數(shù)學(xué)建模報(bào)告公司的銷售額預(yù)測.doc

公司的銷售額預(yù)測一、問題重述某公司想用全行業(yè)的銷售額作為自變量來預(yù)測公司的銷售量,下表給出了19771981年公司的銷售額和行業(yè)銷售額的分季度數(shù)據(jù)(單位：百萬元)年季公司銷售額行業(yè)銷售額年季公司銷售額行業(yè)銷售額19771120.96127.3197931124.54148.32221.40130.041224.30146.43321.96132.7198011325.00150.24421.52129.421425.64153.119781522.39135.031526.36157.32622.76137.141626.98160.73723.48141.2198111727.52164.24823.66142.821827.78165.619791924.10145.531928.24168.721024.01145.342028.78171.7表1 公司的公司銷售額和行業(yè)銷售額的分季度數(shù)據(jù)(單位：百萬元)(1)畫出數(shù)據(jù)的散點(diǎn)圖,觀察用線性回歸模型擬合是否合適.(2)建立公司銷售額對全行業(yè)的回歸模型,并用DW檢驗(yàn)診斷隨機(jī)誤差項(xiàng)的自相關(guān)性.(3) 建立消除了隨機(jī)誤差項(xiàng)自相關(guān)性之后的回歸模型. 二、問題分析與假設(shè)銷售收入預(yù)測的方法主要有時(shí)間序列法、因果分析法和本量利分析法等.時(shí)間序列法,是按照時(shí)間的順序,通過對過去幾期實(shí)際數(shù)據(jù)的計(jì)算分析,確定預(yù)測期產(chǎn)品銷售收入的預(yù)測值.表1 的數(shù)據(jù)是以時(shí)間順序?yàn)樾蛄械?稱為時(shí)間序列.由于公司銷售額和行業(yè)銷售額等經(jīng)濟(jì)變量均有一定的滯后性,因此,在這樣的時(shí)間序列數(shù)據(jù)中,同一變量的順序觀測值之間出現(xiàn)相關(guān)現(xiàn)象是很自然的.然而,一旦數(shù)據(jù)中存在這種自相關(guān)序列,如果仍采用普通的回歸模型直接處理,將會出現(xiàn)不良后果,其觀測也會失去意義,為此,我們必須先來檢驗(yàn)數(shù)據(jù)是否存在自相關(guān),一旦存在,就要考慮自相關(guān)關(guān)系,建立新的模型.定義與符號說明行業(yè)銷售額公司銷售額公司銷售額的估計(jì)值三、模型建立與求解一、基本統(tǒng)計(jì)回歸模型建立以行業(yè)銷售額為自變量、以公司銷售額為因變量的散點(diǎn)圖,其中圖1 對的散點(diǎn)圖 從圖1可以看出,隨著行業(yè)銷售額的增加,公司銷售額也增加,而且兩者有很強(qiáng)的線性關(guān)系,因此可以建立線性回歸模型,為隨機(jī)誤差 假設(shè)與是相互獨(dú)立的,且服從均值為零的正態(tài)分布.由表1的數(shù)據(jù)以及上述線性回歸模型的假設(shè),進(jìn)行數(shù)據(jù)處理,得到回歸系數(shù)估計(jì)值及其置信區(qū)間和檢驗(yàn)統(tǒng)計(jì)量,見表2.參數(shù)參數(shù)估計(jì)值置信區(qū)間-1.45475-1.90465 -1.004850.1762830.173248 0.179318 表2 模型的計(jì)算結(jié)果將參數(shù)估計(jì)值代入得到, 由表2知,幾乎處處可由確定.用作出其交互式畫面,由此可以給出不同水平下的預(yù)測值及其置信區(qū)間,通過左方的下拉式菜單,可以輸出模型的統(tǒng)計(jì)結(jié)果,見圖2.圖2 回歸分析中的交互式畫面二、自相關(guān)性的判別我們可以看到模型的擬合度很高(),即可認(rèn)為可由模型確定.但此模型并未考慮到我們的數(shù)據(jù)是一個(gè)時(shí)間序列.在對時(shí)間序列數(shù)據(jù)做回歸分析時(shí),模型的隨機(jī)誤差項(xiàng)可能存在相關(guān)性,違背于模型對獨(dú)立的基本假設(shè).現(xiàn)在我們考慮如下模型： 其中是自相關(guān)系數(shù),相互獨(dú)立且服從均值為0的正態(tài)分布.模型中,若,則退化為普通的回歸模型;若,則隨機(jī)誤差存在正的自相關(guān);若,則隨機(jī)誤差存在負(fù)的自相關(guān).大多數(shù)與經(jīng)濟(jì)有關(guān)的時(shí)間序列數(shù)據(jù),在經(jīng)濟(jì)規(guī)律作用下,一般隨著時(shí)間的推移有一種向上或向下的變動趨勢,其隨機(jī)誤差表現(xiàn)出正相關(guān)性.檢驗(yàn)是一種常用的診斷自相關(guān)現(xiàn)象的統(tǒng)計(jì)方法.首先根據(jù)模型得到的殘差,計(jì)算統(tǒng)計(jì)量如下： 其中是觀察值個(gè)數(shù),殘差為隨機(jī)誤差項(xiàng)的估計(jì)值.