The $k$-means problem is to compute a set of $k$ centers (points) that minimizes the sum of squared distances to a given set of $n$ points in a metric space. Arguably, the most common algorithm to solve it is $k$-means++ which is easy to implement, and provides a provably small approximation factor in time that is linear in $n$. We generalize $k$-means++ to support: (i) non-metric spaces and any pseudo-distance function. In particular, it supports M-estimators functions that handle outliers, e.g. where the distance $\mathrm{dist}(p,x)$ between a pair of points is replaced by $\min {\mathrm{dist}(p,x),1}$. (ii) $k$-means clustering with $m\geq 1$ outliers, i.e., where the $m$ farthest points from the $k$ centers are excluded from the total sum of distances. This is the first algorithm whose running time is linear in $n$ and polynomial in $k$ and $m$.

A critical analysis of the classical definition of outsiders is given. Some examples show that this notion is not universal and has at least two drawbacks. Particularly, the set of outsiders may have the probability arbitrary close to one half. On the other hand, the deleting of the set of outsiders may dramatically change the value of the median.

It is a common practice to detect outliers in a financial time series in order to avoid the adverse effect of additive outliers. This paper investigated the performance of GARCH family models (sGARCH; gjrGARCH; iGARCH; TGARCH and NGARCH) in the presence of different sizes of outliers (small, medium and large) for different time series lengths (250, 500, 750, 1000, 1250 and 1500) using root mean square error (RMSE) and mean absolute error (MAE) to adjudge the models. In a simulation iteration of 1000 times in R environment using rugarch package, results revealed that for small size of outliers, irrespective of the length of time series, iGARCH dominated, for medium size of outliers, it was sGARCH and gjrGARCH that dominated irrespective of time series length, while for large size of outliers, irrespective of time series length, gjrGARCH dominated. The study further leveled that in the presence of additive outliers on time series analysis, both RMSE and MAE increased as the time series length increased.

The chain ladder method is a popular technique to estimate the future reserves needed to handle claims that are not fully settled. Since the predictions of the aggregate portfolio (consisting of different subportfolios) in general differ from the sum of the predictions of the subportfolios, a general multivariate chain ladder (GMCL) method has already been proposed. However, the GMCL method is based on the seemingly unrelated regression (SUR) technique which makes it very sensitive to outliers. To address this issue a robust alternative is introduced which estimates the SUR parameters in a more outlier resistant way. With the robust methodology it is possible to detect which claims have an abnormally large influence on the reserve estimates. We introduce a simulation design to generate artificial multivariate run-off triangles based on the GMCL model and illustrate the importance of taking into account contemporaneous correlations and structural connections between the run-off triangles. By adding contamination to these artificial datasets, the sensitivity of the traditional GMCL method and the good performance of the robust GMCL method is shown. From the analysis of a portfolio from practice it is clear that the robust GMCL method can provide better insight in the structure of the data.

In this paper, the set-membership affine projection (SM-AP) algorithm is utilized to censor non-informative data in big data applications. To this end, the probability distribution of the additive noise signal and the excess of mean-squared error (EMSE) in steady-state are employed in order to estimate the threshold parameter of the single threshold SM-AP (ST-SM-AP) algorithm aiming at attaining the desired update rate. Furthermore, by defining an acceptable range for the error signal, the double threshold SM-AP (DT-SM-AP) algorithm is proposed to detect very large errors due to the irrelevant data such as outliers. The DT-SM-AP algorithm can censor non-informative and irrelevant data in big data applications, and it can improve misalignment and convergence rate of the learning process with high computational efficiency. The simulation and numerical results corroborate the superiority of the proposed algorithms over traditional algorithms.

One of the most destructive natural disasters is the earthquake which brings enormous risks to humankind. The objective of the current study was to determine the Earthquake’s remote sensing multiparameter (i.e. land surface temperature (LST), air temperature, specific humidity, precipitation and wind speed) spatiotemporal anomaly of many earthquake samples occurred during 2018 around the world. In this research 11 earthquake (M > 6:0) studied (4 samples selected in a land with transparent sky situations, 3 samples in land within cloudy situations and 4 samples in marine earthquakes). The interquartile range (IQR) and mean ± 2σ methods utilized to improve the efficiency of anomalous differences. As a result, based on the IQR method, negative anomaly before the event detected during the daytime in Mexico and during the nighttime in Afghanistan. In addition, a negative outlier of brightness temperature (BT) detected in Alaska before, after and during the event. In contrast, based on IQR and mean ± 2σ positive anomaly detected in precipitation before and after the event in all investigated examples. According to mean ± 2σ, negative anomaly LST, specific humidity, sea surface temperature (SST_100) and wind detected in most examined earthquake samples. In contrast, positive SST_0 anomaly observed in Fiji and Honduras after the earthquake. Our results suggested in marine earthquakes, for earthquake forecasting we can merge a prior negative anomaly in the wind speed and SST_100. Regarding the in land cloudy sky earthquakes, merging anomaly parameters could be the negative prior anomaly in BT, skin temperature, in contrast, a positive anomaly in precipitation. In land transparent sky earthquake, usually negative prior anomalies in air temperature, specific humidity and LST.