Ways to Handle Non-linearities

There are a number of methods to help us deal with NLNG Problem Statement.

1. Linearization as shown with Extended Kalman Filter
2. Monte Carlo Method “brute force” method that randomly samples values in the input distribution, run them through the non-linearity, characterize the output distribution.
   1. Pretty dumb but its actually really accurate (law of large numbers says that you will get to the exact distribution as you reach a infinite number of samples)
   2. Used as an evaluation method
   3. Works with any PDF, not just Gaussians
3. Sigmapoint (Unscented) Transformation Its kinda seen as a compromise between full linearization and the monte carlo method

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EXAMPLE Say we have a 1D non-linearity $f(x)=x^2$ and the prior density is $\mathcal{N}({\mu }_x,{\sigma }_x^2)$

Using Monte Carlo Method There is a closed form, exact answer to this solution, so we don’t need to randomly sample.

$$x_i={\mu }_x+\delta x_i,\delta x_i\sim \mathcal{N}(0,{\sigma }_x^2)$$

Transforming it through the non-linearity we get:

$$y_i=({\mu }_x+\delta x_i{)}^2={\mu }_x^2+2{\mu }_x\delta x_i+\delta x_i^2$$

Grabbing the characteristics:

$${\mu }_y=E[y_i]={\mu }_x^2+2{\mu }_x\underset{0}{\underbrace{E[\delta x_i]}}+\underset{{\sigma }_x^2}{\underbrace{E[\delta x_i^2]}}={\mu }_x^2+{\sigma }_x^2$$ $${\sigma }_y^2=E[(y_i-{\mu }_y{)}^2]=2{\mu }_x^2{\sigma }_x^2+2{\sigma }_x^4$$

In truth the resulting output density is not a Gaussian, but it can resultantly be approximated as one with $\mathcal{N}({\mu }_y,{\sigma }_y^2)$

Using Linearization

$$y_i=f({\mu }_x+\delta x_i)\simeq \underset{{\mu }_{x^2}}{\underbrace{f({\mu }_x)}}+\underset{2{\mu }_x}{\underbrace{\frac{\partial f}{\partial x}{|}_{{\mu }_x}}}\delta x_i={\mu }_x^2+2{\mu }_x\delta x_i$$ $${\mu }_y=E[y_i]={\mu }_x^2+2{\mu }_x\underset{0}{\underbrace{E[\delta x_i]}}={\mu }_x^2$$ $${\sigma }_x^2=E[(y_i-{\mu }_i^2)]=E[(2{\mu }_x\delta x_i{)}^2]=4{\mu }_x^2{\sigma }_x^2$$

Which as you can see already has some descrepancies with our Monte Carlo Method. The linearized mean has a bias and the variance is too small.

Using Sigmapoint Transformation Its 1D so we need $2L+1=3$ sigmapoints.

$$x_0={\mu }_x,x_1={\mu }_x+\sqrt{1+\kappa }{\sigma }_x,x_1={\mu }_x-\sqrt{1+\kappa }{\sigma }_x$$

We can send these points through the nonlinearity.

$$y_o=f(x_o)={\mu }_x^2$$ $$y_1=f(x_1)=({\mu }_x+\sqrt{1+\kappa }{\sigma }_x{)}^2={\mu }_x^2+2{\mu }_x\sqrt{1+\kappa }{\sigma }_x+(1+\kappa ){\sigma }_x^2$$ $$y_2=f(x_2)=({\mu }_x-\sqrt{1+\kappa }{\sigma }_x{)}^2={\mu }_x^2-2{\mu }_x\sqrt{1+\kappa }{\sigma }_x+(1+\kappa ){\sigma }_x^2$$

Mean is given by

$${\mu }_y=\frac{1}{1+\kappa }\left(\kappa y_0+\frac{1}{2}\sum _{i=1}^2{y}_i\right)={\mu }_x^2+{\sigma }_x^2$$

Variance is given by

$${\sigma }_y^2=\frac{1}{1+\kappa }\left(\kappa (y_0-{\mu }_y{)}^2+\frac{1}{2}\sum _{i=1}^2(y_i-{\mu }_y{)}^2\right)=4{\mu }_x^2{\sigma }_x^2+\kappa {\sigma }_x^2$$

Which means that by having the user set $\kappa =2$ we end up with the correct mean and covariance of the output.