\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fswiss\fprq2 Helvetica;}{\f5\fmodern\fprq1 Courier New;}{\f6\froman\fprq2 Times New Roman;}} {\colortbl\red0\green0\blue0;\red0\green128\blue0;\red0\green0\blue255;\red128\green0\blue0;\red255\green255\blue255;\red255\green0\blue0;} \deflang1031\pard\ri4\plain\f5\fs20\cf0 ________________________________________________________________________________ \par \par Inhalt....: Einf\'fchrung Ausgleichsrechnung \par Kategorie.: Unterrichtsmaterial \par Mathematik: Statistik \par MuPAD.....: 3.1.0 \par Datum.....: 2003-03-04 \par Autoren...: Alessandro Dell'Aere \par Funktionen: |, plot, plot::Point2d, plot::Function2d, solve, assign \par Funktionen: is, delete, sum, plotfunc2d, PointSize, YRange \par ________________________________________________________________________________ \par \plain\f3\fs28\cf0 \par \plain\f3\fs36\cf0\b Eine Einf\'fchrung in die Ausgleichsrechnung \par \plain\f3\fs28\cf0 \par \plain\f3\fs24\cf1 Das folgende Arbeitsblatt basiert auf der Unterrichtseinheit "Ausgleichsrechnung im Grundkurs \par Mathematik" von J\'fcrgen Rau und Wolfgang Riemer. \par \par Gegenstand des Arbeitsblattes ist eine Einf\'fchrung in die Technik der Ausgleichsrechnung. In \par diesem Rahmen werden lineare und quadratische Funktionen gewinnbringend miteinander \par verkoppelt. Im neuen Lehrplan der Sekundarstufe II NRW wird Ausgleichsrechnung (Regression \par und Korrelation) im Rahmen beschreibender Statistik verbindlich vorgeschrieben. Es wird eine \par erprobte Sequenz zur Erarbeitung der Thematik vorgestellt. \plain\f3\fs28\cf0 \par \par \plain\f3\fs24\cf3\b Unterrichtliche Voraussetzungen \par \plain\f3\fs24\cf3 \par Eine Unterrichtsreihe mit dem in diesem Notebook beschriebenen Inhalt \par setzt folgende Vorbereitung voraus: \par \par \pard\li500\ri4\plain\f3\fs24\cf3 - den Begriff der linearen Funktion \par \par - den Begriff der quadratischen Funktion \par \par - die Berechnung von Nullstellen und Minimalstellen \par \par - evtl. Grundkenntnisse des Differenzierens \par \pard\ri4\plain\f3\fs28\cf3 \par \par \plain\f3\fs24\cf3\b M\'f6gliche Durchf\'fchrung der Unterrichtseinheit und didaktische Vorbemerkungen \par \plain\f3\fs24\cf3 \par Es werden beide Ausgleichsgeraden-Typen (von x auf y einerseits und von y auf x andererseits) \par parallel betrachtet, um deutlich zu machen, dass die Optimalit\'e4t von den verwendeten Kriterien \par abh\'e4ngt. \par \par Durch den Einsatz von MuPAD lassen sich die verschiedenen Abweichungsma\'dfe funktional deuten \par (positive Funktionen mit linksgekr\'fcmmten Graphen, die sich im ersten Fall als Parabeln entpuppen). \par \par Erst durch den Einsatz von MuPAD wird die experimentelle Untersuchung verschiedener \par Punktwolken praktikabel ("Handlungsorientierung"), ebenso die Untersuchung verschiedener \par Streuungsma\'dfe. \par \par Teilweise ist der Einsatz von MuPAD prinzipiell nicht notwendig. Man kann aber methodisch so \par vorgehen, dass man die komplizierten Ergebnisse der Termumformungen des CAS pr\'e4sentiert \par und dann nach den abgelaufenen Rechenschritten fragt. \par \par Wirklich sinnvoll ist der Einsatz von MuPAD bei Verarbeitung realer Me\'dfdaten, insbesondere \par beim Erstellen von Prognosen. \par \par Die experimentelle Variation des vorgegebenen Punktes in (5) ist ohne MuPAD nur schwer \par praktikabel. Das gilt ebenso f\'fcr die Auswertung der experimentellen Daten in (6). \par \par Die Teilbereiche (1) bis (6) sollten in chronologischer Reihenfolge abgearbeitet werden. \par \par \par \plain\f6\fs22\cf0 \par \plain\f3\fs28\cf0\b (1) Die experimentelle Untersuchung von \par Ursprungs-Regressions-Geraden: \par \par \plain\f3\fs28\cf0 Es seien die drei Punkte \par \pard\li3000\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw3324\pich745\picscalex99\picscaley99\picwgoal1899\pichgoal425 