In Chapter 4 we introduced different two-sample tests for the equality of PEMs.
In this section we will simulate their performance for rGP models. In a first step we will simulate their type I and type II error rate under the hypothesis that no gait similarities are perturbing the data. In a second step we will introduce perturbations by an element ofI0 SO(3)
and remove it as explained in Section 4 by using the estimators introduced in Section 3. A third step will include perturbation by an element from S. However, since the latter simulations are extremely time consuming due to the estimation of the temporal alignment we present a relatively small simulation study.
Error processes for two-sample test simulations. The rGP processes used in this simulation are constructed from the processes (5.4) and the center curves γ0λ(t), λ∈R, given by
αλx t
= 80t2−80t+ 20 +λe−12
t−0.5 0.08
2 0.08√
2π −35 αy t
= 70tsin 4πt0.7 + 5 αz t
= 10 cos 13π ,
whereαλx, αy, αz are the Euler angles representation in degrees (see Section A.2) of γ0λ(t). For λ∈ {0,0.5,1,2,2.5} these curves are shown in Figure D.1.
In all simulations of the type I and II error of our two-sample tests from Section 4 we will use the following five processes, which are selected from the processes (5.5) used for the simulations of the covering rates of our simultaneous confidence sets,
γA0 =γ00(t)Exp ι
A1,1,1,0.05t
!
γBλ =γ0λ(t)Exp ι
A2,3,2,0.05t
!
for λ∈ {0.5,1,2,2.5}.
(5.7)
Examples of simulations from these processes are shown in Figure D.2, D.3 and D.4.
5.2.1 Performances Without Pertubation by Gait Similar-ities
Design of the two-sample test in X simulations. Let γ and η each be distributed according to one of the processes given in (5.7). We simulateγ1, ..., γN and η1, ..., ηN, N ∈ {10,15,30}, realizations on the equidistant time grid T with
∆t = 0.01. We then apply the OCST, HotellingT2-test, ILLPerm and MILLPerm (β = 0.9 for the OCST and α = 0.05 for the other tests) to the two samples γ1, ..., γN and η1, ..., ηN.
We performM = 2000simulations of this type and report the acceptance rate.
If the distributions ofγ andηare equal, then one minus the acceptance rate gives an approximation of the type I error, else, if the distributions are different, the acceptance rate is an approximation of the type II error. For the permutation approaches we useMperm = 5000permutations.
Results of the two-sample test inX simulations. The results of this simu-lation are given in Tables 5.1 and 5.2. ILLPerm, MILLPerm and the simultaneous HotellingT2 test seem to achieve the correct significance level α= 0.05. For the considered error processes the best power is offered by the HotellingT2-test. This is in accordance with the observation that the permutation tests are distribution free, while the Hotelling T2 test is specifically designed for Gaussian processes.
As expected, the overlap of the simultaneous90%-confidence sets (OCST) pro-duces a very conservative test, which has also a higher type II error than the other considered tests.
Note that for small sample sizes relatively small perturbations of the tested distributions do lead to a high type II error, which explains the high values on the second diagonal of Tables 5.1 and 5.2. This is mainly due to the small sample sizes. Thus, as expected the type II error always decreases if the sample size increases.
Table 5.1: Acceptance rate in percentage ofH0 of OCST (left) and simultaneous HotellingT2 (right).
Table 5.2: Acceptance rate in percentage ofH0 of ILLPerm (left) and MILLPerm (right) without application of P and Qto the η sample
5.2.2 Performances Including Perturbation by I
0SO(3)
Design of the two-sample test in X/I0 SO(3)
simulations. Letγ and η each be distributed according to one of the processes given in (5.7). We simulate γ1, ..., γN andη1, ..., ηN,N ∈ {10,15,30}, realizations on the equidistant time grid T with ∆t = 0.01. Contrary to the simulation discussed in the last paragraph, we compute the realization P η1QT, ..., P ηNQT with a rotation P ∈ SO(3) with Euler angles αx = −0.5◦, αy = 13◦, αz = −9◦ and a rotations Q ∈ SO(3) with Euler anglesαx = 12◦,αy = 0◦,αz = 5◦. Denote this rotated sample byη˜1, ...,η˜N. The application of P and Q to the simulated samples is performed in order to explore the robustness of the proposed testing procedures, if one has to correct for the perturbation by an element from I0 SO(3)
. In terms of our biomechanical application it mimics the robustness of the test procedure against different marker placements. To this end we apply the OCST, HotellingT2 test and ILLPerm with the preprocessing step of estimatingPˆandQˆas described in Preprocessing 4.0.16 (β = 0.9 for OCST and α = 0.05 for the other tests). Note that we replace φˆ by id[0,1] in Step 3 of Preprocessing 4.0.16, since temporal registration is not necessary for the data. MILLPerm (α = 0.05) is applied to the two samples γ1, ..., γN and η˜1, ...,η˜N. Note MILLPerm does not require a preprocessing step, since it is constructed to test equality even under a perturbations byI0 SO(3)
. For the same reason mentioned before we replace within MILLPerm the estimator φˆbyid[0,1].
