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Âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèèÐàññìîòðèì âûáîðêó îáú¼ìà n èç ãåíåðàëüíîé ñîâîêóïíîñòè çíà÷åíèé äâóìåðíîé ñëó÷àéíîé âåëè÷èíû (ξ; ζ), ò.å. n ïàð íàáëþäåíèé(xi ; yi ). Ïîñêîëüêó ìíîãèå çíà÷åíèÿ â ýòîé âûáîðêå ìîãóò ïîâòîðÿòüñÿ, èõ çàíîñÿò â òàê íàçûâàåìóþ êîððåëÿöèîííóþ òàáëèöó (òàáë. 94.4). ïåðâîì ñòîëáöå ýòîé òàáëèöû ïåðå÷èñëåíû çíà÷åíèÿ xi , âî âòîðîì yi â âèäå âàðèàöèîííûõ ðÿäîâ. Íà ïåðåñå÷åíèè i-é ñòðîêèÒàáëèöà 94.4Êîððåëÿöèîííàÿ òàáëèöàξ\ζ y1 y2ys ni·x1 n11 n12n1s n1·x2 n21 n22n2s n2·xkn·jnk1 nk2n·1 n·2nksn·snk·nè j -ãî ñòîëáöà ñîîòâåòñòâóþùàÿ ÷àñòîòà nij , ò.å.
êîëè÷åñòâî ðàç, êîòîðîå íàáëþäåíèå (xi ; yj ) âñòðåòèëîñü â âûáîðêå. Ïðè îáðàáîòêå êîððåëÿöèîííîé òàáëèöû â ïîñëåäíåì ñòîëáöå óêàçûâàþò ñóììó ÷àñòîòs∑ïî ñòðîêàì ni· =nij , à â ïîñëåäíåé ñòðîêå ñóììó ÷àñòîò ïîj=1ñòîëáöàì n·j =k∑nij . Ñóììà âñåõ ýëåìåíòîâ ïîñëåäíåãî ñòîëáöà èëèi=1ñòðîêè äàñò îáú¼ì âûáîðêèn=k ∑s∑nij =i=1 j=1k∑ni· =i=1s∑n·j .j=1Ïåðâûé è ïîñëåäíèé ñòîëáöû êîððåëÿöèîííîé òàáëèöû îáðàçóþòñòàòèñòè÷åñêîå ðàñïðåäåëåíèå âûáîðêè ñëó÷àéíîé âåëè÷èíû ξ , à ïåðâàÿ è ïîñëåäíÿÿ ñòðîêè îáðàçóþò âûáîðêó ñëó÷àéíîé âåëè÷èíû ζ .Îáðàáîòàâ èõ, êàê îïèñàíî â ï.
96.4 ïðåäûäóùåé ëåêöèè, ïîëó÷èì÷èñëîâûå õàðàêòåðèñòèêèk∑x=k∑ni· xii=1n,x2 =ni· x2ii=1n,Sx2 = x2 − x2 ,42Ëåêöèÿ 94. Ðåãðåññèîííûé àíàëèçs∑y=j=1Îïðåäåëåíèåíàçûâàåòñÿ:s∑n·j yjy2 =,n·j yj2j=1Sy2 = y 2 − y 2 .,nn∗94.1. Âûáîðî÷íûì êîýôôèöèåíòîì êîððåëÿöèè rxy∗rxy=xy − x · y,Sx · Syk ∑s∑ãäå(94.5)nij xi yji=1 j=1.(94.6)nÂûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè ÿâëÿåòñÿ ñòàòèñòè÷åñêîéîöåíêîé êîýôôèöèåíòà êîððåëÿöèè, ðàññìîòðåííîãî â ëåêöèè 95, èîí îáëàäàåò ñëåäóþùèìè ñâîéñòâàìè, êîòîðûå ìû ïðèâåäåì áåç äîêàçàòåëüñòâà:∗∗1) rxy= ryx;2) Âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè íàõîäèòñÿ â ïðåäåëàõ îò∗−1 äî 1: −1 6 rxy6 1;∗3) |rxy | = 1 òîãäà è òîëüêî òîãäà, êîãäà ìåæäó çíà÷åíèÿìè xi è yi∗èìååòñÿ ëèíåéíàÿ çàâèñèìîñòü. ×åì áëèæå rxyê íóëþ, òåì õóæå ýòàçàâèñèìîñòü àïïðîêñèìèðóåòñÿ ëèíåéíîé.xy =94.2. Óñëîâíûì ñðåäíèì yx íàçûâàþò ñðåäíåå àðèôìåòè÷åñêîå çíà÷åíèé ζ ïðè ôèêñèðîâàííîì çíà÷åíèè ξ = x.Äëÿ êîððåëÿöèîííîé òàáëèöû 94.4 óñëîâíîå ñðåäíåå yx ïîëó÷àåòñÿ óñðåäíåíèåì çíà÷åíèé ζ ïî ñòðîêå, ñîîòâåòñòâóþùåé ξ = x.S∑yj n1jÎïðåäåëåíèåÒàê, íàïðèìåð, yx1 =ñðåäíåå xy .j=1n1 ..
