Functional responses, functional covariates and the concurrent model

Transcription

1 Functional responses, functional covariates and the concurrent model Page 1 of 14

2 1. Predicting precipitation profiles from temperature curves Precipitation is much harder to predict than temperature. It comes in two main forms: Drizzle: Large low pressure zones drop moisture over many hours or days. Storms: Convective, short violent storms with a lot of precipitation in a hurry, and spatially localized. Precipitation tends to be seasonal; more in the spring and fall than in the summer and winter. Page 2 of 14

3 A model We can assume that climate zone is important. We will predict log precipitation; logging stabilizes variance and eliminates the positivity constraint. We will use the difference TempRes mg (t) between a temperature profile and the mean for the climate zone as a function covariate. We can extend the functional ANOVA model to log[prec mg (t)] = µ(t)+α g (t)+tempres mg (t)β(t)+ɛ mg (t) Page 3 of 14 We call this model concurrent because it assumes that the temperature today affects today s precipitation.

4 The functional data Where precipitation was recorded as 0 mm, we changed it to 0.05 mm, half the minimum positive value. We used 11 Fourier series basis functions for precipitation with no roughness penalty. We used 21 Fourier series basis functions for temperature with no roughness penalty. Page 4 of 14

5 Log precipitation profiles Page 5 of 14

6 The fitting criterion and some results The fitting criterion was the unpenalized error sum of squares N k,g LMSSE(µ, α g, β) = [LogPrec kg (t) µ(t) α g (t) TempRes kg (t)β(t)] 2 dt Page 6 of 14 The resulting root mean squared residual was 0.19 mm. When we dropped TempRes(t) from the model, this increased to 0.20 mm. As we see in the following plot, the only place where temperature appears to make a contribution is in mid winter.

7 The estimated regression function β 6 (t) Page 7 of 14

8 The fit to Vancouver s data Page 8 of 14

9 A probe for the winter effect The confidence limits are point wise; we need a measure of the temperature influence accumulated over the winter months. Here is a probe that works: cos[2π(t 64.5)/365]β 6 (t) dt = 2.32, The estimated standard error of this probe is 0.77, giving a t-ratio of 3.0. It appears that elevated temperatures in mid-winter go along with increased precipitation. Page 9 of 14

10 2. Fitting the concurrent model Here is a general statement of the current functional/functional model: q y i (t) = z ij (t)β j (t) + ɛ i (t). j=1 or in matrix notation: y(t) = Z(t)β(t) + ɛ(t), We will use a penalized error sum of squares criterion: Page 10 of 14 LMSSE(β) = [y(t) Z(t)β(t)] [y(t) Z(t)β(t)] dt p + λ j [L j β j (t)] 2 dt. j

11 The basis function expansions for β j (s) Let regression function β j (s) have the expansion β j (s) = b jθ j (s) in terms of K j basis functions θ jk (s). Some of the independent variables can be scalar; in this case the basis for their β j (s) s is the constant basis; θ j1 (s) = 1 Page 11 of 14

12 Defining K β = q j K j, we construct vector b of length K β by stacking the coefficient vectors vertically, that is, b = (b 1, b 2,..., b q). Now assemble q by K β matrix function Θ as follows: θ Θ = 0 θ θ q Page 12 of 14 We can now express our model as y(t) = Z(t)Θ(t)b + ɛ(t).

13 We also need to arrange the order K j penalty matrices λ j R j = λ j Lθ j (t)lθ j(t) dt roughness into the symmetric block diagonal matrix R of order K β : λ 1 R R = 0 λ 2 R (1) 0 0 λ q R q Page 13 of 14

14 The normal equations [ Θ (t)z (t)z(t)θ(t) dt + R]b = [ Θ (t)z (t)y(t) dt] The numerical integration in these equations is not as difficult as it seems. The scalar functions ω jl (t) = N z ij (t)z il (t) i play the role of weighting functions for the functional inner products θ j (t)θ l(t)ω jl (t) dt, j, l = 1,..., q. Page 14 of 14