We used M = 2000 simulations and report the acceptance rate. If the dis-tribution of γ and η is equal, then one minus the acceptance rate approximates the type I error rate, else, if the distributions are different, the acceptance rate is an approximation of the type II error. For the permutation approaches we use Mperm= 5000permutations.
Results of the two-sample test in X/I0 SO(3)
simulations. The results of this simulation are given in Tables 5.3 and 5.5. Interestingly, under the pertur-bation by (P, Q) only MILLPerm seems to achieve the correct significance level α = 0.05 and its power is similar to the power without perturbation, compare Table 5.2. The type I error of the HotellingT2 test and the OCST seem to depend on the chosen error process and is sometimes higher and sometimes lower than the5%. Even worse it increases, if the sample size increases. We suspect that this behavior is due to the observation that the estimator forQˆ and Pˆ are never ex-act and hence the null hypothesis thatP γˆ 1QˆT, ...,P γˆ NQˆT andη˜1, ...,η˜N have the same PEM is never true for the two samples and finiteN ∈N. For the Hotelling T2 test this is in accordance with the observation that it has a good power and hence is able to detect such a difference even better with increasing N. For the OSCT this is unsuspected, since it is a very conservative procedure without a perturbation fromI0 SO(3)
and we have no simple explanation. ILLPerm does perform relatively well in our simulations. However, for the tested error processes it has a lower type I error than expected, which is preferable, but also a higher type II error, which is not preferable.
Therefore we conclude that MILLPerm is the best choice within the compared
methods, if the data is corrupted by a perturbation of an element ofI0 SO(3)
Table 5.3: Acceptance rate in percentage ofH0 of OCST (left) and simultaneous HotellingT2 (right). The samples are perturbed by an element of I0 SO(3)
.
Table 5.4: Acceptance rate in percentage ofH0 of ILLPerm (left) and MILLPerm (right). The samples are perturbed by an element of I0 SO(3)
.
5.2.3 Performances Including Pertubation by S = I
0SO(3)
× Diff
+[0, 1]
Design of the two-sample test in X/S simulations. In this simulation we want to asses the influence of the temporal registration on the performance of ILLPerm and MILLPerm. The used error processes are identical to the error processes, which we used in the simulations with perturbation by I0 SO(3)
. However, we additionally introduce a time warping of the sample η˜1, ...,η˜N i.e., we simulate the sampleη˜1◦φ, ...,η˜N ◦φ, where
Before application of the ILLPerm the perturbation by φ and (P, Q) must be estimated and removed. This is done as described in 4.0.16. In principle, we could apply MILLPerm as described in Test 4.0.22 to deal with both the pertubation byI0 SO(3)
and Diff+[0,1] simultaneously. Estimating the the inverse of φ is,
however, computationally costly and therefore we apply MILLPerm only to the samplesγ1, ..., γN and η˜1◦φ◦φˆ−1, ...,η˜N ◦φ◦φˆ−1, where φˆ−1 is estimated using Preprocessing 4.0.16.
We perform M = 600 simulations with Mperm= 2000and report as before the acceptance rates.
Results of the two-sample test in X/S simulations. We observe the same behavior, but less drastically, as in the case of the Hotelling T2 test and the OCST and and a pertubation from I0 SO(3)
. The type I error increases with increasing sample size. However, recall that due to computational complexity we can not simulate the version of MILLPerm, which takes also the variance of the temporal alignment into account. We believe that the complete MILLPerm would perform well also in the setup of this simulation. This motivates further research.
Table 5.5: Acceptance rate in percentage ofH0 of ILLPerm (left) and MILLPerm correcting forI0 SO(3)
(right). The samples are perturbed by an element ofS.
Applications to Biomechanical Gait Data
“The theory of induction is the despair of philosophy – and yet all our activities are based upon it.”
– Alfred N. Whitehead In this chapter we apply previously developed statistical procedures to real gait data. The data within this thesis was collected in an experiment, designed and carried out by Michael Pierrynowski and Jodi Gallant, McMaster University, Canada. We organize this chapter as follows: first we will introduce the reader to the experimental protocol in Section 6.1 such that we can connect the results of our statistical analysis to the experimental setup. In Section 6.2 we explain the data processing, since we could not use all of the recorded data, and preprocessing steps like extraction of gait cycles are necessary. In Section 6.3 we discuss the results of our statistical analysis of the data and draw conclusions, among others, on identifiability of individuals (especially, after marker replacement), the influ-ence of self-selected walking speeds and the effect of kneeling prior to recording walking trials.