{"id":73545,"date":"2021-01-25T13:15:28","date_gmt":"2021-01-25T18:15:28","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=73545"},"modified":"2022-11-21T09:46:58","modified_gmt":"2022-11-21T14:46:58","slug":"trend-following-filters-part-2-2","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-2-2\/","title":{"rendered":"Trend-Following Filters \u2013 Part 2\/2"},"content":{"rendered":"\n<p><em>The post &#8220;Trend-Following Filters \u2013 Part 2\/2&#8221; first appeared on <a href=\"https:\/\/alphaarchitect.com\/2021\/01\/21\/trend-following-filters-part-2-2\/\">Alpha Architect Blog<\/a>.<\/em><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-1-introduction\">1. Introduction<\/h2>\n\n\n\n<p>Part 1 of this analysis, which is available&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2020\/12\/29\/trend-following-filters-part-1-2\/\" target=\"_blank\" rel=\"noreferrer noopener\">here<\/a>, 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&nbsp;<em>a<\/em>&nbsp;and a linear trend&nbsp;<em>b<\/em>&nbsp;which is contaminated with random normally distributed noise \u03b5(t) where \u03b5(t) ~ N(0, \u03c3<sub>\u03b5<\/sub><sup>2<\/sup>):<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>second-order process \u2013 mean&nbsp;<em>a<\/em>&nbsp;and linear trend&nbsp;<em>b<\/em>: &nbsp; x(t) =&nbsp;<em>a<\/em>&nbsp;+&nbsp;<em>b<\/em>*t + \u03b5(t)<\/li><\/ul>\n\n\n\n<p>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&nbsp;<em>a<\/em>, a linear trend&nbsp;<em>b<\/em>, and a quadratic trend&nbsp;<em>c<\/em>&nbsp;which is contaminated with random normally distributed noise \u03b5(t):<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>third-order process \u2013 mean&nbsp;<em>a<\/em>, linear trend&nbsp;<em>b<\/em>, and quadratic trend&nbsp;<em>c<\/em>: &nbsp; x(t) =&nbsp;<em>a<\/em>&nbsp;+&nbsp;<em>b<\/em>*t + \u00bd*<em>c<\/em>*t<sup>2<\/sup>&nbsp;+ \u03b5(t)<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"600\" height=\"396\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter.png\" alt=\"\" class=\"wp-image-73551 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter.png 600w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-300x198.png 300w\" data-sizes=\"(max-width: 600px) 100vw, 600px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 600px; aspect-ratio: 600\/396;\" \/><\/figure>\n\n\n\n<p>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 \u201cAn Introduction to Digital Signal Processing for Trend Following\u201d, which is available&nbsp;<a href=\"https:\/\/alphaarchitect.com\/2020\/08\/13\/an-introduction-to-digital-signal-processing-for-trend-following\/\" target=\"_blank\" rel=\"noreferrer noopener\">here<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-2-triple-moving-average-tma\">2. Triple Moving Average (TMA)<\/h2>\n\n\n\n<p>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\u00a0<a href=\"https:\/\/alphaarchitect.com\/2021\/01\/21\/trend-following-filters-part-2-2\/#note-60324-1\"><sup>1<\/sup><\/a>. The triple moving average set of equations is:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1100\" height=\"429\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2-1100x429.png\" alt=\"Trend-Following Filters\n\" class=\"wp-image-73552 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2-1100x429.png 1100w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2-700x273.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2-300x117.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2-768x300.png 768w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2-1536x599.png 1536w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-2.png 1600w\" data-sizes=\"(max-width: 1100px) 100vw, 1100px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 1100px; aspect-ratio: 1100\/429;\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p>where N is the number of input data points, i.e., the moving average length N (N &gt; 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.<\/p>\n\n\n\n<p>The triple moving average generates five main outputs: an estimate of the mean y<sub>0<\/sub>(t) at time step t, an estimate of the linear trend y<sub>1<\/sub>(t) at time step t, an estimate of the quadratic trend y<sub>2<\/sub>(t) at time step t, a mean expectation y<sub>0<\/sub>^(t) made at time step t for the next time step t+1, and a linear trend expectation y<sub>1<\/sub>^(t) made at time step t for the next time step t+1. The quadratic trend expectation y<sub>2<\/sub>^(t) for the next time step t+1 is the same as the quadratic trend estimate y<sub>2<\/sub>(t) since a triple moving average does not model cubic and higher-order trends.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"800\" height=\"450\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-3.png\" alt=\"\" class=\"wp-image-73554 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-3.png 800w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-3-700x394.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-3-300x169.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2021\/01\/alpha-architect-trend-filter-3-768x432.png 768w\" data-sizes=\"(max-width: 800px) 100vw, 800px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 800px; aspect-ratio: 800\/450;\" \/><\/figure>\n\n\n\n<p><em>Visit Alpha Architect to learn more about Triple Moving Average Mean Filter Frequency Response (N = 10)<\/em>.<\/p>\n\n\n\n<p>Notes:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>Brown, R. G.,&nbsp;<em>Smoothing, Forecasting, and Prediction of Discrete Time Series<\/em>, Prentice Hall, 1962.&nbsp;<\/li><li>Brown, R. G., Smoothing, Forecasting, and Prediction of Discrete Time Series, Prentice Hall, 1962.&nbsp;<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Learn more about filters modeled on second-order processes from a digital signal processing (DSP) perspective with this feature by Henry Stern &#8211; via Alpha Architect Blog.<\/p>\n","protected":false},"author":575,"featured_media":48516,"comment_status":"closed","ping_status":"open","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341,352,344],"tags":[9116,5761,9117],"contributors-categories":[13651],"class_list":{"0":"post-73545","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"category-quant-north-america","11":"category-quant-regions","12":"tag-digital-signal-processing-dsp","13":"tag-trend-following","14":"tag-trend-following-filters","15":"contributors-categories-alpha-architect"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.4) - 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