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Can someone please help me with this exercise and explain clearly to me? For me personally, when I look at this exercise, the matrix (2) stand out obviously, as it is a isotropic matrix so, it must be the dataset 2. The matrix (1) is a diagonal matrix, so it should be the dataset 3. For the dataset 1 should have a positive covariance, so it should be the matrix (4). Is that right? Thanks

enter image description here

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    $\begingroup$ There are many ways to approach this. One easy way is to begin with the obvious. Take dataset 3, for instance, which is qualitatively so different from the others. Estimate the variances: which matrix comes anywhere near those estimates? Among the remaining datasets, now dataset 2 stands out with one component and circular symmetry. Which among the remaining matrices represents a symmetric distribution? That leaves datrasets 1 and 4, each looking approximately like a 180 degree rotation of the other. Thus, one has positive covariance and the other has negative covariance. $\endgroup$
    – whuber
    Commented Mar 28 at 13:51
  • $\begingroup$ @whuber Do you think dataset 3 has marginal standard deviation around 50, so variances around 2500? I do not see such a diagonal element among the choices, though… $\endgroup$
    – Dave
    Commented Mar 28 at 13:53
  • $\begingroup$ @Dave If you're uncertain, create a dataset like Dataset 3 and compute its covariance matrix ;-). (There really is just one very obvious candidate among matrices (1) through (4).) $\endgroup$
    – whuber
    Commented Mar 28 at 13:54
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    $\begingroup$ First of all, thanks for your answers, guys. For me personally, when I look at this exercise, the matrix (2) stand out obviously, as it is a isotropic matrix so, it must be the dataset 2. The matrix (1) is a diagonal matrix, so it should be the dataset 3. And like @whuber said, dataset 1 should have a positive covariance, so it should be the matrix (4). Is that right? $\endgroup$
    – Thai Mai
    Commented Mar 28 at 14:07

1 Answer 1

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My thought process is as follows.

Datasets 2 and 3 have, more-or-less, uncorrelated marginal distributions, so their covariance matrices should be diagonal.

This means that matrices 3 and 4 must correspond to datasets 1 and 4.

Dataset 1 has a positive correlation, which means that its covariance matrix has positive off-diagonal elements, so matrix 4.

Dataset 4 has a negative correlation, which means that its covariance matrix has negative off-diagonal elements, so matrix 3.

Dataset 2 looks like it is jointly Gaussian and that each margin has a standard deviation of about 1 (about 95% of the points are between -2 and 2). Then each variance is 1, so its covariance matrix has 1s along the diagonal with 0s elsewhere by the lack of correlation. This is covariance matrix 2.

That leaves dataset 3 with covariance matrix 1, though that does not quite look correct. For instance, I mimicked that plot in a simulation and got a wildly different covariance matrix. The off-diagonals do not concern me, but the variance values along the diagonal do. Nonetheless, matrix 1 is the least bad of the options for dataset 3.

set.seed(2024)
N0 <- 1000
N1 <- 1000
N2 <- 1000
x0 <- runif(N0, -100, 100)
y0 <- runif(N0, -3, 3)
x1 <- runif(N1, -3, 3)
y1 <- runif(N1, -100, 100)
x2 <- rnorm(N2, 0, 30)
y2 <- rnorm(N2, 0, 30)
x <- c(x0, x1, x2)
y <- c(y0, y1, y2)
plot(x, y)
cov(cbind(x, y))

Mimic dataset 3

           x          y
x 1384.89114  -16.17137
y  -16.17137 1412.33919

Overall

Dataset 1 <--> Covariance Matrix 4

Dataset 2 <--> Covariance Matrix 2

Dataset 3 <--> Covariance Matrix 1

Dataset 4 <--> Covariance Matrix 3

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    $\begingroup$ Thanks for your answer! through your explanation I could totally understand the thinking process for this type of exercise! $\endgroup$
    – Thai Mai
    Commented Apr 1 at 15:46

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