第0章时间序列预测习题答案

1、第10章时间序列预测教材习题答案10.1下表是1981年一1999年国家财政用于农业的支出额数据年份支出额（亿元）年份支出额（亿元）1981110.211991347.571982120.491992376.021983132.871993440.451984141.291994532.981985153.621995574.931986184.21996700.431987195.721997766.391988214.0719981154.761989265.9419991085.761990307.84（1）绘制时间序列图描述其形态。（2）计算年平均增长率。（3）根据年平均增长率预测20

2、00年的支出额详细答案：（1）时间序列图如下：140012001000%lE00支4QQ'2000138119阴13851987198319311333199513971999从时间序列图可以看出，国家财政用于农业的支出额大体上呈指数上升趋势。(2)年平均增长率为：G=J108576-1=113.55%-1=13.55%彳托V110.21(3)名qqq=1057拆冥(1+134%)=122*210.2下表是1981年一2000年我国油彩油菜籽单位面积产量数据(单位：kg/hm2)年份单位面积产量年份单位面积产量19811451199112151982137219921281198311

3、68199313091984123219941296198512451995141619861200199613671987126019971479198810201998127219891095199914691990126020001519(1) 绘制时间序列图描述其形态(2) 用5期移动平均法预测2001年的单位面积产量。(3)采用指数平滑法，分别用平滑系数a=0.3和a=0.5预测2001年的单位面积产量，分析预测误差，说明用哪一个平滑系数预测更合适？详细答案：(1)时间序列图如下：（2）2001年的预测值为：严1367+1479+1272+1469+15197106小ooi=1421

4、.2I(3)由Excel输出的指数平滑预测值如下表:指数平滑预测指数平滑预测年份单位面积产量误差平方误差平方a=0.3a=0.519811451198213721451.06241.01451.06241.0198311681427.367236.51411.559292.3198412321349.513808.61289.83335.1198512451314.34796.51260.9252.0198612001293.58738.51252.92802.4198712601265.429.51226.51124.3198810201263.859441.01243.249833.619

5、8910951190.79151.51131.61340.8199012601162.09611.01113.321518.4199112151191.4558.11186.7803.5199212811198.56812.41200.86427.7199313091223.27357.61240.94635.8199412961249.02213.11275.0442.8199514161263.123387.71285.517035.9199613671308.93369.91350.7264.4199714791326.423297.71358.914431.3199812721372.

6、210031.01418.921589.8199914691342.116101.51345.515260.3200015191380.219272.11407.212491.7合计291455.2239123.02001年a=0.3时的预测值为:耳冋二叱+1_氐）啓=0.3x1519+（1-0.3）x1380.2=1421.8a=0.5时的预测值为：嘔=曲+（1_閃绡=0.5x1519+（1-0.5）x1407.1=1463.1比较误差平方可知，a=0.5更合适。10.3下面是一家旅馆过去18个月的营业额数据月份营业额（万元）月份营业额（万元）129510473228311470332212

7、481435513449528614544637915601738116587843117644942418660（1）用3期移动平均法预测第19个月的营业额。（2）采用指数平滑法，分别用平滑系数a=0.3、a=0.4和a=0.5预测各月的营业额，分析预测误差，说明用哪一个平滑系数预测更合适？（3）建立一个趋势方程预测各月的营业额，计算出估计标准误差。详细答案:(1)第19个月的3期移动平均预测值为:587+644+66018=630.333(2)月份营业额预测a=0.3误差平方预测a=0.4误差平方预测a=0.5误差平方12952283295.0144.0295.0144.0295.0144

8、.03322291.4936.4290.21011.2289.01089.04355300.62961.5302.92712.3305.52450.35286316.9955.2323.81425.2330.31958.16379307.65093.1308.74949.0308.15023.37381329.02699.4336.81954.5343.61401.68431344.67459.6354.55856.2362.34722.39424370.52857.8385.11514.4396.6748.510473386.67468.6400.75234.4410.33928.71147

