\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fmodern\fprq1 Courier New;}{\f4\fswiss\fprq2 Arial;}{\f5\froman\fcharset1 Times New Roman;}{\f6\fswiss\fprq2\fcharset1 Arial;}{\f7\fswiss Arial;}{\f8\froman\fprq2 Times New Roman;}} {\colortbl\red0\green0\blue0;\red0\green128\blue0;\red128\green0\blue128;\red0\green128\blue128;\red255\green0\blue0;\red128\green128\blue128;\red0\green0\blue255;} \deflang1031\pard\ri4\plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f3\fs20\cf0 Inhalt....: Regression \par Kategorie.: Arbeitsblatt \par Mathematik: Stochastik, Statistik \par MuPAD.....: 3.1.1 \par Datum.....: 2005-07-04 \par Autoren...: Anke Kusterer \par Autoren...: Anna Lena Stahr \par Funktionen: stats::linReg, stats::reg, plot::Scatterplot\plain\f3\fs20\cf0\b \par ________________________________________________________________________________ \par \plain\f5\fs22\cf0 \par \plain\f4\fs40\cf0\b Regression\plain\f4\fs36\cf0\b \par \plain\f5\fs22\cf0 \par \plain\f4\fs24\cf1\b Dieses Notebook f\'fchrt in die lineare und allgemeine Regression mit MuPAD ein.\plain\f5\fs22\cf0 \par \plain\f6\fs28\cf0 \par Wir betrachten ein Zufallsexperiment, dessen Ergebnis Y nicht nur vom Zufall, \par sondern auch vom Wert x einer sogenannten erkl\'e4renden Variablen abh\'e4ngt, \par und fragen nach dem Zusammenhang zwischen den Werten der erkl\'e4renden \par Variablen und dem Ergebnis Y des Zufallsexperiments. Zum Beispiel k\'f6nnten \par wir den Einfluss des K\'f6rpergewichts auf den Blutdruck oder den Einfluss der \par K\'f6rpergr\'f6\'dfe auf das K\'f6rpergewicht untersuchen.\plain\f4\fs28 Nat\'fcrlich haben nicht alle \par Menschen mit einer festen K\'f6rpergr\'f6\'dfe x dasselbe K\'f6rpergewicht. Das \par K\'f6rpergewicht h\'e4ngt also nicht nur von der K\'f6rpergr\'f6\'dfe x, sondern auch von \par der zuf\'e4llig ausgew\'e4hlten Person ab. In diesem Sinne ist das K\'f6rpergewicht Y \par als eine Zufallsvariable zu verstehen. \par Wird der Zusammenhang zwischen der erkl\'e4renden Variable und dem Ausgang \par des Zufallsexperiments durch eine Gleichung der Form Y=a+bx+N beschrieben \par - hierbei sind a und b reelle Zahlen und N ist eine normalverteilte Zufallsvariable \par mit Erwartungswert 0 und Varianz \plain\f6\fs28\cf0 s\plain\f4\fs19\b\up14 2 \plain\f4\fs28\b \plain\f4\fs28 - so spricht man von einem\plain\f8\fs24 \plain\f7\fs28\cf2\b\i linearen \par Regressionsmodell.\plain\f7\fs28 \plain\f6\fs28\cf0 Zwischen der erkl\'e4renden Variable x und dem \par Erwartungswert der Zufallsvariablen Y besteht dann eine lineare Abh\'e4ngigkeit: \par E(Y)=a+bx. Die Beeinflussung des Ergebnisses durch den Zufall wird also durch \par die Zufallsvariable N wiedergegeben. \par \par Die Parameter a und b werden \plain\f6\fs28\cf2\b\i Regressionskoeffizienten\plain\f6\fs28\cf0 genannt, die durch \par y=a+bx beschriebene Gerade hei\'dft \plain\f6\fs28\cf2\b\i Regressionsgerade\plain\f6\fs28\cf0 . \par Wie aber bestimmt man die Regressionskoeffizienten? \par \par Dazu f\'fchren wir bei verschiedenen Werten \plain\f4\fs28 x\plain\f4\fs19\dn5 1\plain\f4\fs28 ,\'85,x\plain\f4\fs19\dn5 n\plain\f6\fs28\cf0 der erkl\'e4renden Variablen \par unabh\'e4ngige Experimente durch und benennen die zugeh\'f6rigen Zufallsvariablen \par mit \plain\lang1036\f4\fs28 