# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 2 Structure of the Dataset and Preliminary Analysis
The _outlier detection by visual screening_ and LMS regression with Bonferroni correction give comparable results;
* Many robust methods (including those implemented in FSDA) are based on the assumption that the error component of the data analyzed is normally distributed. As the actual distribution can of course deviate from this common assumption, the confidence intervals are just 'nominal' in that the actual proportion of outliers detected can differ from the expected one. * median | > treshold then discard satellites in current epoch. would have not detected the obvious outliers. We now exploit, on the same dataset, two other estimators in addition to LMS above: least trimmed squares (LTS) and forward search (FS) as implemented in the FSDA toolkit. A detailed description of these methods with their actual implementation in FSDA can be found in [18]. The three robust regression methods build a _linear model_ upon a set of data points composed of a vector of _independent variables_ and one _dependent variable_. In our case, the data points would be structured, as shown in Table 1, as a vector of independent variables containing at least (epoch, satellite, elevation) and with the pre-fit as the dependent variable. This would lead to a multi-variate distribution. We adopted a simplification, by considering the dataset as a sample from a uni-variate distribution allowing, in each epoch, for the repetition of the observations. While computing the _linear model_, we will not make distinctions among the individual satellites and we will not consider the elevation and other information included in the complete satellite's messages. The formal expression of the _linear model_ (with intercept) is: \(y=x\theta_{1}+\theta_{2}+\varepsilon\), where \(\varepsilon\) represents a normally distributed error and \(x\) represents now the pairs (epoch, satellite). Given \(n\) as the cardinality of the dataset, the set of data points will be \(\{x_{i},y_{i}\}_{1\leq i\leq n}\) with \(y_{i}\) the pre-fit and \(x_{i}\) representing a distinct pair (epoch, satellite). The model is then computed by estimating with \(\hat{\theta}_{i}\) the parameters \(\theta_{i}\) with respect to an appropriate objective function of the _residuals_: \(r_{i}=y_{i}-(x_{i}\hat{\theta}_{1}+\hat{\theta}_{2})\). The objective functions to minimize are, respectively, the sum of the squares of the _residuals_ for OLS, the median of the squares of the residuals for LMS and the sum of the squares of a selected subset of the residuals for LTS [16]:  
\[\min_{\theta}\sum_{i}^{n}r_{i}^{2}\qquad\qquad\min_{\theta}\operatorname*{ median}_{i}r_{i}^{2}\qquad\qquad\min_{\theta}\sum_{i}^{h}r_{i:n}^{2} \tag{5}\]  
where \(r_{i:n}^{2}\) is the ordered set of the squared residuals \(r_{1:n}^{2}\leq r_{2:n}^{2}\leq\ldots\leq r_{n:n}^{2}\) and \(h\) decides the appropriate subset of the points considered. Typically, for LTS, \(h\) is chosen with a _breakdown point_ of 50% (\(h=\lceil 0.5\cdot n\rceil\)) but it can be automatically increased according to the estimate of the residuals variance [16]. This _re-weighting_ step allows keeping the desired robustness while increasing the efficiency of the estimator \(\hat{\theta}_{i}\). The _forward search_ further exploits the re-weighting idea by building a sequence of data subsets of increasing size, and fitting a sequence of estimators \(\hat{\theta}_{i}^{m},\ m_{0}\leq m\leq n\), on them. More precisely, the forward process starts with a first small subset of \(m_{0}\) data points, where \(\hat{\theta}_{i}^{m_{0}}\) is estimated with LMS or LTS. Note that \(m_{0}\) can just be equal to the number of parameters, so the computational burden required by the initial LMS or LTS application is very limited. Then, the process continues by repeatedly computing \(\hat{\theta}_{i}^{m}\) with the OLS on the subsequent subsets of cardinality \(m_{0}<m\leq n\).