Add MMAM_LM 02.07
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9 changed files with 311 additions and 8 deletions
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@ -62,7 +62,10 @@
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\setcounter{section}{8}
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\input{section9}\newpage
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\input{section10}\newpage
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\input{section11}%\newpage
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\input{section11}\newpage
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\input{section12}%\newpage
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\input{chapter3}
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\input{section13}%\newpage
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\appendix
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%\input{anhang}
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28
MMAM_LM_SoSe2019/chapter3.tex
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28
MMAM_LM_SoSe2019/chapter3.tex
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% !TEX root = MMAM_LM.tex
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% This work is licensed under the Creative Commons
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% Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy
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% of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ or
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% send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
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\chapter{Varianzanalyse}
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\setcounter{section}{12}
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Erinnere an lineare Regressionsanalyse:
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\begin{align*}
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Y&=f\klammern{x_1,\ldots,x_m}+\varepsilon
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\end{align*}
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Also:
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\index{Einflussgrößen}
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\begin{itemize}
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\item Einflussgrößen $x_i$ sind quantierbar (z.b. Länge, Temperatur, Einkommen, ...)
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\item Es gibt einen funktionalen Zusammenhang spezifiziert durch die Regressionsfunktion $f$
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\end{itemize}
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Dagegen in der Varianzanalyse:
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\begin{itemize}
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\item Einflussgrößen $x_i$ können auch \betone{qualitativ} sein (z.B. Geschlecht, Automarke, geografische Lage, ...)
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Diese Einflussgrößen heißen jetzt auch \define{Faktoren}
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\index{Faktoren}
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\item Es wird kein funktionaler Zusammenhang spezifiziert.
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\end{itemize}
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Ziel der Varianzanalyse (analysis of variance / ANOVA):
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Untersuchung, ob und wie gewisse Faktoren $x_1,\ldots,x_m$ im Mittel einen Einfluss auf die Zielgröße $Y$ haben.
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@ -104,7 +104,12 @@ Haben die $\hat{\varepsilon}_i$ deutlich andere als diese Eigenschaften, so weis
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\Cov(A\mal X,B\mal Y)=A\mal\Cov(X,Y)\mal B'
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\end{align*}
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\begin{proof}
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siehe Kopie %TODO
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\begin{align*}
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\Cov\klammern{A\mal X,B\mal Y}
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&=\E\eckigeKlammern[\Big]{\klammern[\big]{A\mal X-\E\eckigeKlammern{A\mal X}}\mal\klammern[\big]{B\mal Y-\E\eckigeKlammern{B \mal Y}}'}\\
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&=\E\eckigeKlammern[\Big]{A\mal\klammern[\big]{X-\E\eckigeKlammern{X}}\mal\klammern[\big]{Y-\E\eckigeKlammern{Y}}'\mal B'}\\
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&=A\mal\Cov\klammern{X,Y}\mal B'
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\end{align*}
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\end{proof}
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Damit folgt:
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\begin{align*}
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@ -27,9 +27,10 @@ Also:
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=V^{-\frac{1}{2}}\mal V^{-\frac{1}{2}}
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\end{align*}
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Damit (Multiplikation der Gleichung in \eqref{eq:sec:11:WLM} mit $V^{-\frac{1}{2}}$):
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\begin{align*}
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\begin{align}\nonumber
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\eqref{eq:sec:11:WLM} \overset{\Def}&{\iff}
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Y=X\mal\beta+\varepsilon\\
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\nonumber
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&\iff\underbrace{V^{-\frac{1}{2}}\mal Y}_{
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=_Y^*
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}=\underbrace{V^{-\frac{1}{2}}\mal X}_{
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@ -37,9 +38,11 @@ Damit (Multiplikation der Gleichung in \eqref{eq:sec:11:WLM} mit $V^{-\frac{1}{2
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}\mal\beta+\underbrace{V^{-\frac{1}{2}}\mal\varepsilon}_{
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=:\varepsilon^*
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}\\
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\nonumber
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&\iff Y^*=X^*\mal\beta+\varepsilon^*\\
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&\iff\colon(\text{WLM})^*,
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\end{align*}
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&~\Longleftrightarrow\vcentcolon(\text{WLM})^*,
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\label{eq:WLM:Stern}\tag{WLM*}
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\end{align}
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wobei
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\begin{align*}
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\E\eckigeKlammern{\varepsilon^*}
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@ -58,10 +61,10 @@ wobei
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\mal\underbrace{V^{\frac{1}{2}}\mal\klammern{V^{\frac{1}{2}}}^{-1}}_{=I_n}\\
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&=\sigma^2\mal I_n
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\end{align*}
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Daraus folgt, dass (WLM$)^*$ ein lineares Modell ist mit Eigenschaften \eqref{eq:3.2}, \eqref{eq:3.3} und \eqref{eq:3.4}, also unser "altes" lineares Modell.
