3.2 Spatial Alignment: Removing of MP Effect
5.1.1 Small Sample Behavior of Multiplier Bootstrap, Asymp-
Confidence Bands
The aim of this section is to analyze the small sample behavior of different meth-ods for estimating simultaneous confidence bands for the pointwise mean function in functional data. These simulations are included in this thesis, since they justify the use of the GKF as our tool for computing theα-quantile of the maximum of the Hotelling T2 statistic of rGP models. Let µ : [0,1] → R be a deterministic function. We consider in this section the following data model
X(t) =µ(t) +Zt, t∈I (5.1)
where {Zt}t∈I is a real valued Gaussian process on I = [0,1]with almost surely C2-sample paths, E[Zt] = 0, Var[Zt] =σ2t >0and Zt/σt fulfills the Assumptions (GKF 3) and (GKF 4). Moreover, we assume that we observe the process X only at times 0 =t1 < t2 < ... < tK = 1.
LetX1, . . . , XN be a sample from model (5.1), then we want to find two func-tions lN, uN : [0,1]→R depending on the sample such that
P lN(t)≤µ(t)≤uN(t)for all t ∈I
≥1−α .
using the observed values Xn(tk) for k ∈ {1, ..., K} and n ∈ {1, ..., N}. This can be achieved using the stochastic process
Tt =
√ N
X(t)¯ −µ(t) ˆ
σt , (5.2)
where we define
X(t) =¯ N−1
N
X
n=1
Xn(t) ˆ
σ2t = 1 N −1
N
X
n=1
Rn(t)2 Rn(t) =Xn(t)−X(t)¯ .
Note that this process is as shown in Adler and Taylor [2009, Section 15.10.3, p.430] well-defined for all N ≥2.
Now, given hα ∈R>0 such that P max
t∈[0,1]|Tt|> hα
!
≤α.
we obtain that the collection of intervals hX(t)¯ −hα√σˆt
N,X(t) +¯ hα√σˆt
N
i
, fort ∈[0,1]
form a simultaneous (1−α)·100% confidence band for µi.e., P µ(t)∈h
X(t)¯ −hα√ˆσt
N,X(t) +¯ hα√ˆσt
N
i
for all t∈I
!
≥1−α .
We will now describe three different methods for estimating the threshold hα and explore their small sample performance using simulations.
Naive bootstrap approach. The first method, which we will use in our com-parison, is proposed in Degras [2011] called the naive bootstrap. The main result of Degras [2011] is a functional asymptotic normality result for the local linear estimator for dense functional data. Although this result allows for constructing (asymptotically correct) (1−α)% confidence bands of the mean curve, Degras proposes to use the naive bootstrap for small sample sizes. The naive bootstrap works as follows.
1. Resample with replacement from a sample X1, ..., XN of model (5.1) to produce a bootstrap sample X1,∗, ..., XN,∗.
2. Compute the pointwise empirical mean X¯∗ and variance (ˆσ∗)2 functions of the bootstrap sample X1,∗, ..., XN,∗.
3. Compute Z∗ =√
Nmaxt∈Ik( ¯X∗−X)/ˆ¯ σ∗k.
4. Repeat steps 1 to 3 many times to approximate the conditional law L∗ = L Z∗| X1, ..., XN
and take the (1−α)·100% quantile of L∗ to estimate hα.
Note that in Degras [2011] instead ofX¯ the local linear estimator is used, which smooths the data. This simplification can be done, since we do not include an additional observation error in model (5.1) as done in Degras [2011] and therefore smoothing is not necessary.
Multiplier bootstrap. The second method builds on a version of the multi-plier (or Wild) bootstrap (e.g., Mammen [1993]) designed for the maximum of sums ofN independent random variables in high dimensions as discussed in detail by Chernozhukov et al. [2013]. More precisely, letY1, ..., YN be independent ran-dom vectors inRK,N, K ∈N withE
Yn
= 0 and finite covarianceE YnYnT these assumptions it is shown in Chernozhukov et al. [2013, Theorem 3.1] that the quantiles of the distribution of
max
can be asymptotically consistently estimated by the quantiles of the multiplier bootstrap i.e., by the distribution of
max
In order to apply Chernozhukov et al. [2013, Theorem 3.1], note that we can rewrite assump-tions of Chernozhukov et al. [2013, Theorem 3.1] and therefore the multiplier bootstrap is applicable to estimate quantiles of the distribution of the maximum of the random vector T = Tt1, ..., TtK
.