010009000003CE05000009001C0000000000050000000B0200000000050000000C02E902FC0C03 0000001E00050000000C02EE02160D050000000B0200000000030000001E00050000000C02F202 1F0D050000000B0200000000050000000B0200000000030000001E00050000000C02F7023A0D05 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C02FA02420D050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C0201035F0D050000000B0200000000050000000B02 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Diese liegen nicht alle auf einer Geraden, denn konstruiert man die \par Gleichung der eindeutig bestimmten Geraden \par \par \pard\li3000\ri4\plain\f6\fs28\cf4\i ___\plain\f6\fs28\cf2\i y = mx + b\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 \par durch zwei dieser Punkte, so\plain\f3\fs28\cf0\b \par \plain\f3\fs28\cf0 wird der dritte Punkt diese Gleichung nicht erf\'fcllen: \par \plain\f3\fs28\cf0\b \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [3,6,9]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [1,6,12]: \par S := solve(\{y[1] = m*x[1] + b, y[2] = m*x[2] + b\}, [m, b]): \par assign(S[1]); \par is(m*x[3] + b = y[3])\plain\f5\fs22\cf5 \par \par \par \pard\ri4\plain\f3\fs28\cf0 Es soll nun eine Ursprungsgerade \plain\f6\fs28\cf2\i y = mx\plain\f3\fs28\cf0 gezeichnet werden, die nach \par Augenma\'df m\'f6glichst genau an den genannten Punkten vorbeigeht. \par Wir w\'e4hlen hierf\'fcr \plain\f6\fs28\cf2\i m = 1\plain\f3\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plot1 := plot::Point2d([x[i], y[i]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Color = RGB::Black, \par PointSize = 2 * unit::mm) \par $ i = 1..3: \par plot2 := plot::Function2d(1*z, z = 0..10): \par plot(plot1,plot2);\plain\f5\fs22\cf5 \par \par \pard\ri4\plain\f3\fs28\cf0 Die Streuung der Punkte in vertikaler Richtung um diese Gerade misst man \par durch \par \pard\li2500\ri4\plain\f6\fs22\cf0 {\pict\wmetafile8\picw8676\pich806\picscalex99\picscaley98\picwgoal4922\pichgoal462 0100090000035D07000009001C0000000000050000000B0200000000050000000C022603E42103 0000001E00050000000C023003EB21050000000B0200000000030000001E00050000000C023903 3D22050000000B0200000000050000000B0200000000030000001E00050000000C023B03442205 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 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}\plain\f6\fs22\cf0 \par \pard\ri4\plain\f3\fs28\cf0 und die Streuung der Punkte in horizentaler Richtung um diese Gerade misst \par man durch \par \pard\li2500\ri4\plain\f6\fs22\cf0 {\pict\wmetafile8\picw7841\pich1045\picscalex99\picscaley99\picwgoal4450\pichgoal596 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00002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF03 0000000000 }\plain\f6\fs22\cf0 \par \pard\ri4\plain\f3\fs28\cf0 Es sind also \plain\f6\fs28\cf2\i v\plain\f3\fs28\cf0 und \plain\f6\fs28\cf2\i h\plain\f3\fs28\cf0 als Funktionen der Steigung \plain\f6\fs28\cf2\i m \plain\f3\fs28\cf0 einer Geraden \par aufzufassen. Wir wollen nun \plain\f6\fs28\cf2\i v\plain\f3\fs28\cf0 und \plain\f6\fs28\cf2\i h\plain\f3\fs28\cf0 f\'fcr unsere obige Sch\'e4tzung \plain\f6\fs28\cf2\i m = 1\plain\f3\fs28\cf0 \par berechnen:\plain\f6\fs22\cf0 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}delete m: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 v := m -> sum((m*x[i] - y[i])^2, i = 1..3): \par h := m -> sum((1/m*y[i] - x[i])^2, i = 1..3): \par v(1); \par h(1);\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Mit Hilfe der Funktionsgraphen von \plain\f6\fs28\cf2\i v \plain\f3\fs28\cf0 und \plain\f6\fs28\cf2\i h \plain\f3\fs28\cf0 sch\'e4tzen wir, f\'fcr welche \plain\f6\fs28\cf2\i m\plain\f3\fs28\cf0 die \par Streuungen minimal werden. Dazu verkleinern wir schrittweise den \par Zeichenbereich, so dass wir die Minima recht genau ablesen