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App. Stats. sLM fertig
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@ -75,7 +75,7 @@
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Y_i &\overset{i.i.d}{\sim}Normal(\beta_0+\beta_1x_i,\sigma) \notag \\
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Y &\sim Normal(X\beta,\mathbbm{1}\sigma) \notag
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\end{align}
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Since the mean of the response is a linear function of the explanatory variables, there are often called simple linear models.
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Since the mean of the response is a linear function of the explanatory variables, there are often called \begriff{simple linear models}.
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\end{enumerate}
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\end{definition}
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@ -129,4 +129,34 @@ where $SS_{xx}=\sum(x_i-\bar{x})^2$ and $s^2=\frac{\sum (Y_i-\hat{Y_i})^2}{n-2}$
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\begin{align}
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t = \frac{\hat{\beta_1}-\text{hypothesised value}}{\frac{s}{\sqrt{SS_{xx}}}} \notag
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\end{align}
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with degrees of freedom based on the number of data points and model parameters. In practice, software will does this for us!
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with degrees of freedom based on the number of data points and model parameters. In practice, software will does this for us!
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As an alternative for interference on parameter estimates, we can also compute confidence intervals. The $(1-\alpha)100\%$ confidence interval for the gradient $\beta_1$ is
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\begin{align}
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\hat{\beta_1}\pm t_{\nicefrac{\alpha}{2}}s_{\hat{\beta_1}}\notag
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\end{align}
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where $s_{\hat{\beta_1}} = \frac{s}{\sqrt{SS_{xx}}}$ and $t_{\nicefrac{\alpha}{2}}$ is based on $n-2$ degrees of freedom. $t_{\nicefrac{\alpha}{2}}$ is obtained from the \person{Stundent}'s t-distribution in the usual way. Some statistical software provides these values by default, but often (e.g. in MATLAB), only the standard errors for parameter estimates are provided.
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\subsection{Estimation and prediction for simple linear models}
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\begin{definition}[estimation]
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\begriff[simple linear models!]{Estimation}: estimate mean value of $Y$ over many data points
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\end{definition}
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\begin{definition}[prediction]
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\begriff[simple linear models!]{Prediction}: predict $Y$ for a particular value of $x$. This leads to higher error bounds (add error in mean to variation around mean).
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\end{definition}
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\input{./TeX_files/materials/estimation_prediction_slms}
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\textcolor{red}{Be careful with predictions far away from mean of explanatory variable or outside of region covered by data.}
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It's good practise to always check the fit visually\footnote{If you don't believe that have a look at \url{https://en.wikipedia.org/wiki/Anscombe\%27s_quartet}}. Looking at residual plots is also useful.
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\begin{definition}[Coefficient of Determination]
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The \begriff{Coefficient of Determination}, $R^2$, is defined as
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\begin{align}
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R^2 = \frac{SS_{yy}-SSE}{SS_{yy}} \notag
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\end{align}
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where $SS_{yy}=\sum(Y_i-\bar{Y})^2$ and $SSE=\sum(Y_i-\hat{Y_i})^2$. $R^2$ can be interpreted as the proportion of the variance in the response variable that is explained by (or attributed to) the explanatory variable. There are other measures, similar to $R^2$ and there are a few issues making it problematic for assessing goodness of fit. We'll revisit this later in the course.
