# Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data
## 5 Quantitative Analysis of Noise Models  
In this section, we present and discuss the quantitative performance of the previously described noise models. For each model, the parameters were chosen to best fit the location data. The process of selecting which of these models best describes the data, called model selection, is a large and complex field in statistics; thus, we leave detailed explanation to the technical literature, e.g., [55] and references therein. There are three properties of model selection criteria: consistency, efficiency, and parsimony. Consistency means that the criterion selects the true model with probability approaching unity with increasing sample size provided the true model is among those considered. However, nothing guarantees that the true model is simple and tractable; thus, it is also necessary to measure how much error one might make by accepting a model knowing that it is not the true model. Efficiency is a measure of this property; as the sample size becomes large, the error in predictions using the selected model becomes indistinguishable from the error obtained using the best model among the candidates. Parsimony dictates that if all measures are equal for two competing statistical models, the preferred one should be the simpler model. The most common model selection criteria use the likelihood function, \(L\), and introduce some penalty proportional to the model complexity. Akaike's Information Criterion (AIC) is defined as \(\text{AIC}=-2\ln(L)+2k\), where \(k\) is the number of fit parameters in the model [56; 57]. The best model has the smallest AIC value--it encodes the most information using the least number of fitted parameters. The AIC has a definite minimum value with respect to \(k\), and the appropriate order of the model is determined by the value of \(k\) at this minimum. The AIC is biased for small samples, and a bias-corrected version (AICc) is gaining popularity [43]. Here, we analyse time series with \(n>10^{5}\) and \(k\approx 10\), so we can ignore this correction. Other criteria have been developed, e.g., Bayesian Information Criterion (BIC), which more severely penalises the introduction of extra model parameters by replacing the \(2k\) term with \(k+k\ln{(n)}\). Regarding the model selection criteria, AIC and AICc are both inconsistent, while BIC is only asymptotically consistent. Meanwhile, AIC is efficient, while BIC is not. Therefore there is no clear choice between AIC and BIC. The latter is asymptotically consistent as the sample size approaches infinity, while the former selects models that have acceptable predictive power [58]. We use AIC, as it is almost universally accepted. While it does not validate a statistical model, it determines relative probabilities between alternative models. We compare AIC values for a selection of models. Table 2 shows AIC scores for model fits of latitude, longitude, and altitude at four geographic locations. For each location, we individually fitted different noise models using the same observation datasets. For the AR(\(p\)), MA(\(q\)), and the ARMA(\(p\), \(q\)), we increased the number of parameters involved. Occasionally the statistics routine was unable to converge for the MA(\(q\)) processes. In these cases, the corresponding entry in Table 2 is left empty. Table 2 clearly shows that the Gaussian white noise model, irrespective of the geographic location or the coordinate, is the worst-performing noise model. All of its AIC values are positive, indicating a very low likelihood that the observations are described by the model. The MA models also seem a poor fit to the data, although one may increase the order of the model, \(q\), to achieve lower AIC scores. Our standard numerical routine, based on the Yule-Walker method [37], could not converge. In contrast, an AR model with increasing order fit best among the noise models examined.