Y\plain\lang1036\f4\fs19\dn5 1\plain\lang1036\f4\fs28 ,\'85,Y\plain\lang1036\f4\fs19\dn5 n\plain\f6\fs28\cf0 . Ihre Ausg\'e4nge bezeichnen wir entsprechend mit \plain\f4\fs28\cf0 y\plain\f4\fs19\cf0\dn5 1\plain\f4\fs28\cf0 ,\'85,y\plain\f4\fs19\cf0\dn5 n\plain\f4\fs19\dn5 . \plain\f6\fs28\cf0 F\'fcr das \par Gesamtexperiment erhalten wir dann das Modell \plain\f4\fs28\cf0 Y\plain\f4\fs19\cf0\dn5 i\plain\f4\fs28\cf0 = a + bx\plain\f4\fs19\cf0\dn5 i\plain\f4\fs28\cf0 + N\plain\f4\fs19\cf0\dn5 i\plain\f6\fs28\cf0 , i=1,...,n, wobei \par \plain\f4\fs28\cf0 N\plain\f4\fs19\cf0\dn5 i\plain\f6\fs28\cf0 unabh\'e4ngige, normalverteilte Zufallsvariablen mit Erwartungswert 0 und \par Varianz s\plain\f4\fs19\b\up14 2 \plain\f6\fs28\cf0 sind. Die Zufallsvariablen \plain\lang1036\f4\fs28 Y\plain\lang1036\f4\fs19\dn5 1\plain\lang1036\f4\fs28 ,\'85,Y\plain\lang1036\f4\fs19\dn5 n\plain\f6\fs28\cf0 sind somit unabh\'e4ngige, aber nicht \par identisch verteilte Zufallsvariablen. Wir stellen die Datenpaare \plain\f4\fs28 (x\plain\f4\fs19\dn5 i\plain\f4\fs28 , y\plain\f4\fs19\dn5 i\plain\f4\fs28 )\plain\f6\fs28\cf0 als \par Punktwolke in einem zweidimensionalen Koordinatensystem dar. Wegen des \par Zufallseinflusses auf das Ergebnis des Experiments liegen die Punkte nicht alle \par auf der Regressionsgeraden, sondern um diese herum verstreut: \par \par \par {\object\objemb{\*\objclass VCam.Graphics.3}{\*\objname Object1}\objw6809\objh4529\objscalex83\objscaley83{\*\objdata 01050000 02000000 10000000 5643616D2E47726170686963732E3300 00000000 00000000 00340000 D0CF11E0A1B11AE1000000000000000000000000000000003E000300FEFF090006000000000000 0000000000010000000100000000000000001000000200000001000000FEFFFFFF000000000000 0000FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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}}}\plain\f6\fs28\cf0 \par Unser Problem besteht nun darin, aus den Datenpaaren \plain\f4\fs28 (x\plain\f4\fs19\dn5 i\plain\f4\fs28 , y\plain\f4\fs19\dn5 i\plain\f4\fs28 ),\plain\f4\fs28\cf3 \plain\f4\fs28\cf0 i=1,...,n,\plain\f6\fs28\cf0 die \par Regressionsgerade zu sch\'e4tzen. Dazu versuchen wir, die Summe der Quadrate \par der vertikalen Abst\'e4nde zwischen den Datenpunkten und der \par Regressiongeraden y=a+bx zu minimieren: \par {\pict\wmetafile8\picw5093\pich825\picscalex99\picscaley99\picwgoal2889\pichgoal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}\plain\f6\fs28\cf0 \par Diese Methode wird \plain\f4\fs28\cf2\b\i Kleinste-Quadrate-Methode\plain\f6\fs28\cf0 genannt. \par \par Die Werte f\'fcr die Regressionskoeffizienten erhalten wir, indem wir die partiellen \par Ableitungen von Q(a,b) nach a und b gleich 0 setzen. 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0008000000FA0200000000000000000000040000002D0102001C000000FB021000070000000000 BC02000000000102022253797374656D00003C0C0A8214ED1200D89FF177E19FF1772020F37736 1C6690040000002D01030004000000F001010004000000F0010000040000002701FFFF04000000 2701FFFF040000002701FFFF030000000000 }\plain\f6\fs28\cf0 \par \plain\f5\fs22\cf0 \par \par \plain\f6\fs28\cf0\b Im Folgenden betrachten wir ein \plain\f6\fs28\cf3\b Beispiel\plain\f6\fs28\cf0\b : \par \plain\f6\fs28\cf0 \par Beim Auswerten von Studien ist man h\'e4ufig daran interessiert, den Zusammen- \par hang