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.

 

[tacman]

Treating propagation of errors in transformations from $R^n$ to $R^m$ can be a complex task, but there are some general principles that can guide the process. One approach is to use the Jacobian matrix, which accounts for the change in variables in the transformation. This can be seen in the formula for the characteristic function and inverse Fourier transform, where the Jacobian appears as a factor.

In general, when dealing with transformations from $R^n$ to $R^m$, it is important to consider the dimensionality of the variables involved. For example, in the case of two independent variables taken from the same distribution, the resulting pdf can be calculated using convolution, which is a known result. However, when dealing with vector variables, such as in the case of $R^n$ to $R^m$, the dirac delta function becomes a vector dirac delta function, which may require a different approach.

One possible approach is to use the formula found on Wikipedia for the vector dirac delta function, which involves integrating over the boundary of the region. However, it may be helpful to seek out additional resources or references that explain this formula in more detail. Alternatively, there may be other methods or techniques that can be used to handle vector dirac delta functions in the context of transformations from $R^n$ to $R^m$.

Overall, it is important to carefully consider the specific variables and dimensions involved in the transformation and to seek out additional resources or guidance when necessary.