9、0412.53305.6429.61632.9441.7803.112481429.82626.2445.81242.3455.8633.513449445.115.0459.9117.8468.4376.914544446.39547.4455.57830.2458.77274.815601475.615724.5490.912120.5501.49929.416587513.25443.2534.92709.8551.21283.317644535.411803.7555.87785.2569.15611.718660567.98473.4591.14752.7606.52857.5合计8

10、7514.762992.550236由Excel输出的指数平滑预测值如下表：a=0.3时的预测值:竝=0.3価0+(1-0.3)山19=刃豊，误差均方=87514.7。a=0.4时的预测值：=0.4x660+(1-0.4)x591.1=618.7,误差均方=62992.5.。a=0.5时的预测值：理=宀豌+(1一°”亦4=633.3，误差均方=50236。比较各误差平方可知，a=0.5更合适。(3)根据最小二乘法，利用Excel输出的回归结果如下：方差分析回归分析1232982.5232982.5232.39445.99E-11残差1616040.491002.53总计Interce

11、ptdf17Coefficients239.73203SS249022.9标准误差15.57055MStStat15.3965P-value5.16E-11SignificanceFLower95%206.7239Upper95%272.7401XVariable121.9287931.43847415.244495.99E-1118.8793624.97822*二239.73+21.9238。估计标准误差旳=31.662810.4下表是1981年一2000年我国财政用于文教、科技、卫生事业费指出额数据年份支出（万元）年份支出（万元）1981171.361991708.001982196.96

12、1992792.961983223.541993957.771984263.1719941278.181985316.7019951467.061986379.9319961704.251987402.7519971903.591988486.1019982154.381989553.3319992408.061990617.2920002736.88(1)绘制时间序列图描述其趋势。(2) 选择一条适合的趋势线拟合数据，并根据趋势线预测2001年的支出额。详细答案：(1)趋势图如下：(2)从趋势图可以看出，我国财政用于文教、科技、卫生事业费指出额呈现指数增长趋势，因此，选择指数曲线。经线性变换

13、后，利用Excel输出的回归结果如下：方差分析回归分析12.7876162.7876165694.8855.68E-24残差180.0088110.000489总计InterceptXVariable1df19Coefficients2.1636990.064745log血）=2.163699俎=145.78SS2.796427标准误差0.0102780.000858MStStat210.526975.46446P-value5.55E-325.68E-24SignificanceFLower95%2.1421060.062942Upper95%2.1852910.066547=侈，俎二1.1

14、亦。所以，指数曲线方程为:Yt=145.78x1.1608:2001年的预测值为:4oi=145-78x116Os21=3338910.5我国1964年1999年的纱产量数据如下（单位：万吨）：年份纱产量年份纱产量年份纱产量196497.01976196.01988465.71965130.01977223.01989476.71966156.51978238.21990462.61967135.21979263.51991460.81968137.71980292.61992501.81969180.51981317.01993501.51970205.21982335.41994489.5

15、1971190.01983327.01995542.31972188.61984321.91996512.21973196.71985353.51997559.81974180.31986397.81998542.01975210.81987436.81999567.0（1）绘制时间序列图描述其趋势。（2）选择一条适合的趋势线拟合数据，并根据趋势线预测2000年的产量。详细答案：（1）趋势图如下：196496197097397697398985983991994997年份（2）从图中可以看出，纱产量具有明显的线性趋势。用Excel求得的线性趋势方程为:X=69.5202+13.9495

16、3;2000年预测值为:忌皿宓2+曲宓心"注=585.65（万吨）。4小213.6对下面的数据分别拟合线性趋势线打=俎+知、二阶曲线和阶次曲线42孑。并对结果进行比较。时间t观测值Y时间t观测值Y1372193602370203573374213564375223525377233486377243537374253568372263569373273561037228359113692936012367303571336731357143653235515363333561635934363173583536518359详细答案:在求二阶曲线和三阶曲线时，首先将其线性化，然后用最小