zwischen zwei oder mehreren ermittelten Variablen zu betrachten. \par Graphisch k\'f6nnen zwei solche Gr\'f6\'dfen (hier K\'f6rpergewicht und Systolischer \par Blutdruck) als Punktwolke dargestellt werden. Dazu definieren wir die zwei Listen \par "Gewicht" und "Blutdruck", in denen der i. Eintrag jeweils dem Gewicht bzw. dem \par Blutdruck des i. Patienten entspricht: \par \plain\f5\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Gewicht:=[54.5, 77.0, 78.5, 48.0, 90.0, 86.5, 54.6, 61.0, 66.0, 54.0, 85.0, 80.0, 80.5, 96.7, \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 68.0, 50.0, 71.5, 55.0, 78.5, 94.5, 68.7, 97.2, 53.0, 84.0]: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Blutdruck:=[128, 154, 180, 96, 142, 170, 122, 130, 118, 98, 172, 149, 150, 181, 170, 109, 140, \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 150, 139, 157, 121, 160, 91, 161]: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}N:=nops(Gewicht);nops(Blutdruck) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Z:=[[Gewicht[i],Blutdruck[i]]$i=1..N] \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Wolke:=plot::PointList2d(Z,Color=RGB::DarkBlue, \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 AxesTitles=["K\'f6rpergewicht", \par "Systolischer Blutdruck"]): \par plot(Wolke) \par \pard\ri4\plain\f5\fs28\cf0 \par \plain\f6\fs28\cf0\b Berechnung der Regressionsgeraden mit MuPAD \par \plain\f6\fs28\cf0 \par Durch diese Punktwolke (das sogenannte Scatter-Diagramm), kann nun eine \par Regressionsgerade gelegt werden. Diese kann auf zwei Arten ermittelt werden: \par a) Mit dem Befehl \plain\f4\fs28\cf5 stats::linReg \par \plain\f6\fs28\cf0 b) Mit Hilfe von \plain\f4\fs28\cf5 plot::Scatterplot\plain\f6\fs28\cf0 \par \par zu a) \par Der Befehl \plain\f4\fs28\cf5 stats::linReg\plain\f6\fs28\cf0 berechnet unter Annahme einer linearen Beziehung \par \plain\f4\fs28\cf0 y\plain\f4\fs19\cf0\dn5 i\plain\f4\fs28\cf0 = a + bx\plain\f4\fs19\cf0\dn5 i\plain\f4\fs28\cf0 + N\plain\f4\fs19\cf0\dn5 i\plain\f6\fs28\cf0 zwischen den Datenpaaren \plain\f4\fs28 (x\plain\f4\fs19\dn5 i\plain\f4\fs28 , y\plain\f4\fs19\dn5 i\plain\f4\fs28 )\plain\f6\fs28\cf0 die Koeffizienten a und \plain\f4\fs28 b\plain\f6\fs28\cf0 mit \par der Methode der kleinsten Quadrate (d.h., dass die Summe der quadratischen \par Fehler minimiert wird). \plain\f4\fs28\cf5 stats::linReg\plain\f6\fs28\cf0 gibt folgende Werte der Regressionsgerade \par aus: [[y-Achsen-Abschnitt, Steigung], minimale Summe der Quadrate] \par \plain\f6\fs28\cf0\ul \par \plain\f6\fs28\cf0 F\'fcr unser K\'f6rpergewicht-Blutdruck-Beispiel von oben sieht dies wie folgt aus: \par \plain\f5\fs22\cf0\ul \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}A:=stats::linReg(Gewicht,Blutdruck) \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 \par \pard\ri4\plain\f6\fs28\cf0 Die oben erzeugte Punktwolke und die so ermittelte Regressionsgerade in einer \par Grafik: \par \plain\f5\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}RG:= x -> A[1][1]+x*A[1][2]: \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 RG_Fkt:=plot::Function2d(RG,x=30..100,Color=RGB::Green): \par plot(RG_Fkt,Wolke,AxesTitles=["K\'f6rpergewicht", \par "Systolischer Blutdruck"]) \par \pard\ri4\plain\f6\fs28\cf0 Im n\'e4chsten Diagramm zeichnen wir zus\'e4tzlich die vertikalen Abst\'e4nde zwischen \par den Datenpunkten und der Regressiongeraden ein.