
Submitted by
Assigned_Reviewer_2
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
The paper introduced a novel independency test for
time series, which is based on reproducing Hilbert kernel theory (Kernel
Cross Spectral Density estimation). In particular, the key object is the
reproducing kernel, so the method can be applied on complex time series
such as nonnumerical time series. Both theoretical study and empirical
results are presented.
This is certainly one of those technically
dense papers. It involves several aspects of applied math in one way or
another: signal processing, statistical test, RKHS, graphical models,
Markov process, function operators, time series, and, of course at last,
neural science. More importantly, all these techniques are involved in a
nontrivial way.
The key step of using RKHS in analyzing time
series is to connect the covariance matrix and reproducing kernel, and
further extend to operators in functional spaces. The paper did a good job
make each step work nicely together. I did not read the supplementary
materials, but I think the technical content seems fine. Overall, this is
a well executed paper in many aspects.
Several questions and
comments:
1, the paper seems a little too crowed for the content.
But given the page limit, it is a conflict. I am sure the authors will
have a longer version.
2, several details:
line 91, “it’s”
? should it be “its” ? line 9596, “a couple of”, should it be a
“coupled”? line 290, topleft? Should it be topright? In Fig.1,
Why the estimate of a norm can be negative? Does it come from the left
hand side of Corollary 8? A bit explanation would be great.
line
418, “is in accordance with other studies”: it would be great to have
relevant references here. In fact, this can be put at the very beginning
of the paper as one motivation.
Q2: Please
summarize your review in 12 sentences
This paper involves many technical areas, and all
these techniques are involved in a nontrivial way. Overall, this is a
well executed paper in many aspects. Submitted by
Assigned_Reviewer_4
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
Detection of potential interdependencies between
timeseries is an important problem. This paper introduces an analysis of
kernel cumulant methods and via estimation of higherorder crossspectra,
the paper links to particular forms of independence testing.
There
is a vast literature on higherorder methods in signal processing, from
higherorder spectra to mixed norm methods of signal separation and
coupling analysis. I doubt space permitted the authors to acknowledge much
of this domain, but the introduction to the paper does make clear that
much of the work lies in the linear coupling domain.
The
derivations in the paper are sound  I was able to rederive and follow
the math. I have some minor comments and qns as below: 1) The issues
surrounding *any* higher order cumulants surely don't disappear with a
kernel trick : namely that very large numbers of iid samples are required
for good estimation. Kurtotic cumulants in particular require very large
sample sets. I can find no discussion of this. 2) Although based on
generic linear models, nonGaussian [generalized] MAR models are widely
used for assessing higherorder spectra and crosscoupling, with the
nonGaussian excitation being inferred using generalizations of EM with a
GMM. Links with this work? 3) The latter models link neatly with ICA
style approaches, which [for certain assumed heavytailed density models]
allow for independence to be related to negentropy and hence higherorder
cumulants in the multivar space. There are clear links with this work.
4) Fig 3. Is the blue curve under the green in the 50100Hz region?
Q2: Please summarize your review in 12
sentences
A fairly wellwritten paper, which details a more
extensive theoretical treatment of kernel higherorder crossspectra and
coupling. The paper is sound in the math, and goes some way to provide an
underpinning for kernel coupling approaches. The choice of realworld
example does not do much to highlight the method. Submitted by
Assigned_Reviewer_5
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
The paper considers kernel crossspectral density
(KCSD) as a way to determine interactions among time series in a better
way than methods developed under the i.i.d assumption, which is typically
violated for time series data. Such a framework was originally proposed in
[Besserve et al, ICASSP, 2011]. The main contribution here is to
characterize cases where KCSD can be used to test independence, and to
propose and study a way to estimate the properties of crossspectral
density operators from finite samples. The method is compared to the
HilbertSchmidt Independence Criterion test under the same kernels on
simulated and real neural data.
Developing independence test for
time series data that are able to cope with nonlinear interactions is an
important area of research. The present paper provides a sound and
wellmotivated approach.
Many applications, however, consider a
large number of time series data. Even though the proposed approach
accommodate very general forms of dependencies, it is a *pairwise*
independence test, and thus suffers from the limitations inherent to
pairwise testing. It would be interesting to discuss whether the proposed
method could be extended to testing simultaneously multiple time series
(each being potentially multivariate).
The methodology and results
depend on the choice of kernel, and various kernels might lead to
contradictory conclusions. This should be discussed.
Kernel
methods seem very promising candidates in capturing dependencies in time
series, in the present setting and beyond. For instance, it would also be
relevant to mention the recent work by Sindhwani et al, Scalable
Matrixvalued Kernel Learning for Highdimensional Nonlinear Multivariate
Regression and Granger Causality, UAI 2013, which uses kernel methods to
capture nonlinearity in causal inference, via a generalized form of
sparse vector autoregression. Q2: Please summarize your
review in 12 sentences
Developing independence tests that are tailored to
time series data and can accommodate nonlinearity is an important
research topic. This paper proposes a sound and wellmotivated approach.
