2.80 3.20 3.70
13.39 16.53 21.20 2.8 3.2 3.6 13.3 16.5 20.2 3.2 3.7 0.5 0.5 6-12 炼焦炉的焦化时间y与炉宽x1及烟道管相对温度x2的数据如下：
y/min 6.40 15.05 18.75 30.25 44.85 48.94 51.55 61.50 100.44 111.42 x1/m 1.32 2.69 3.56 4.41 5.35 6.20 7.12 8.87 9.80 10.65 x2 1.15 3.40 4.10 8.75 14.82 15.15 15.32 18.18 35.19 40.40 求回归方程y?b0?b1x?b2x2，检验显著性，并讨论x1，x2对y的影响。 解：利用最小二乘法：
所求的回归方程为：y?b0?b1x?b2x2
式中的b0，b1，b2是回归方程的回归系数。对每一组的x1，x2 可以确定一个回归值yt，实际测得值yt与回归值yt之差就是残余误差vt， 利用矩阵形式：令 ?6.40??11.32?15.05??12.69????18.75??13.56???30.25???14.41?44.85??15.35Y??? X???48.94??16.20?51.55??17.12????61.50??18.87???100.44???19.88???110.65?111.42??1.15??v1??v?3.40??2???v3?4.10????8.75??v4?b0??v5?14.82?? V???? b? b＝??1?v15.15??6???b2???v?15.32??7??18.18??v8????35.19??v9??40.40???v10??????则误差方程化成矩阵形式为V=Y-Xb
根据最小二乘原理，回归系数的矩阵解为
-1Tb＝（XTX）XY=CB
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??N??NTA=XX???x1t?t＝1?N??x2t?t＝1?x1tt＝1NN?xt＝121t?xt＝1N1t2tx?x?2t?t＝1?N?x1tx2t??t＝1?N?2x?2t?t＝1?1.15?3.40??4.10??8.75?14.82??15.15?15.32??18.18??35.19?40.40??N?11.32?12.69??13.56?14.41111111111???1?15.35??＝?1.322.693.564.415.356.207.128.879.8010.65???16.20??1.153.404.108.7514.8215.1515.3218.1835.1940.40???17.12??18.87??19.88??110.6559.97156.46??10?＝?59.97446.99651282.5215????156.461282.52153991.1208??
?3?y?t???t＝1?3??B???x1tyt?＝XTY?t＝1??3?xy??2tt??t＝1??6.40??15.05????18.75???30.25??111111111??1?44.85??? ??1.322.693.564.415.356.207.128.879.8010.65??? 48.94????1.153.404.108.7514.8215.1515.3218.1835.1940.40???51.55????61.50????100.44???111.42???489.15?? ??3875.9365????11749.8781?? 27
C=A-1 ?11059.97156.4659.97446.99651282.5215156.461282.52153991.1208?1.3916?105－1.3915?105?41.5431?104?－3.8648?10?6.9757?103?3.4423?103?6.9757?105???3.4423?103?873.5641???0.8532?0.23720.0428?? ＝??0.23720.0946?0.0211????0.0428?0.02110.0054??b=CB?0.8532?0.23720.0428??489.15???3875.9365? ???0.23720.0946?0.0211??????0.0428?0.02110.0054????11749.8781???0.5800????2.7122????2.0497??所求回归方程为：
y＝0.5800＋2.7122x1＋2.0497x2检验回归方程的显著性：
??0＝M?2?Q?N?M?1?10?2?1?7U??(y－y)＝?bLjy＝b1l1y＋b2l2y2tj＝1?M ＝2.6997?942.5040＋2.0453?4096.6372 ＝10923.3301Q?Lyy－U?10956.9961?10923.3301＝33.6660F?U/M10923.3301/2??1135.6162Q/(N?M?1)33.6660/7
由于 F?1135.6162?F0.01(2.7)?12.25因此回归方程高度显著。 b12b122.66972P???77.02791?2C11l22(l11l22?l12)0.09462F1?P2?F2?P77.02791/1??16.02Q/(N?M?1)33.6660/7bb2.0453???781.04032C22l11/(l11l22?l12)0.005356P2/1781.0403??162.40Q/(N?M?1)33.6660/722222
F2?162.04?F0.01(1,7)?12.25故x1，由于 F1?16.02?F0.01(1,7)?12.25，x2对y均为主要因
素。
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