\plain\f5\fs22\cf0 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Strecken:=plot::Line2d([Gewicht[i],RG(Gewicht[i])], \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 [Gewicht[i],Blutdruck[i]], \par Color=RGB::CadmiumRedLight)$i=1..N: \par plot(Wolke, RG_Fkt, Strecken, \par AxesTitles=["K\'f6rpergewicht","Systolischer Blutdruck"]) \par \pard\ri4\plain\f6\fs28\cf0 Durch die folgende Berechnung wird die Summe der Quadrate der vertikalen \par Abst\'e4nde ermittelt. Diese entspricht der minimalen Quadratsumme, die MuPAD \par oben alks zweiten Operanden von \plain\f6\fs28\cf4 A\plain\f6\fs28\cf0 ausgegeben hat.\plain\f6\fs28\cf3 \par \plain\f5\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}_plus((RG(Gewicht[i])-Blutdruck[i])^2 $ i=1..N) \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 \par \par \pard\ri4\plain\f6\fs28\cf0 \par \pard\plain\f6\fs28\cf0 zu b) \par Die Funktion \plain\f4\fs28\cf5 plot::Scatterplot \plain\f6\fs28\cf0 erzeugt eine Punktwolke zweier diskreter \par \pard\ri4\plain\f6\fs28\cf0 Datens\'e4tze \plain\f4\fs28 [x\plain\f4\fs19\dn5 1\plain\f4\fs28 , x\plain\f4\fs19\dn5 2\plain\f4\fs28 ,\'85,x\plain\f4\fs19\dn5 n\plain\f4\fs28 ] und [y\plain\f4\fs19\dn5 1\plain\f4\fs28 , y\plain\f4\fs19\dn5 2\plain\f4\fs28 ,\'85,y\plain\f4\fs19\dn5 n\plain\f4\fs28 ]. Zus\'e4tzlich wird eine Regressionsgerade \par \plain\f4\fs28\cf0 y=a+bx\plain\f4\fs19\cf0\dn5 \plain\f4\fs28 aus den gegebenen Datenpaaren [x\plain\f4\fs19\dn5 1\plain\f4\fs28 , y\plain\f4\fs19\dn5 1\plain\f4\fs28 ], ..., [x\plain\f4\fs19\dn5 n\plain\f4\fs28 , y\plain\f4\fs19\dn5 n\plain\f4\fs28 ] berechnet und \par gezeichnet. Die Werte f\'fcr die Regressionskoeffizienten werden intern mit der oben \par bereits beschriebenen Funktion \plain\f4\fs28\cf5 stats::linReg\plain\f4\fs28 berechnet. \plain\f8\fs28 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}plot(plot::Scatterplot(Gewicht, Blutdruck), \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 AxesTitles=["K\'f6rpergewicht","Systolischer Blutdruck"]): \par \par \pard\ri4\plain\f5\fs22\cf0 \par \plain\f3\fs22\cf4 \par \plain\f4\fs28\cf2\b Allgemeine Regression\plain\f4\fs28\cf0\b \par \plain\f4\fs28\cf0 \par Bei komplexeren Zusammenh\'e4ngen ist die Methode der linearen Regression \par oft nicht anwendbar.\plain\f4\fs28\cf3 \plain\f4\fs28\cf0 Ein Beispiel f\'fcr so einen komplexen Zusammenhang ist \par das Wachstum von Pflanzen oder Bakterienpopulationen, bei dem die \par Gr\'f6\'dfe der Pflanze bzw. der Population exponentiell mit der Zeit zunimmt. \par Faktoren wie die Lichteinstrahlung, Temperatur, usw. bewirken den \par Zufallseinfluss auf das Experiment. \par Einige Vorg\'e4nge lassen sich auch durch Regressionskurven h\'f6herer als erster \par Ordnung beschreiben. Wir wollen im Folgenden ein Beispiel f\'fcr einen \par quadratischen Zusammenhang untersuchen. \par \tab \plain\f4\fs28 \par \plain\f4\fs28\cf3\b Beispiel f\'fcr ein quadratisches Regressionsmodell:\plain\f4\fs28\cf0\b \par \plain\f4\fs28\cf0 \par In einem Test wurde der Bremsweg eines Fahrzeugs in Abh\'e4ngigkeit von der \par Geschwindigkeit gemessen. Das Fahrzeug fuhr dazu mit Geschwindigkeiten \par von 30 km/h bis 150 km/h auf die Teststrecke zu: \par \plain\f3\fs22\cf4 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Geschwindigkeit:=[i$i=30..150]: \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 \par \pard\ri4\plain\f4\fs28\cf0 Es wurden jeweils die folgenden Bremswege gemessen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Bremsweg:= \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 [3.97891745, 21.061060588, 17.06037225, 24.01629131, 12.43126793, 9.8094236134, \par \par 19.651019548, 16.98468274, 19.84417492, 12.73642417, 25.14649198, \par \par 21.44683376, 15.46205634, 24.29414796, 15.392030178, 33.71120709, \par \par 23.16474327, 18.57062022, 23.36230183, 29.21325307, 28.75010144, 15.9006755, \par \par 27.74729473, 15.00564141, 30.59248623, 45.93836398, 35.95137232, \par \par 33.46136017, 36.38423997, 41.13594597, 40.59400644, 50.59150549, \par \par 25.15378406, 35.81940001, 41.77130433, 49.61868359, 37.45803546, \par \par 47.73291978, 41.18957922, 45.20238232, 49.92015362, 61.77461266, \par \par 50.31902651, 41.14787663, 63.87246223, 60.0820335, 64.78429958, 69.47030006, \par \par 65.52763979, 70.98651649, 63.6321556, 60.82405466, 69.01625347, 62.98062434, \par \par 75.9652618, 82.48404609, 64.5017531, 68.7538461, 77.1799272, 77.75655815, \par \par 75.73198462, 78.69878317, 92.20989107, 91.38679956, 89.82156977, \par \par 93.93645958, 82.52517617, 94.26709463, 102.0333729, 89.90982932, \par \par 105.9134785, 103.6583097, 113.5349407, 109.0112797, 115.3997244, \par \par 104.9828971, 108.8500553, 102.6576437, 117.0859873, 124.6961986, \par \par 115.9476952, 124.0723524, 125.0712362, 126.6447242, 130.7179894, \par \par 128.1730612, 134.0034832, 144.0433438, 125.5202107, 140.737293, 133.4085889, \par \par 145.1791798, 153.2579911, 140.0843198, 152.3572198, 150.8776044, \par \par 163.6669613, 152.0843418, 169.1345325, 167.9793435, 170.0085124, \par \par 182.4530317, 163.730378, 173.7505832, 174.3549419, 190.2999208, 189.7779234, \par \par 196.9631235, 187.3460035, 201.4765792, 206.7704613, 189.4863833, \par \par 202.3327058, 217.4426568, 210.3733267, 203.0912722, 210.4641933, \par \par 225.4045869, 235.6651301, 222.3858326, 226.2468696]: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}M:=nops(Geschwindigkeit);nops(Bremsweg) \par \pard\ri4\plain\f4\fs28\cf0 Wir stellen die Datenpaare wieder in einer Punktwolke dar: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}Autotest:=[[Geschwindigkeit[i],Bremsweg[i]]$i=1..M]: \par {\pntext\f1\'b7\tab}test:=plot::PointList2d(Autotest,Color=RGB::MidnightBlue): \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 plot(test,AxesTitles=["Geschwindigkeit in km/h", \par "Bremsweg in m"]) \par \par \pard\ri4\plain\f3\fs22\cf4 \par \plain\f4\fs28\cf0 Wendet man nun den Befehl \plain\f4\fs28\cf5 plot::Scatterplot\plain\f4\fs28\cf0 an, erh\'e4lt man folgende Grafik: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}plot(plot::Scatterplot(Geschwindigkeit,Bremsweg), \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 AxesTitles=["Geschwindigkeit in km/h","Bremsweg in m"]): \par \par \pard\ri4\plain\f4\fs28\cf0 Die lineare Regression liefert hier eine unbefriedigende L\'f6sung. F\'fcr \par kleine Geschwindigkeiten sagt sie einen negativen