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Full title: Trend Analysis of Fragmented Time Series: Hypothesis Testing Based Adaptive Spline Filtering Method.

Missing data present significant challenges to trend analysis of time series. Straightforward approaches consisting of supplementing missing data with constant or zero values or with linear trends can severely degrade the quality of the trend analysis, which significantly reduces the reliability of the trend analysis. 

We present a robust adaptive approach to discover the trends from fragmented time series. The approach proposed in this paper is based on the HASF (Hypothesis-testing-based Adaptive Spline Filtering) trend analysis algorithm, which can accommodate non-uniform sampling and is therefore inherently robust to missing data. 

HASF adapts the nodes of the spline based on hypothesis testing and variance minimization, which adds to its robustness. Further improvement is obtained by filling gaps by data estimated in an earlier trend analysis, provided by HASF itself. Three variants for filling the gaps of missing data are considered, the best of which seems to consist of filling significantly large gaps with linear splines matched for continuity and smoothness with cubic splines covering data-dense regions. Small gaps are ignored and addressed by the underlying cubic spline fitting. 

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