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Daraus folgt, dass \eqref{eq:WLM:Stern} ein lineares Modell ist mit Eigenschaften \eqref{eq:3.2}, \eqref{eq:3.3} und \eqref{eq:3.4}, also unser "altes" lineares Modell.
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Damit die "alte Theorie" anwendbar (auf $Y^*$ und $X^*$) ist, sei
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\begin{align}\label{eq:11.3}\tag{11.3}
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\hat{\beta}_V:=\text{ MQS für $\beta$ in (WLM})^*
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\hat{\beta}_V:=\text{ MQS für $\beta$ in }\eqref{eq:WLM:Stern}
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\end{align}
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Da
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\begin{align*}
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@ -78,3 +81,101 @@ folgt mit Satz \ref{satz3.3} (angewendet auf $X^*,Y^*$ anstelle von $X,Y$):
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\end{align*}
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Dieser Schätzer heißt \define{Aitken-Schätzer}.
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\index{Aitken-Schätzer}
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\begin{align}\label{eq:11.4}\tag{11.4}
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\hat{\sigma}_V^2&:=\S^2\text{-Schätzer für $\sigma$ in } \eqref{eq:WLM:Stern}\\
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\nonumber
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\overset{\Def}&{=}
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\frac{1}{n-p}\mal\norm{Y^*-X^*\mal\hat{\beta}_V}^2\\
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\nonumber
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&=\frac{1}{n-p}\mal\norm{V^{-\frac{1}{2}}\mal\klammern{Y-X\mal\hat{\beta}_V}}^2\\
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\nonumber
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&=\frac{1}{n-p}\mal\klammern{Y-X\mal\hat{\beta}_V}'\mal\underbrace{V^{-\frac{1}{2}}\mal V^{-\frac{1}{2}}}_{=V^{-1}}\mal\klammern{Y-X\mal\hat{\beta}_V}\\
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\nonumber
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&=\frac{1}{n-p}\mal\norm{Y-X\mal\hat{\beta}_V}_V^2,
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\end{align}
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wobei
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\begin{align*}
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\norm{x}_V:=x'\mal V^{-1}\mal x\overset{!}{=}\norm{V^{-\frac{1}{2}}\mal x}
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\end{align*}
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eine Norm auf dem $\R^n$ ist.
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\begin{satz}\label{satz11.1}
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Es gelte das \eqref{eq:sec:11:WLM}.
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Dann gilt:
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\begin{enumerate}[label=(\arabic*)]
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\item $\begin{aligned}
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\E\eckigeKlammern{\hat{\beta}_V}=\beta
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\end{aligned}$
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\label{item:satz:11.1:1}
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\item $\begin{aligned}
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\Var\klammern{\hat{\beta}_V}=\sigma^2\mal\klammern{X'\mal V^{-1}\mal X}^{-1}
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\end{aligned}$
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\label{item:satz:11.1:2}
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\item $\begin{aligned}
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\E\eckigeKlammern{\hat{\sigma}_V^2}=\sigma^2
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\end{aligned}$
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\label{item:satz:11.1:3}
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\item $\begin{aligned}
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\norm{Y-X\mal\hat{\beta}_V}_V=\min\set{\norm{Y-X\mal\beta}_V:\beta\in\R^p},
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\end{aligned}$\\
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d.h. $\hat{\beta}_V$ ist MQS für $\beta$ im \eqref{eq:sec:11:WLM} bzgl. der Norm $\norm{\cdot}_V$.
