Functional Outlier Detection With Application

Functional data analysis has attracted an increasing number of researchers during the last two decades,however,not enough consideration has been given to outlier detection of functional data,and the potential outliers will affect the statistical analysis of the data.Here,we propose three algorithms for one-dimensional functional outlier detection.Since the functional outlier detection algorithm based on the modified band depth and the modified epigraph index is fail to detect the isolate outliers,we propose a deformation anomaly,which can measure the information of the curve shape,and improve the outlier detection algorithm based on the deformation anomaly and the modified band depth.Simulation studies indicate that the improved outlier algorithm can effectively identify the isolate outliers,and it also improves the shape outliers' detection rates.Inspired by the data depth and deformation anomaly,we propose the position outlyingness and shape outlyingness in order to provide the position order and the shape order of the functional data.We also propose the comprehensive outlyingness to cover the data information of position and shape.Based on the above three outlyingness,we put forward a outlier detection algorithm.Simulation studies indicate that the outlier detection algorithm can effectively identify the magnitude outlier and shape outlier.We propose a robust principal component analysis to detect functional outliers based on the comprehensive depth.The data matrix has been sorted and filtered with comprehensive depth,then we apply the principal component analysis to the data matrix in order to find the data's best projection subspace,and the outlier detection algorithm is based on the robust scores distance and orthogonal distance.Simulation studies prove the effectiveness of this method.Finally,we apply the Spanish weather data to the three outlier detection algorithm,and compare their detection result's visualization.