k\'f6nnen:\plain\f6\fs22\cf0 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plotfunc2d(v(m), h(m), m = 0.5..2) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plotfunc2d(v(m), h(m), m = 1..1.5) \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf5 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plotfunc2d(v(m), h(m), m = 1.1..1.3) \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Wir lesen f\'fcr \plain\f6\fs28\cf2\i v \plain\f3\fs28\cf0 ein Minimum bei \plain\f6\fs28\cf2\i m = 1.17\plain\f3\fs28\cf0 und f\'fcr \plain\f6\fs28\cf2\i h \plain\f3\fs28\cf0 ein Minimum bei \plain\f6\fs28\cf2\i m = 1.23\plain\f3\fs28\cf0 ab. \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}m1 := 1.17: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 plot2 := plot::Function2d(m1*z, z = 0..10, \par Color = RGB::Red): \par m2 := 1.23: \par plot3 := plot::Function2d(m2*z, z = 0..10, \par Color = RGB::Blue): \par plot(plot1, plot2, plot3)\plain\f5\fs24\cf5 \par \par \pard\ri4\plain\f3\fs28\cf0 Die Gerade, f\'fcr welche die vertikale Streuung minimal wird, hei\'dft \par \par \pard\li1500\ri4\plain\f3\fs28\cf0\b "Regressionsgerade von y auf x". \par \plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 Die Gerade, f\'fcr welche die horizentale Streuung minimal wird, hei\'dft \par \par \pard\li1500\ri4\plain\f3\fs28\cf0\b "Regressionsgerade von x auf y". \par \pard\ri4\plain\f3\fs28\cf0 \par Den Quotienten der Steigungen \par \pard\li3500\ri4\plain\f6\fs22\cf0 {\pict\wmetafile8\picw839\pich1132\picscalex99\picscaley99\picwgoal479\pichgoal644 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}\plain\f6\fs22\cf0 \par \pard\ri4\plain\f3\fs28\cf0 nennt man \par \pard\li1500\ri4\plain\f3\fs28\cf0\b "Bestimmtheitsma\'df". \par \plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 Er ist das Quadrat des sog. \par \par \pard\li1500\ri4\plain\f3\fs28\cf0\b "Korrelationskoeffizienten". \par \par \pard\ri4\plain\f3\fs28\cf0 Wenn nichts anderes gesagt wird, \par meinen wir mit Regressionsgerade die Regressionsgerade von y auf x. \par \par Als n\'e4chstes wollen wir einsehen, dass die Steigungen der beiden \par Regressionsgeraden umso mehr voneinander abweichen, \par je mehr die Punkte streuen. Zu diesem Zweck definieren wir neue \par Punktwolken: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [1, 2, 3]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [1, 2, 3]: \par plot(plot::Point2d([0,0], PointSize = 2.5 * unit::mm), \par plot::Point2d([x[i], y[i]], Color = RGB::Black, \par PointSize = 2.5 * unit::mm) \par $i = 1..3): \par plotfunc2d(v(m), h(m), m = 0.5..2, \par ViewingBox = [0.5..2, 0..15]) \par \plain\f5\fs24\cf5 \par \par \plain\f5\fs22\cf5 \par \par \pard\ri4\plain\f3\fs28\cf0 Hierbei liegen alle Punkte auf einer Ursprungsgeraden. \par Die Funktionen \plain\f6\fs28\cf2\i v\plain\f3\fs28\cf0 und \plain\f6\fs28\cf2\i h\plain\f3\fs28\cf0 haben beide bei \plain\f6\fs28\cf2\i m = 0\plain\f3\fs28\cf0 ihr Minimum. \par Nun lassen wir die Punkte von diesem Sonderfall abweichen, \par d.h. sie werden gestreut: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [1, 2, 3]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [0, 2, 4]: \par plot(plot::Point2d([0,0], PointSize = 2.5 * unit::mm), \par plot::Point2d([x[i], y[i]], \par Color = RGB::Black, \par PointSize = 2.5 * unit::mm) \par $i = 1..3): \par plotfunc2d(v(m), h(m), m = 0.5..2, \par ViewingBox=[0.5..2,0..15]) \par \plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Wir sehen, dass die Minima der Funktionen \plain\f6\fs28\cf2\i v\plain\f3\fs28\cf0 und \plain\f6\fs28\cf2\i h \plain\f3\fs28\cf0 nun voneinander \par abweichen. Wir werden nun die Punkte nun noch weiter auseinander streuen: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [1, 2, 3]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [-2, 