17、二乘法按线性回归进行求解。用Excel求得的趋势直线、二阶曲线和三阶曲线的系数如下：直线二阶曲线三阶曲线Intercept374.1613Intercept381.6442Intercept372.5617XVariable1-0.6137XVariable1-1.8272XVariable11.0030XVariable20.0337XVariable2-0.1601XVariable30.0036各趋势方程为：线性趋势：=374.1613-0.6137二阶曲线：=381.6442-1.8272+0.0337?三阶曲线：?=372.5617+1.0030-0.160U2+0.0036?根据趋

18、势方程求得的预测值和预测误差如下表：时间t观测值Y直线二阶曲线三阶曲线预测误差平方预测误差平方预测误差平方1372373.52.4379.961.6373.42.02370372.98.6378.166.0374.015.63374372.32.8376.56.1374.20.14375371.710.8374.90.0374.20.65377371.134.9373.413.3374.08.96377370.542.5371.926.1373.611.67374369.917.1370.512.2373.01.18372369.37.6369.27.9372.20.09373368.619.

19、0367.925.7371.23.110372368.015.8366.727.6370.23.311369367.42.5365.611.4369.00.012367366.80.0364.65.9367.70.613367366.20.7363.611.6366.40.314365365.60.3362.75.4365.10.015363365.03.8361.81.4363.70.516359364.328.5361.04.2362.311.117358363.732.8360.35.4361.08.918359363.116.9359.70.5359.70.519360362.56.3

20、359.10.8358.42.420357361.923.9358.62.5357.30.121356361.327.8358.14.6356.30.122352360.775.0357.833.2355.411.323348360.0145.1357.589.3354.643.724353359.441.4357.217.7354.01.125356358.87.9357.01.1353.75.526356358.24.9356.90.9353.56.327356357.62.5356.90.8353.65.928359357.04.1356.94.4353.925.829360356.41

21、3.2357.09.0354.529.830357355.71.6357.20.0355.52.331357355.13.5357.40.2356.70.132355354.50.2357.77.2358.311.033356353.94.4358.14.2360.318.434363353.394.2358.520.4362.70.135365352.7151.8359.036.2365.40.2合计854.9524.7232.1不同趋势线预测的标准误差如下:二阶曲线:三阶曲线:i-ln-m2321=2.7435-4比较各预测误差可知，直线的误差最大，三阶曲线的误差最小。从不同趋势方程的预测

22、图也可以看出，三阶曲线与原序列的拟合最好。390380370玖360测廿350现测恒Y直线二阶曲歩匸阶曲些¥340111I1111111111111111113301471013161333£5283134时问10.7下表是19812000年我国的原煤产量数据年份原煤产量(亿吨)年份原煤产量(亿吨)19816.22199110.8719826.66199211.1619837.15199311.5019847.89199412.4019858.72199513.6119868.94199613.9719879.28199713.7319889.80199812.501989

23、10.54199910.45199010.8020009.98(1) 绘制时间序列图描述其趋势。(2) 选择一条适合的趋势线拟合数据，并根据趋势线预测2001年的产量详细答案：(1)原煤产量趋势图如下:从趋势图可以看出，拟合二阶曲线比较合适。(2)用Excel求得的二阶曲线趋势方程为：=4.5824+0.9674-0.03092001年的预测值为：r2001=4.5824+0.9674x21-0.0309x212=11.2710.8一家贸易公司主要经营产品的外销业务，为了合理地组织货源，需要了解外销订单的变化状况。下表是19972001年各月份的外销定单金额(单位：万元)。年/月1997199

24、8199920002001154.349.156.764.461.1246.650.452.054.569.4362.659.361.768.076.5458.258.561.471.971.6557.460.062.469.474.6656.655.663.667.769.9756.158.063.268.071.4852.955.863.966.372.7954.655.863.267.869.91051.359.863.471.574.21154.859.464.470.572.71252.155.563.869.472.5(1) 根据各年的月份数据绘制趋势图，说明该时间序列的特点。(2

