时间序列分析第二章王燕第一到第三题习题解答

时间序列分析习题解答第二章P.332.3习题2.1考虑序列{1，2，3，4，5，…，20}：(1)判断该序列是否平稳；(2)计算该序列的样本自相关系数k^(k=1,2,…,6);(3)绘制该样本自相关图，并解释该图形。解：(1)由于不存在常数，使,tEXtT，所以该序列不是平稳序列。显然，该序列是按等步长1单调增加的序列。(2)1^=0.850002^=0.701503^=0.556024^=0.415045^=0.280086^=0.15263(3)样本自相关图该图横轴表示自相关系数，纵轴表示延迟时期数。该图的自相关系数递减的速度缓慢，在6期的延迟时期里，自相关系数一直为正，说明该序列是有单调趋势的非平稳序列。附：SAS程序如下：dataex2_1;inputfreq@@;cards;1234567891011121314151617181920;procarimadata=ex2_1;identifyvar=freqNlag=6;run;可得到上图的自相关图等内容,更多结果被省略。2.21975-1980年夏威夷岛莫那罗亚火山（MaunaLoa）每月释放的CO2数据如下（单位：ppm）见下表。330.45330.97331.64332.87333.61333.55331.90330.05328.58328.31329.41330.63331.63332.46333.36334.45334.82334.32333.05330.87329.24328.87330.18331.50332.81333.23334.55335.82336.44335.99334.65332.41331.32330.73332.05333.53334.66335.07336.33337.39337.65337.57336.25334.39332.44332.25333.59334.76335.89336.44337.63338.54339.06338.95337.41335.71333.68333.69335.05336.53337.81338.16339.88340.57341.19340.87339.25337.19335.49336.63337.74338.36（1）绘制该序列时序图，并判断该序列是否平稳；（2）计算该序列的样本自相关系数k^(k=1,2,…,24);（3）绘制该样本自相关图，并解释该图形。解：(1)该序列的时序图：CO2328329330331332333334335336337338339340341342time01JAN7501MAY7501SEP7501JAN7601MAY7601SEP7601JAN7701MAY7701SEP7701JAN7801MAY7801SEP7801JAN7901MAY7901SEP7901JAN8001MAY8001SEP8001JAN81由上图可以看出，CO2排量总体逐步上升，且以年为周期呈现出一定的周期性。故该序列是呈现带周期性的单调上升趋势，该序列不平稳。(2)该序列的样本自相关系数k^(k=1,2,…,24)如下图第三列Lag=1到24所示。(3)样本自相关图如上。解释如下：上图第三列数可看出样本自相关系数k^(k=0，1,2,…,24)。该图形显示序列自相关系数长期位于零轴的一边，这是具有单调趋势序列的典型特征，同时自相关图呈现出明显的正弦波动规律，这是具有周期变化规律的非平稳序列的典型特征。自相关图显示出带长期递增趋势的周期性质。附SAS程序如下：dataex2_2;inputCO2@@;time=intnx('month','01jan1975'd,_n_-1);formattimedate.;cards;330.45330.97331.64332.87333.61333.55331.90330.05328.58328.31329.41330.63331.63332.46333.36334.45334.82334.32333.05330.87329.24328.87330.18331.50332.81333.23334.55335.82336.44335.99334.65332.41331.32330.73332.05333.53334.66335.07336.33337.39337.65337.57336.25334.39332.44332.25333.59334.76335.89336.44337.63338.54339.06338.95337.41335.71333.68333.69335.05336.53337.81338.16339.88340.57341.19340.87339.25337.19335.49336.63337.74338.36;procgplotdata=ex2_2;plotCO2*time=1;symbol1c=blackv=stari=join;run;procarimadata=ex2_2;identifyvar=CO2Nlag=24;run;2.31945-1950年费城月度降雨量数据如下（单位：mm）见下表。——————————————————————————————————69.380.040.974.984.6101.1225.095.3100.648.3144.5128.338.452.368.637.1148.6218.7131.6112.881.831.047.570.196.861.555.6171.7220.5119.463.2181.673.964.8166.948.0137.780.5105.289.9174.8124.086.4136.931.535.3112.3143.0160.897.080.562.5158.27.6165.9106.792.263.226.277.052.3105.4144.349.5116.154.1148.6159.385.367.3112.859.4____________________________________________________________________（1）计算该序列的样本自相关系数k^(k=1,2,…,24)；（2）判断该序列的平稳性；（3）判断该序列的纯随机性。解：(1)该序列的样本自相关系数k^(k=1,2,…,24)如下图第三列Lag=1到24所示。(2)该序列基本平稳，时序图如下：rainfall0100200300timeJAN75APR75JUL75OCT75JAN76APR76JUL76OCT76JAN77APR77JUL77OCT77JAN78APR78JUL78OCT78JAN79APR79JUL79OCT79JAN80APR80JUL80OCT80JAN81(3)白噪声检验输出结果为：观察上面结果，由于延迟6，12，18，24时，0.14P0.37，所以该序列为非白噪声序列，但相关性不够显著。附SAS程序（画时序图、计算相关系数和白噪声检验）如下：dataex2_3;inputrainfall@@;time=intnx('month','01jan1975'd,_n_-1);formattimeMONYY5.;cards;69.380.040.974.984.6101.1225.095.3100.648.3144.5128.338.452.368.637.1148.6218.7131.6112.881.831.047.570.196.861.555.6171.7220.5119.463.2181.673.964.8166.948.0137.780.5105.289.9174.8124.086.4136.931.535.3112.3143.0160.897.080.562.5158.27.6165.9106.792.263.226.277.052.3105.4144.349.5116.154.1148.6159.385.367.3112.859.4;procgplot;plotrainfall*time=1;symbol1c=blackv=stari=join;run;procarima;identifyvar=rainfallnlag=24;run;