第章时间序列预测习题答案

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)1981110.211991347.5711982120.491992376.0211983132.871993440.451984141.291994532.981985153.621995574.93■1986184.21996700.431987195.721997766.391988214.0719981154.761989265.9419991085.761990307.84(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2000年的支出额详细答案：(1)时间序列图如下：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：

(3) 。10.2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1981145119911215198213721992128119831168199313091984123219941296198512451995141619861200199613671987126019971479198810201998127219891095199914691 11990126020001519(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：I(3)由Excel输出的指数平滑预测值如下表:年份单位面积产量指数平滑预测a=0.3误差平方指数平滑预测a=0.5误差平方19811451

198213721451.06241.01451.06241.0198311681427.367236.51411.559292.3198412321349.513808.61289.83335.1198512451314.34796.51260.9252.0198612001293.58738.51252.92802.4198712601265.429.51226.51124.3198810201263.859441.01243.249833.6198910951190.79151.51131.61340.8199012601162.09611.01113.321518.4199112151191.4558.11186.7803.5199212811198.56812.41200.86427.7199313091223.27357.61240.94635.8199412961249.02213.11275.0442.8199514161263.123387.71285.517035.9199613671308.93369.91350.7264.4199714791326.423297.71358.914431.3199812721372.210031.01418.921589.8199914691342.116101.51345.515260.3200015191380.219272.11407.212491.7合计291455.2239123.02001年a=0.3时的预测值为：a=0.5时的预测值为：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）129510473228311470332212481435513449528614544

1一637915601738116587843117644942418660(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：(2)1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12952283295.0144.0295.0144.0295.0144.0

3322291.4936.4290.21011.2289.01089.04355300.62961.5302.92712.3305.52450.35286316.9955.2323.81425.2330.31958.16379307.65093.1308.74949.0308.15023.37381329.02699.4336.81954.5343.61401.68431344.67459.6354.55856.2362.34722.39424370.52857.8385.11514.4396.6748.510473386.67468.6400.75234.4410.33928.711470412.53305.6429.61632.9441.7803.112481429.82626.2445.81242.3455.8633.513449445.115.0459.9117.8468.4376.914544446.39547.4455.57830.2458.77274.815601475.615724.5490.912120.5501.49929.416587513.25443.2534.92709.8551.21283.317644535.411803.7555.87785.2569.15611.718660567.98473.4591.14752.7606.52857.5合计87514.762992.550236由Excel输出的指数平滑预测值如下表：a=0.3时的预测值:,误差均方=87514.7a=0.4时的预测值：,误差均方=62992.5.a=0.5时的预测值:

,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：1回归统计MultipleR0.9673RSquare0.93561AdjustedRSquare0.9316标准误差31.66281 观测值181方差分析dfSSMSFSignificanceF1回归分析1232982.5232982.5232.39445.99E-11残差1616040.491002.531-总计17249022.91Coefficients标准误差tStatP-valueLower95%Upper95%Intercept239.7320315.5705515.39655.16E-11206.7239272.74011XVariable121.9287931.43847415.244495.99E-1118.8793624.97822估计标准误差10.4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）1981171.361991708.001982196.961992792.961983223.541993957.771984263.1719941278.181985316.7019951467.061986379.9319961704.251987402.7519971903.591988486.1019982154.381989553.3319992408.06

1990617.2920002736.88(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。经线性变换后，利用Excel输出的回归结果如下：回归统计MultipleR0.998423RSquare0.996849AdjustedRSquare0.9966741标准误差0.022125观测值20

方差分析dfSSMSFSignificanceF回归分析12.7876162.7876165694.8855.68E-24残差180.0088110.000489总计192.796427Coefficients标准误差tStatP-valueLower95%Upper95%1Intercept2.1636990.010278210.52695.55E-322.1421062.185291XVariable10.0647450.00085875.464465.68E-240.0629420.0665475 >曲线方程为：2001年的预测值为：10.5我国1964年〜1999年的纱产量数据如下（单位：万吨）：所以，指数年份1964纱产量年份纱产量年份纱产量97.01976196.01988465.71965130.01977223.01989476.719661967156.51978238.21990462.6135.21979263.51991460.81968137.71980292.61992501.81969180.51981317.01993501.51970205.21982335.41994489.51971190.01983327.01995542.31972188.61984321.91996512.21973196.71985353.51997559.81974180.31986397.81998542.011975210.81987436.81999567.0(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:(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:2000年预测值为：=585.65（万吨）13.6对下面的数据分别拟合线性趋势线、二阶曲线和阶次曲线对结果进行比较。时间t观测值Y时间t观测值Y137219360237020357337421356437522352537723348637724353737425356

