预测与决策：第三讲 时间序列平滑预测法

第三讲

时间序列平滑预测法时间序列的构成移动平均法指数平滑法时间序列的构成

所谓时间序列，是指各种经济、社会、自然现象的数量指标按照时间顺序排列起来的统计数据。时间序列的构成因素：长期趋势、季节变动、循环变动、随机变动

1、移动平均法在预测实践中,人们有时采用一种朴素的预测方法,即简单地用最近一期数值作为下一期的预测值。如果时间序列资料包含有大量的随机成分,采用这种方法将产生较大的偏差。为了消除这些随机变动因素的影响,人们对朴素预测法做了适当的改进,从而产生了移动平均法。什么是移动平均法？

移动平均法是根据时间序列数据，逐项推移，依次计算包含一定项数的时序平均值，以反映长期趋势的方法。移动平均法的种类

一次移动平均法加权移动平均法二次移动平均法

1.1

一次移动平均法一次移动平均法是收集一组观察值，计算这组观察值的均值，利用这一均值作为下一期的预测值。在移动平均值的计算中包括的过去观察值的实际个数，必须一开始就明确规定。

例1

分析预测我国平板玻璃月产量。时间序号实际观测值三个月移动平均值五个月移动平均值

2009.12009.22009.32009.42009.52009.62009.72009.82009.92009.102009.112009.12123456789101112203.8214.1229.9223.7220.7198.4207.8228.5206.5226.8247.8259.5---215.9222.6224.8214.6209.0211.6214.3220.6227.0-----218.4217.4216.1215.8212.4213.6223.5下表是我国2009-2010年平板玻璃月产量，试选用N=3和N=5用一次移动平均法进行预测。计算结果列入表中。一次移动平均法计算公式：式中：t≥N，为时间序列的数据，为t期一次移动平均值，N为移动平均的项数如果时间序列没有明显的周期变化和趋势变化，可用t期的一次移动平均值作为t+1期的预测值，即预测模型为每出现一个新观察值，就要从移动平均中减去一个最早观察值，再加上一个最新观察值，计算移动平均值，这一新的移动平均值就作为下一期的预测值。一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.45513.86612.97714.08814.49915.3101014.7111116.5121214.72009113预测值一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.45513.86612.97714.08814.49915.3101014.7111116.5121214.72009113预测值=(15.4+16.5+14.7)/3一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.86612.97714.08814.49915.3101014.7111116.5121214.72009113预测值=16.2-15.4一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.86612.97714.08814.49915.3101014.7111116.5121214.72009113预测值=(16.5+14.7+16.2)/3一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.97714.08814.49915.3101014.7111116.5121214.72009113预测值=13.8-15.8一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.07714.08814.49915.3101014.7111116.5121214.72009113预测值一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.07714.014.3-0.38814.49915.3101014.7111116.5121214.72009113预测值一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.07714.014.3-0.38814.413.60.89915.313.81.5101014.714.60.1111116.514.81.7121214.715.5-0.82009113预测值一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.07714.014.3-0.38814.413.60.89915.313.81.5101014.714.60.1111116.514.81.7121214.715.5-0.82009113预测值15.3一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.015.2-2.37714.014.3-0.38814.413.60.89915.313.81.5101014.714.60.1111116.514.81.7121214.715.5-0.82009113预测值15.3=(15.4+16.5+14.7+16.2+13.8)/5=12.9-15.2一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.015.2-2.37714.014.3-0.314.8-0.88814.413.60.89915.313.81.5101014.714.60.1111116.514.81.7121214.715.5-0.82009113预测值15.3一次移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.015.2-2.37714.014.3-0.314.8-0.88814.413.60.814.3-0.19915.313.81.514.31.0101014.714.60.114.10.6111116.514.81.714.32.2121214.715.5-0.815.0-0.32009113预测值15.315.1一次移动平均法MSE(3)=1.68MSE(5)=1.75年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.015.2-2.37714.014.3-0.314.8-0.88814.413.60.814.3-0.19915.313.81.514.31.0101014.714.60.114.10.6111116.514.81.714.32.2121214.715.5-0.815.0-0.32009113预测值15.315.1一次移动平均法MSE(3)=1.68MSE(5)=1.75MSE(3)=1.68MSE(5)=1.75年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.015.2-2.37714.014.3-0.314.8-0.88814.413.60.814.3-0.19915.313.81.514.31.0101014.714.60.114.10.6111116.514.81.714.32.2121214.715.5-0.815.0-0.32009113预测值15.315.1一次移动平均法MSE(3)=1.68MSE(5)=1.75年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.9-2.015.2-2.37714.014.3-0.314.8-0.88814.413.60.814.3-0.19915.313.81.514.31.0101014.714.60.114.10.6111116.514.81.714.32.2121214.715.5-0.815.0-0.32009113预测值15.315.1一般情况下,移动平均的项数N取得越大,对原序列修匀的程度也越大,移动平均序列较平坦。反之,移动平均序列中,保留原始序列的特征较多。如何选择N？移动平均项数N的大小反映移动平均数的修匀性强弱和对趋势的灵敏度。选取时有两种情形：（1）如果时间序列的总趋势比较平稳，波动不大，则N应取大一点，使平滑的效果更显著。（2）如果时间序列的总趋势不稳定，波动较大，则N应取小一点，使之更适应当前变化的趋势。在实用上，常用几个N值进行试算，比较它们的均方误差MSE，选取均方误差较小的那个N。