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\label{item:satz:11.1:4}
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\item Falls $\varepsilon\verteilt\Nor_n\klammern{0,\sigma^2\mal V}$, so gilt, dass $\hat{\beta}_V$ und $\hat{\sigma}_V^2$ unabhängig sind.
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\label{item:satz:11.1:5}
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\end{enumerate}
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\end{satz}
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\begin{proof}
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\ref{item:satz:11.1:1} und \ref{item:satz:11.1:3} folgen aus Satz \ref{satz3.4} bzw. Satz \ref{satz3.11} angewendet auf \eqref{eq:WLM:Stern}.\nl
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\betone{Zeige \ref{item:satz:11.1:2}:}
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\begin{align*}
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\Var\klammern{\hat{\beta}_V}
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\overset{\ref{satz3.4}}&{=}
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\sigma^2\mal \klammern{{X^*}'\mal X^*}
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=\sigma^2\mal\klammern[\big]{X'\mal\underbrace{V^{-\frac{1}{2}}\mal V^{-\frac{1}{2}}}_{=V^{-1}}\mal X}^{-1}
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\end{align*}
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\betone{Zeige \ref{item:satz:11.1:4}:}
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\begin{align*}
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\norm{Y-X\mal\hat{\beta}_V}_V
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\overset{\eqref{eq:11.4}}&{=}
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\norm{Y^*-{X^*}'\mal\hat{\beta}_V}\\
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\overset{\ref{satz3.3}}&{=}
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\min\set{\norm{Y^*-X^*\mal\beta}:\beta\in\R^p}\\
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\overset{\eqref{eq:11.4}}&{=}
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\min\set{\norm{Y-X\mal\beta}_V:\beta\in\R^p}
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\end{align*}
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\betone{Zeige \ref{item:satz:11.1:5}:}
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\begin{align*}
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\varepsilon^*
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&=V^{-\frac{1}{2}}\mal\varepsilon
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\overset{\ref{satz5.8}}{\verteilt}
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\Nor_n\klammern[\big]{0,\sigma^2\mal\underbrace{V^{-\frac{1}{2}}\mal V\mal V^{-\frac{1}{2}}}_{=I_n}}
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\end{align*}
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Anwendung von Satz \ref{satz6.1}\ref{item:satz6.1_3} und Definition von $\hat{\beta}_V$ und $\hat{\sigma}_V^2$ liefert die Behauptung.
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\end{proof}
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\begin{bemerkungnr}\label{bem:11.2}
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Beachte: \eqref{eq:sec:11:WLM}$\iff$\eqref{eq:WLM:Stern} und \eqref{eq:WLM:Stern} ist "altes" LM (bzw. NLM).
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Da gemäß Definition $\hat{\beta}_V$ und $\hat{\sigma}_V^2$ die "alten" Schätzer sind, lassen sich \betone{alle} Ergebnisse aus den Abschnitten \ref{sec:6} bis \ref{sec:9} auf \eqref{eq:sec:11:WLM} übertragen.
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Man hat nur $Y$ und $X$ dort durch $Y^*=V^{-\frac{1}{2}}\mal Y$ und $X^*=V^{-\frac{1}{2}}\mal X$ zu ersetzen.
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\end{bemerkungnr}
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\begin{beispiel}[Lineares Regressionsmodell bei Heteroskedastizität]\label{beisp:11.3LinearesRegressionsModellBeiHeteroskedastizitaet}\enter
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Sei
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\begin{align*}
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Y_i=\alpha+\beta_1\mal x_{i,1}+\ldots+\beta_m\mal x_{i,m}+\underbrace{\sigma_i\mal\eta_i}_{=\varepsilon_i}
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\qquad\forall i\in\set{1,\ldots,n}
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\end{align*}
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wobei $\eta_1,\ldots,\eta_n$ i.i.d. $\verteilt\Nor\klammern{0,\sigma^2}$ mit $\sigma^2>0$ unbekannt und $\sigma_i^2$ \betone{bekannt} für alle $i$ (d.h. \betone{ungleiche} Varianzen). (Für $\sigma_1=\ldots=\sigma_n=1$ sind wir im bekannten Fall).
