{"id":124294,"date":"2022-02-16T11:36:55","date_gmt":"2022-02-16T16:36:55","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=124294"},"modified":"2022-11-21T09:51:49","modified_gmt":"2022-11-21T14:51:49","slug":"trend-following-filters-part-5","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/trend-following-filters-part-5\/","title":{"rendered":"Trend-Following Filters \u2013 Part 5"},"content":{"rendered":"\n

The article “Trend-Following Filters \u2013 Part 5” first appeared on Alpha Architect Blog<\/a>. <\/em><\/p>\n\n\n\n

Excerpt<\/em><\/strong><\/p>\n\n\n\n

1. Introduction<\/h2>\n\n\n\n

Previous articles in this series examine, from a digital signal processing (DSP) frequency domain perspective, various types of digital filters used by quantitative analysts and market technicians to analyze and transform financial time series for trend-following<\/a> purposes.<\/p>\n\n\n\n

  1. An Introduction to Digital Signal Processing for Trend Following<\/a><\/li>
  2. Trend-Following Filters \u2013 Part 1<\/a><\/li>
  3. Trend-Following Filters \u2013 Part 2<\/a><\/li>
  4. Trend-Following Filters \u2013 Part 3<\/a><\/li>
  5. Trend-Following Filters \u2013 Part 4<\/a><\/li><\/ol>\n\n\n\n

    Part 4<\/a>\u00a0of this series, along with this article, examines the application of the Kalman filter to financial time series. The Kalman filter is a statistics-based algorithm used to perform estimation of random processes(1)<\/sup><\/a>. Estimation of processes in noisy environments is a critical task in many fields, such as communications, process control, track-while-scan radar systems, robotics, and aeronautical, missile, and space vehicle guidance. The Kalman filter is a real-time algorithm that operates iteratively in the discrete-time domain and is used in a variety of complex random process estimation applications.<\/p>\n\n\n\n

    There are two general types of Kalman filter models: steady-state and adaptive. A steady-state filter assumes that the statistics of the process under consideration are constant over time, resulting in fixed, time-invariant filter gains. The gains of an adaptive filter, on the other hand, are able to adjust to processes that have time-varying dynamics, such as financial time series which typically display volatility and non-stationarity.<\/p>\n\n\n\n

    2. Two State Noise-Adaptive Kalman Filter<\/strong><\/h2>\n\n\n\n

    This article assumes familiarity with Part 4, which includes an overview of the Kalman filter and describes an adaptive covariance matching method originally proposed by Myers and Tapley (2)<\/sup><\/a> and re-stated with some modification by Stengel (3)<\/sup><\/a>. The Stengel version is employed in both articles. The method estimates the process and measurement noise covariances directly from the Kalman filter algorithm, replacing, for example, the need for assumed or externally-derived noise covariance estimates.<\/p>\n\n\n\n

    To more easily illustrate adaptive filter concepts and calculations, the Kalman filter algorithm described in Part 4 has one state variable, x(t), modeled on a first-order process that has a locally constant mean value <\/em>a<\/em> contaminated by additive random noise \u03b5(t)~<\/sup>N(0, \u03c3\u03b5<\/sub>2<\/sup>):<\/p>\n\n\n\n

    y(t) = a<\/em> + \u03b5(t)<\/p>\n\n\n\n

    The filter described in this article has two state variables, x1<\/sub>(t) and x2<\/sub>(t), modeled on a second-order process that has mean value a<\/em> and locally constant linear trend b<\/em> contaminated by additive random noise:<\/p>\n\n\n\n

    y(t) = a<\/em> + b<\/em>*t + \u03b5(t)<\/p>\n\n\n\n

    More specifically, the process models in both articles are based on the assumption that financial asset prices follow the general Newtonian equation of motion (4)<\/sup><\/a>:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n

    where p is the position (analogous to a financial asset price), v is the velocity (analogous to what is commonly called \u201cmomentum\u201d by quantitative analysts and market technicians), a is the acceleration (change in velocity), j is the jerk (change in acceleration), etc. State variable x1<\/sub>(t) is equivalent to the second-order process mean value <\/em><\/em>a<\/em> as well as to Newtonian position p. State variable x2<\/sub>(t) is equivalent to second-order process linear trend b<\/em> as well as to Newtonian velocity v (i.e., it is assumed that acceleration a = 0, jerk j = 0, etc.).<\/p>\n\n\n\n

    The two-state noise-adaptive Kalman filter process and measurement model equations, variable initialization, algorithm steps, and noise estimate calculations are shown in Appendix 1 where:<\/p>\n\n\n\n