2, 4]: \par plot(plot::Point2d([0,0], PointSize = 2.5 * unit::mm ), \par plot::Point2d([x[i], y[i]], \par Color = RGB::Black, \par PointSize = 2.5 * unit::mm) \par $i = 1..3); \par plotfunc2d(v(m), h(m), m = 0.5..2, \par ViewingBox=[0.5..2,0..15])\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Nun sind die Minima der Funktionen \plain\f6\fs28\cf2\i v\plain\f3\fs28\cf0 und \plain\f6\fs28\cf2\i h \plain\f3\fs28\cf0 noch weiter voneinander \par abgewichen. \par \par \plain\f3\fs28\cf0\b (2) Experimentelle Erforschung weiterer Abweichungsma\'dfe: \par \plain\f3\fs28\cf0 \par Wir betrachten wieder die folgende Punktwolke: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [3,6,9]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [1,6,12]:\plain\f5\fs24\cf5 \par \par \pard\ri4\plain\f3\fs28\cf0 Ein Abweichungsma\'df kann auch alternativ definiert werden, \par indem man die Differenzen \par \pard\li4000\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw1843\pich754\picscalex99\picscaley99\picwgoal1052\pichgoal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}\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 auf andere Art und Weise verarbeitet. Einige wichtige Abweichungsma\'dfe \par ergeben sich zum Beispiel, wenn man zur Berechnung von v \par \par \pard\li500\ri4\plain\f3\fs28\cf0 a) die Differenzen nicht quadriert \par b) die Betr\'e4ge der Differenzen verwendet \par c) die dritte Potenz der Differenzen verwendet \par d) die vierte Potenz der Differenzen verwendet \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}v1 := m -> sum((m*x[i] - y[i]), i=1..3): \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 v2 := m -> sum(abs(m*x[i] - y[i]), i=1..3): \par v3 := m -> sum((m*x[i] - y[i])^3, i = 1..3): \par v4 := m -> sum((m*x[i] - y[i])^4, i = 1..3): \par plot(plot::Function2d(v1(m), m = 0.9..1.4)): \par plot(plot::Function2d(v2(m), m = 0.9..1.4)): \par plot(plot::Function2d(v3(m), m = 0.9..1.4)): \par plot(plot::Function2d(v4(m), m = 0.9..1.4)):\plain\f5\fs24\cf5 \par \par \pard\ri4\plain\f5\fs24\cf2 \par \plain\f3\fs28\cf0 Wie aus den Zeichnungen zu erkennen ist, ist der Optimalwert f\'fcr \plain\f6\fs28\cf2\i m\plain\f3\fs28\cf0 vom \par jeweiligen Abweichungsma\'df abh\'e4ngig. \par \par \plain\f3\fs28\cf0\b (3) Herleitung der Formel f\'fcr die Ursprungs-Regressionsgerade: \par \plain\f3\fs28\cf0 \par Wenn man die quadratische vertikale Abweichung \par \pard\li1500\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw8415\pich841\picscalex99\picscaley99\picwgoal4772\pichgoal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797374656D0000F1110A7438E91200D89FF177E19FF1772020F377D91166B4040000002D010800 040000002701FFFF04000000F001000004000000F001010004000000F001020004000000F00103 0004000000F001040004000000F001050004000000F0010600040000002701FFFF040000002701 FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF0400000027 01FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF04000000 2701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 in Abh\'e4ngigkeit von der Steigung \plain\f6\fs28\cf2\i m\plain\f3\fs28\cf0 graphisch darstellt, \par vermutet man eine Parabel. Wir wollen nun analytisch vorgehen und so \par den Scheitelpunkt dieser Parabel exakt bestimmen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}expand(v(m)); \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 msv := solve(v'(m), m)[1]: \par Scheitel := plot::Point2d(msv, v(msv), \par Color = RGB::Blue, \par PointSize = 2.5 * unit::mm)\plain\f5\fs24\cf5 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plot(plot::Function2d(v(m), m = 0..2), Scheitel): \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Wir wollen nun f\'fcr unsere Punkte Platzhalter verwenden, um eine allgemeine \par Formel zur Bestimmung der Ursprungs-Regressionsgeraden