, in the context of functional magnetic resonance imaging, study confidence statements for estimators of the mean functionfrom a sample of a signal plus noise model , where
is a stochastic error process with variance function, where
is a spatial index. This requires estimation of the quantiles of the maximum of a limiting Gaussian processes. The quantiles are estimated using standardized residuals from the estimated mean function either through a multiplier bootstrap(Chang and Ogden, 2009; Chang et al., 2017) or the Gaussian kinematic formula (Worsley et al., 2004; Adler and Taylor, 2009). These methods successfully approximate the quantiles, since the standardized residuals asymptotically have the same covariance structure as the limiting Gaussian process.
However, this approach no longer works when the object of interest is a nonlinear transformation of the parameters. In order to guarantee comparability between different scanners, Bowring et al. (2020) extends the work of Bowring et al. (2019) to the population Cohen’s , i.e., , rather than the mean function . This causes a new conceptional problem. While the standard residuals capture the covariance structure for the limiting Gaussian process in estimation of the mean, this no longer holds true for Cohen’s as we show in Corollary 1. We visualize this effect in Figures 1 and 2. In particular, Figure 1 shows samples of Cohen’s residuals approximating the correct covariance structure.
In this paper, we use the functional delta method to construct random processes, called functional delta residuals, which can be used for obtaining distributional properties of the limiting process whenever the object of inference is a non linear transformation of the functional parameters. The proposed delta residuals are necessary because the nonlinearity not only affects the variance of the limiting transformed process but also its covariance function. As an application, we here use delta residuals and the quantiles of the maximum of the limiting process for construction of simultaneous confidence bands, a problem commonly found in functional data analysis(Degras, 2011; Cao et al., 2012; Cao and others, 2014; Chang et al., 2017; Wang et al., 2019), for the Cohen’s
parameter. Its extension to an application to spatial inference using coverage probability excursion sets for the Cohen’sparameter can be found in Bowring et al. (2020).
Given a functional central limit theorem (fCLT) and a parameter transformation, the construction of the delta residuals is obtained by linearisation in relation to the functional delta method. Our main result, Theorem1, shows that delta residuals have asymptotically the covariance structure of the limiting process of the transformed parameters. In Section 3 we apply the general theory to the functional Cohen’s statistic, prove the necessary fCLT in Theorem 2, derive the corresponding delta residuals in Section 3.2 and prove, in Theorem 3, a multiplier functional limit theorem for the delta residuals based on Chang and Ogden (2009). We use these results to construct simultaneous confidence bands and study the accuracy of their covering rate and the effect of using the wrong residuals in a simulation study in Section 4.
The methods for simultaneous confidence bands for Cohen’s are implemented in the R-packge SCBfda available under https://github.com/ftelschow/SCBfda and code reproducing the presented simulation results are available under https://github.com/ftelschow/DeltaResiduals.
samples of a Gaussian process with square exponential covariance function having a scaled and horizontally shifted Gaussian kernel with standard deviation 0.05 as mean.Middle: the standard residuals of this process. Right: samples from the delta residuals of Cohen’s of the same process as given in Corollary 1.
2 Functional Delta Residuals
In this section we introduce the construction of functional delta residuals. We develop the idea in the framework of the Banach space of continuous functions with values in over a compact domain , however the concept can also be generalized to other Banach spaces. The norm on is the maximum norm , where denotes the standard norm on . For ease of readability will be denoted by .
Since a purely formal treatment hides the basic idea of delta residuals, we motivate them with a special case. Let be a functional population parameter and let be estimators of . Further, assume that their average satisfies a fCLT, i.e.,
where is a tight zero mean Gaussian process in with covariance function and ”” denotes weak convergence in . We call the processes with values in standard residuals since, by the fCLT, their empirical covariance function converges to the covariance of , i.e.,
Here the convergence is in probability and
denotes the transpose of a vector. Almost sure convergence would require addtional regularity conditions on the standard residuals . We discuss sufficient conditions in the case of Cohen’s in Section 3.
which we call functional delta residuals, can be used to approximate the covariance structure of , which is the limiting process from the delta method, in the sense that
with convergence again being in probability.
For illustrative purposes, consider the following more concrete example. Let be a triangular array of random processes in independent and identically distributed as with and . Let so that , and suppose that weakly in for with being a tight, zero mean Gaussian process with covariance . Then are standard residuals satisfying (2). For continuously differentiable, the delta residuals are given by , which can be used to approximate the covariance function . To be even more concrete, let . Thus, we can define . Say we are interested in the asymptotic behavior of the product of the sample means of the two components of the process. Then and the delta residuals are given by . These delta residuals can be used to approximate the covariance , where and .
