time series
We test the AR order detection methodology introduced in Section2.3.3on simulated fMRI voxel time series, validating that nominal PCER and FDR significance levels are achieved. Recall that, for PCER thresholding, the significance level δ is also is equal to Pr(ˆp > p|ˆp ≥ p),
(a) LRT, PCER (b) LRT, FDR
(c) PACF, PCER (d) PACF, FDR
Figure 3.1 Images of the detected AR orders ˆp for the dataset, using the complex-valued model order detection methodology described in Section 2.3.3, under PACF/LR test statistics and PCER/FDR thresholding.
the probability of over-detection given that under-detection has not occurred. To validate this property, we detect orders with δ = 0.05 for 100,000 complex-valued voxel time series simulated from model (2.2), with non-AR parameters as in Table 2.1 and AR(1) parameters α1 = 0.00, 0.05, 0.10, 0.15, 0.20. The previous property is shown to hold for both LRT and
PACF statistics in the last columns of Table 3.2. Specifically, approximately 5 percent of the AR(0) time series have detected ˆp greater than 0, and approximately 5 percent of the AR(1) time series not detecting ˆp = 0 (not under-detecting) have detected ˆp > 1. Further, as we should expect, ˆp = 1 is detected more frequently as |α1| increases.
We also show that FDR levels are controlled in the order detection procedures by validating two properties of the FDR-controlling procedure inBenjamini and Hochberg(1995). The first such property is that when H0 is true for all tests, the false discovery rate is equal to the
(a) ˆβ0 (b) ˆβ1 (c) ˆβ2
(d) ˆσ (e) ˆθ (f) SNR (g) CNR
(h) ˆα1 (i) ˆα2 (j) ˆα3 (k) ˆα4
Figure 3.2 Images of complex-valued AR(4) model MLEs for the finger-tapping data set.
the order detection context, we illustrate this property by simulating AR(0) time series (i.e. H0: α1= 0 is true) and forming blocks of them (of constant size, say, 5); then, we calculate the
FWER as the proportion of blocks in which, under FDR thresholding, ˆp ≥ 1 (i.e. H0 : α1= 0
is rejected) at least once. For an FDR level of q∗ = 0.05, simulations showed an FWER of 0.055 for the LRT statistic and 0.053 for the PACF statistic. Second, the Benjamini and Hochberg (1995) FDR controlling procedure has expected FDR equal to the nominal level q∗ times the proportion of tests in which H0 is true. We chose a proportion of one-half, simulating 100,000
AR(0) and AR(1) time series, and calculated FDR as the proportion of time series detecting ˆ
p ≥ 1 (rejecting H0 : α1 = 0) for which p = 0 (H0 : α1 = 0 is true). Results for q∗ = 0.05
showed observed FDRs at approximately the expected level of 0.025: they were 0.027 and 0.026 for the LRT and PACF statistics, respectively.
Pr(ˆp = 0) Pr(ˆp = 1) Pr(ˆp > 1) Pr(ˆp > p|ˆp ≥ p)
p α1 LRT PACF LRT PACF LRT PACF LRT PACF
0 0.00 0.947 0.948 0.050 0.049 0.003 0.003 0.053 0.052
1 0.05 0.842 0.847 0.150 0.144 0.008 0.009 0.050 0.058
1 0.10 0.458 0.467 0.512 0.504 0.030 0.029 0.055 0.054
1 0.15 0.109 0.113 0.841 0.837 0.049 0.049 0.055 0.056
1 0.20 0.009 0.010 0.937 0.936 0.053 0.054 0.054 0.055
Table 3.2 Proportion of simulated AR(0) and AR(1) complex-valued time series detecting orders ˆp = 0, 1, and greater than 1 for different values of α1 under the LRT/PACF
order detection procedures with PCER level δ = 0.05.
3.5 Diagnostics for checking model assumptions
An assumption of the AR(p) model for complex-valued fMRI time series in Section 2.3.1 is that the real and imaginary errors have the same autoregressive dependence structure. We check this assumption for the finger-tapping dataset with images of the PACFs computed from the real and imaginary residuals, which are shown in figure3.3. These residuals ˆηRand ˆηI are computed under independence; that is, they are
ˆ
ηR= yR− X ˆβ cos ˆθ, ˆηI = yI− X ˆβ sin ˆθ, (3.4)
where ˆβ and ˆθ are as in Section 2.3.1 with ˆR−1n = In. Because these images of real and
imaginary PACFs look similar at each lag, we are willing to accept this model assumption. To check whether the AR(p) model sufficiently removes the temporal dependence from the complex-valued voxel time series in the finger-tapping dataset, we compute Box-Pierce Q- statistics (Box and Pierce,1970). The Box-Pierce statistic QK for a (real-valued) time series
of length n is defined as nPK
k=1ρˆ2(k), where ˆρ2(k) is the lag-k sample autocorrelation of the
model fit residuals. For truly AR(p) time series, QK is asymptotically χ2K−p. The value of K is
chosen somewhat arbitrarily, but typically K = 20 (Shumway and Stoffer,2006), which we use here. We compute Q20 for residuals under independent and AR(p) model fits, which gives us a
measure of the reduction in autocorrelation due to the AR(p) model fit. Under independence, the residuals are as in (3.4); the AR(p)-model-fit residuals are the (n − p)-vectors ˆηR and ˆηI, where ˆeRt = ˆηR,t+p−Ppk=1αkηˆR,t+p−k and ˆeIt = ˆηI,t+p−Ppk=1αkηˆI,t+p−k, t = 1, . . . , n − p,
Real 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Imaginary 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
Figure 3.3 Images of the PACFs computed from the real and imaginary residuals of the fin- ger-tapping dataset for lags 1 to 20.
where ˆηRt and ˆηIt are entries of residual vectors in (3.4). For each model fit residuals, two
Q-statistics are computed: one for each of the real and imaginary residuals.
We compute Q20 statistics for independent- and AR(4)-model residuals for voxels inside
the brain (2916 in all), where ˆp = 4 is the order detected for the majority of in-brain voxels in Section3.3. We also simulated truly AR(4) complex-valued voxel time series, with parameters as in table2.1, and calculated Q20 statistics for the same two kinds of residuals. The resulting
four Q-statistics are compared to a random sample from the null distribution of Q20 under
an AR(4) model, χ216, in the quantile-quantile plot in figure 3.4. The AR(4)-model residual Q-statistics for the simulated data are close to the χ216 null distribution and are well-below the AR(4)-model residual Q-statistics for the dataset. This indicates the dataset still contains substantial autocorrelation after the AR model fit, and perhaps more complex methods are needed to remove the autocorrelation, such as incorporating moving average, integrated, or seasonal components in the time series model. However, the AR(4)-model residual Q-statistics
10 20 30 40 0 500 1000 1500 2000 Chi squared (df=16) Bo x−Pierce Statistic
Empirical indep residuals Empirical AR(4) residuals Simulated indep residuals Simulated AR(4) residuals Chi squared (df=16)
Figure 3.4 Quantile-quantile plot of Box-Pierce Q20-statistics for independent- and
AR(4)-model-fit residuals of the simulated and finger-tapping data (“empirical”) versus a random sample from the null distribution of Q20 under an AR(4) model,
χ2 16.
for the dataset are much smaller than its independent-model counterparts, indicating that the AR model greatly reduces temporal autocorrelation.