Functional data analysis
Encyclopedia
Functional data analysis is a branch of statistics
that analyzes data providing information about curves, surfaces or anything else varying over a continuum. The continuum is often time, but may also be spatial location, wavelength, probability, etc.
The data may be so accurate that error can be ignored, may be subject to substantial measurement error, or even have a complex indirect relationship to the curve that they define. For example, measurements of the heights of children over a wide range of ages have an error level so small as to be ignorable for many purposes, but daily records of precipitation
at a weather station
are so variable as to require careful and sophisticated analyses in order to extract something like a mean precipitation curve.
However these curves are estimated
, it is the assumption that they are intrinsically smooth that often defines a functional data analysis. In particular, functional data analyses often make use of the information in the slopes and curvatures of curves, as reflected in their derivatives. Plots of first and second derivatives as functions of t, or plots of second derivative values as functions of first derivative values, may reveal important aspects of the processes generating the data. As a consequence, curve estimation methods designed to yield good derivative estimates can play a critical role in functional data analysis.
Models for functional data and methods for their analysis may resemble those for conventional multivariate data, including linear and nonlinear regression models, principal components analysis
, and many others. But the possibility of using derivative information greatly extends the power of these methods, and also leads to purely functional models such as those defined by differential equation
s, often called dynamical systems.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
that analyzes data providing information about curves, surfaces or anything else varying over a continuum. The continuum is often time, but may also be spatial location, wavelength, probability, etc.
The data may be so accurate that error can be ignored, may be subject to substantial measurement error, or even have a complex indirect relationship to the curve that they define. For example, measurements of the heights of children over a wide range of ages have an error level so small as to be ignorable for many purposes, but daily records of precipitation
Precipitation (meteorology)
In meteorology, precipitation In meteorology, precipitation In meteorology, precipitation (also known as one of the classes of hydrometeors, which are atmospheric water phenomena is any product of the condensation of atmospheric water vapor that falls under gravity. The main forms of precipitation...
at a weather station
Weather station
A weather station is a facility, either on land or sea, with instruments and equipment for observing atmospheric conditions to provide information for weather forecasts and to study the weather and climate. The measurements taken include temperature, barometric pressure, humidity, wind speed, wind...
are so variable as to require careful and sophisticated analyses in order to extract something like a mean precipitation curve.
However these curves are estimated
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...
, it is the assumption that they are intrinsically smooth that often defines a functional data analysis. In particular, functional data analyses often make use of the information in the slopes and curvatures of curves, as reflected in their derivatives. Plots of first and second derivatives as functions of t, or plots of second derivative values as functions of first derivative values, may reveal important aspects of the processes generating the data. As a consequence, curve estimation methods designed to yield good derivative estimates can play a critical role in functional data analysis.
Models for functional data and methods for their analysis may resemble those for conventional multivariate data, including linear and nonlinear regression models, principal components analysis
Principal components analysis
Principal component analysis is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. The number of principal components is less than or equal to...
, and many others. But the possibility of using derivative information greatly extends the power of these methods, and also leads to purely functional models such as those defined by differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, often called dynamical systems.
Further reading
- Ramsay, J. O. and Silverman, B.W.Bernard SilvermanBernard Silverman FRS is a British statistician. He was Master of St Peter's College, Oxford from 1 October 2003 to 31 December 2009...
(2002) Applied functional data analysis : methods and case studies, Springer series in statistics, New York ; London : Springer, ISBN 0-387-95414-7 - Ramsay, J. O. and Silverman, B.W.Bernard SilvermanBernard Silverman FRS is a British statistician. He was Master of St Peter's College, Oxford from 1 October 2003 to 31 December 2009...
(2005) Functional data analysis, 2nd ed., New York : Springer, ISBN 0-387-40080-X