However, the proposed approach can only deal with pairwise tests, while
many problems involve dependence involving multiple time series, and
relies on a prespecified kernel choice.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We are grateful to the reviewers for their careful
reading of our manuscript and their suggestions. Responses follow.
I. Novelty and impact. KCSD is a complex object to study and
an important aspect of our contribution is to estimate its properties with
good asymptotic result under mild conditions. We introduce an unbiased
estimate and a statistical test using fast algorithms which are easily
applicable to many datasets. Our results cannot be found elsewhere and
further work can build on our mathematical treatment to assess statistical
properties of kernel methods for stationary data. Most importantly, this
contribution aims at bringing results from kernel methods to communities
that are in important need for general time series analysis techniques
with good statistical properties. Measures describing the dependency
structure of the data without model assumptions, such as the linear
crossspectrum, became standard in applications such as Neurophysiology.
Our approach provides a nonlinear generalization of this quantity, which
enables a model free statistical assessment of the dependency between time
series using minimal assumptions on the system (Theorem 1, Proposition 2).
We believe this measure can become a new reference in many applications
related to time series. In particular, nonlinear interactions are
ubiquitous in brain signals and our approach provides a simple way to map
these interactions in the frequency domain.
II. Links to time
series models. Reviewer 4 suggests interesting links with the use of
higher order statistics in multivariate time series models and system
identification techniques (in a broad sense). We included more references
related to this topic by adding the following text on line 38 after
“specific contexts”: “and have been extensively used in system
identification, causal inference and blind source separation (see for
example [Giannakis 1989; Cardoso 1999;Hyvarinen 2009])”. While the
present paper focuses on the study of the kernel dependency measure in
itself, it can be connected to time series model estimation. Indeed, most
time series models rely on the assumption of i.i.d. innovations (or
residuals). These assumptions are key to estimate model parameters and to
validate the model. As a consequence, several methods rest on testing or
maximizing independence in order to fit a model [Hyvarinen 2008;Peters].
Our independence measure, which is robust to non i.i.d. samples, can be
used in similar frameworks to improve these techniques. In particular, it
can be combined with recent kernel regression techniques suggested by
Reviewer 5. We added the following related sentence to line 431:
“Following [Hyvarinen 2008;Peters], our independence test can be
combined to recent developments in kernel time series prediction
techniques [Sindhwani 2013] to define more general and reliable
multivariate causal inference techniques.”
III. Dependency between
multiple time series. We agree with Reviewer 5 that pairwise
independence does not capture all the dependency structure in case more
than two time series are involved. However, using the faithfulness
assumption, it is possible to combine pairwise independence tests with
multivariate regression techniques to fully characterize this dependency
structure (see [Peters] and references therein). As mentioned in the
previous paragraph, our method can thus be used to validate or fit models
involving more than two time series, for example by applying it to the
residuals of multivariate regressions.
IV. Choice of the kernel
and connections to higher order statistics. As mentioned by the
reviewers, the choice of the kernel can affect the outcome of the analysis
and can depend on the number of samples available. As mentioned in the
paper, ability to detect any dependency will depend on whether the kernels
are characteristic or not. However, in relation to difficulties in
estimating higher order statistics, reliable estimation with a
characteristic kernel might require more samples, so simpler kernels can
be used first to capture the most obvious dependencies in the data. Kernel
selection has been studied in a related context in [Sriperumbudur
2009;Gretton 2012]. We added this sentence on line 241: “In
general, the choice of the kernel is a tradeoff between the ability to
detect complex dependencies (a characteristic kernel being more
sensitive), and the convergence rate of the estimate (simpler kernels
related to lower order statistics usually require less samples). Related
theoretical analysis can be found in [Sriperumbudur 2009;Gretton 2012].”
Detailed comments from Reviewer 2. Regarding negative
values of our estimate we added the sentence on line 296: “The observed
negative values are also a direct consequence of the unbiased property of
our estimate (Corollary 8).” Also, we fixed the mentioned typos (lines 91,
95 and 290). Finally, we added the reference [Whittingstal 2009] on line
418.
Reviewer 4, question 4: Yes, on Fig. 3 curves are
superimposed in the gamma band.
References: Cardoso,
Highorder contrasts for independent component analysis. Neu. Comput.
1999. Giannakis et al., Identification of nonminimum phase systems
using higher order statistics. IEEE TSP 1989. Gretton et al., Optimal
kernel choice for largescale twosample tests. NIPS 2012. Peters et
al., Causal Inference on Time Series using Structural Equation Models.
arXiv. Hyvarinen et al., Causal modeling combining instantaneous and
lagged effects: an identifiable model based on nongaussianity. ICML 2008.
Fukumizu et al., Kernel choice and classifiability for RKHS embeddings
of probability distributions. NIPS 2009. Whittingstall et al.,
Frequencyband coupling in surface EEG reflects spiking activity in monkey
visual cortex. Neuron 2009.
 