Functional linear models

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# Functional linear models - PowerPoint PPT Presentation

Functional linear models. Three types of linear model to consider:. Response is a function; covariates are multivariate. Response is scalar or multivariate; covariates are functional. Both response and covariates are functional. Functional response with multivariate covariates.

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## PowerPoint Slideshow about 'Functional linear models' - jacob

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Presentation Transcript
Three types of linear model to consider:
• Response is a function; covariates are multivariate.
• Response is scalar or multivariate; covariates are functional.
• Both response and covariates are functional.
Functional response with multivariate covariates
• Response: yi(t), i=1,…,N
• Covariate: xi1,…, xip
• Model:
How does daily temperature depend on climate zone?
• 35 Canadian temperature stations, divided into four zones: Atlantic, Pacific, Continental, and Arctic.
• Response is 30-year average daily temperature.
• A functional one-way analysis of variance, set up to have a main effect, and zone effects summing to zero.
Analyzing the data
• This is straightforward.
• If Y(t) is the N-vector of response functions, β(t) is the 5-vector of regression functions (main effect + zone effects), then the LS estimate is
• β(t) = (X’X)-1X’ Y(t) .
Assessing effects
• We probably want to assess effects pointwise: For what times t is an effect substantial?
• This can be done using F-ratios conditional on t, pointwise confidence bands, etc.
• The multiple comparison problem is especially challenging here.
• Response: yi , i=1,…,N
• Covariate: xi (t)
• Model:
We have to smooth!
• The technical and conceptual issues become much more interesting when the covariate is functional.
• A functional covariate is effectively an infinite-dimensional predictor for a finite set of N responses. We can fit the data exactly!
• Smoothing becomes essential; without it, β(t) will be unacceptably rough, and we won’t learn anything useful.
• Can we determine how much precipitation a weather station will receive from the shape of the temperature profile?
• What roughness penalty should we use to smooth β(t) ?
• We penalize the size of (2π/365)2Dβ+D3β,

the harmonic acceleration of β(t) . This smooths towards a shifted sinusoid.

The smoothed regression function
• Annual precipitation is determined by: (1) spring temperature, and (2) by the contrast between late summer and fall temperatures.
The fit to the data
• The fit is good.
• We see clusters of hi-precip. marine stations, and of continential stations.
• Arctic stations have the least precip.
What about both the response and covariate being functional?
• Response y(t), covariate x(s) or x(s,t).

Here we have a lot of possibilities. We can predict y(t) using the shape of x(s,t) over:

• all of s, especially for periodic data,
• only at s = t, concurrent influence only, or for some delay s = t – δ,
• s  t, no feed forward,
• some region Ωt depending on t.
• The model is:

In this case we have to smooth β(s,t) with respect to both s and t.

The concurrent model
• This time, we’ll only use temperature at time t to predict precipitation at time t:
The regression functions

The influence of temperature is nearly constant over the year.

Let’s see how the two fits compare.

The historical linear model
• When the functions are not periodic, it may not be reasonable to assume that x(s) can influence y(t) when s > t.
• The historical linear model is described in Applied Functional Data Analysis, and in talk at this conference by Nicole Malfait.
The concurrent model and differential equations
• One important extension of the concurrent model is to the fitting of data by a differential equation.
• A simple example is