Applied Statistics Correlations Troels C. Petersen (NBI) “Statistics is merely a quantization of common sense” Sunday, August 31, 14 1 Correlation Are there any correlations here? 2 Sunday, August 31, 14 Correlation Are there any correlations here? 3 Sunday, August 31, 14 Correlation Recall the definition of the Variance, V: n X 1 2 V = = (xi µ)2 = E[(x N i µ)2 ] = E[x2 ] Likewise, one defines the Covariance, Vxy: n X Vxy 1 = N (xi µx )(yi µy ) = E[(xi µx )(yi µ2 µy )] i “Normalizing” by the widths, gives the (linear) correlation: ⇢xy = Sunday, August 31, 14 1 < ⇢xy < 1 Vxy x y (⇢) ' r 1 (1 n ⇢2 )2 + O(n 2) 4 Correlation Correlations in 2D are in the Gaussian case the “degree of ovalness”! Note how ALL of the bottom distributions have ! = 0, despite obvious correlations! 5 Sunday, August 31, 14 Correlation The correlation matrix Vxy explicitly looks as: Vxy 2 6 6 =6 4 2 1 2 21 .. . 2 12 2 22 .. . ... ... .. . 2 1N 2 2N 2 N 2 N2 ... 2 NN .. . Very specifically, the calculations behind are: 2 E[(X1 6 6 6 E[(X2 6 6 V =6 6 6 6 4 E[(Xn Sunday, August 31, 14 3 7 7 7 5 µ1 )(X1 µ1 )] E[(X1 µ1 )(X2 µ2 )] ··· E[(X1 µ1 )(Xn µ2 )(X1 µ1 )] E[(X2 µ2 )(X2 µ2 )] ··· E[(X2 µ2 )(Xn .. . µn )(X1 .. . µ1 )] E[(Xn µn )(X2 .. µ2 )] .. . . ··· E[(Xn µn )(Xn µn )] 3 7 7 µn )] 7 7 7 7. 7 7 7 5 µn )] 6 Correlation and Information Correlations influence results in complex ways! They need to be taken into account, for example in Error Propagation! Correlations may contain a significant amount of information. We will consider this more when we play with multivariate analysis. 7 Sunday, August 31, 14 Correlation example 8 Sunday, August 31, 14 Planck example 9 Sunday, August 31, 14 Correlation Vs. Causation “Com hoc ergo propter hoc” (with this, therefore because of this) 10 Sunday, August 31, 14 Correlation Vs. Causation “Com hoc ergo propter hoc” (with this, therefore because of this) 11 Sunday, August 31, 14
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