Bremsweg voraus. \par Au\'dferdem wissen wir aus der Physik bereits, dass sich bei doppelter \par Geschwindigkeit der Bremsweg vervierfacht. \par \par MuPAD bietet f\'fcr das allgemeine Regressionsmodell die Funktion \plain\f4\fs28\cf5 stats::reg\plain\f4\fs28\cf0 . \par \plain\f4\fs28\cf5 stats::reg\plain\f4\fs28\cf0 berechnet die Regressionskoeffizienten f\'fcr die allgemeine \par Regression. Dazu muss der funktionale Zusammenhang angegeben werden. \par F\'fcr unseren quadratischen Fall bedeutet dies: \par \pard\plain\f4\fs28 \tab ax\plain\f4\fs19\b\up14 2\plain\f4\fs28 +bx+c \par \pard\ri4\plain\f4\fs28 Es k\'f6nnte jedoch auch jeder andere Zusammenhang eingegeben werden. \par \plain\f4\fs28\cf5 stats::reg\plain\f6\fs28\cf0 gibt die Regressionskoeffizienten in der in der Eingabe angegeben \par Reihenfolge und als letzten Wert die minimale Summe der Quadrate aus.\plain\f4\fs28 \par \plain\f4\fs28\cf6 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}C:=stats::reg(Geschwindigkeit,Bremsweg,a*x^2+b*x+c,[x],[a,b,c]) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}w:=plot::Function2d(C[1][1]*x^2+C[1][2]*x+C[1][3], \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 x=30..150,Color=RGB::Pink): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}plot(w,test) \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 \par \pard\ri4\plain\f4\fs28\cf0 Zuletzt vergleichen wir noch die Summe der Quadrate der vertikalen Abst\'e4nde \par bei linearer und bei quadratischer Regression: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs24\cf4 {\pntext\f1\'b7\tab}stats::linReg(Geschwindigkeit, Bremsweg); \par \pard\li600\ri1\fi-300\plain\f3\fs24\cf4 stats::reg(Geschwindigkeit,Bremsweg,a*x^2+b*x+c,[x],[a,b,c]) \par \par \pard\ri4\plain\f4\fs28\cf0 Die quadratische Regression liefert also tats\'e4chlich eine bessere Beschreibung \par des funktionalen Zusammenhangs zwischen der Geschwindigkeit und dem \par Bremsweg. \par \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf0 \par \plain\f4\fs24\cf6\b Aufgabe:\plain\f4\fs24\cf6 \par 1. \plain\f4\fs20\cf6 \plain\f4\fs24\cf6 An Versuchspersonen wurden K\'f6rpergr\'f6\'dfe [cm] (erster Eintrag) und K\'f6rpergewicht [kg] \par (zweiter Eintrag) gemessen: \par (188,5;88,5), (177,5;86,5), (183;102), (182;93), (170;81,5), (185,5;83,5), \par (175,5;82,5), (175,5;96), (183;87,5), (173;79,5) \par Stellen Sie die Ergebnisse der Untersuchung in einer Punktwolke dar, berechnen Sie die \par Regressionskoeffizienten und plotten Sie die Regressionsgerade. \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs22\cf0 \par \plain\f4\fs24\cf1\b Anmerkungen:\plain\f4\fs24\cf1 \par 1. F\'fcr die wahrscheinlichkeitstheoretischen Grundlagen sei beispielweise auf das Buch "Einf\'fchrung in \par die Wahrscheinlichkeitstheorie und Statistik" von Dehling, H. & Haupt, B., Springer-Verlag \par Berlin Heidelberg 2003, verwiesen. \par \par 2. Weitere Anregungen zum Einsatz von MuPAD in der Lehre finden Sie auf unserem WebPortal \par \plain\f4\fs24\cf1\i MuPAD in Schule und Studium\plain\f4\fs24\cf1 unter: \plain\f4\fs24\cf6 http://schule.mupad.de\plain\f4\fs24\cf1 bzw. \plain\f4\fs24\cf6 http://studium.mupad.de\plain\f4\fs24\cf1 . \par \plain\f3\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f5\fs22\cf0 \par }