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\end{definition}
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@ -0,0 +1,823 @@
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\begin{center}
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\begin{tikzpicture}
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\begin{axis}[
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xmin=0, xmax=400, xlabel=$x$,
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ymin=0, ymax=14, ylabel=$y$,
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axis x line=middle,
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axis y line=middle,
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domain=0:400,
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samples=400
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]
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\addplot[thick, blue] {0.014*x+4.8};
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\addlegendentry {fit};
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(367.00,8.81)
|
||||
(368.00,8.82)
|
||||
(369.00,8.83)
|
||||
(370.00,8.84)
|
||||
(371.00,8.85)
|
||||
(372.00,8.86)
|
||||
(373.00,8.87)
|
||||
(374.00,8.88)
|
||||
(375.00,8.89)
|
||||
(376.00,8.90)
|
||||
(377.00,8.91)
|
||||
(378.00,8.92)
|
||||
(379.00,8.93)
|
||||
(380.00,8.94)
|
||||
(381.00,8.95)
|
||||
(382.00,8.95)
|
||||
(383.00,8.96)
|
||||
(384.00,8.97)
|
||||
(385.00,8.98)
|
||||
(386.00,8.99)
|
||||
(387.00,9.00)
|
||||
(388.00,9.01)
|
||||
(389.00,9.02)
|
||||
(390.00,9.03)
|
||||
(391.00,9.04)
|
||||
(392.00,9.05)
|
||||
(393.00,9.06)
|
||||
(394.00,9.07)
|
||||
(395.00,9.08)
|
||||
(396.00,9.08)
|
||||
(397.00,9.09)
|
||||
(398.00,9.10)
|
||||
(399.00,9.11)
|
||||
};
|
||||
\addlegendentry {95\% estimation limits};
|
||||
\addplot[green,dotted] {x*-5.7*10^-6*x+0.017*x+1.4};
|
||||
\addlegendentry {95\% prediction limits};
|
||||
\addplot[red,dashed] coordinates {
|
||||
(0.00,6.21)
|
||||
(1.00,6.22)
|
||||
(2.00,6.23)
|
||||
(3.00,6.24)
|
||||
(4.00,6.25)
|
||||
(5.00,6.26)
|
||||
(6.00,6.27)
|
||||
(7.00,6.27)
|
||||
(8.00,6.28)
|
||||
(9.00,6.29)
|
||||
(10.00,6.30)
|
||||
(11.00,6.31)
|
||||
(12.00,6.32)
|
||||
(13.00,6.33)
|
||||
(14.00,6.34)
|
||||
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|
||||
(16.00,6.36)
|
||||
(17.00,6.37)
|
||||
(18.00,6.38)
|
||||
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|
||||
(20.00,6.40)
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
(30.00,6.49)
|
||||
(31.00,6.50)
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
(68.00,6.86)
|
||||
(69.00,6.87)
|
||||
(70.00,6.88)
|
||||
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|
||||
(72.00,6.90)
|
||||
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|
||||
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|
||||
(75.00,6.93)
|
||||
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|
||||
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|
||||
(78.00,6.96)
|
||||
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|
||||
(80.00,6.98)
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
(86.00,7.04)
|
||||
(87.00,7.05)
|
||||
(88.00,7.06)
|
||||
(89.00,7.07)
|
||||
(90.00,7.08)
|
||||
(91.00,7.09)
|
||||
(92.00,7.10)
|
||||
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|
||||
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|
||||
(95.00,7.13)
|
||||