Àíàëîãè÷íî îïðåäåëÿåòñÿ óñëîâíîå ëåêöèè 95 áûëî ââåäåíî ïîíÿòèå ðåãðåññèè ζ íà ξ :M (ζ/ξ = x) = fζ/ξ (x) è ξ íà ζ : M (ξ/ζ = y) = Ψξ/ζ (y) è ïîëó÷åíûôîðìóëû ( ) è ( ) äëÿ ïðÿìûõ ñðåäíåêâàäðàòè÷åñêîé ðåãðåññèè.Íèæå áóäóò ââåäåíû èõ ñòàòèñòè÷åñêèå àíàëîãè.????Ëåêöèÿ 94. Ðåãðåññèîííûé àíàëèç4394.3. Âûáîðî÷íûå óðàâíåíèÿ ïðÿìûõñðåäíåêâàäðàòè÷åñêîé ðåãðåññèèÏî äàííûì íàáëþäåíèé íàä äâóìåðíîé ñëó÷àéíîé âåëè÷èíîé, ïðåäñòàâëåííûõ â êîððåëÿöèîííîé òàáëèöå 94.4, íàéä¼ì ìåòîäîì íàèìåíüøèõ êâàäðàòîâ âûáîðî÷íîå óðàâíåíèå ïðÿìîé ëèíèè ñðåäíåêâàäðàòè÷åñêîé ðåãðåññèè ζ íà ξ :yx = ρ∗ζ/ξ · x + b∗ . ñîîòâåòñòâèè ñ ýòèì ìåòîäîì êîýôôèöèåíòû ïîäáèðàþòñÿ òàê,n∑∗∗÷òîáû îáåñïå÷èòü ìèíèìóì ôóíêöèè Φ(ρζ/ξ ; b ) =(ρ∗ζ/ξ ·xi +b∗ −yi )2 ,i=1âûðàæàþùåé ñóììó êâàäðàòîâ îòêëîíåíèé yi îò f (xi ). Ïðèðàâíÿâ ê∂Φ∂Φíóëþ ÷àñòíûå ïðîèçâîäíûåè, ïîëó÷èì ñèñòåìó ëèíåéíûõ∗∂ρζ/ξ∂b∗óðàâíåíèé (94.3) äëÿ íàõîæäåíèÿ êîýôôèöèåíòîâ ρ∗ζ/ξ è b∗ , îáåñïå÷èâàþùèõ ìèíèìóì äàííîé ôóíêöèè: ∂Φ ∗ 2= 0,∗ ∂ρ∗ζ/ξ ρζ/ξ x + b x = xi yi ,⇐⇒⇐⇒ ∗∗ρζ/ξ x + b n = yi , ∂Φ = 0,∗∂bxy − x · y∗, ρζ/ξ = 2x − x2(94.7) b∗ = y − ρ∗ x.ζ/ξ∗Êîýôôèöèåíò ρζ/ξ íàçûâàåòñÿ âûáîðî÷íûì êîýôôèöèåíòîì ðåãðåññèè ζ íà ξ .