The next result is immediate, yet generalizes the previous concept of functional delta residuals to estimators , which are not averages.
Let and be an estimator such that as
weakly in , where denotes a tight mean zero Gaussian process on with mean zero and covariance function . Let be a continuously differentiable function. Moreover, let be a triangular array of random processes satisfying uniformly in probability that
Then the functional delta methods yields
with being a zero mean Gaussian process with covariance . Furthermore, the functional delta residuals , , satisfy
uniformly in probability.
By a simple Taylor expansion argument considered as a function of is Hadamard differentiable tangential to and therefore (Kosorok, 2008, Theorem 2.8) implies that the functional delta method is applicable.
Two observations are noteworthy. First, the factors in equation (4) and in equation (5) can be replaced by general factors tending to infinity and zero respectively. We only keep these simple factors for notational simplicity. Secondly, if for all , then the delta residuals can be identically equal to zero. Here an assumption of higher differentiability of can be used to establish a similar result using a second order delta method.
3 Functional delta residuals for Cohen’s
In this section we show how to apply Theorem 1 to the pointwise Cohen’s statistic for processes with -Hölder continuous paths, see Definition 1 below. This special continuity condition is needed to ensure that the sample mean and the sample variance satisfy a fCLT, which is necessesary to obtain the functional delta residuals of Cohen’s . As a second step, we establish a multiplier bootstrap result for the delta residuals. This result implies that the quantiles of the maximum of the limiting process of the functional delta method can be estimated consistently.
The purpose of these considerations is to provide a theoretical basis for the approach taken in Bowring et al. (2020). Neuroimaging data is typically smoothed with a Gaussian kernel and therefore the assumption on the sample paths is satisfied for the smoothed process provided that the observed data at the voxels has finite
th moment. To circumvent technicalities from the application inBowring et al. (2020) we will demonstrate the usefulness of the delta residuals for the task of constructing simultaneous confidence bands for the functional Cohen’s parameter.
Hereafter, we assume that is an i.i.d. sample in . The pointwise population Cohen’s is the function defined by
with . The Cohen’s parameter is estimated using its corresponding sample counterpart
The biased variance estimator is used in the denominator, since the delta residuals will be simpler.
Two observations are noteworthy here.
b.) It will be obvious from the proofs that the theorems on delta residuals for Cohen’s hold true not only for but any .
3.1 A Functional Central Limit Theorem
We want to apply Theorem 1 to the function . As such we need to establish a fCLT for the process , which takes values in . The following sample path property will be our main assumption on the process to prove the fCLT.
Let be a process in . Given , we say that has -Hölder continuous paths, if
for a positive random variable
for a positive random variablewith and .
-Hölder continuous paths ensure that satisfies a fCLT, i.e., for iid processes in , the sum converges weakly to a tight mean zero Gaussian process which has the same covariance structure as , see Jain and Marcus (1975, Theorem 1).
The following Lemma states useful properties of processes with -Hölder continuous paths.
Let and be iid processes in having -Hölder continuous paths with over a compact set , and assume that there exists such that is finite. Then for all and uniformly almost surely. If , then also uniformly almost surely.
First claim: Using the convexity of and we have
where is the random variable from the -Hölder property. This yields for all .
We now apply the generic uniform convergence result in Davidson (1994, Theorem 21.8)
. Since pointwise convergence is obvious by the strong law of large numbers, we only need to establish strong stochastical equicontinuity of the random function. This is established using Davidson (1994, Theorem 21.10 (ii)), since
for all . Here iid denote the random variables from the -Hölder paths of the ’s and . Hence the random variable converges almost surely to the constant by the strong law of large numbers.
Second claim: With the same strategy and assuming w.l.o.g. , we compute
where iid denote the random variables from the -Hölder property of the ’s and . Again by the strong law of large numbers the random Hölder constant converges almost surely and is finite. ∎
Since we are dealing with vector-valued random processes, we need the next lemma in our proof of Theorem 2. It states simple conditions for obtaining weak convergence of a vector-valued process from its components.
Let be -valued random variables on the probability space such that and . If the finite dimensional distributions of converge to those of , we have in .
Since and in and is complete and separable, the sequences are tight and so for each , there exist compact such that for all , and . This implies
The latter is true, since in general , if and , and since
and similarly . This holds for all and so the sequence is tight. Tightness implies relative compactness by Prohorov’s theorem.
