# role of chi2 in SLAM back-end optimization

All-most all SLAM back-end implementation compute chi2 estimates. Chi2 computation is usually used to compute the best-fitness score of model to the data. How it is related to optimization framework for SLAM?

Rgds Nitin

• – CroCo Jan 30 '15 at 5:09

Specifically, the Chi-Square Distribution(or Chi2, $\chi^2$, or equivalently $\chi^2_1$) is used to model the probability of the absolute value of the deviation of the measurement from it's expected value. This calculation is vital to tackle the measurement origin uncertainty problem. It can also be used to determine the "correctness" of a multi-hypothesis estimate using a similar idea, but I won't touch on that specifically.
Note that the $\chi^2_k$ distribution has $k$ degrees of freedom. Why $\chi^2_k$? Any time you assume Gaussian (e.g., $\mathcal{N}(0,1)$) measurement noise, you will encounter the $\chi^2_1$ distribution because the two-norm squared of a $n$-dimensional vector of $\mathcal{N}(0,1)$ variables is equivalent to a $\chi_{n}^2$ pdf. [see: Applications of Chi-squared].