Îí âûðàæàåòñÿ ÷åðåç âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè ïî ôîðìóëå:∗ Sy.(94.8)ρ∗ζ/ξ = rxySxÒàêèì îáðàçîì, óðàâíåíèå ïðÿìîé ïîëó÷èëîñü ñëåäóþùèì:∗yx = rxySy(x − x) + y .Sx(94.9)Îïðåäåëåíèå 94.3. Óðàâíåíèå(94.9) íàçûâàåòñÿ âûáîðî÷íûìóðàâíåíèåì ïðÿìîé ðåãðåññèè ζ íà ξ .44Ëåêöèÿ 94. Ðåãðåññèîííûé àíàëèçÇàìåòèì, ÷òî çíàê êîýôôèöèåíòà ðåãðåññèè ñîâïàäàåò ñî çíàêîì∗êîýôôèöèåíòà êîððåëÿöèè: åñëè rxy> 0, òî ëèíåéíàÿ ôóíêöèÿ ðå∗ãðåññèè ζ íà ξ âîçðàñòàåò, åñëè rxy < 0 óáûâàåò.Àíàëîãè÷íî ìîæíî ïîëó÷èòü âûáîðî÷íîå óðàâíåíèå ïðÿìîé ðåãðåññèè ξ íà ζ :∗ Sxxy = rxy(y − y) + x.(94.10)SyÎáà óðàâíåíèÿ äîïóñêàþò ñëåäóþùóþ ñèììåòðè÷íóþ ôîðìó çàïèñè:∗ Sy(x − x) ,yx − y = rxySx∗ Sxxy − x = rxy(y − y) .SyÎáå ïðÿìûå ïðîõîäÿò ÷åðåç òî÷êó (x; y), íàçûâàåìóþ öåíòðîìðàñïðåäåëåíèÿ.∗Îáå ïðÿìûå ñîâïàäàþò òîãäà è òîëüêî òîãäà, êîãäà |rxy| = 1.
Âýòîì ñëó÷àå, êàê óæå óïîìèíàëîñü, ìåæäó çíà÷åíèÿìè xi è yi èìååòñÿëèíåéíàÿ çàâèñèìîñòü. Ïðÿìàÿ ðåãðåññèè è ÿâëÿåòñÿ ýòîé çàâèñèìîñòüþ.94.3. Íàéòè âûáîðî÷íûå óðàâíåíèÿ ïðÿìûõ ðåãðåññèè èïîñòðîèòü èõ ãðàôèê äëÿ äàííûõ êîððåëÿöèîííîé òàáëèöû:Ïðèìåðξ\ζ6080100120n·j40 60 80 ni·5 3 8 2 19 217 6 1314 4 1826 12 22 60Ð å ø å í è å: Îáú¼ì âûáîðêè n = 60. Ïî ôîðìóëàì (93.2), (93.4),(94.5) îïðåäåëÿåì õàðàêòåðèñòèêè âûáîðêè:x=x2 =5620≈ 93,7;60552400≈ 9206,7;60Sx2 ≈ 433,2;y=3520≈ 58,6;60y2 =225600≈ 3760;60Sy2 ≈ 318,2;Ëåêöèÿ 94.
Ðåãðåññèîííûé àíàëèçSx ≈ 20,8;∗rxy=45Sy ≈ 17,8; xy =xy − x y≈ −0,55;Sx Sy∗ρ∗ξ/ζ = rxy·ρ∗ζ/ξ317600≈ 5293,3;60sy∗= rxy·≈ −0,47;sxsx≈ −0,64.syÂûáîðî÷íûå óðàâíåíèÿ ïðÿìûõ ðåãðåññèè íàõîäèì ïî ôîðìóëàì(94.9), (94.10):a)yx = −0,47x + 102,3,b)xy = −0,64y + 131,3.Èõ ãðàôèêè ïðèâåäåíû íà ðèñ. 9y58,7ab0x93,7Ðèñ. 9. Ïðÿìûå ðåãðåññèèÇàìå÷àíèå 94.1. Àíàëîãè÷íî ìåòîäîì íàèìåíüøèõ êâàäðàòîâìîæíî íàéòè êîýôôèöèåíòû êâàäðàòè÷íîé ðåãðåññèè ζ íà ξ , ò.å.ïîäîáðàòü êîýôôèöèåíòû êâàäðàòè÷íîé ôóíêöèè f (x) = ax2 + bx + cn∑òàê, ÷òîáû îáåñïå÷èòü ìèíèìóì ôóíêöèè Φ = (ax2i + bxi + c − yi )2 .i=1Çíà÷åíèÿ êîýôôèöèåíòîâ âûáîðî÷íîé êâàäðàòè÷íîé ðåãðåññèè îïðåäåëÿþòñÿ èç ñèñòåìû (94.4).46Ëåêöèÿ 94. Ðåãðåññèîííûé àíàëèçÀíàëîãè÷íûì îáðàçîì ìîæíî îïðåäåëèòü êîýôôèöèåíòû ïàðàáîëè÷åñêîé êîððåëÿöèè òðåòüåãî (ò.å., f (x) = ax3 + bx2 + cx + d) è áîëååâûñîêèõ ïîðÿäêîâ, ÷òî, âïðî÷åì, èñïîëüçóåòñÿ ðåäêî.Èíîãäà íà ïðàêòèêå ïðèõîäèòñÿ èññëåäîâàòü äëÿ äàííîé òð¼õìåðíîé âûáîðêè çàâèñèìîñòü z = f (x; y) òàêóþ, ÷òî çíà÷åíèÿ zi∗ = f (xi ; yi ),âû÷èñëåííûå äëÿ íàáëþäàåìûõ çíà÷åíèé xi è yi , áëèçêè â ñìûñëåìèíèìóìà ñóììû êâàäðàòîâ îòêëîíåíèé ê íàáëþäàåìûì âûáîðî÷íûì çíà÷åíèÿì zi (òàê íàçûâàåìàÿ ìíîæåñòâåííàÿ ðåãðåññèÿ).