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\begin{align*}
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Y=X\mal\beta+\varepsilon
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\mit
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\varepsilon=\klammern{\sigma_1\mal\eta_1,\ldots,\sigma_n\mal\eta_n}'\verteilt\Nor_n\klammern{0,\sigma^2\mal V}
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\und
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V=\Diag\klammern{\sigma_1^2,\ldots,\sigma_n^2}\\
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\implies
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V^{-1}=\Diag\klammern{\frac{1}{\sigma_1^2},\ldots,\frac{1}{\sigma_n^2}},\qquad
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V^{-\frac{1}{2}}=\Diag\klammern{\frac{1}{\sigma_1},\ldots,\frac{1}{\sigma_n}}
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\end{align*}
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\end{beispiel}
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84
MMAM_LM_SoSe2019/section12.tex
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84
MMAM_LM_SoSe2019/section12.tex
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@ -0,0 +1,84 @@
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% !TEX root = MMAM_LM.tex
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% This work is licensed under the Creative Commons
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% Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy
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% of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ or
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% send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
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\section{Polynomiale Regression}
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\label{sec:12}
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Wir betrachten den Output $Y\in\R$ und einen Input $x\in\R$ mit Regressionsfunktion $f$, d.h.
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\index{Regressionsfunktion}
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$
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Y=f(x)+\varepsilon
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$.
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Angenommen unsere Daten
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\begin{align*}
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Y_i&=f(x_i)+\varepsilon_i
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\qquad\forall i\in\set{1,\ldots,n}
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\end{align*}
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liefert einen Scatterplot der $\klammern{x_i,Y_i}_{1\leq i\leq n}$ wie folgt:
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%TODO hässlich
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\begin{figure}[H]
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\begin{center}
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\input{./tikz/polyReg}
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\caption{polynomiale Regression}
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\label{Abb:polyReg}
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\end{center}
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\end{figure}
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Hier ist offensichtlich eine lineare Regressionfunktion $f(x)=\alpha+\beta\mal x$ ungeeignet.
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Besser wäre z.B.
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\begin{align*}
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f(x)=\alpha+\beta_1\mal x+\beta_2\mal x^2+\beta_3\mal x^2
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\end{align*}
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Wir betrachten deshalb allgemein das \define{Polynomiale Regressionsmodell:}
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\index{polynomiales Regressionsmodell}
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\begin{align*}
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Y_i&=\alpha+\beta_1\mal x_i+\beta_2\mal x_i^2+\ldots+\beta_m\mal x_i^m+\varepsilon_i
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\qquad\forall i\in\set{1,\ldots,n}
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\end{align*}
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Die Funktion
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\index{polynomiale Regressionsfunktion}
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\begin{align*}
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f(x)&=\alpha+\sum\limits_{j=1}^{m}\beta_j\mal x^j
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\end{align*}
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heißt \define{polynomiale Regressionsfunktion}.
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Die zugehörige Designmatrix ist
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\begin{align*}
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X=\begin{pmatrix}
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1 & x_1^2 & x_1^2 & \hdots & x_1^m\\
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\vdots & \vdots & \vdots & \ddots & \vdots\\
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1 & x_i^1 & x_i^2 & \hdots & x_i^m\\
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\vdots & \vdots & \vdots & \ddots & \vdots\\
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1 & x_n^2 & x_n^2 & \hdots & x_n^m\\
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\end{pmatrix}
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\end{align*}
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Es gilt (siehe \cite{fischerLinAlg}, Seite 188)
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\index{Vandermonde-Matrix}
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\index{Vandermonde-Determinante}
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\begin{align*}
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\det\underbrace{\begin{pmatrix}
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1 & x_1 & \hdots & x_1^{n-1}\\
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\vdots & \vdots & \ddots & \vdots\\
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1 & x_n & \hdots & x_n^{n-1}
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\end{pmatrix}}_{
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=:V
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}\in M(n\times n)
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=\prod\limits_{1\leq i<j\leq n}(x_j-x_i)
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=:\Delta_n
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\end{align*}
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Hierbei ist $V$ die \define{Vandermonde-Matrix} und die $\det(V)=\Delta_n$ die\\ \define{Vandermonde-Determinante}.