herzuleiten. \par Zun\'e4chst \'fcberzeugen wir uns davon, dass v auch i.A. eine Parabel ist: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}delete x,y: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 collect(v(m),m)\plain\f5\fs24\cf5 \par \par \pard\ri4\plain\f3\fs28\cf0 Nun berechnen wir die Steigung \plain\f6\fs28\cf2\i m\plain\f3\fs28\cf0 als Minimum von \plain\f6\fs28\cf2\i v \plain\f3\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}solve(diff(v(m), m), m)[1]\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 \par Nat\'fcrlich k\'f6nnen wir \par \pard\li2500\ri4\plain\f6\fs22\cf0 {\pict\wmetafile8\picw4606\pich806\picscalex99\picscaley98\picwgoal2635\pichgoal462 010009000003A504000008001C0000000000050000000B0200000000050000000C022603FE1103 0000001E00050000000C022F032812050000000B0200000000030000001E00050000000C023803 2E12050000000B0200000000050000000B0200000000030000001E00050000000C024103581205 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 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00000B0200000000030000001E00050000000C027C03F512050000000B0200000000050000000B 0200000000050000000B0200000000050000000B0200000000050000000B020000000005000000 0B0200000000050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C0285032213050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000050000000B 0200000000050000000B0200000000050000000B0200000000050000000B020000000005000000 0B0200000000050000000B0200000000030000001E00050000000C0290032813050000000B0200 000000050000000B0200000000050000000B0200000000050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000050000000B 0200000000050000000B0200000000050000000B0200000000050000000B020000000003000000 1E00030000001E00050000000C020502DC0A050000000B0200000000050000000B020000000005 0000000B0200000000050000000B0200000000050000000B0200000000050000000B0200000000 050000000B0200000000050000000B0200000000050000000B0200000000050000000B02000000 00050000000B0200000000050000000B0200000000050000000B020000000008000000FA020000 0000000000000000040000002D0100001C000000FB0238FF000000000000900100000001070000 00417269616C000000D0120A4F38E91200D89FF177E19FF1772020F377251266BE040000002D01 010005000000020101000000050000000102FFFFFF00050000002E011800000005000000090200 00000004000000080100001C000000FB0210FF0000000000009001000000010700000054696D65 73204E657720526F6D616E00D89FF177E19FF1772020F377251266BE040000002D0102000B0000 0026060F000C004D6174685479706500007D001C000000FB0210FF000000000000900101000001 0700000054696D6573204E657720526F6D616E00D89FF177E19FF1772020F377251266BE040000 002D0103001C000000FB0256FF0000000000009001000000010700000054696D6573204E657720 526F6D616E00D89FF177E19FF1772020F377251266BE040000002D0104001C000000FB0210FF00 00000000009001000000020700000053796D626F6C000054110A5E38E91200D89FF177E19FF177 2020F377251266BE040000002D010500040000002D010200040000002D010300040000002D0104 00040000002D010500040000002D010200040000002D010300040000002D010400040000002D01 0500040000002D010200040000002D010500040000002D010200040000002D0105000500000009 020000FF00040000002D0102000700000021050100320068016400040000002D01050007000000 21050100D70068010C01040000002D010200040000002D010400040000002D0103000700000021 050100780068017F01040000002D010400070000002105010031009F01EA010700000021050100 3200EE003F02040000002D01050007000000210501002B006801D002040000002D010200070000 0021050100320068019003040000002D0105000700000021050100D70068013804040000002D01 0200040000002D010400040000002D010300070000002105010078006801AB04040000002D0104 00070000002105010032009F01160507000000210501003200EE006B05040000002D0105000700 0000210501002B006801FC05040000002D010200070000002105010032006801BC06040000002D 0105000700000021050100D70068016407040000002D010200040000002D010400040000002D01 0300070000002105010078006801D707040000002D010400070000002105010033009F01420807 