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I have a signal x that I want to classify in one of the classes A and B in which the means are

Ma=[0.5,0.6] and Mb=[2,2]

and with variances 

Va=1 and Vb=2.

And to classify the signal I have to use the Mahalanobis distance which requires the covariance S that those classes share and have in common. Can I find it if there is only given the mean and variance between those classes?

P.S I know this looks like a homework exercise but I have been looking everywhere and I could not find anything

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    $\begingroup$ The two subclauses of your 'P.S.' are not logically connected. Is this a homework exercise? If so, please add the [self-study] tag & read its wiki. $\endgroup$
    – gung - Reinstate Monica
    Jan 13, 2016 at 16:30
  • $\begingroup$ Thanks you for adding the tag. What would you do if you had the full covariance matrices? $\endgroup$
    – gung - Reinstate Monica
    Jan 13, 2016 at 16:46
  • $\begingroup$ The mahalanobis distance formula requires the covariance matrice to be able to calculate the distances between the signal x to the classes A and B $\endgroup$
    – Izanagi
    Jan 13, 2016 at 16:49
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    $\begingroup$ The problem states that the two classes have a common covariance matrix that was produced by independent characteristics with variances Va=1 and Vb=2. $\endgroup$
    – Izanagi
    Jan 16, 2016 at 16:03
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    $\begingroup$ If we got the cov(A,B) right then I think I can solve it .Anyway thank you so much for your time that you spent helping me . $\endgroup$
    – Izanagi
    Jan 16, 2016 at 16:22

1 Answer 1

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You have two classes of objects that exist within a two dimensional space defined by two characteristics $(x_1, x_2)$. The issue here is how the data covary within that space. That is, there is some value for the correlation between $x_1$ and $x_2$. If you know that $x_1$ and $x_2$ are independent, then you know the correlation that corresponds to independence. From there, you can calculate the covariance using the definition of correlation: $$ {\rm Cor}(x_1, x_2) = \frac{{\rm Cov(x_1, x_2)}}{{\rm Var(x_1)}{\rm Var(x_2)}} $$ Remember that a correlation matrix is symmetrical. That is, $s_{12}=s_{21}$. With that last piece, you should be able to construct the correlation matrix and solve the problem.

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  • $\begingroup$ But what do you do if you don't have the correlation? I mean the original question, you have only mean and variance. $\endgroup$
    – xkcd
    Aug 30, 2018 at 0:39
  • $\begingroup$ @xkcd, if you know x1 & X2 are independent, you can infer the correlation from that. If you didn't know whether they were independent, you could use background knowledge to impute a distribution of possible correlations (like a prior) & integrate over that. $\endgroup$
    – gung - Reinstate Monica
    Aug 30, 2018 at 0:51

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