當(dāng)較大時(shí), 而式的右端正是自相關(guān)系數(shù)的估計(jì)值,于是 由于,所以,并且若在0附近時(shí),則在2附近,的自相關(guān)性很弱(或不存在自相關(guān)性);若在附近時(shí),則在0或4附近,的自相關(guān)性很強(qiáng).要根據(jù)的具體數(shù)值來確定是否存在自相關(guān)性,應(yīng)該在給定的檢驗(yàn)水平下依照樣本容量和回歸變量數(shù)目,查分布表,得到檢驗(yàn)的臨界和,然后由表3中所在的區(qū)間來決定.2正自相關(guān)不能確定無自相關(guān)不能確定負(fù)自相關(guān)表3 與值相對應(yīng)的自相關(guān)狀態(tài)三、加入自相關(guān)后的模型根據(jù)式可計(jì)算出,對于顯著性水平,查分布表,得到檢驗(yàn)的臨界值和,現(xiàn)在,由表3可以認(rèn)為隨機(jī)誤差存在正自相關(guān),且的估計(jì)值可由式得.作變換, 則模型化為 ,其中, 以的估計(jì)值代入式作變換,利用變換后的數(shù)據(jù)、估計(jì)模型的參數(shù),得到的結(jié)果見表4,可以得到其剩余標(biāo)準(zhǔn)差為0.067.變換變量和回歸系數(shù)估計(jì)回歸系數(shù)置信區(qū)間-0.74396 -0.038870.16748 0.18000 表4 模型的計(jì)算結(jié)果對模型也做一次自相關(guān)檢驗(yàn),即診斷隨機(jī)誤差是否還存在自相關(guān),從模型的殘差可計(jì)算出,對于顯著性水平以及時(shí),檢驗(yàn)的臨界值為,故,所以可以認(rèn)為隨機(jī)誤差不存在自相關(guān).因此經(jīng)變換得到的回歸模型是適用的.最后,將模型中的和還原為原始變量和,得到結(jié)果為： 四、結(jié)果分析與預(yù)測從機(jī)理上看,對于帶滯后性的經(jīng)濟(jì)規(guī)律作用下的時(shí)間序列數(shù)據(jù),加入自相關(guān)的模型更為合理,而且在本例中,衡量與實(shí)際數(shù)據(jù)擬合程序的指標(biāo)剩余標(biāo)準(zhǔn)差從模型的0.081減少到0.0671.當(dāng)用模型對公司的銷售額作預(yù)測時(shí),先估計(jì)未來的全行業(yè)銷售額,比如,設(shè)t=21時(shí),=174.1,容易由模型得到=29.1860.四 、模型的評價(jià)一、模型的優(yōu)點(diǎn)經(jīng)檢驗(yàn)認(rèn)為普通回歸模型的隨機(jī)誤差存在自相關(guān),由,式估計(jì)出自相關(guān)系數(shù)后,采用變換的方法得到模型,成稱為廣義差分法.這種方法消除了原模型隨機(jī)誤差的自相關(guān)性,得到的式是一階自相關(guān)模型.二、模型的缺點(diǎn)檢驗(yàn)和廣義差分法在經(jīng)濟(jì)數(shù)據(jù)建模中有著廣泛的應(yīng)用,但是也存在著明顯的不足：若的數(shù)值落在無法確定自相關(guān)性的區(qū)間,則只能設(shè)法增加數(shù)據(jù)量,或選用其他方法;如果原始數(shù)據(jù)序列存在高階自相關(guān)性,則需要反復(fù)使用檢驗(yàn)和廣義差分,直至判定不存在自相關(guān)為止.另外,分布表中數(shù)據(jù)容量的下限是15.參考文獻(xiàn)1 徐金明,張孟喜,丁濤,北京:清華大學(xué)出版社;北京交通大學(xué)出版社,2005.7(2007.8重印).2. 姜啟源,謝金星、葉俊,數(shù)學(xué)模型(第四版),北京:高等教育出版社,20011.1(2012.5重印).附錄1. 散點(diǎn)圖的程序clear;x=127.3 130.0 132.7 129.4 135.0 137.1 141.2 142.8 145.5 145.3 148.3 146.4 150.2 153.1 157.3 160.7 164.2 165.6 168.7 171.7;y=20.96 21.40 21.96 21.52 22.39 22.76 23.48 23.66 24.10 24.01 24.54 24.30 25.00 25.64 26.36 26.98 27.52 27.78 28.24 28.78;plot(x,y,*)2.模型(1)的計(jì)算程序clear;x=127.3 130.0 132.7 129.4 135.0 137.1 141.2 142.8 145.5 145.3 148.3 146.4 150.2 153.1 157.3 160.7 164.2 165.6 168.7 171.7;y=20.96 21.40 21.96 21.52 22.39 22.76 23.48 23.66 24.10 24.01 24.54 24.30 25.00 25.64 26.36 26.98 27.52 27.78 28.24 28.78;x=ones(20,1),x;b,bint,r,rint,stats=regress(y,x)輸出結(jié)果:b = -1.45475004139634 0.176282811457384bint = -1.90465420468789 -1.00484587810479 0.173247525367995 0.179318097546773r = -0.0260518571286674 -0.0620154480636046 0.0220209610014628 0.163754238810824 0.046570494649476 0.0463765905889701 0.043617063613695 -0.058435434718124 -0.0943990256530576 -0.149142463361581 -0.147990897733738 -0.0530535559647056 -0.0229282395027646 0.105851607270822 0.0854637991498031 0.106102240194701 0.0291124000938581 0.0423164640535205 -0.0441602514643726 -0.0330086858365206rint = -0.195385871759251 0.143282157501916 -0.231871886037631 0.107840989910421 -0.152853753861225 0.19689567586415 0.0131634673388031 0.314345010282846 -0.128820728276878 0.22196171757583 -0.130374632908482 0.223127814086423 -0.135221944162416 0.222456071389806 -0.236666572408848 0.119795702972601 -0.26910531130757 0.0803072600014548 -0.313617420748614 0.0153324940254506 -0.31288219186159 0.0169003963941133 -0.232314787682902 0.126207675753491 -0.203701178032592 0.157844699027063 -0.0664228030147936 0.278126017556437 -0.0879446576592735 0.25887225595888 -0.0620923905849254 0.274296870974328 -0.144034956949209 0.202259757136925 -0.128748679311955 0.213381607418996 -0.211614739705066 0.123294236776321 -0.197152337494454 0.131134965821413stats = Columns 1 through 3 0.998792444207198 14888.1435565111 1.01315527327091e-027 Column 4 0.007405683407079543. 散點(diǎn)圖的交互式程序clear;x=127.3 130.0 132.7 129.4 135.0 137.1 141.2 142.8 145.5 145.3 148.3 146.4 150.2 153.1 157.3 160.7 164.2 165.6 168.7 171.7;y=20.96 21.40 21.96 21.52 22.39 22.76 23.48 23.66 24.10 24.01 24.54 24.30 25.00 25.64 26.36 26.98 27.52 27.78 28.24 28.78;rstool(x,y,linear)4.模型(2)的殘差x=127.3 130.0 132.7 129.4 135.0 137.1 141.2 142.8 145.5 145.3 148.3 146.4 150.2 153.1 157.3 160.7 164.2 165.6 168.7 171.7;y=20.96 21.40 21.96 21.52 22.39 22.76 23.48 23.66 24.10 24.01 24.54 24.30 25.00 25.64 26.36 26.98 27.52 27.78 28.24 28.78;for i=1:1:20z(i)=-1.45475+0.17628*x(i);e(i)=y(i)-z(i);endze輸出結(jié)果:z = Columns 1 through 3 20.985694 21.46165 21.937606 Columns 4 through 6 21.355882 22.34305 22.713238 Columns 7 through 9 23.435986 23.718034 24.19399 Columns 10 through 12 24.158734 24.687574 24.352642 Columns 13 through 15 25.022506 25.533718 26.274094 Columns 16 through 18 26.873446 27.490426 27.737218 Columns 19 through 20 28.283686 28.812526e = Columns 1 through 3 -0.0256939999999979 -0.0616500000000002 0.0223940000000056 Columns 4 through 6 0.164118000000002 0.0469500000000025 0.0467620000000046 Columns 7 through 9 0.0440140000000042 -0.0580339999999993 -0.093989999999998 Columns 10 through 12 -0.148733999999997 -0.147574000000002 -0.0526419999999987 Columns 13 through 15 -0.0225059999999964 0.106282000000004 0.0859059999999978 Columns 16 through 18 0.106554000000003 0.0295740000000038 0.0427820000000025 Columns 19 through 20 -0.0436859999999974 -0.03252599999999365.計(jì)算DW和e =-0.0256939999999979 -0.0616500000000002 0.02239400000000560.164118000000002 0.0469500000000025 0.04676200000000460.0440140000000042 -0.0580339999999993 -0.093989999999998-0.148733999999997 -0.147574000000002 -0.0526419999999987-0.0225059999999964 0.106282000000004 0.08590599999999780.106554000000003 0.0295740000000038 0.0427820000000025-0.0436859999999974 -0.0325259999999936;s=0;for t=2:1:20s=s+(e(t)-e(t-1)2;endm=0;for i=1:1:20m=m+e(i)2;endDW=s/mp=1-1/2*DW輸出結(jié)果:DW = 0.734645539224993p = 0.6326772303875036.模型(3)中的數(shù)據(jù)變換x=127.3 130.0 132.7 129.4 135.0 137.1 141.2 142.8 145.5 145.3 148.3 146.4 150.2 153.1 157.3 160.7 164.2 165.6 168.7 171.7;y=20.96 21.40 21.96 21.52 22.39 22.76 23.48 23.66 24.10 24.01 24.54 24.30 25.00 25.64 26.36 26.98 27.52 27.78 28.24 28.78;p=0.63268for t=1:1:19y1(t)=y(t+1)-p*y(t);x1(t)=x(t+1)-p*x(t);endy1x1輸出結(jié)果:p = 0.63268y1 = Columns 1 through 3 8.1390272 8.420648 7.6263472 Columns 4 through 6 8.7747264 8.5942948 9.0802032 Columns 7 through 9 8.8046736 9.1307912 8.762412 Columns 10 through 12 9.3493532 8.7740328 9.625876 Columns 13 through 15 9.823 10.1380848 10.3025552 Columns 16 through 18 10.4502936 10.3686464 10.6641496 Column 19 10.9131168x1 = Columns 1 through 3 49.459836 50.4516 45.443364 Columns 4 through 6 53.131208 51.6882 54.459572 Columns 7 through 9 53.465584 55.153296 53.24506 Columns 10 through 12 56.371596 52.573556 57.575648 Columns 13 through 15 58.071464 60.436692 61.179436 Columns 16 through 18 62.528324 61.713944 63.928192 Column 19 64.9668847.