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Somit gilt:
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\begin{align*}
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x_1,\ldots,x_n\text{ paarweise verschieden}&\iff\Delta_n\neq0\\
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&\iff V\text{ invertierbar}\\
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\overset{\ref{satz:2.11}\ref{item:satz2.11(e)}}&{\iff}
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\Rg(V)=n\\
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\overset{\ref{satz:2.11}\ref{item:satz2.11(b)}}&{\implies}
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\Rg(X)=m+1,
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\end{align*}
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falls wie üblich $n>m+1$ angenommen wird.
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Also hat $X$ Vollrang.
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Somit ist die allgemeine Theorie anwendbar.\\
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(Randbemerkung: Äquidistantes Gitter $x_1,\ldots,x_n$ ist schlechte Wahl)
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21
MMAM_LM_SoSe2019/section13.tex
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21
MMAM_LM_SoSe2019/section13.tex
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% !TEX root = MMAM_LM.tex
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% This work is licensed under the Creative Commons
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% Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy
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% of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ or
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% send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
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\section{Das Einfaktor-Modell} %13
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\label{sec:13}
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Wir betrachten \betone{einen} Faktor $x_1=x$ $(m=1)$ mit $k$ möglichen Ausprägungen (so genannte \define{Faktorstufen}).
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\index{Faktorstufen}
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\begin{beispiel}\label{beisp:13.1}
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Der Faktor $x$ sei die verwendete Düngermethode $(M_1,M_2,M_3)$.
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Die Zielgröße $Y$ sei der Ertrag abhängig von der gewählten Düngermethode bei sonst gleichen Versuchsbedingungen (Klima, Boden, ...).\nl
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\betone{Fragen:}
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\begin{enumerate}[label=(\arabic*)]
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\item Ist der (mittlere) Ertrag bei allen drei Methoden gleich?
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\item Welche mittleren Erträge sind signifikant verschieden?
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\end{enumerate}
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\end{beispiel}
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@ -5,6 +5,7 @@
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% send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
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\section{Verteilungstheorie im linearen Modell mit normalverteilten Fehlern}
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\label{sec:6}
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Sei $Y=X\mal\beta+\varepsilon$ mit
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\begin{align}\label{eq:6.1}\tag{6.1}
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@ -5,6 +5,7 @@
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% send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
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\section{Das lineare Regressionsmodell}
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\label{sec:9}
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Betrachten wir eine reelle Zufallsgröße $Y$ (response), die \betone{linear} von Einflussgrößen (inputs) $x_1,\ldots,x_m$ abhängt und die einer zufälligen Schwankung unterliegt.
