I’ve learn this two articles concerning the “normal deviation” of bi (and multi) variate gaussians: newer, older

So lets have the next gaussian:

$$f(x,y)=Aexpbigg[-frac{1}{2(1-rho^2)}bigg(frac{x^2}{sigma_x^2}+frac{y^2}{sigma_y^2}-frac{2rho xy}{sigma_xsigma_y}bigg)bigg]$$

(The gaussian is centered on the origin, $rho$ is the correlation coefficient, and $sigma_{x,y}$ are the usual deviations alongside $X$ and $Y$ axes.)

They intruduced the usual deviation curve (SDC) as:

$$(x^2+y^2)^2=sigma_x^2x^2+sigma_y^2y^2+2rhosigma_xsigma_yxy$$

(This got here from the case, when $sigma$ is calculated in some rotated coordinate system and the outcomes are reworked again to the unique coordinate system. So far as i may perceive.)

And the usual deviation ellipse (SDE) is outlined with the Mahalanobis distance (if I get it proper). So one can get the next ellipse (for an arbitary distance $d$):

$$pm d=sqrt{frac{1}{1-rho^2}bigg(frac{x^2}{sigma_x^2}+frac{y^2}{sigma_y^2}-frac{2rho xy}{sigma_xsigma_y}bigg)}$$

Utilizing the covariance matrix:

start{bmatrix}

sigma_x^2 & rhosigma_xsigma_y

rhosigma_xsigma_y & sigma_y^2

finish{bmatrix}

Additionally this ellipse defines the factors $f(x_d,y_d)=c$, the place $d^2=-2ln(c/A)$.

And we will write the next 1D gaussian (or half gaussian, if we assume $d>0$):

$$f(d)=Aexp[-d^2/2]$$

what defines the factors the place the density operate of the 2D gaussian has some fixed $c$ worth. So the (“$1 sigma$“) SDE can be $d=1$.

So for me the SDE appears to be the $(pm)sigma$ factors of all of the 1D gaussians, which can be a “central part” of the unique 2D gaussian (i.e. $f(x,y(x))$ the place $y(x)=ax$ with any $a$). Whereas the SDC appears to be the $(pm)sigma$ factors of all of the 1D gaussians, which can be a “central projection” of the unique 2D gaussian (i.e. projecting to $y(x)=ax$ with any $a$).

So basecally my query is, if these are good interpretations? (And if that’s the case, how may one generalize SDC for different values than “$1sigma$“?)