# What is the relationship between covariance matrix and contour plot?

From this video,

I deduced that here, $a==b$ and $c<0$.

Here are my questions,

(1) What does it mean by "equal loss values of errors"?

(2) What happens when a-priori probabilities of $C_1$ and $C_2$ are equal or unequal?

• Can you describe or copy the description of the plot/image? What do the ellipses we are seeing represent? Also, my guess for "equal loss values of errors" would be an output that gives the same error-value regardless of the actual label. So if $o$ is our output and $E$ is our error-function/cost-function: $E(o, C_1)==E(o,C_2)$ Nov 17 '16 at 11:27
• @dimpol, see the edit. Nov 17 '16 at 11:36
• If the horizontal and vertical axes are assumed to be drawn on the same scale, then it is not the case that $a=b$, because these describe the spreads of the data along those axes and the spreads clearly differ.
– whuber
Nov 17 '16 at 20:10

Assuming that both axes are on the same scale, you have $$a and $$c<0$$. The contour plot for a bivariate normal distribution with the specified covariance matrix is an ellipse of the form:
\begin{align} \text{const} &= \begin{bmatrix} x_a - \mu_a & x_b - \mu_b \end{bmatrix} \begin{bmatrix} a & c \\ c & b \end{bmatrix}^{-1} \begin{bmatrix} x_a - \mu_a \\ x_b - \mu_b \end{bmatrix} \\[12pt] &= \frac{1}{ab-c^2} \begin{bmatrix} x_a - \mu_a & x_b - \mu_b \end{bmatrix} \begin{bmatrix} b & -c \\ -c & a \end{bmatrix} \begin{bmatrix} x_a - \mu_a \\ x_b - \mu_b \end{bmatrix} \\[12pt] &= \frac{b (x_a - \mu_a)^2 - 2c (x_a - \mu_a) (x_b - \mu_b) + a (x_b - \mu_b)^2}{ab-c^2}, \\[6pt] \end{align}
where $$\mu_a$$ and $$\mu_b$$ are the means of the two variables. You can tell that $$c<0$$ because the ellipse slopes in the negative direction (i.e., from top-left to bottom-right). You can tell that $$a because the ellipse is narrower on the horizontal axis (for variable $$x_a$$) than on the vertical axis (for variable $$x_b$$).
• Kindly, write something on the relationship between $P(C_1)$ and $P(C_2)$. What happens when they are equal or unequal? Jul 22 at 17:02