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I'm trying to learn up some matlab to do some basic computer exercises and I have a small doubt in a question, the problem first arises in my very little knowledge in the subject of pattern recognition owing to my school schedule. I hope I don't come out as a total noob, but just help me out with parts B and C from the below image

the question

For the 2D asked in part b i just added x1+x2 for one data set, x2+x3 for the second and x1+x3 for the third. Is that the right way of doing it ? If not please tell me how the different pairing datasets should be formed.


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up vote 1 down vote accepted

When you say "added", do you mean you found the actual sums, as follows? $$ \begin{array}{r} 0.42 - 0.087 \\ -0.2-3.3 \\ 1.3-0.32 \\ 0.39 + 0.71 \\ \vdots\quad\qquad{} \end{array} $$ If so, that doesn't get you the estimates for the two-dimensional Gaussian distribution. What you need is the estimated mean and variance, and the variance is a $2\times 2$ matrix, sometimes called the "covariance matrix" because its entries are covariances (in particular, its diagonal entries are variances). Notice the pair $(\mu,\Sigma)$. The expected value $\mu$ is normally (no pun intended) thought of as a $2\times1$ column vector. Its entries will be just the average for the first column in your table and the average for the second column. The matrix $$ \Sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{12} & \sigma_{22} \end{bmatrix} $$ has as entries the variance $\sigma_{11}$ of the first scalar-valued random variable, the variance $\sigma_{22}$ of the second one, and the covariance $\sigma_{12}$ between them.

The MLE for $\sigma_{11}$ is $$ \frac{1}{10}\sum_{i=1}^{10} (x_{1i} - \bar{x}_{1\bullet})^2 $$ where $$\bar{x}_{1\bullet}= \frac{1}{10}\sum_{i=1}^{10} x_{1i}$$ is the sample average for the first column. (This differs from the conventional unbiased estimate in that the denominator is $10$ rather than $10-1$) The MLE for $\sigma_{22}$ is found similarly by using the second column. The MLE for $\sigma_{12}$ is $$ \frac{1}{10}\sum_{i=1}^{10} (x_{1i}-\bar{x}_{1\bullet})(x_{2i}-\bar{x}_{2\bullet}). $$ See this section of a Wikipedia article.

The article Estimation_of_covariance_matrices.

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yes i did add them up, and got a single dimensional dataset, thank you for your answer, the idea is now clear! – Aadi Droid Nov 1 '11 at 4:31

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