Dynamic functional principal components. (18th July 2014)

• Record Type: Journal Article
• Authors: Hörmann, Siegfried; Kidziński, Łukasz; Hallin, Marc
• Is Part Of: Journal of the Royal Statistical Society. Volume 77:Number 2(2015:Mar.)
• Page Start: 319
• Page End: 348
• Publication Date: 2014-07-18
• Subjects: Statistics -- Periodicals; Great Britain -- Statistics -- Periodicals; 519.2
• DOI: 10.1111/rssb.12076
• Languages: English
• ISSNs: 1369-7412

Abstract

<abstract abstract-type="main" id="rssb12076-abs-0001"> <title>Summary</title> <p>We address the problem of dimension reduction for time series of functional data <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3p44mhpv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math altimg="urn:x-wiley:13697412:media:rssb12076:rssb12076-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo lspace="0pt">:</mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Such <italic>functional time series</italic> frequently arise, for example, when a continuous time process is segmented into some smaller natural units, such as days. Then each <italic>X</italic><sub><italic>t</italic></sub> represents one intraday curve. We argue that functional principal component analysis, though a key technique in the field and a benchmark for any competitor, does not provide an adequate dimension reduction in a time series setting. Functional principal component analysis indeed is a <italic>static</italic> procedure which ignores the essential information that is provided by the serial dependence structure of the functional data under study. Therefore, inspired by Brillinger's theory of <italic>dynamic principal components</italic>, we propose a <italic>dynamic</italic> version of functional principal component analysis which is based on a frequency domain approach. By means of a simulation study and an empirical illustration, we show the considerable improvement that the dynamic approach entails when compared with the usual static procedure.</p> </abstract>