000000210501003300EC009708040000002D0105000700000021050100B9006801340904000000 2D010200070000002105010030006801000A08000000FA0200000000000000000000040000002D 0106001C000000FB021000070000000000BC02000000000102022253797374656D00000E0F0A98 38E91200D89FF177E19FF1772020F377251266BE040000002D010700040000002701FFFF040000 00F001000004000000F001010004000000F001020004000000F001030004000000F00104000400 0000F0010500040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF04 0000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF 040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f6\fs22\cf0 \par \pard\ri4\plain\f3\fs28\cf0 annehmen, denn anderenfalls w\'fcrde alle Punkte auf der Ordinaten-Achse \par liegen und dann w\'fcrde unsere Aufgabenstellung nicht sinnvoll sein. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}Minimum := (x[1]*y[1]+x[2]*y[2]+x[3]*y[3]) \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 /(x[1]^2 + x[2]^2 + x[3]^3) \par \plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Somit erhalten wir als Formel f\'fcr die Ursprungs-Regressionsgerade von y auf x: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}y = Minimum*x \par \pard\li600\ri1\fi-300\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0\b (4) Reale Datens\'e4tze: \par \plain\f3\fs28\cf0 \par Wir wollen unsere Erkenntnisse nun an einem realistischen Beispiel anwenden: \par Ein kalter Wasserkessel (aus dem K\'fchlschrank) wird aufgeheizt. Bei 17\'b0C ist der \par Aufheizprozess voll im Gange. Wir w\'e4hlen diesen Zeit-Temperatur-Datenwert \par als Ursprungspunkt. Es wurden folgende Messwerte beobachtet: \par \par \pard\li1000\ri4\plain\f3\fs28\cf0 Nr.\plain\f3\fs28\cf4 _____\plain\f3\fs28\cf0 x (Zeit / Minuten) \plain\f3\fs28\cf4 ______\plain\f3\fs28\cf0 y (Temperatur / \'b0C) \par \plain\f3\fs22\cf0 \par \plain\f3\fs28\cf0 0\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 0,0\plain\f3\fs28\cf4 ___________________\plain\f3\fs28\cf0 0,00 \par 1\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 0,2\plain\f3\fs28\cf4 ___________________\plain\f3\fs28\cf0 4,62 \par 2\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 0,4\plain\f3\fs28\cf4 ___________________\plain\f3\fs28\cf0 9,52 \par 3\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 0,6\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 15,70 \par 4\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 0,8\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 21,90 \par 5\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 1,0\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 29,22 \par 6\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 1,2\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 34,43 \par 7\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 1,4\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 39,98 \par 8\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 1,6\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 44,88 \par 9\plain\f3\fs28\cf4 _________\plain\f3\fs28\cf0 1,8\plain\f3\fs28\cf4 __________________\plain\f3\fs28\cf0 49,92 \par 10\plain\f3\fs28\cf4 ___-_____\plain\f3\fs28\cf0 2,0\plain\f3\fs28\cf4 ______--___________\plain\f3\fs28\cf0 53,91 \par \par \pard\ri4\plain\f3\fs28\cf0 Benutze \par \par \pard\li500\ri4\plain\f3\fs28\cf0 a) die Messwerte 1 bis 3, \par b) die Messwerte 8 bis 10, \par c) alle Messwerte, \par \pard\ri4\plain\f3\fs28\cf0 \par um die Regressionsgeraden durch den Ursprung zu berechnen. \par Welche Prognosen ergeben sich jeweils f\'fcr den Zeitpunkt des Kochens \par (83\'b0C \'fcber der Starttemperatur) ? \par \par Zun\'e4chst schreiben wir die Daten wieder in Listen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [4.62, 9.52, 15.70, 21.90, 29.22, \par 34.43, 39.98, 44.88, 49.92, 53.91]: \par \plain\f5\fs24\cf5 \par \par \par \pard\ri4\plain\f3\fs28\cf0 a) Berechnung der Regressionsgeraden unter Verwendung der \par Messwerte 1 bis 3: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}v := m -> sum((m*x[i] - y[i])^2, i = 1..3): \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Steigung := solve(diff(v(m), m), m)[1]\plain\f5\fs24\cf5 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plot1 := plot::Point2d([x[i], y[i]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Color = RGB::Black, \par PointSize = 2.5 * unit::mm) \par $i = 1..3, \par plot::Point2d([x[i], y[i]], \par Color = RGB::Grey, \par PointSize = 2.5 * unit::mm) \par $i=4..10: \par plot2 := plot::Function2d(Steigung*z, z = 0..2): \par plot(plot1, plot2)\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 Wie man an der Zeichnung sieht, bekommt man in diesem Fall f\'fcr die Steigung \par der Regressionsgeraden einen sehr ungenauen Wert. \par \par b) Berechnung der Regressionsgeraden unter Verwendung der \par Messwerte 8 bis 10: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}v := m -> sum((m*x[i] - y[i])^2, i = 8..10): \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Steigung := solve(diff(v(m), m), m)[1]\plain\f5\fs24\cf5 \par \par \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plot1 := plot::Point2d([x[i], y[i]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Color = RGB::Grey, \par PointSize = 2.5 * unit::mm) \par $i=1..7, \par plot::Point2d([x[i], y[i]], \par Color = RGB::Black, \par PointSize = 2.5 * unit::mm) \par $i = 8..10: \par \par plot2 := plot::Function2d(Steigung*z, z = 0..2): \par plot(plot1, plot2)\plain\f5\fs24\cf5 \par \pard\ri4\plain\f3\fs28\cf0 c) Berechnung der Regressionsgeraden unter Verwendung der \par Messwerte 1 bis 10: \par \plain\f6\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}v := m -> sum((m*x[i] - y[i])^2, i = 1..10): \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Steigung := solve(diff(v(m), m), m)[1]\plain\f5\fs24\cf5 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}plot1 := plot::Point2d([x[i], y[i]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 Color = RGB::Black, \par PointSize = 2.5 * unit::mm) \par $i = 1..10: \par plot2 := plot::Function2d(Steigung*z, z = 0..2): \par plot(plot1, plot2) \par \pard\ri4\plain\f3\fs28\cf0\b (5) Allgemeine Regressionsgeraden: \par \par \plain\f3\fs28\cf0 Wir betrachten wieder die drei Datenpunkte von oben: \par \plain\f3\fs28\cf0\b \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf5 {\pntext\f1\'b7\tab}x := [3,6,9]: \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf5 y := [1,6,12]: \par \plain\f5\fs24\cf5 \par \pard\ri4\plain\f6\fs22\cf0 \par \plain\f3\fs28\cf0 Statt durch den Ursprung soll die Regressionsgerade aber nun durch einen \par vorgegebenen Punkt \par \pard\li3000\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw1950\pich751\picscalex99\picscaley98\picwgoal1109\pichgoal430 0100090000031F04000009001C0000000000050000000B0200000000050000000C02EF029E0703 0000001E00050000000C02F802A507050000000B0200000000030000001E00050000000C020003 B307050000000B0200000000050000000B0200000000030000001E00050000000C020903BA0705 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 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2701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000 002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF0300 00000000 }\plain\f3\fs28\cf0 \par \pard\li4000\ri4\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 bzw. durch den Schwerpunkt der Punktwolke verlaufen. 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Man kann diesen Fall auf den oben analysierten \par zur\'fcckf\'fchren: Dazu denkt man sich den Ursprung des Koordinatensystems \par nach \plain\f6\fs28\cf2\i M\plain\f3\fs28\cf0 verschoben. 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