(X).The countrys real population in 2016 is 323.1.Question:Its relative error is 1.24%.Whether we ca

9、n and how to narrow this error?polyval(p,2016)ans=327.0964In the above example,we have used the polyfit function to predict the population of acountry in 2020.Whether the forecasting result is accurate?We cannot verify the datafor 2020,but the countrys population for 2016 have been counted.So lets f

10、orecastthe countrys population for 2016.Analysis:According to the research,the population growth of a country has the following features:(1)The more stable the country development is,the more regular the population growth is.(2)When the economy development reaches a certain level,the population grow

11、th rate hasdeclined instead.In other words,there are different population growth laws at different environments anddifferent economic levels.Conclusion:In the population growth data fitting,we should separate the data of the period after theSecond World War from previous data.x=1950:10:2010;y=150.7,

12、179.3,203.2,226.5,248.7,281.4,308.7;p=polyfit(x,y,3)p=1.0e+06*0.0000 -0.0000 0.0034 -2.2305 p=polyfit(x,y,2);plot(x,y,*,x,polyval(p,x)polyval(p,2016)ans=325.1696 polyval(p,2020)ans=336.7857The degree of polynomial fitting is too high.The relative error was successfully reduced to 0.64%!Conclusions:(

13、1)We should make a careful analysis on the problems background.(2)For curve fitting,more sample points dont mean better.We can appropriately reduce the number of sampling points to fit in sections.x0.61.01.41.82.22.63.03.43.84y0.080.220.310.40.480.560.670.750.81.0 x=0.6,1.0,1.4,1.8,2.2,2.6,3.0,3.4,3

14、.8,4;y=0.08,0.22,0.31,0.4,0.48,0.56,0.67,0.75,0.8,1.0;p=polyfit(x,y,1)p=0.2390 -0.0418Empirical formula:y=0.239x-0.04184.Practical 4.Practical application:household application:household savings lawsavings lawThis is a set of survey data between household income x and household savings y in one district(Unit:ten thousand).Try to establish the empirical formula of linear function of x and y.plot(x,y,*,x,polyval(p,x)Three functions of curve fitting:Estimate dataForecast trendsConclude laws