Copula理論及MATLAB應(yīng)用實(shí)例

1、%Copula理論及應(yīng)用實(shí)例%-%*讀取數(shù)據(jù)*%從文件hushi.xls中讀取數(shù)據(jù)hushi = xlsread(hushi.xls);%提取矩陣hushi的第5列數(shù)據(jù)，即滬市的日收益率數(shù)據(jù)X = hushi(:,5);%從文件shenshi.xls中讀取數(shù)據(jù)shenshi = xlsread(shenshi.xls);%提取矩陣shenshi的第5列數(shù)據(jù)，即深市的日收益率數(shù)據(jù)Y = shenshi(:,5);%*繪制頻率直方圖*%調(diào)用ecdf函數(shù)和ecdfhist函數(shù)繪制滬、深兩市日收益率的頻率直方圖fx, xc = ecdf(X);figure;ecdfhist(fx, xc, 30);x

2、label(滬市日收益率)；%為X軸加標(biāo)簽ylabel(f(x); %為Y軸加標(biāo)簽fy, yc = ecdf(Y);figure;ecdfhist(fy, yc, 30);xlabel(深市日收益率)；%為X軸加標(biāo)簽ylabel(f(y); %為Y軸加標(biāo)簽%*%計(jì)算X和Y的偏度xs = skewness(X)ys = skewness(Y)%計(jì)算X和Y的峰度kx = kurtosis(X)ky = kurtosis(Y)%*正態(tài)性檢驗(yàn)*%分別調(diào)用jbtest、kstest和lillietest函數(shù)對(duì)X進(jìn)行正態(tài)性檢驗(yàn)h,p = jbtest(X) % Jarque-Bera檢驗(yàn)h,p = kst

3、est(X,X,normcdf(X,mean(X),std(X) % Kolmogorov-Smirnov檢驗(yàn)h, p = lillietest(X) % Lilliefors檢驗(yàn)計(jì)算偏度和峰度*%分別調(diào)用jbtest、kstest和lillietest函數(shù)對(duì)Y進(jìn)行正態(tài)性檢驗(yàn)h,p = jbtest(Y) % Jarque-Bera檢驗(yàn)h,p = kstest(Y,Y,normcdf(Y ,mean(Y),std(Y) % Kolmogorov-Smirnov檢驗(yàn)h, p = lillietest(Y) % Lilliefors檢驗(yàn)%*求經(jīng)驗(yàn)分布函數(shù)值*%調(diào)用ecdf函數(shù)求X和Y的經(jīng)驗(yàn)分布函數(shù)

4、fx, Xsort = ecdf(X);fy, Ysort = ecdf(Y);%調(diào)用spline函數(shù)，利用樣條插值法求原始樣本點(diǎn)處的經(jīng)驗(yàn)分布函數(shù)值U1 = spline(Xsort(2:end),fx(2:end),X);V1 = spline(Ysort(2:end),fy(2:end),Y);%調(diào)用ecdf函數(shù)求X和Y的經(jīng)驗(yàn)分布函數(shù)fx, Xsort = ecdf(X);fy, Ysort = ecdf(Y);%提取fx和fy的第2個(gè)至最后一個(gè)元素，即排序后樣本點(diǎn)處的經(jīng)驗(yàn)分布函數(shù)值fx = fx(2:end);fy = fy(2:end);%通過(guò)排序和反排序恢復(fù)原始樣本點(diǎn)處的經(jīng)驗(yàn)分布函數(shù)

5、值U1和V1Xsort,id = sort(X);idsort,id = sort(id);U1 = fx(id);Ysort,id = sort(Y);idsort,id = sort(id);V1 = fy(id);%調(diào)用ksdensity函數(shù)分別計(jì)算原始樣本X和Y處的核分布估計(jì)值U2 = ksdensity(X,X,function,cdf);V2 = ksdensity(Y,Y ,function,cdf);% *繪制經(jīng)驗(yàn)分布函數(shù)圖和核分布估計(jì)圖*Xsort,id = sort(X); %為了作圖的需要，對(duì)X進(jìn)行排序figure; %新建一個(gè)圖形窗口plot(Xsort,U1(id),