25、) 要寻找各月份的预测值，你认为应该采取什么方法？(3) 选择你认为合适的方法预测2002年1月份的外销订单金额详细答案：(1)趋势图如下：时间从趋势图可以看出，每一年的各月份数据没有趋势存在，但从19972001年的变化看，订单金额存在一定的线性趋势。（2）由于是预测各月份的订单金额，因此采用移动平均法或指数平滑法比较合适。（3）用Excel采用12项移动平均法预测的结果为：耳眦门=714。用Excel采用指数平滑法（a=0.4）预测的预测结果为：也眦门=化3。10.919932000年我国社会消费品零售总额数据如下（单位：亿元）月/年1993199419951996199719981999

26、20001977.51192.21602.21909.12288.52549.52662.12774.72892.51162.71491.51911.22213.52306.42538.42805.03942.31167.51533.31860.12130.92279.72403.12627.04941.31170.41548.71854.82100.52252.72356.82572.05962.21213.71585.41898.32108.22265.22364.02637.061005.71281.11639.71966.02164.72326.02428.82645.07963.81

27、251.51623.61888.72102.52286.12380.32597.08959.81286.01637.11916.42104.42314.62410.92636.091023.31396.21756.02083.52239.62443.12604.32854.0101051.11444.11818.02148.32348.02536.02743.93029.0111102.01553.81935.22290.12454.92652.22781.53108.0121415.51932.22389.52848.62881.73131.43405.73680.0（1）绘制时间序列线图，

28、说明该序列的特点。（2）利用分解预测法预测2001年各月份的社会消费品零售总额。详细答案：(1)趋势图如下:4000?50030002500零2000售1500駆1000500°193313105127294116£00(1肘间从趋势图可以看出，我国社会消费品零售总额的变具有明显的季节变动和趋势。(2)利用分解法预测的结果如下：2001年/月时间编号季节指数回归预测值最终预测值1971.04393056.303190.482980.99393077.503058.873990.95933098.712972.4841000.93983119.922931.9951010.9

29、4393141.132964.8861020.95893162.333032.3071030.92873183.542956.4381040.92613204.752967.8691050.98143225.963166.05101061.00753247.163271.51111071.04723268.373422.77121081.26943289.584175.9510.101995年2000年北京市月平均气温数据如下(单位:叱):月/年1995199619971998199920001-0.7-2.2-3.8-3.9-1.6-6.422.1-0.41.32.42.2-1.537.76

30、.28.77.64.88.1414.714.314.515.014.414.6519.821.620.019.919.520.4624.325.424.623.625.426.7725.925.528.226.528.129.6825.423.926.625.125.625.7919.020.718.622.220.921.81014.512.814.014.813.012.6117.74.25.44.05.93.012-0.40.9-1.50.1-0.6-0.6(1) 绘制年度折叠时间序列图，判断时间序列的类型。(2) 用季节性多元回归模型预测2001年各月份的平均气温详细答案：(1)年度折

31、叠时间序列图如下：从年度折叠时间序列图可以看出，北京市月平均气温具有明显的季节变动。由于折线图中有交叉，表明该序列不存在趋势。（2）季节性多元回归模型为：设月份为岖°=1忆,12）。则季节性多元回归模型为：?=俎+bxt+b2MY+b3M2+b5M+b6M5+Eg就'了呂+3四艇9+B姻"Q+I虚拟变量为:第1月份其他月份1第2月份0其他月份1第11月份0其他月份由Excel输出的回归结果如下:M21.3365M37.5062M414.9092M520.5289M625.3319M727.6349M825.7213M920.8743M1013.9606M115.3803季节性多元回归方程为：¥=-0.2233-0.0030-2.7832+.3365M2+7.5062A>f3+14.9092A>f4+20.52S9A>f5+25.3319M6+21.6349M7+25.7213AS+20.8743M9+13.9606Mlo+5.33Mn2001年各月份平均气温的预测值如下：7790000001000027.28800000000100025.39810000000010020.410