 ïðîñòåéøåì ñëó÷àå îïðåäåëÿþòñÿ êîýôôèöèåíòû ëèíåéíîé çàâèñèìîñòèz = ax + by + c. Ìåòîä íàèìåíüøèõ êâàäðàòîâ â ýòîì ñëó÷àå äà¼òñëåäóþùèå ôîðìóëû äëÿ êîýôôèöèåíòîâ:∗∗∗∗∗∗ryz− rxz· rxyrxz− ryz· rxySzSz,b=,··a=22∗∗SxSy1 − rxy1 − rxyc = z − ax − by.94.4. Íàéäåì âûáîðî÷íûå óðàâíåíèÿ ïðÿìûõ ðåãðåññèè ñïîìîùüþ ïðîãðàìì Maxima è Mathcad. Âî èçáåæàíèå ïóòàíèöû ñ ïåðåìåííûìè â óðàâíåíèÿõ ðåãðåññèè îáîçíà÷èì íàáëþäåíèÿ áîëüøèìèáóêâàìè X è Y .ÏðèìåðÊîððåëÿöèîííàÿ òàáëèöà äàíà äëÿ 4-õ çíà÷åíèé Xj è 3-õ çíà÷åíèé Yi .Maxima-ïðîãðàììà:(%i1) numer : true$ fpprintprec : 6$ nx : 4$ ny : 3$(%i5) X : makelist(40 + 20 * i, i, 1, nx);(%o5) [60, 80,100, 120](%i6) Y : makelist(20+20*j, j, 1, ny);(%o6)[40,60,80](%i7) m : matrix([5,0,3], [0,2,19], [7,6,0], [14, 4, 0]);(%i8) n : sum(sum(m[i,j], j, 1, ny), i, 1, nx);(%i9) nj : makelist(sum(m[i,j], j, 1, ny), i, 1, nx);(%o9)[8, 21, 13, 18](%i10) ni : makelist(sum(m[i,j], i, 1, nx), j, 1, ny);(%o10)[26, 12, 22](%i11) Mx : sum(nj[i]*X[i], i, 1, nx)/n;(%o11)93.6667(%i12) My : sum(ni[j]*Y[j], j, 1, ny)/n;(%i13) Sx2 : sum(nj[i]*(X[i]-Mx)^2, i, 1, nx)/n;(%o13)433.222(%i14) Sx : sqrt(Sx2);Ëåêöèÿ 94.
Ðåãðåññèîííûé àíàëèç47(%o14) 20.814(%i15) Sy2 : sum(ni[j]*(Y[j]-My)^2, j, 1, ny)/n;(%o15) 318.222(%i16) Sy : sqrt(Sy2);(%o16) 17.8388(%i17) xy : sum(sum(X[i]*Y[j]*m[i,j], j, 1, ny), i, 1, nx)/n;(%o17) 5293.33(%i18) R : (xy-Mx*My)/(Sx*Sy);(%o18) -0.5434(%i19) Rxy : R*Sy/Sx; Ryx:R*Sx/Sy;(%o19) -0.4658(%o20) -0.6341(%i21) yx(x) := Rxy*(x-Mx)+My$(%i22) xy(y):= Ryx*(y-My)+Mx$(%i23) wxplot2d([yx(x) ,xy(x)], [x, 50, 200],[y, 0, 80],[gnuplot_preamble, "set grid;"])$MathCad-ïðîãðàììà:ORIGIN := 1nx := 4ny := 3i := 1..nxj := 1..nyXi := 40 + 20 · iYj := 20 + 20 · jTTX = ( 60 80 100 120)Y = ( 40 60 80 )/*Ñîîòâåòñòâóþùèå ÷àñòîòû mi,j ïðèâåäåíû â ìàòðèöå m:*/5 0m := 7140 32 19 6 0 4 0/*Íàéäåì îáú¼ì âûáîðêè n è ñóììû ïî ñòðîêàì nj è ïî ñòîëáöàìni: */ ∑ ∑n :=mi,jn = 60jnji :=∑jimi,j8 21 nj = 13 18/*Íàéäåì âûáîðî÷íûå ñðåäíèå: */nij :=∑imi,j26ni = 12 2248Ëåêöèÿ 94.
