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# Trend-Following Filters – Part 2/2

###### Posted January 25, 2021 at 1:15 pm
Henry Stern
Alpha Architect

The post “Trend-Following Filters – Part 2/2” first appeared on Alpha Architect Blog.

## 1. Introduction

Part 1 of this analysis, which is available here, examines filters modeled on second-order processes from a digital signal processing (DSP) perspective to illustrate their properties and limitations. To briefly recap, a time series based on a second-order process consists of a mean a and a linear trend b which is contaminated with random normally distributed noise ε(t) where ε(t) ~ N(0, σε2):

• second-order process – mean a and linear trend b:   x(t) = a + b*t + ε(t)

The filters analyzed in Part 1 include double moving average, double linear weighted moving average, double exponential smoothing, and alpha-beta tracking filters. Part 2 extends the analysis to filters modeled on third-order processes. A third-order process consists of a mean a, a linear trend b, and a quadratic trend c which is contaminated with random normally distributed noise ε(t):

• third-order process – mean a, linear trend b, and quadratic trend c:   x(t) = a + b*t + ½*c*t2 + ε(t)

The filters analyzed are triple moving average, triple linear weighted moving average, triple exponential smoothing, and alpha-beta-gamma tracking filters. Note: This article assumes familiarity with Part 1 and also with the characteristics of financial time series and the digital signal processing concepts discussed in “An Introduction to Digital Signal Processing for Trend Following”, which is available here.

## 2. Triple Moving Average (TMA)

Triple moving average (TMA) is a time series estimation and process control method that uses three single moving averages to estimate time series that contain linear and quadratic trends 1. The triple moving average set of equations is:

where N is the number of input data points, i.e., the moving average length N (N > 1), included in the three single moving averages used to calculate the triple moving average, and x(t) represents the price at integer time t.

The triple moving average generates five main outputs: an estimate of the mean y0(t) at time step t, an estimate of the linear trend y1(t) at time step t, an estimate of the quadratic trend y2(t) at time step t, a mean expectation y0^(t) made at time step t for the next time step t+1, and a linear trend expectation y1^(t) made at time step t for the next time step t+1. The quadratic trend expectation y2^(t) for the next time step t+1 is the same as the quadratic trend estimate y2(t) since a triple moving average does not model cubic and higher-order trends.

Visit Alpha Architect to learn more about Triple Moving Average Mean Filter Frequency Response (N = 10).

Notes:

1. Brown, R. G., Smoothing, Forecasting, and Prediction of Discrete Time Series, Prentice Hall, 1962.
2. Brown, R. G., Smoothing, Forecasting, and Prediction of Discrete Time Series, Prentice Hall, 1962.
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