# 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

## 1 Answer

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].

In SLAM it is commonly used to model the probability of a certain landmark providing a certain measurement.

• Can you point to some literature how it gets related to landmark observations? – nkd Jan 28 '15 at 1:27
• Yes, but honestly just googling "data association chi-square" or "data association mahalanobis distance" will provide a ton of references. It is standard practice. For a nice introduction, see Bar-Shalom's "Estimation with Applications to Tracking and Navigation" – Josh Vander Hook Jan 28 '15 at 16:13