# 1 Trend Filtering

@article{Kim20091TF,
title={1 Trend Filtering},
author={Seung-Jean Kim and Kwangmoo Koh and Stephen P. Boyd and Dimitry M. Gorinevsky},
journal={SIAM Rev.},
year={2009},
volume={51},
pages={339-360}
}
The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on Hodrick-Prescott (H-P) filtering, a widely used method for trend estimation. The proposed $\ell_1$ trend filtering method substitutes a sum of absolute values (i.e., $\ell_1$ norm) for the sum of squares used in H-P filtering to penalize variations in the estimated trend. The $\ell_1$ trend filtering method produces trend estimates that are piecewise linear… Expand
435 Citations
HP Trend Filtering Using Gaussian Mixture Model Weighted Heuristic
• Mathematics, Computer Science
• 2014 IEEE 26th International Conference on Tools with Artificial Intelligence
• 2014
A modified version of HP weighted heuristic is presented that provides the best trend according to the abovementioned criteria and Gaussian Mixture Models on the preliminary estimated trend are used in the weighted HP heuristic to decrease the penalty in the objective function for turning-point intervals. Expand
Adaptive piecewise polynomial estimation via trend filtering
We study trend filtering, a recently proposed tool of Kim et al. [SIAM Rev. 51 (2009) 339-360] for nonparametric regression. The trend filtering estimate is defined as the minimizer of a penalizedExpand
Trend Filtering: Empirical Mode Decompositions versus ℓ1 and Hodrick-Prescott
• Computer Science
• 2011
This work proposes a novel approach based on empirical mode decomposition (EMD), called EMD trend filtering, which offers the possibility of estimating a trend of arbitrary shape as a sum of low-frequency intrinsic mode functions produced by the EMD. Expand
Business Cycles, Trend Elimination, and the HP Filter
• Mathematics
• 2015
We analyze trend elimination methods and business cycle estimation by data filtering of the type introduced by Whittaker (1923) and popularized in economics in a particular form by Hodrick andExpand
$$\ell _{1}$$ Common Trend Filtering
• 2021
The $$\ell _{1}$$ trend filtering enables us to estimate a continuous piecewise linear trend of univariate time series. This filter and its variants have subsequently been applied in various fields,Expand
Accurate Changing Point Detection for ℓ1 Mean Filtering
• Computer Science
• IEEE Signal Process. Lett.
• 2016
The main contribution in this paper is incorporating a technique to remove false changing points to a fast mean filtering algorithm, referred to as the taut-string method, resulting in an efficient procedure with accurate change point detection and thus the removal of the stair-case effect. Expand
Enhancements of Moving Trend Based Filters Aimed at Time Series Prediction
• Computer Science
• ICSS
• 2013
MTFs properties in frequency domain are considered, from seasonal time series decomposition, smoothing and prediction efficiency perspectives, and a number of MTFs enhancements is proposed, involving different approximating polynomials and final section signal corrections. Expand
Accurate Changing Point Detection for ${\ell _1}$ Mean Filtering
• IEEE Signal Processing Letters
• 2016
It is often desirable to find the underlying trends in time series data. This is a well known signal processing problem that has many applications in areas such as financial data analysis,Expand
Wasserstein total variation filtering
• Engineering, Computer Science
• ArXiv
• 2019
A globally optimal algorithm for efficiently estimating the filtered signal under a Wasserstein finite differences operator is introduced and the efficacy of the proposed algorithm in preserving spatiotemporal trends in time series video is demonstrated in both simulated and fluorescent microscopy videos. Expand
Trend Filtering Methods for Momentum Strategies
• Engineering
• 2011
This paper studies trend filtering methods. These methods are widely used in momentum strategies, which correspond to an investment style based only on the history of past prices. For example, theExpand

#### References

SHOWING 1-10 OF 157 REFERENCES
The Use of Butterworth Filters for Trend and Cycle Estimation in Economic Time Series
Long-term trends and business cycles are usually estimated by applying the Hodrick and Prescott (HP) filter to X-11 seasonally adjusted data. A two-stage procedure is proposed in this article toExpand
A Bayesian Time Series Model of Multiple Structural Changes in Level, Trend, and Variance
• Mathematics
• 2000
We consider a deterministically trending dynamic time series model in which multiple structural changes in level, trend, and error variance are modeled explicitly and the number, but not the timing,Expand
Local trend estimation and seasonal adjustment of economic and social time series (with discussion)
• Economics, Geography
• 1982
The first main purpose of this paper is to compare the performance of a number of methods of estimating local trends levels in a sample of 23 monthly economic and social time series. This is done forExpand
Jump process for the trend estimation of time series
• Computer Science, Mathematics
• Comput. Stat. Data Anal.
• 2003
The proposed jump process estimator can locally minimize two important features of a trend, the smoothness and fidelity, and explicitly balance the fundamental tradeoff between them. Expand
On trend estimation of time-series: a simple linear programming approach
• Mathematics
• 1997
This paper presents a simple linear programme for the solution of the trend-estimation problem in time-series. We studied the trade-off between two important properties of the trend component:Expand
Trend assessment in a long memory dependence model using the discrete wavelet transform
• Mathematics
• 2004
In this article we consider trend to be smooth deterministic changes over long scales, and tackle the problem of trend estimation in the presence of long memory errors (slowly decayingExpand
Estimation of Quasi-Linear Trend and Seasonal Variation
Abstract Given a series of quarterly data, estimates may be obtained for both trend and seasonal variation by minimising the sum, or more generally a linear combination, of two sums of squares, oneExpand
General Model-Based Filters for Extracting Cycles and Trends in Economic Time Series
• Economics
• Review of Economics and Statistics
• 2003
A class of model-based filters for extracting trends and cycles in economic time series is presented. These lowpass and bandpass filters are derived in a mutually consistent manner as the jointExpand
A Simple Method of Trend Construction
SUMMARY The principle adopted here in the construction of a trend for a time series consists in minimizing a linear combination of two sums of squares, of which one refers to the second differencesExpand
Trend estimation and de-trending via rational square-wave filters
This paper gives an account of some techniques of linear filtering which can be used for extracting trends from economic time series of limited duration and for generating de-trended series. A familyExpand