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Functional linear modelsPowerPoint Presentation

Functional linear models

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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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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 is scalar, Covariate is a single functional variable

- Response: yi , i=1,…,N
- Covariate: xi (t)
- Model:

We have to smooth! variable

- 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.

Predicting log annual precipitation from the temperature profiles

- 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 profiles

- Annual precipitation is determined by: (1) spring temperature, and (2) by the contrast between late summer and fall temperatures.

The fit to the data profiles

- 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? profiles

- 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.

Predicting the precipitation profile from the temperature profile

- The model is:

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

The regression function profile

The concurrent model profile

- This time, we’ll only use temperature at time t to predict precipitation at time t:

The regression functions profile

The influence of temperature is nearly constant over the year.

Let’s see how the two fits compare.

The historical linear model profile

- 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 profile

- One important extension of the concurrent model is to the fitting of data by a differential equation.
- A simple example is

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