Small Fraction Of Outliers Research Articles

Robust model selection in linear regression models using information complexity

In recent years, in the literature of linear regression models, robust model selection methods have received increasing attention when the datasets contain even a small fraction of outliers. Outliers can have a serious impact on statistical inference and the choice of models using model selection criteria. Most of the existing robust information-based model selection methods are confined to robust AIC and robust AIC-type criteria. However, the penalty function used in AIC-type criteria happens to be insufficient to measure the overall model complexity in the presence of outliers and when the model is misspecified. Furthermore, in the literature, there is not much attention paid to the overall model complexity despite its importance. To overcome these problems, this paper proposes a robust version of the information-theoretic measure of complexity (ICOMP) criterion due to Bozdogan based on robust alternatives to the maximum likelihood (ML) method. Unlike the AIC criterion, ICOMP penalizes not only the number of free parameters but also the profusion of the model complexity to reduce the effect of outliers. The proposed robust ICOMP criteria are based on robust M, S, and MM estimation methods. Numerical examples are shown using a large-scale Monte Carlo simulation to study the performance of the proposed criteria. Also, a real numerical example is provided to detect outliers and to choose the best subset of predictors. Our results show the flexibility and the versatility of the proposed new approach.

A Critical Study of Usefulness of Selected Functional Classifiers in Economics

In this paper we conduct a critical analysis of the most popular functional classifiers. Moreover, we propose a new classifier for functional data. Some robustness properties of the functional classifiers are discussed as well. We can use an approach worked out in this paper to predict the expected state of the economy from aggregated Consumer Confidence Index (CCI, measuring consumers optimism) and Industrial Price Index (IPI, reflecting a degree of optimism in industry sector) exploiting not only scalar values of the indices but also the trajectories/shapes of functions describing the indices. Thus our considerations may be helpful in constructing a better economic barometer. As far as we know, this is the first comparison of functional classifiers with respect to a criterion of their usefulness in economic applications. The main result of the paper is a presentation of how a small fraction of outliers in a training sample, which are linearly independent from the training sample, consisting of almost linearly dependent functions, corrupt all analysed classifiers.

Fama-French 1992 Redux with Robust Statistics

Robust statistical methods provide estimates that are not much influenced by a small percentage of outliers but perform in a nearly optimal manner for normally distributed data. Unfortunately, such methods are rarely used in quantitative finance research despite their potential utility, particularly in empirical asset pricing studies. As a means of stimulating the use of robust statistical methods in such studies and in quantitative finance in general, we demonstrate the efficacy of using a theoretically well-justified robust regression method in the cross-sectional regressions often used in empirical asset pricing studies. We compare the results of using both least squares and robust regression methods for the models presented in Fama and French (1992) (FF92), as well as some extensions to these models, over the time period 1963–2015 and subsets thereof. Our analysis clearly demonstrates that a very small fraction of outliers, in the returns and/or the factors, often distorts least squares cross-sectional regression estimates sufficiently enough to result in misleading conclusions as to whether a risk factor is priced. We reconfirm previously reported robust regression results demonstrating a positive relationship between average equity returns and firm size during the period 1963–1990 of the FF92 study, and show that this relationship continues to hold through 2015. Furthermore, we show that, once the impact of extreme outliers is eliminated by use of robust regression, the size effect is significant in most months, not just in January as was previously shown by other researchers. We confirm and extend the FF92 results and other previous work demonstrating a positive relationship between average returns and the book-to-market ratio, and clarify that the relationship is driven largely by small stocks. Contrary to FF92 and other previous empirical studies, we find a significant and negative relationship between average returns and beta for most U.S. equities between 1963 and 2015 when the influence of a small fraction of outliers is eliminated by robust regression. Furthermore, we show that there exists a highly significant interaction between size and beta that is needed to fully explain the dependence of returns on size and beta. Unlike FF92, who focused on positive earnings-to-price, we find that total earnings-to-price factor is highly significant in explaining the cross-section of returns for small to moderately-sized stocks both alone and when included along with the size and book-to-market factors. Finally, we also show that the time series of cross-sectional regression slopes contain large outliers, and that a robust location estimate provides a better representation of the “typical” slope than the sample mean.