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\begin{align*}
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Y&=\alpha+\beta_1\mal x_1+\ldots+\beta_m\mal x_m+\varepsilon
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|||
59
MMAM_LM_SoSe2019/tikz/polyReg.tex
Normal file
59
MMAM_LM_SoSe2019/tikz/polyReg.tex
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
|
||||
|
||||
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
|
||||
|
||||
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]
|
||||
%uncomment if require: \path (0,300); %set diagram left start at 0, and has height of 300
|
||||
|
||||
%Shape: Axis 2D [id:dp2616136336954519]
|
||||
\draw (95,168.68) -- (487,168.68)(281,54) -- (281,242) (480,163.68) -- (487,168.68) -- (480,173.68) (276,61) -- (281,54) -- (286,61) ;
|
||||
%Straight Lines [id:da7824841050780786]
|
||||
\draw (245,160.93) -- (245,174.93) ;
|
||||
|
||||
|
||||
%Straight Lines [id:da0816979054092829]
|
||||
\draw (276,133.93) -- (286,133.93) ;
|
||||
|
||||
|
||||
%Shape: Circle [id:dp5596995214968037]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (100.53,208.47) .. controls (100.53,206.26) and (102.32,204.47) .. (104.53,204.47) .. controls (106.74,204.47) and (108.53,206.26) .. (108.53,208.47) .. controls (108.53,210.68) and (106.74,212.47) .. (104.53,212.47) .. controls (102.32,212.47) and (100.53,210.68) .. (100.53,208.47) -- cycle ;
|
||||
%Shape: Circle [id:dp7400363804883632]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (129.53,179.47) .. controls (129.53,177.26) and (131.32,175.47) .. (133.53,175.47) .. controls (135.74,175.47) and (137.53,177.26) .. (137.53,179.47) .. controls (137.53,181.68) and (135.74,183.47) .. (133.53,183.47) .. controls (131.32,183.47) and (129.53,181.68) .. (129.53,179.47) -- cycle ;
|
||||
%Shape: Circle [id:dp8006090336443885]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (407.53,148.47) .. controls (407.53,146.26) and (409.32,144.47) .. (411.53,144.47) .. controls (413.74,144.47) and (415.53,146.26) .. (415.53,148.47) .. controls (415.53,150.68) and (413.74,152.47) .. (411.53,152.47) .. controls (409.32,152.47) and (407.53,150.68) .. (407.53,148.47) -- cycle ;
|
||||
%Shape: Circle [id:dp43438427405806523]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (148.53,146.47) .. controls (148.53,144.26) and (150.32,142.47) .. (152.53,142.47) .. controls (154.74,142.47) and (156.53,144.26) .. (156.53,146.47) .. controls (156.53,148.68) and (154.74,150.47) .. (152.53,150.47) .. controls (150.32,150.47) and (148.53,148.68) .. (148.53,146.47) -- cycle ;
|
||||
%Shape: Circle [id:dp9865668392659667]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (184.53,134.47) .. controls (184.53,132.26) and (186.32,130.47) .. (188.53,130.47) .. controls (190.74,130.47) and (192.53,132.26) .. (192.53,134.47) .. controls (192.53,136.68) and (190.74,138.47) .. (188.53,138.47) .. controls (186.32,138.47) and (184.53,136.68) .. (184.53,134.47) -- cycle ;
|
||||
%Shape: Circle [id:dp3331791788072267]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (293.53,196.47) .. controls (293.53,194.26) and (295.32,192.47) .. (297.53,192.47) .. controls (299.74,192.47) and (301.53,194.26) .. (301.53,196.47) .. controls (301.53,198.68) and (299.74,200.47) .. (297.53,200.47) .. controls (295.32,200.47) and (293.53,198.68) .. (293.53,196.47) -- cycle ;
|
||||
%Shape: Circle [id:dp007417163026953721]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (320.53,204.47) .. controls (320.53,202.26) and (322.32,200.47) .. (324.53,200.47) .. controls (326.74,200.47) and (328.53,202.26) .. (328.53,204.47) .. controls (328.53,206.68) and (326.74,208.47) .. (324.53,208.47) .. controls (322.32,208.47) and (320.53,206.68) .. (320.53,204.47) -- cycle ;
|
||||