1一83722635693732735610372283591136929360123673035713367313571436532355153633335616359343631735835365118359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：1直线二阶曲线三阶曲线1Intercept374.1613Intercept381.6442Intercept372.56171XVariable1-0.6137XVariable1-1.8272XVariable11.00301XVariable20.0337XVariable2-0.16011XVariable30.0036各趋势方程为：线性趋势:阶曲线:三阶曲线：根据趋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11372373.52.4379.961.6373.42.02370372.98.6378.166.0374.015.63374372.32.8376.56.1374.20.14375371.710.8374.90.0374.20.65377371.134.9373.413.3374.08.96377370.542.5371.926.1373.611.67374369.917.1370.512.2373.01.18372369.37.6369.27.9372.20.09373368.619.0367.925.7371.23.110372368.015.8366.727.6370.23.311369367.42.5365.611.4369.00.012367366.80.0364.65.9367.70.6■13367366.20.7363.611.6366.40.314365365.60.3362.75.4365.10.015363365.03.8361.81.4363.70.516359364.328.5361.04.2362.311.117358363.732.8360.35.4361.08.918359363.116.9359.70.5359.70.519360362.56.3359.10.8358.42.420357361.923.9358.62.5357.30.121356361.327.8358.14.6356.30.122352360.775.0357.833.2355.411.3i23348360.0145.1357.589.3354.643.724353359.441.4357.217.7354.01.1

25356358.87.9357.01.1353.75.526356358.24.9356.90.9353.56.327356357.62.5356.90.8353.65.928359357.04.1356.94.4353.925.829360356.413.2357.09.0354.529.830357355.71.6357.20.0355.52.331357355.13.5357.40.2356.70.132355354.50.2357.77.2358.311.033356353.94.4358.14.2360.318.434363353.394.2358.520.4362.70.135365352.7151.8359.036.2365.40.2.合计854.9524.7232.1阶曲线:三阶曲线：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图也可以看出，三阶曲线与原序列的拟合最好

10.7下表是1981—2000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)19816.22199110.8719826.66199211.1619837.15199311.5019847.89199412.4019858.72199513.6119868.94199613.9719879.28199713.7319889.80199812.50198910.54199910.45199010.8020009.98(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：2001年的预测值为：10.8一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1997—2001年各月份的外销定单金额(单位：万元)。年/月199719981999200020011154.349.156.764.461.1246.650.452.054.569.4362.659.361.768.076.5458.258.561.471.971.6557.460.062.469.474.6656.655.663.667.769.9i756.158.063.268.071.4852.955.863.966.372.7954.655.863.267.869.91051.359.863.471.574.21154.859.464.470.572.71252.155.563.869.472.5(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)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：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1997—2001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：用Excel采用指数平滑法(a=0.4)预测的预测结果为：10.91993—2000年我国社会消费品零售总额数据如下(单位：亿元)月/年199319941995199619971998199920001977.51192.21602.21909.12288.52549.52662.12774.72892.51162.71491.51911.22213.52306.42538.42805.03942.31167.51533.31860.12130.92279.72403.12627.04941.31170.41548.71854.82100.52252.72356.82572.05962.21213.71585.41898.32108.22265.22364.02637.061005.71281.11639.71966.02164.72326.02428.82645.07963.81251.51623.61888.72102.52286.12380.32597.08959.81286.01637.11916.42104.42314.62410.92636.091023.31396.21756.02083.52239.62443.12604.32854.0101051.11444.11818.02148.32348.02536.02743.93029.0111102.01553.81935.22290.12454.92652.22781.53108.0121415.51932.22389.52848.62881.73131.43405.73680.0(1)绘制时间序列线图，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: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1971.04393056.303190.482980.99393077.503058.873990.95933098.712972.4841000.93983119.922931.9951010.94393141.132964.8861020.95893162.333032.3071030.92873183.542956.4381040.92613204.752967.8691050.98143225.963166.05101061.00753247.163271.51111071.04723268.373422.77

121081.26943289.584175.9510.101995年〜2000年北京市月平均气温数据如下(单位:月/年199519961997199819992000L111-0.7-2.2-3.8-3.9-1.6-6.422.1-0.41.32.42.2-1.537.76.28.77.64.88.1414.714.314.515.014.414.6519.821.620.019.919.520.4624.325.424.623.625.426.7「一725.925.528.226.528.129.6825.423.926.625.125.625.7919.020.718.622.220.921.81014.512.814.014.813.012.6117.74.25.44.05.93.012-0.40.9-1.50.1-0.6-0.6(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：设月份为则季节性多元回归模型为：设月份为虚拟变量为：

由Excel输出的回归结果如下：系数b0-0.2233bl-0.0030M1-2.7832M21.3365M37.5062M414.9092M520.5289M625.3319M727.6349M825.7213M920.8743M1013.9606M115.3803季节性多元回归方程为：

2001年各月份平均气温的预测值如下：年/月时间虚拟变量预测M1M2M3M4M5M6M7M8M9M10M1117310000000000-3.2274010000000000.9广一375001000000007.14760001000000014.55770000100000020.16780000010000024.97790000001000027.28800000000100025.39810000000010020.410820000000001013.51183000000000014.9128400000000