With these preparatory results we are now able to prove the main theorem of this section.
Let be a compact space and be an i.i.d. sample in satisfying and having -Hölder continuous paths. Then
where is a 2D mean zero Gaussian process with covariance matrix function given by
with , and .
Since for iid random variables with mean we have that
where as , we can w.l.o.g replace by in the definition of from equation (9) and further, for simplicity, assume for all .
For and any , applying the multivariate CLT to the sequence of vectors
yields convergence to the finite dimensional distributions of from the statement of the theorem. Hence the finite dimensional distributions of converge to those of . Since the process is -Hölder continuous, we have -Hölder bounds on and as shown in the proof of Theorem in Telschow and Schwartzman (2019). Thus, by Jain and Marcus (1975) both satisfy the CLT in . In particular, by Lemma 2 we obtain the fCLT for (, ). ∎
The functional delta method yields the following corollary.
Under the assumptions of Theorem 2 we have that with covariance structure given by
Moreover, if is a Gaussian process, then simplifies to
By Theorem 2 and the fact that is continuously differentiable, we can apply Theorem 1. Note that the denominator in the fCLT will be nonzero with probability for all , if , by Adler and Taylor (2009, Lemma 11.2.10).
For the Gaussian situation the asymptotic covariance structure simplifies significantly. To show this, we define
and use the fact from the moments of multivariate normal distributions, better known as Isserlis’ theorem, cf. Theorem 1 inVignat (2012),
for all to compute
Finally, we note that
yielding the simplified version of the limiting covariance structure. ∎
The above corollary shows why linear residuals fail to capture the asymptotic correlation structure of Cohen’s and therefore are unsuitable for building inferential tools. For simplicity, suppose that is Gaussian. Then the standardized linear residuals yield
which is not equal to (13).
3.2 Functional Delta Residuals
converges to a tight mean zero Gaussian process. This is an average estimator as discussed in Section 2. Hence we can identify with standard residuals
The functional delta residuals for therefore are
We call these residuals Cohen’s residuals. It is easy to show that . To prove that these residuals satisfy Theorem 1, the following result is necessary.
Suppose has -Hölder continuous paths and for some , then the standard residuals (14) are componentwise -Hölder continuous.
For the component it is clear that the process has -Hölder continuous paths. Therefore we only prove the claim for . Note that for all
Here each term can be bounded in a similar manner. As such we only provide the bound for the middle term, which is
Applying the inequality , using that by Lemma 1 and shows that . ∎
3.3 A Multiplier Bootstrap Functional Limit Theorem
Our main application of delta residuals is to approximate statistics that depend on the limiting process in (6) or quantiles thereof such as quantiles of the maximum of the process. The latter are used in Bowring et al. (2020) to construct coverage probability excursion sets for Cohen’s . In order to justify their construction we establish weak conditional convergence for the multiplier process based on the delta residuals. For , the multiplier bootstrap process is defined by
where is an iid triangular array of multipliers satisfying and . Moreover, the multipliers are assumed to be independent of the ’s and thereby independent of the delta residuals defined in equation (15).
The following theorem is based on the proofs from Chang and Ogden (2009) and implies that the multiplier bootstrap process conditioned on the delta residuals asymptotically has similar sample path properties as the limiting process from the delta method. As such it can, for example, be used to estimate quantiles of the maximum, see Remark 4.
Under the assumptions of Theorem 2 the following statements hold
Here convergence in is in outer probability, is the expectation over conditional on and is the set of all such that and for all .
It suffices to prove the result for the multiplier bootstrap process defined by the standard residuals with weak convergence towards . This can be seen as follows: converges to uniformly almost surely by Theorem 2 and the continuous mapping theorem. Here and . Thus, applying Theorem 18.10(v) from Van der Vaart (2000), we obtain the weak convergence
Let us define the unobservable iid samples
where and . By definition these samples satisfy . Since has -Hölder continuous paths and , both components of divided by satisfy (A), (B), (C) and (D) from Chang and Ogden (2009) meaning their Theorem 1 and 2 are applicable. In particular, applying Lemma 2, this means and converge weakly to . Simple algebra shows that is a random process converging uniformly to zero as tends to infinity. Thus, converge weakly to in the space of bounded functions over . Since and all are assumed to be continuous processes, the convergence is also in by Van Der Vaart and Wellner (1996, Theorem 1.3.10). This finishes the proof of part .