一次移动平均法的三个主要限制:限制一：计算移动平均必须具有N个过去观察值，当需要预测大量的数值时，就必须存储大量数据；

限制二：简单移动平均法只能做一步预测,且仅适用于基本趋势呈平稳发展的序列,用于具有明显上升(或下降)趋势的时间序列作预测,会产生滞后误差。限制三：N个过去观察值中每一个权数都相等，而早于（t-N+1）期的观察值的权数等于0，而实际上往往是最新观察值包含更多信息，应具有更大权重。1.2加权移动平均法各期数据所包含的信息量并不相同。考虑到近期的水平要比远期的水平对未来趋势有更大的影响,可以采用加权移动平均的方式来计算移动平均值,即给近期数据以较大权数,而远期数据以较小的权数。

加权移动平均法式中：

，为t期的加权移动平均值； 为权数，它体现了相应的y在加权移动平均值中的重要程度。实际中常选用：

若以第t期的加权移动平均值作为第t+1期的预测值，则预测模型为：

加权移动平均法

(权数分别为3,2,1)年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.4(1)2216.5(2)3314.7(3)4416.215.40.85513.86612.97714.08814.49915.3101014.7111116.5121214.72009113预测值=(15.4*1+16.5*2+14.7*3)/6加权移动平均法年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.5(1)3314.7(2)4416.2(3)15.40.85513.815.8-2.06612.97714.08814.49915.3101014.7111116.5121214.72009113预测值=(16.5*1+14.7*2+16.2*3)/6加权移动平均法

年月t销售额N=3N=5预测值预测误差预测值预测误差20081115.42216.53314.74416.215.40.85513.815.8-2.06612.914.8-1.97714.013.80.28814.413.60.89915.314.01.3101014.7(1)14.8-0.1111116.5(2)14.91.6121214.7(3)15.7-1.02009113预测值15.3=(14.7*1+16.5*2+14.7*3)/6

1.3线性二次移动平均法基本原理为了避免利用移动平均法预测有线性趋势的数据时产生系统误差，发展了线性二次移动平均法。这种方法的基础是在对实际值进行一次移动平均的基础上，再进行一次移动平均。二次移动平均法二次移动平均法适用于时间序列具有线性趋势与周期波动的情况。一次移动平均值二次移动平均值预测模型其中：（τ为预测超前期数）二次移动平均法（N=4）年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.1199565019.8199675729.5199786531.8199896970.81999107498.52000118493.52001128922.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.8199675729.5199786531.8199896970.81999107498.52000118493.52001128922.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.5199786531.8199896970.81999107498.52000118493.52001128922.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.8199786531.85353.8199896970.86063.01999107498.56682.72000118493.57373.72001128922.77971.4200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.3199786531.85353.8199896970.86063.01999107498.56682.72000118493.57373.72001128922.77971.4200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.3199786531.85353.84033.5199896970.86063.01999107498.56682.72000118493.57373.72001128922.77971.4200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3816.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3816.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.7200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3816.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.78920.1632.47200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3816.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.78920.1632.47200213预测值200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3816.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.78920.1632.47200213预测值9552.9200314二次移动平均法年份t人均生活费支出199011646.1199121860.2199232134.7199342939.62145.2199454134.12767.2199565019.83557.1199675729.54455.83231.35680.3816.3199786531.85353.84033.5199896970.86063.04857.41999107498.56682.75638.82000118493.57373.76368.32001128922.77971.47022.78920.1632.47200213预测值9552.920031410185.0例2:某建材商店玻璃销售量如下表所示,试预测该店第二年2月份的销售量(取移动平均项数n=3)月份(t)销售量yt(箱)St’St’’atbt150____245____35249.0___45350.0___54851.050.052.01.065251.050.751.30.375451.351.151.50.285052.051.452.60.695553.052.153.90.9105653.752.954.50.8115154.053.654.40.4125855.054.255.80.8解:用线性二次移动平均法作预测。现在要预测第二年2月份的玻璃销售量,即预测t=14期的预测量。2、指数平滑法