%Shape: Circle [id:dp9349012357145514]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (339.53,194.47) .. controls (339.53,192.26) and (341.32,190.47) .. (343.53,190.47) .. controls (345.74,190.47) and (347.53,192.26) .. (347.53,194.47) .. controls (347.53,196.68) and (345.74,198.47) .. (343.53,198.47) .. controls (341.32,198.47) and (339.53,196.68) .. (339.53,194.47) -- cycle ;
|
||||
%Shape: Circle [id:dp868294969954776]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (362.53,200.47) .. controls (362.53,198.26) and (364.32,196.47) .. (366.53,196.47) .. controls (368.74,196.47) and (370.53,198.26) .. (370.53,200.47) .. controls (370.53,202.68) and (368.74,204.47) .. (366.53,204.47) .. controls (364.32,204.47) and (362.53,202.68) .. (362.53,200.47) -- cycle ;
|
||||
%Shape: Circle [id:dp0029409808230549395]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (390.53,175.47) .. controls (390.53,173.26) and (392.32,171.47) .. (394.53,171.47) .. controls (396.74,171.47) and (398.53,173.26) .. (398.53,175.47) .. controls (398.53,177.68) and (396.74,179.47) .. (394.53,179.47) .. controls (392.32,179.47) and (390.53,177.68) .. (390.53,175.47) -- cycle ;
|
||||
%Shape: Circle [id:dp4694852901579938]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (243.53,134.47) .. controls (243.53,132.26) and (245.32,130.47) .. (247.53,130.47) .. controls (249.74,130.47) and (251.53,132.26) .. (251.53,134.47) .. controls (251.53,136.68) and (249.74,138.47) .. (247.53,138.47) .. controls (245.32,138.47) and (243.53,136.68) .. (243.53,134.47) -- cycle ;
|
||||
%Shape: Circle [id:dp4658705606615454]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (424.53,120.47) .. controls (424.53,118.26) and (426.32,116.47) .. (428.53,116.47) .. controls (430.74,116.47) and (432.53,118.26) .. (432.53,120.47) .. controls (432.53,122.68) and (430.74,124.47) .. (428.53,124.47) .. controls (426.32,124.47) and (424.53,122.68) .. (424.53,120.47) -- cycle ;
|
||||
%Shape: Circle [id:dp857872830065371]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (423.53,140.47) .. controls (423.53,138.26) and (425.32,136.47) .. (427.53,136.47) .. controls (429.74,136.47) and (431.53,138.26) .. (431.53,140.47) .. controls (431.53,142.68) and (429.74,144.47) .. (427.53,144.47) .. controls (425.32,144.47) and (423.53,142.68) .. (423.53,140.47) -- cycle ;
|
||||
%Shape: Circle [id:dp9962277489337455]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (378.53,191.47) .. controls (378.53,189.26) and (380.32,187.47) .. (382.53,187.47) .. controls (384.74,187.47) and (386.53,189.26) .. (386.53,191.47) .. controls (386.53,193.68) and (384.74,195.47) .. (382.53,195.47) .. controls (380.32,195.47) and (378.53,193.68) .. (378.53,191.47) -- cycle ;
|
||||
%Shape: Polynomial [id:dp4930854613858482]
|
||||
\draw (97,236) .. controls (212,-18) and (327,363) .. (442,109) ;
|
||||
%Shape: Circle [id:dp3421671618148874]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (211.53,154.47) .. controls (211.53,152.26) and (213.32,150.47) .. (215.53,150.47) .. controls (217.74,150.47) and (219.53,152.26) .. (219.53,154.47) .. controls (219.53,156.68) and (217.74,158.47) .. (215.53,158.47) .. controls (213.32,158.47) and (211.53,156.68) .. (211.53,154.47) -- cycle ;
|
||||
%Shape: Circle [id:dp9618779121865014]
|
||||
\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (257.53,164.47) .. controls (257.53,162.26) and (259.32,160.47) .. (261.53,160.47) .. controls (263.74,160.47) and (265.53,162.26) .. (265.53,164.47) .. controls (265.53,166.68) and (263.74,168.47) .. (261.53,168.47) .. controls (259.32,168.47) and (257.53,166.68) .. (257.53,164.47) -- cycle ;
|
||||
|
||||
% Text Node
|
||||
\draw (245,184) node {$x_{i}$};
|
||||
% Text Node
|
||||
\draw (294,134) node {$Y_{i}$};
|
||||
|
||||
|
||||
\end{tikzpicture}
|
||||
Loading…
Reference in a new issue