(96.00,7.14)
|
||||
(97.00,7.15)
|
||||
(98.00,7.16)
|
||||
(99.00,7.17)
|
||||
(100.00,7.18)
|
||||
(101.00,7.20)
|
||||
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|
||||
(103.00,7.22)
|
||||
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|
||||
(105.00,7.24)
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
(111.00,7.30)
|
||||
(112.00,7.31)
|
||||
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|
||||
(114.00,7.33)
|
||||
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|
||||
(116.00,7.36)
|
||||
(117.00,7.37)
|
||||
(118.00,7.38)
|
||||
(119.00,7.39)
|
||||
(120.00,7.40)
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
(134.00,7.56)
|
||||
(135.00,7.57)
|
||||
(136.00,7.58)
|
||||
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|
||||
(138.00,7.60)
|
||||
(139.00,7.62)
|
||||
(140.00,7.63)
|
||||
(141.00,7.64)
|
||||
(142.00,7.65)
|
||||
(143.00,7.66)
|
||||
(144.00,7.68)
|
||||
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|
||||
(146.00,7.70)
|
||||
(147.00,7.71)
|
||||
(148.00,7.72)
|
||||
(149.00,7.74)
|
||||
(150.00,7.75)
|
||||
(151.00,7.76)
|
||||
(152.00,7.77)
|
||||
(153.00,7.78)
|
||||
(154.00,7.80)
|
||||
(155.00,7.81)
|
||||
(156.00,7.82)
|
||||
(157.00,7.83)
|
||||
(158.00,7.85)
|
||||
(159.00,7.86)
|
||||
(160.00,7.87)
|
||||
(161.00,7.88)
|
||||
(162.00,7.90)
|
||||
(163.00,7.91)
|
||||
(164.00,7.92)
|
||||
(165.00,7.93)
|
||||
(166.00,7.95)
|
||||
(167.00,7.96)
|
||||
(168.00,7.97)
|
||||
(169.00,7.98)
|
||||
(170.00,8.00)
|
||||
(171.00,8.01)
|
||||
(172.00,8.02)
|
||||
(173.00,8.04)
|
||||
(174.00,8.05)
|
||||
(175.00,8.06)
|
||||
(176.00,8.08)
|
||||
(177.00,8.09)
|
||||
(178.00,8.10)
|
||||
(179.00,8.12)
|
||||
(180.00,8.13)
|
||||
(181.00,8.14)
|
||||
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|
||||
(183.00,8.17)
|
||||
(184.00,8.18)
|
||||
(185.00,8.20)
|
||||
(186.00,8.21)
|
||||
(187.00,8.22)
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
(192.00,8.29)
|
||||
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|
||||
(194.00,8.32)
|
||||
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|
||||
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|
||||
(197.00,8.36)
|
||||
(198.00,8.38)
|
||||
(199.00,8.39)
|
||||
(200.00,8.41)
|
||||
(201.00,8.42)
|
||||
(202.00,8.44)
|
||||
(203.00,8.45)
|
||||
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|
||||
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|
||||
(206.00,8.49)
|
||||
(207.00,8.51)
|
||||
(208.00,8.52)
|
||||
(209.00,8.54)
|
||||
(210.00,8.55)
|
||||
(211.00,8.57)
|
||||
(212.00,8.58)
|
||||
(213.00,8.60)
|
||||
(214.00,8.61)
|
||||
(215.00,8.63)
|
||||
(216.00,8.64)
|
||||
(217.00,8.66)
|
||||
(218.00,8.67)
|
||||
(219.00,8.69)
|
||||
(220.00,8.71)
|
||||
(221.00,8.72)
|
||||
(222.00,8.74)
|
||||
(223.00,8.75)
|
||||
(224.00,8.77)
|
||||
(225.00,8.78)
|
||||
(226.00,8.80)
|
||||
(227.00,8.81)
|
||||
(228.00,8.83)
|
||||
(229.00,8.85)
|
||||
(230.00,8.86)
|
||||
(231.00,8.88)
|
||||
(232.00,8.89)
|
||||
(233.00,8.91)
|
||||
(234.00,8.93)
|
||||
(235.00,8.94)
|
||||
(236.00,8.96)
|
||||
(237.00,8.97)
|
||||
(238.00,8.99)
|
||||
(239.00,9.01)
|
||||
(240.00,9.02)
|
||||
(241.00,9.04)
|
||||
(242.00,9.06)
|
||||
(243.00,9.07)
|
||||
(244.00,9.09)
|
||||
(245.00,9.10)
|
||||
(246.00,9.12)
|
||||
(247.00,9.14)
|
||||
(248.00,9.15)
|
||||
(249.00,9.17)
|
||||
(250.00,9.19)
|
||||
(251.00,9.20)
|
||||