Ðåãðåññèîííûé àíàëèçM x :=1 ∑·nji ·Xin iM x = 93.667 M y :=1 ∑·nij ·Yjn jM y = 58.667/*è âûáîðî÷íûå ñðåäíèå êâàäðàòè÷åñêèå îòêëîíåíèÿ: /√1 ∑Sx2 := ·nji ·(Xi −M x)2 Sx2 = 433.222 Sx := Sx2 Sx = 20.814n i√1 ∑nij ·(Yj −M y)2 Sy2 = 318.222 Sy := Sy2 Sy = 17.839Sy2 := ·n j/*Íàéäåì âûáîðî÷íûé êîýôôèöèåíò êîððåëÿöèè:*/xy − M x · M y1 ∑∑Xi ·Yj ·mi,j xy = 5293 R :=xy := ·n i jSx · SyR · SyR · Sxρξζ :=ρζξ = −0.466SxSy/*è âûáîðî÷íûå óðàâíåíèÿ ïðÿìûõ ðåãðåññèè:*/ρζξ :=yx(x) := ρζξ · (x − M x) + M yR = −0.543ρξζ = −0.634xy(y) := ρξζ · (y − M y) + M x/*Ïîñòðîèì èõ ãðàôèêè: */5041.67yx(x)xy(y)33.332516.678.33100130160190Ðèñ.
10. Ïðÿìûå ðåãðåññèè äëÿ ïðèìåðà 94.4220Ïðàêòè÷åñêîå çàíÿòèå 94. Òî÷å÷íûå è èíòåðâàëüíûå îöåíêè49Ïðàêòè÷åñêîå çàíÿòèå 94. Òî÷å÷íûåè èíòåðâàëüíûå îöåíêè ïàðàìåòðîâ ðàñïðåäåëåíèÿ94.1. Âûáîðêà çàäàíà â âèäå ðàñïðåäåëåíèÿ ÷àñòîò:xi 3 5 8 10 11ni 20 25 30 15 10Íàéòè ðàñïðåäåëåíèå îòíîñèòåëüíûõ ÷àñòîò.ÏðèìåðÐ å ø å í è å: Çäåñü îáú¼ì âûáîðêèn = 20 + 25 + 30 + 15 + 10 = 100.Íàéäåì îòíîñèòåëüíûå ÷àñòîòû:P1∗ = 20/100 = 1/5,P2∗ = 25/100 = 1/4,, P3∗ = 30/100 = 3/10,P4∗ = 15/100 = 3/20,P5∗ = 10/100 = 1/10.Òîãäà ðàñïðåäåëåíèå îòíîñèòåëüíûõ ÷àñòîò ïðèìåò âèä:xi3581011Pi∗ 1/5 1/4 3/10 3/20 1/10Èç ýòîé òàáëèöû íåòðóäíî óáåäèòüñÿ, ÷òî5∑Pi∗ = 1.i=194.2. Èç ãåíåðàëüíîé ñîâîêóïíîñòè èçâëå÷åíà âûáîðêàîáú¼ìà n = 80:xi 0,9 1 1,2 1,4 1,5ni 10 25 20 15 10Íàéòè íåñìåù¼ííóþ îöåíêó ãåíåðàëüíîãî ñðåäíåãî, ìàòåìàòè÷åñêîãî îæèäàíèÿ, âûáîðî÷íóþ äèñïåðñèþ, à òàêæå âûáîðî÷íîå ñðåäíååêâàäðàòè÷åñêîå îòêëîíåíèå.ÏðèìåðÐ å ø å í è å: Íåñìåù¼ííîé îöåíêîé ãåíåðàëüíîãî ñðåäíåãî ÿâëÿåòñÿ âûáîðî÷íîå ñðåäíåå.