Treating Propagation of Errors in $R^n$ to $R^m$ Transformations

[MaartenB]

I want to know how to treat propagation of errors in general.
When the transformation of variables is a transformation of $R^n$ to $R^n$ it
simply involves a jacobian:
$$g(\vec{y}) = f(x(\vec{y}))|J|$$
With
$$J_{ij} = \frac{\partial x_i}{\partial y_j}$$
(see http://pdg.lbl.gov/2005/reviews/probrpp.pdf for instance)

But there are also situation of $R^n$ to $R^m$ possible.

This is how far I got:
(Much of this can be found in http://arxiv.org/abs/hep-ex/0002056 appendix A)

The characteristic function (CF) is defined as:
$$\phi_X(t) = E[e^{itX}] = \int e^{itx} f(x) dx$$
The inverse (FT):
$$f(x) = \frac{1}{2\pi} \int e^{-itX} \phi_X(t) dt$$
If we have a function Y=g(X) the pdf according to Kendal and Stuart (1943) is:
$$\phi_Y(t) = \int e^{itg(x)} f(x) dx$$
Taking the inverse, and some rewriting
$$f(y) = \frac{1}{2\pi} \int e^{-ity} \phi_Y(t) dt = \int f(x)\delta(y-g(x))dx$$
In vector form
$$f(y) = \int f(\vec{x})\delta(y-g(\vec{x}))d\vec{x}$$
Take for instance two independent variables taken from the same distribution:
$$Y = g(\vec{X}) = g(X_1, X_2) = X_1 + X_2$$
Then the resulting pdf is with:
$$f(y) = \int f(x_1,x_2)\delta(y-x_1-x_2)dx_1 dx_2 = \int f(x_1) f(y - x_1 - u)\delta(u)dx_1 du = \int f(x_1) f(y - x_1) dx_1$$
Using:
$$u = y - x_1 - x_2$$
$$x_2 = y - x_1 - u$$
$$dx_2 = du$$
Which is a simply a convolution, which is a known result,
see for instance: http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
Or for $R^1$ to $R^1$:
$$f(y) = \int f(x)\delta(y-g(x))dx = \int f(g^{-1}(y-u))\frac{1}{g'(x)}\delta(u)du = f(g^{-1}(y))\frac{1}{g'(x)} = f(x(y)) \frac{dx}{dy}$$
Using:
$$u = y - g(x)$$
$$du = \frac{dg(x)}{dx}dx$$
$$dx = \frac{1}{g'(x)}du$$
$$x = g^{-1}(y-u)$$
Also a general result, just a change of variable.

But how to do this for the case of $R^n$ to $R^m$?, I don't know how to evaluate
the dirac delta function in general.

I did find on wikipedia (http://en.wikipedia.org/wiki/Dirac_delta_function)
$$\int_V f(\mathbf{r}) \, \delta(g(\mathbf{r})) \, d^nr = \int_{\partial V}\frac{f(\mathbf{r})}{|\mathbf{\nabla}g|}\,d^{n-1}r$$
But I couldn't find a reference where this is explained.

But then again, if y is a vector, how to solve this?:
$$f(\vec{y}) = \int f(\vec{x})\vec{\delta}(\vec{y}-g(\vec{x}))d\vec{x}$$

Anyone got some hints? Or am I going the wrong direction with this?

 

[EnumaElish]

When you are transforming from R^n to R^m and m < n, you can define n - m new random variables and set them equal to an identical number of the existing variables. Example: given transformation Y = X1 + X2, you can define Y1 = Y and Y2 = X2, at which point you will have a square matrix. I hope this is helpful.