(252.00,9.22)
|
||||
(253.00,9.24)
|
||||
(254.00,9.25)
|
||||
(255.00,9.27)
|
||||
(256.00,9.29)
|
||||
(257.00,9.31)
|
||||
(258.00,9.32)
|
||||
(259.00,9.34)
|
||||
(260.00,9.36)
|
||||
(261.00,9.37)
|
||||
(262.00,9.39)
|
||||
(263.00,9.41)
|
||||
(264.00,9.42)
|
||||
(265.00,9.44)
|
||||
(266.00,9.46)
|
||||
(267.00,9.48)
|
||||
(268.00,9.49)
|
||||
(269.00,9.51)
|
||||
(270.00,9.53)
|
||||
(271.00,9.55)
|
||||
(272.00,9.56)
|
||||
(273.00,9.58)
|
||||
(274.00,9.60)
|
||||
(275.00,9.62)
|
||||
(276.00,9.63)
|
||||
(277.00,9.65)
|
||||
(278.00,9.67)
|
||||
(279.00,9.69)
|
||||
(280.00,9.70)
|
||||
(281.00,9.72)
|
||||
(282.00,9.74)
|
||||
(283.00,9.76)
|
||||
(284.00,9.78)
|
||||
(285.00,9.79)
|
||||
(286.00,9.81)
|
||||
(287.00,9.83)
|
||||
(288.00,9.85)
|
||||
(289.00,9.87)
|
||||
(290.00,9.88)
|
||||
(291.00,9.90)
|
||||
(292.00,9.92)
|
||||
(293.00,9.94)
|
||||
(294.00,9.96)
|
||||
(295.00,9.97)
|
||||
(296.00,9.99)
|
||||
(297.00,10.01)
|
||||
(298.00,10.03)
|
||||
(299.00,10.05)
|
||||
(300.00,10.06)
|
||||
(301.00,10.08)
|
||||
(302.00,10.10)
|
||||
(303.00,10.12)
|
||||
(304.00,10.14)
|
||||
(305.00,10.16)
|
||||
(306.00,10.17)
|
||||
(307.00,10.19)
|
||||
(308.00,10.21)
|
||||
(309.00,10.23)
|
||||
(310.00,10.25)
|
||||
(311.00,10.27)
|
||||
(312.00,10.28)
|
||||
(313.00,10.30)
|
||||
(314.00,10.32)
|
||||
(315.00,10.34)
|
||||
(316.00,10.36)
|
||||
(317.00,10.38)
|
||||
(318.00,10.40)
|
||||
(319.00,10.41)
|
||||
(320.00,10.43)
|
||||
(321.00,10.45)
|
||||
(322.00,10.47)
|
||||
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|
||||
(324.00,10.51)
|
||||
(325.00,10.53)
|
||||
(326.00,10.55)
|
||||
(327.00,10.56)
|
||||
(328.00,10.58)
|
||||
(329.00,10.60)
|
||||
(330.00,10.62)
|
||||
(331.00,10.64)
|
||||
(332.00,10.66)
|
||||
(333.00,10.68)
|
||||
(334.00,10.70)
|
||||
(335.00,10.72)
|
||||
(336.00,10.73)
|
||||
(337.00,10.75)
|
||||
(338.00,10.77)
|
||||
(339.00,10.79)
|
||||
(340.00,10.81)
|
||||
(341.00,10.83)
|
||||
(342.00,10.85)
|
||||
(343.00,10.87)
|
||||
(344.00,10.89)
|
||||
(345.00,10.90)
|
||||
(346.00,10.92)
|
||||
(347.00,10.94)
|
||||
(348.00,10.96)
|
||||
(349.00,10.98)
|
||||
(350.00,11.00)
|
||||
(351.00,11.02)
|
||||
(352.00,11.04)
|
||||
(353.00,11.06)
|
||||
(354.00,11.08)
|
||||
(355.00,11.10)
|
||||
(356.00,11.11)
|
||||
(357.00,11.13)
|
||||
(358.00,11.15)
|
||||
(359.00,11.17)
|
||||
(360.00,11.19)
|
||||
(361.00,11.21)
|
||||
(362.00,11.23)
|
||||
(363.00,11.25)
|
||||
(364.00,11.27)
|
||||
(365.00,11.29)
|
||||
(366.00,11.31)
|
||||
(367.00,11.33)
|
||||
(368.00,11.35)
|
||||
(369.00,11.37)
|
||||
(370.00,11.38)
|
||||
(371.00,11.40)
|
||||
(372.00,11.42)
|
||||
(373.00,11.44)
|
||||
(374.00,11.46)
|
||||
(375.00,11.48)
|
||||
(376.00,11.50)
|
||||
(377.00,11.52)
|
||||
(378.00,11.54)
|
||||
(379.00,11.56)
|
||||
(380.00,11.58)
|
||||
(381.00,11.60)
|
||||
(382.00,11.62)
|
||||
(383.00,11.64)
|
||||
(384.00,11.66)
|
||||
(385.00,11.68)
|
||||
(386.00,11.69)
|
||||
(387.00,11.71)
|
||||
(388.00,11.73)
|
||||
(389.00,11.75)
|
||||
(390.00,11.77)
|
||||
(391.00,11.79)
|
||||
(392.00,11.81)
|
||||
(393.00,11.83)
|
||||
(394.00,11.85)
|
||||
(395.00,11.87)
|
||||
(396.00,11.89)
|
||||
(397.00,11.91)
|
||||
(398.00,11.93)
|
||||
(399.00,11.95)
|
||||
};
|
||||
\addplot[green,dotted] {x*5.7*10^-6*x+0.012*x+8.2};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{center}
|
||||
Loading…
Add table
Reference in a new issue