指数平滑法是加权移动平均法的进一步发展和完善,它是由美国经济学家布朗(Brown)于1959年在其《库存管理的统计预测》一书中首先提出来的。指数平滑法是对时间序列由近及远采取具有逐步衰减性质的加权处理，是移动平均法的改进型。指数平滑法根据平滑次数的不同，可分为一次、二次、三次指数平滑法，分别适用于对不同类型的时间序列进行预测。一次指数平滑法是利用作为远期值得到预测的通式，即：令得可见这一方法是“自适应”的,且它通过现期的预测误差,自动修正下一期的预测值,且平滑常数α体现了修正的幅度。2.1一次指数平滑法一次指数平滑法是一种加权预测，权数为α。它既不需要存储全部历史数据，也不需要存储一组数据，从而可以大大减少数据存储问题，甚至有时只需一个最新观察值、最新预测值和α值，就可以进行预测。它提供的预测值是前一期预测值加上前期预测值中产生的误差的修正值。递推得到由此可见,第t+1期的预测值实际上是以前各期的观测值及初始预测值的加权平均值。所以,它是加权移动平均的推广。由于所加的一串权数均呈指数形式,并逐渐衰减,α越大衰减越快,反之越慢;而且这种平均方法具有修匀或平滑一系列观测值的作用,故称之为指数平滑法。使用指数平滑法进行预测,要解决好两个问题:一是平滑常数α的选择;二是初始预测值的确定。指数平滑系数α的确定当时间序列波动不大、较为平稳时，可取较小的α值。当时间序列具有明显的变动趋势时，可取较大的α值。实际应用中，可多取几个α值进行试算，选取使均方误差最小的α作为加权系数。(1)如果时间序列具有迅速而明显的变化倾向,则α宜取较大值(一般取0.3~0.6),加大近期数据的作用,使新近变化较强地反映在预测值中。(2)如果时间序列虽有不规则的起伏变化,但其长期变动趋势却较缓慢时,则需取较小的α值(一般取0.05~0.2)使较远期数据的影响也能比较充分地体现在观测值当中。(3)

在不容易判断时,可分别选用几个不同的α值进行试算,选用预测误差较小的α值。计算预测误差可选下述两个公式中的任何一个:

①平均绝对误差:

②均方标准差:初始值的确定当时间序列的样本容量n＞20时，初始值对预测结果影响很小，可选取时间序列第一期数据作为初始值。当时间序列的样本容量n≤20时，初始值对预测结果影响较大，应选取时间序列最初几期数据的均值作为初始值。一次指数平滑法年、月tya=0.2a=0.52008.11152216.53314.74416.25513.86612.977148814.49915.3101014.7111116.5121214.72009.113合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.92216.53314.74416.25513.86612.977148814.49915.3101014.7111116.5121214.72009.113合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.92216.514.923314.74416.25513.86612.977148814.49915.3101014.7111116.5121214.72009.113合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.90.012216.514.923314.715.244416.25513.86612.977148814.49915.3101014.7111116.5121214.72009.113合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.92216.514.923314.715.244416.215.135513.815.346612.915.03771414.68814.414.489915.314.46101014.714.63111116.514.64121214.715.012009.113预测值合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.92216.514.923314.715.244416.215.135513.815.346612.915.03771414.68814.414.489915.314.46101014.714.63111116.514.64121214.715.012009.113预测值14.95合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.90.012216.514.923314.715.244416.215.135513.815.346612.915.03771414.68814.414.489915.314.46101014.714.63111116.514.64121214.715.012009.113预测值14.95合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.90.012216.514.922.49643314.715.244416.215.135513.815.346612.915.03771414.68814.414.489915.314.46101014.714.63111116.514.64121214.715.012009.113预测值14.95合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.90.012216.514.922.49643314.715.240.29164416.215.131.14495513.815.342.37166612.915.034.5369771414.60.36098814.414.480.00649915.314.460.7056101014.714.630.0049111116.514.643.4596121214.715.010.09612009.113预测值14.95合计平均一次指数平滑法年、月tya=0.2a=0.52008.111514.90.012216.514.922.49643314.715.240.29164416.215.131.14495513.815.342.37166612.915.034.5369771414.60.36098814.414.480.00649915.314.460.7056101014.714.630.0049111116.514.643.4596121214.715.010.09612009.113预测值14.95合计15.484平均1.2903一次指数平滑法年、月tya=0.2a=0.52008.111514.90.0114.92216.514.922.49643314.715.240.29164416.215.131.14495513.815.342.37166612.915.034.5369771414.60.36098814.414.480.00649915.314.460.7056101014.714.630.0049111116.514.643.4596121214.715.010.09612009.113预测值14.95合计15.484平均1.2903一次指数平滑法年、月tya=0.2a=0.52008.111514.90.0114.92216.514.922.496414.973314.715.240.29164416.215.131.14495513.815.342.37166612.915.034.5369771414.60.36098814.414.480.00649915.314.460.7056101014.714.630.0049111116.514.643.4596121214.715.010.09612009.113预测值14.95合计15.484平均1.2903一次指数平滑法年、月tya=0.2a=0.52008.111514.90.0114.92216.514.922.496414.953314.715.240.291615.374416.215.131.144915.225513.815.342.371615.716612.915.034.536914.76771414.60.360913.838814.414.480.006413.929915.314.460.705614.16101014.714.630.004914.73111116.514.643.459614.72121214.715.010.096115.612009.113预测值14.9515.16合计15.484平均1.2903一次指数平滑法年、月tya=0.2a=0.52008.111514.90.0114.90.012216.514.922.496414.952.40253314.715.240.291615.371.06094416.215.131.144915.220.96045513.815.342.371615.713.64816612.915.034.536914.763.4595771414.60.360913.830.02898814.414.480.006413.920.23049915.314.460.705614.161.2996101014.714.630.004914.730.0009111116.514.643.459614.723.1684121214.715.010.096115.610.82812009.113预测值14.9515.16合计15.48417.09平均1.29031.4248例3:甲、乙两工厂的总产值如下表所列,试用一次指数平滑法分别预测这两个工厂第8期的总产值。时期t01234567甲厂2030404248505460乙厂2030402048305440解:分别选用α=0.1,0.3,0.9进行试算。由于数据较少,初始预测值选前三期预测值的平均值,甲、乙两厂前三期数据一样,故均选按公式计算,如α=0.1,甲厂第一期预测值:其余各值仿此计算,结果列于下表中工厂时间t实际值yt平滑预测值误差平滑预测值误差平滑预测值误差甲02030-1030-1030-1013029127-32192402911281229113423012321039344831173513426550331739114736543519421250476037234614546分析:甲厂总产值稳步发展,上升明显,应选用较大的α值,试算结果也证明了这一点。由表中记录的误差值算得预测均方标准差:α=0.1时,S1=16.2;α=0.3时,S2=11.9;α=0.9时,S3=7.6故应选α=0.9作为平滑常数,并预测甲厂第8期总产值为:工厂时间t实际值yt平滑预测值误差平滑预测值误差平滑预测值误差乙02030-1030-1030-1013029127321924029112812291132030-1032-1239-1944829112820222653031-134-445-1565433213321322274035539152-12分析:乙厂总产值忽高忽低,波动较大,长期趋势不明显,宜选用较小的α值,试算结果也证明了这一点。由表中记录的误差值算得预测均方标准差:α=0.1时,S