6、c,LineWidth,5); %繪制滬市日收益率的經(jīng)驗(yàn)分布函數(shù)圖hold onplot(Xsort,U2(id),k-.,LineWidth,2); %繪制滬市日收益率的核分布估計(jì)圖legend(經(jīng)驗(yàn)分布函數(shù),核分布估計(jì)，Location,NorthWest); %加標(biāo)注框%*核分布估計(jì)*xlabel(滬市日收益率)；%為X軸加標(biāo)簽ylabel(F(x); %為Y軸加標(biāo)簽Ysort,id = sort(Y); %為了作圖的需要，對(duì)Y進(jìn)行排序figure; %新建一個(gè)圖形窗口plot(Ysort,V1(id),c,LineWidth,5); %繪制深市日收益率的經(jīng)驗(yàn)分布函數(shù)圖hold onpl

7、ot(Ysort,V2(id),k-.,LineWidth,2); %繪制深市日收益率的核分布估計(jì)圖legend(經(jīng)驗(yàn)分布函數(shù),核分布估計(jì)，Location,NorthWest); %加標(biāo)注框xlabel(深市日收益率)；%為X軸加標(biāo)簽ylabel(F(x); %為Y軸加標(biāo)簽%*繪制二元頻數(shù)直方圖*%調(diào)用ksdensity函數(shù)分別計(jì)算原始樣本X和Y處的核分布估計(jì)值U = ksdensity(X,X,function,cdf);V = ksdensity(Y,Y,function,cdf);figure; %新建一個(gè)圖形窗口%繪制邊緣分布的二元頻數(shù)直方圖，hist3(U(:) V(:),30,3

8、0)xlabel(U(滬市); %為X軸加標(biāo)簽ylabel(V(深市)；為Y軸加標(biāo)簽zlabel(頻數(shù))；為z軸加標(biāo)簽%*繪制二元頻率直方圖*figure; %新建一個(gè)圖形窗口%繪制邊緣分布的二元頻數(shù)直方圖，hist3(U(:) V(:),30,30)h = get(gca, Children); %獲取頻數(shù)直方圖的句柄值cuv = get(h, ZData); %獲取頻數(shù)直方圖的Z軸坐標(biāo)set(h,ZData,cuv*30*30/length(X); %對(duì)頻數(shù)直方圖的Z軸坐標(biāo)作變換xlabel(U(滬市); %為X軸加標(biāo)簽ylabel(V(深市)；為Y軸加標(biāo)簽zlabel(c(u,v); %

9、為z軸加標(biāo)簽%調(diào)用copulafit函數(shù)估計(jì)二元正態(tài)Copula中的線性相關(guān)參數(shù)rho_norm = copulafit(Gaussian,U(:), V(:)%調(diào)用copulafit函數(shù)估計(jì)二元t-Copula中的線性相關(guān)參數(shù)和自由度rho_t,nuhat,nuci = copulafit(t,U(:), V(:)%*求Copula中參數(shù)的估計(jì)值*Udata,Vdata = meshgrid(linspace(0,1,31); %為繪圖需要，產(chǎn)生新的網(wǎng)格數(shù)據(jù)%調(diào)用copulapdf函數(shù)計(jì)算網(wǎng)格點(diǎn)上的二元正態(tài)Copula密度函數(shù)值Cpdf_norm = copulapdf(Gaussian,U

10、data(:), Vdata(:),rho_norm);%調(diào)用copulacdf函數(shù)計(jì)算網(wǎng)格點(diǎn)上的二元正態(tài)Copula分布函數(shù)值Ccdf_norm = copulacdf(Gaussian,Udata(:), Vdata(:),rho_norm);%調(diào)用copulapdf函數(shù)計(jì)算網(wǎng)格點(diǎn)上的二元t-Copula密度函數(shù)值Cpdf_t = copulapdf(t,Udata(:), Vdata(:),rho_t,nuhat);%調(diào)用copulacdf函數(shù)計(jì)算網(wǎng)格點(diǎn)上的二元t-Copula分布函數(shù)值Ccdf_t = copulacdf(t,Udata(:), Vdata(:),rho_t,nuhat

11、);%繪制二元正態(tài)Copula的密度函數(shù)和分布函數(shù)圖figure; %新建圖形窗口surf(Udata,Vdata,reshape(Cpdf_norm,size(Udata); %繪制二元正態(tài)Copula密度函數(shù)圖xlabel(U); %為X軸加標(biāo)簽ylabel(V); %為Y軸加標(biāo)簽zlabel(c(u,v); %為z軸加標(biāo)簽figure; %新建圖形窗口surf(Udata,Vdata,reshape(Ccdf_norm,size(Udata); %繪制二元正態(tài)Copula分布函數(shù)圖xlabel(U); %為X軸加標(biāo)簽ylabel(V); %為Y軸加標(biāo)簽zlabel(C(u,v); %為z