Local Variability of Macular Thickness Measurements With SD-OCT and Influencing Factors.

PurposeTo compare the intrasession variability of spectral-domain optical coherence tomography (SD-OCT)-derived local macular thickness measures and explore influencing factors.MethodsOne hundred two glaucomatous eyes (102 patients) and 21 healthy eyes (21 subjects) with three good quality macular images during the same session were enrolled. Thickness measurements were calculated for 3° superpixels for the inner plexiform (IPL), ganglion cell (GCL), or retinal nerve fiber layers (mRNFL), GC/IPL, ganglion cell complex, and full macular thickness. Spatial distribution and magnitude of measurement errors (ME; differences between the 3 individual superpixel values and their mean) and association between MEs and thickness, age, axial length, and image quality were explored.ResultsMEs had a normal distribution with mostly random noise along with a small fraction of outliers (1.2%–6.6%; highest variability in mRNFL and on the nasal border) based on M-estimation. Boundaries of 95% prediction intervals for variability reached a maximum of 3 μm for all layers and diagnostic groups after exclusion of outliers. Correlation between proportion of outliers and thickness measures varied among various parameters. Age, axial length, or image quality did not influence MEs (P > 0.05 for both groups).ConclusionsLocal variability of macular SD-OCT measurements is low and uniform across the macula. The relationship between superpixel thickness and outlier proportion varied as a function of the parameter of interest.Translational RelevanceGiven the low and uniform variability within and across eyes, definition of an individualized ‘variability space' seems unnecessary. The variability measurements from this study could be used for designing algorithms for detection of glaucoma progression.

Quasi-stellar objects in the ALHAMBRA survey

We characterize the ability of the ALHAMBRA survey to assign accurate photo-z's to BLAGN and QSOs based on their ALHAMBRA very-low-resolution optical-NIR spectroscopy. A sample of 170 spectroscopically identified BLAGN and QSOs have been used together with a library of templates (including SEDs from AGN, normal, starburst galaxies and stars) in order to fit the 23 photometric data points provided by ALHAMBRA in the optical and NIR (20 medium-band optical filters plus the standard JHKs). We find that the ALHAMBRA photometry is able to provide an accurate photo-z and spectral classification for ~88% of the spectroscopic sources over 2.5 deg^2 in different areas of the survey, all of them brighter than m678=23.5 (equivalent to r(SLOAN)~24.0). The derived photo-z accuracy is better than 1% and comparable to the most recent results in other cosmological fields. The fraction of outliers (~12%) is mainly caused by the larger photometric errors for the faintest sources and the intrinsic variability of the BLAGN/QSO population. A small fraction of outliers may have an incorrectly assigned spectroscopic redshift. The definition of the ALHAMBRA survey in terms of the number of filters, filter properties, area coverage and depth is able to provide photometric redshifts for BLAGN/QSOs with a precision similar to any previous survey that makes use of medium-band optical photometry. In agreement with previous literature results, our analysis also reveals that, in the 0<z<4 redshift interval, very accurate photo-z can be obtained without the use of near-IR broadband photometry at the expense of a slight increase of outliers. The NIR importance is expected to increase at higher redshifts (z>4). These results are relevant for the design of future optical follow-ups of surveys with a large fraction of BLAGN, as it is the case for X-rays or radio surveys.

A k-Median Algorithm with Running Time Independent of Data Size

We give a sampling-based algorithm for the k-Median problem, with running time O(k(\frac{k^2}{\epsilon} log k)2 log(\frac{k}{\epsilon} log k)), where k is the desired number of clusters and e is a confidence parameter. This is the first k-Median algorithm with fully polynomial running time that is independent of n, the size of the data set. It gives a solution that is, with high probability, an O(1)-approximation, if each cluster in some optimal solution has Ω(\frac{n\epsilon}{k}) points. We also give weakly-polynomial-time algorithms for this problem and a relaxed version of k-Median in which a small fraction of outliers can be excluded. We give near-matching lower bounds showing that this assumption about cluster size is necessary. We also present a related algorithm for finding a clustering that excludes a small number of outliers.