模型(3)的計(jì)算結(jié)果y1 =8.1390272 8.420648 7.62634728.7747264 8.5942948 9.08020328.8046736 9.1307912 8.7624129.3493532 8.7740328 9.6258769.823 10.1380848 10.302555210.4502936 10.3686464 10.664149610.9131168;x1 =49.459836 50.4516 45.44336453.131208 51.6882 54.45957253.465584 55.153296 53.2450656.371596 52.573556 57.57564858.071464 60.436692 61.17943662.528324 61.713944 63.92819264.966884;x2=ones(19,1),x1;b,bint,r,rint,stats=regress(y1,x2)輸出結(jié)果:b = -0.391413728791626 0.173739484728835bint = -0.743959505289538 -0.0388679522937137 0.167481672152938 0.179997297304732r = -0.0626854926210676 0.0466267410463246 0.122454283086727 -0.0648485721489394 0.00542729423064792 0.00983895095872533 -0.0929956860946302 -0.060100299345299 -0.0969435599642878 -0.0532051135904403 0.0313439989890654 0.0141263123428352 0.125107495982524 0.0292588019963098 0.0646852421508761 -0.0219314639260322 0.0379112976474367 -0.0512878089344184 0.0172175781936197rint = -0.194112737017012 0.0687417517748767 -0.0885336593085636 0.181787141401213 0.0161292648746043 0.228779301298849 -0.201462271956211 0.0717651276583323 -0.133792997199822 0.144647585661118 -0.131797031575754 0.151474933493204 -0.225302669864897 0.0393112976756366 -0.1985101568317 0.0783095581411017 -0.228268383451548 0.0343812635229727 -0.192519727444524 0.0861095002636433 -0.107973506382257 0.170661504360388 -0.127481451736758 0.155734076422428 -0.000122557504840748 0.250337549469889 -0.109224013348295 0.167741617340915 -0.0693312030770827 0.198701687378835 -0.157373208239308 0.113510280387243 -0.0980543981169386 0.173876993411812 -0.181348373667138 0.0787727557983009 -0.112564188002673 0.146999344389912stats = Columns 1 through 3 0.995069834050953 34314.68398711682588e-021 Column 4 0.004508149250504388.模型(3)的殘量clear; y1 =8.1390272 8.420648 7.62634728.7747264 8.5942948 9.08020328.8046736 9.1307912 8.7624129.3493532 8.7740328 9.6258769.823 10.1380848 10.302555210.4502936 10.3686464 10.664149610.9131168;x1 =49.459836 50.4516 45.44336453.131208 51.6882 54.45957253.465584 55.153296 53.2450656.371596 52.573556 57.57564858.071464 60.436692 61.17943662.528324 61.713944 63.92819264.966884;for i=1:1:19z(i)=-0.39141+0.17374*x1(i);e(i)=y1(i)-z(i);endze輸出結(jié)果:z = Columns 1 through 3 8.20174190664 8.374050984 7.50392006136 Columns 4 through 6 8.83960607792 8.588897868 9.07039603928 Columns 7 through 9 8.89770056416 9.19092364704 8.8593867244 Columns 10 through 12 9.40259108904 8.74271961944 9.61178308352 Columns 13 through 15 9.69792615536 10.10886086808 10.23790521064 Colu