Local regression: Difference between revisions

Revision as of 20:57, 15 May 2025

Revision as of 06:56, 17 May 2025

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Solving these equations for the polynomial coefficients yields the graduated value, <math>\hat Y_0 = a</math>.

Henderson went further. In preceding years, many 'summation formula' methods of graduation had been developed, which derived graduation rules based on summation formulae (convolution of the series of obeservations with a chosen set of weights). Two such rules are the 15-point and 21-point rules of [[John Spencer (Actuary)|Spencer]] (1904).<ref>{{citeQ|Q127775139}}</ref>. These graduation rules were carefully designed to have a quadratic-reproducing property: If the ungraduated values exactly follow a quadratic formula, then the graduated values equal the ungraduated values. This is an important property: a simple moving average, by contrast, cannot adequately model peaks and troughs in the data. Henderson's insight was to show that ''any'' such graduation rule can be represented as a local cubic (or quadratic) fit for an appropriate choice of weights.

Further discussions of the historical work on graduation and local polynomial fitting can be found in [[Frederick Macaulay|Maculay]] (1931),<ref name="mac1931">{{citeQ|Q134465853}}</ref>, [[William S. Cleveland|Cleveland]] and [[Catherine Loader|Loader]] (1995);<ref name="slrpm">{{citeQ|Q132138257}}</ref> and [[Lori Murray|Murray]] and [[David Bellhouse (statistician)|Bellhouse]] (2019).<ref>{{cite Q|Q127772934}}</ref>

The [[Savitzky-Golay filter]], introduced by [[Abraham Savitzky]] and [[Marcel J. E. Golay]] (1964)<ref>{{cite Q|Q56769732}}</ref> significantly expanded the method. Like the earlier graduation work, their focus was data with an equally-spaced predictor variable, where (excluding boundary effects) local regression can be represented as a [[convolution]]. Savitzky and Golay published extensive sets of convolution coefficients for different orders of polynomial and smoothing window widths.

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Books substantially covering local regression and extensions:

* Macaulay (1931) "The Smoothing of Time Series",<ref name="mac1931">{{citeQ|Q134465853}}</ref>, discusses graduation methods with several chapters related to local polynomial fitting.

* Katkovnik (1985) "Nonparametric Identification and Smoothing of Data"<ref name="katbook">{{citeQ|Q132129931}}</ref> in Russian.

* Fan and Gijbels (1996) "Local Polynomial Modelling and Its Applications".<ref>{{citeQ|Q134377589}}</ref>

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Book chapters, Reviews:

* "Smoothing by Local Regression: Principles and Methods"<ref name="slrpm">{{citeQ|Q132138257}}</ref>

* "Local Regression and Likelihood", Chapter 13 of ''Observed Brain Dynamics'', Mitra and Bokil (2007)<ref>{{citeQ|Q57575432}}</ref>

* [[Rafael Irizarry (scientist)|Rafael Irizarry]], "Local Regression". Chapter 3 of "Applied Nonparametric and Modern Statistics".<ref>{{cite web|last=Irizarry|first=Rafael|title=Applied Nonparametric and Modern Statistics|url=https://rafalab.dfci.harvard.edu/pages/754/|access-date=2025-05-16}}</ref>