Since Chernozhukov and co-authors show that the multiplier bootstrap works also forK N, we apply this method without further theoretical justification to the functional case and the process {Tt}t∈I. The same reasoning as above, then yields that we can use the multiplier bootstrap to estimate hα given by
P max
with
The estimator of hα is then given by ˆhα = inf
h∈R
Pg W ≤h
≥1−α ,
where Pg is the probability measure induced by the multipliers g holding Rn(t) fixed i.e.,Pg W ≤h
=P W ≤h|R1, ..., RN .
Gaussian kinematic formula of T-statistic. Analogously to our approach in Section 2.2 we can estimate the threshold hα using the GKF for the process {Tt}t∈[0,1] given in equation (5.2), which has pointwise at-distribution with (N− 1)-degrees of freedom. By the expected Euler characteristic heuristic and the Gaussian kinematic formula (see Adler and Taylor [2009, Theorem 15.10.3.]), we obtain HereFN−1denotes the cumulative distribution function of a Student’s t-distribution with(N−1)-degrees of freedom and the first equality is due to the fact that the processes{Tt}t∈I and {−Tt}t∈I have the same distribution; hence
P
By equation 5.3 we only have to estimate the Lipschitz killing curvature from the observations to construct simultaneous confidence bands. The Lipschitz killing curvature is given by
which can be found in Taylor and Worsley [2007, Section 3.3]. Given an i.i.d.
sample X1, ..., XN of model (5.1) evaluated on a partition 0 = t1 < t2 < ... <
tK = 1 we use a discretized version of L1 [0,1]
by replacing the integral by its Riemann sum and the derivative by finite differences. Moreover, we use that
Varh
This yields the estimator
Error processes for 1D confidence bands simulation. In the simulations of the covering rate of 1D confidence bands constructed using the methods proposed above, we assume for simplicity that µ(t) = 0 for allt ∈[0,1]. The performance of the presented methods is tested using for the error processes Z in model 5.1 the processes Note that the processes satisfy Varh
εν,lt i
=fl(t)2for allt∈[0,1]andν ∈ {1,2,3}.
Moreover, the sample paths of the processes ε1,l and ε2,l have C∞-sample paths, whereas the sample paths of ε3,l, which is a Ornstein-Uhlenbeck process (e.g., Iacus [2008, p.43]) are only continuous implying that the GKF is not applicable for this process. However, since the estimator of the Lipschitz killing curvature is computable also for the Ornstein-Uhlenbeck process, we studied also confidence sets using the GKF approach for the Ornstein-Uhlenbeck process. We expect, that this does not work well, since the estimation of the Lipschitz killing curvature relies on the estimation of the variance of the derivative of the process, which does not exists in this case.
Design of 1D confidence bands simulation. We use the proposed meth-ods to construct confidence bands for the mean function µ ≡ 0. To obtain the covering rates for small sample sizes, we do the following: simulate N ∈ {5,10,15,20,30,50} realizations of the process εν,lt for ν, l ∈ {1,2,3} on the equidistant time grid T with ∆t = 0.01 and compute the simultaneous confi-dence band with the selected method at these points. Then check whether µ(t) is contained in the constructed confidence band for all t ∈ T. We repeat this M = 5000 times and the relative frequency between the trials such that µ is always within the constructed confidence band and the number of simulations approximates the true covering rate.
For the bootstrap methods we used 2000 bootstrap replicates.
Results of 1D confidence bands simulation. The results of this simulation are collected in the Tables C.1, C.2 and C.3 in Appendix C. The conclusions are the following: in the case of small sample sizes (≈10-20) the only reliably work-ing method is the Gaussian kinematic formula approach, which is surpriswork-ingly accurate and only systematically overestimates the covering rate for the Ornstein Uhlenbeck error process, which, anyway, does not satisfy the assumptions of the GKF. While the naive bootstrap yields too conservative confidence bands for small samples sizes, we discovered that the multiplier bootstrap underestimates the covering rate. For larger sample sizes (≥ 50) both bootstrap methods start to perform well.
Note that another advantage of the GKF is that it is computational very fast.
Due to these observations we will only use the GKF approach to construct simul-taneous confidence bands of the PEM in rGP models.