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CE408_Lecture_17_101811

# CE408_Lecture_17_101811 - Cov X = E X‘I-l Elﬁn-EC...

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Unformatted text preview: Cov X = E X‘I-l- Elﬁn-EC] = " x y Pug: ' Et w x .. WMM -.- Etxﬂ - MDT} waCX‘X + MM} -: EEXII ., 2 £51 I”! 4. “E I”: '— i -- 3 . O - Voy Pearson's product-moment coefficient 0 ' PM - WINK. Y) — M -mm /ﬁ.ﬁ§%ﬁi%\ t? '65 Several sets of (x, y) points, with the Pearson correlation coefficient of x and y for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Yis zero. X and Y are s.i. :> X and Y uncorrelated X and Y uncorrelated => X and Y are s.i. NOT TRUEI! The converse is not true because the correlation coefficient detects only linear dependencies between two variables. For example, suppose the random variable X is symmetrically distributed about zero, and Y = X2. Then Y is completely determined by X, so that X and Y are perfectly dependent, but their correlation is zero; they are uncorrelated. However, in the special case when X and Y are iointly normal, uncorrelatedness is equivalent to independence. The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. Reference: htt : en.wiki edia.or wiki Correlation and de endence 6) Common misconceptions g N Correlation and causality Main article: Correlation does not imply causation The conventional dictum that "correlation does not imply causation" means that correlation cannot be used to infer a causal relationship between the variablesm This dictum should not be taken to mean that correlations cannot indicate the potential existence of causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautologies), where no causal process exists. Consequently, establishing a correlation between two variables is not a sufﬁcient condition to establish a causal relationship (in either direction). For example, one may observe a correlation between an ordinary alarm clock ringing and daybreak, though there is no direct causal relationship between these events. A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health, or does good health lead to good mood, or both? Or does some other factor underlie both? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be. Correlation and linearity CD trimaran-u— 1: Four sets of data with the same mean (7.5), standard deviation (4.12), correlation (0.816) and regression line (y = 3 + 0.5x). However, as can be seen on the plots. the distribution of the variables is very different. The Pearson correlation coefﬁcient indicates the strength of a linear relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the conditional mean of Ygiven X, denoted E(Y|X), is not linear in X, the correlation coefficient will not fully determine the form of E(Y|X), The image on the right shows scatterglots of Anscombe's guartet, a set of four different pairs of variables created by Francis Anscombe.‘El The four y variables have the same mean (7.5), standard deviation (4.12), correlation (0.816) and regression line (y = 3 + 0.5x). However, as can be seen on the plots, the distribution of the variables is very different. The ﬁrst one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefﬁcient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one Mgr which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefﬁcient, even though the relationship between the two variables is not linear. These examples indicate that the correlation coefﬁcient, as a summary statistic, cannot replace visual examination of the data. Note that the examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is not correct.”1 Reference: htt : en.wiki edia.or wiki Correlation and de endence . WWW , [L .11! ¢ ,‘1 uw'meu Cu 3 FHMJIWS 0L {1.0- 15 [2341106 9.9} .’S M FU‘U-Q-[O‘UE 0.1: ‘ Sagéb 9N- Obfcdwc'. X ) j3x(Y) '0 xbc) ' 7 ...
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CE408_Lecture_17_101811 - Cov X = E X‘I-l Elﬁn-EC...

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