12、軸加標(biāo)簽%繪制二元t-Copula的密度函數(shù)和分布函數(shù)圖figure; %新建圖形窗口surf(Udata,Vdata,reshape(Cpdf_t,size(Udata); %繪制二元t-Copula密度函數(shù)圖xlabel(U); %為X軸加標(biāo)簽ylabel(V); %為Y軸加標(biāo)簽zlabel(c(u,v); %為z軸加標(biāo)簽figure; %新建圖形窗口surf(Udata,Vdata,reshape(Ccdf_t,size(Udata); %繪制二元t-Copula分布函數(shù)圖xlabel(U); %為X軸加標(biāo)簽ylabel(V); %為Y軸加標(biāo)簽zlabel(C(u,v); %為z軸加標(biāo)簽

13、%*求Kendall秩相關(guān)系數(shù)和Spearman秩相關(guān)系數(shù)*%調(diào)用copulastat函數(shù)求二元正態(tài)Copula對(duì)應(yīng)的Kendall秩相關(guān)系數(shù)Kendall_norm = copulastat(Gaussian,rho_norm)%調(diào)用copulastat函數(shù)求二元正態(tài)Copula對(duì)應(yīng)的Spearman秩相關(guān)系數(shù)Spearman_norm = copulastat(Gaussian,rho_norm,type,Spearman)%調(diào)用copulastat函數(shù)求二元t-Copula對(duì)應(yīng)的Kendall秩相關(guān)系數(shù)%*繪制Copula的密度函數(shù)和分布函數(shù)圖*Kendall_t = copulasta

14、t(t,rho_t)%調(diào)用copulastat函數(shù)求二元t-Copula對(duì)應(yīng)的Spearman秩相關(guān)系數(shù)Spearman_t = copulastat(t,rho_t,type,Spearman)%直接根據(jù)滬、深兩市日收益率的原始觀測(cè)數(shù)據(jù)，調(diào)用Kendall = corr(X,Y,type,Kendall)%直接根據(jù)滬、深兩市日收益率的原始觀測(cè)數(shù)據(jù)，調(diào)用Spearman = corr(X,Y,type,Spearman)corr函數(shù)求Kendall秩相關(guān)系數(shù)corr函數(shù)求Spearman秩相關(guān)系數(shù)%*模型評(píng)價(jià)*%調(diào)用ecdf函數(shù)求X和Y的經(jīng)驗(yàn)分布函數(shù)fx, Xsort = ecdf(X);fy

15、, Ysort = ecdf(Y);%調(diào)用spline函數(shù)，利用樣條插值法求原始樣本點(diǎn)處的經(jīng)驗(yàn)分布函數(shù)值U = spline(Xsort(2:end),fx(2:end),X);V = spline(Ysort(2:end),fy(2:end),Y);%定義經(jīng)驗(yàn)Copula函數(shù)C(u,v)C = (u,v)mean(U = u).*(V = v);%為作圖的需要，產(chǎn)生新的網(wǎng)格數(shù)據(jù)Udata,Vdata = meshgrid(linspace(0,1,31);%通過(guò)循環(huán)計(jì)算經(jīng)驗(yàn)Copula函數(shù)在新產(chǎn)生的網(wǎng)格點(diǎn)處的函數(shù)值for i=1:numel(Udata)CopulaEmpirical(i) =C(Udata(i),Vdata(i);endfigure; %新建圖形窗口%繪制經(jīng)驗(yàn)Copula分布函數(shù)圖像surf(Udata,Vdata,reshape(CopulaEmpirical,size(Udata)xlabel(U); %為X軸加標(biāo)簽ylabel(V); %為Y軸加標(biāo)簽zlabel(Empirical Copula C(u,v); %為z軸加標(biāo)簽%通過(guò)循環(huán)計(jì)算經(jīng)驗(yàn)Copula函數(shù)在原始樣本點(diǎn)處的函數(shù)值CUV = zeros(size(U(:);for i=1:numel(U)CUV(i) = C(U(i),V(i);end%計(jì)算線性相關(guān)參數(shù)為0.9264的二元正態(tài)Cop