From: ~Michael_McCourt1
comment: Thank you very much for reading this paper and for your thorough comments.  They are greatly appreciated.  We will incorporate some into this paper, and everything into a future, more thorough, treatment of the topic.

1) We agree that we failed to explain the motivation for the Laguerre polynomials.  The contrast that we were trying to make was that "Chebyshev polynomials are orthogonal on the interval [-1, 1] which is why the kernel only lives on that interval; in the search for a kernel defined on [0, inf) we will choose functions (polynomials) orthogonal on that interval such as the Laguerre polynomials".  We will make an appropriate addition to the article to that effect.

2) Agreed on the topic of the appendix.  We were debating the addition, but were not certain if that would be appropriate given the short format of the workshop papers.  We will be including some of the details in the poster, some in the revised version, and hopefully a great deal more in an upcoming paper where space is less of an issue.

3) The topic of error rates for the asymptotic computations is a very good point.  We included the discussion here primarily to help garner intuition as to the behavior of the kernel with respect to different hyperparameters (in these asymptotic regimes), but error rates would be beneficial in understanding the implications of actually using the asymptotic forms in suitable circumstances.

4) We 100% agree regarding the comment on free parameters and consider this potentially one of the great strengths of this kernel (if we can understand it).  The parameters for this kernel seem to have very tangible impacts on the predictions, which will provide an avenue for making effective predictions if we can fully understand their impacts.  Or, so we hope.

5) You are absolutely correct that we are not investigating the implications of the spatial component here.  It is indisputable that there should be an interaction between the spatial and time component (in circumstances such as this where they are chosen independently).  We are very interested in exploring this in the future, but felt that we had limited space to do so here.

6) The discussion in lines 18-23 was meant only to introduce other literature where kernels are constructed.  It is, however, the case that there is a relationship between weighted Sobolev spaces and Mercer kernels.  In knowing the eigenfunctions of a kernel, the inner product for that kernel can be stated explicitly, meaning that the norm of that weighted Sobolev space (or just generally RKHS) can be stated explicitly.  The theoretical implications of this provide insights as to what norm the kernel interpolant satisfies (which in turn can provide insights on the behavior of the interpolant).  In the absence of these eigenfunctions, this norm (and thus many properties of the kernels and their interpolants) would only be know implicitly.  We will add some discussion to this point in an extended version of this paper.
title: Great help

From: ~Michael_McCourt1
comment: We want to thank you for reading the paper and for your helpful comments.  We will be making edits to the paper leading up to the workshop and will incorporate your thoughts.  Some responses are below.

1) The Laguerre polynomials were chosen for this article because of their orthogonality over the half-plane; this gives us a natural way from which to define eigenfunctions.  One could use the Chebyshev polynomials (or Jacobi polynomials) instead if there were a specific "end-time" or "end condition" at which data would be available, but in the absence of that we thought it would be more natural to use the unbounded kernel.  We will add an explanation of this logic to the paper.

Along a slightly different path, different eigenfunctions could be derived from other functions which satisfy some other orthogonality condition on the half-plane.  We have experimented trying to construct such eigenfunctions from Bessel functions, but that has proved less fruitful than from the Laguerre polynomials (over normalization issues).

2) This comment on the asymptotic form citation is quite astute, and we greatly appreciate your careful reading of this paper.  I think that is actually what was used here and it was mis-cited.  I will confirm in my notes of the derivation and change what is written.

3) We wholeheartedly agree with more numerical experiments with this kernel and are running some which will not make it into this paper (for space considerations) but which we hope will appear in the poster.

4) This is a fantastic reference which we will include in our future comparisons.  Learning about this and other such points of comparison is one of our goals in attending this workshop.
title: Agreed on all points

From: ~Ransalu_Senanayake1
comment: Dear authors,
      
You should now be able to see the reviews. We encourage you to engage in the discussion by commenting on the reviews. Please take this as an opportunity to clarify the issues highlighted by the reviewers.  Thanks!
      
Regards,
Ransalu, Neal, Fabio, and Girish
title: Rebuttal

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper14/AnonReviewer2
relevance: 4: high
impact: 4: high
novelty: 4: high

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper14/AnonReviewer2
evaluation: Brief Summary:
- This paper designs a non-stationary kernel for temporal phenomena.
- First the paper considers the general eigenfunction representation of positive definite kernels through Mercer's theorem.
- The idea is that by designing the eigenfunctions explicitly it can invoke kernels with desired properties.
- The eigenfunctions are designed from Laguerre polynomials.
- Potentially numerically unstable terms within the resulting kernel computation are addressed by considering their asymptotics which are more stable and tractable.
- Intuition, interpretation, and implications of the free parameters are discussed with illustrations.

General Comments:
- The core contribution could benefit from some motivation on why Laguerre polynomials were chosen for the construction of eigenfunctions. What are some of their properties that when used to construct eigenfunctions would translate to a kernel with suitable non-stationary properties in the temporal domain?
- Similarly, the choice of eigenvalues seem nice and sparks curiosity on how it was motivated. Some motivative reasoning could benefit the reader's intuition regarding the resulting kernel.
- The kernel's construction is clearly presented. If possible, the author could consider including an appendix detailing the derivations step by step.
- The brief discussion on safe computation considers asymptotics in both directions to approximate complicated terms. This is an important consideration. In future iterations the discussion could benefit from a presentation on the error rates in both directions to quantify the approximation quality.
- The discussion on free parameters is great for providing some initial intuition for the kernel hyperparameters. Examining kernel properties with respect to those parameters could be a great future direction. Kernel hyperparameter learning is a very important topic in general and its performance could make or break the practicality of such a flexible kernel.

Strengths:
- Clear presentation of kernel construction
- Good initial discussion on important topics on safe computation and interpretation of free parameters
- Good initial visualisations and initial experiments to demonstrate its characteristics and potential
- Good summary of potential future work that can follow

Weaknesses:
- Choices of eigenfunctions and eigenvalues that precede the kernel construction could benefit from clearer motivation
- While the paper proposes a design for a space-time kernel, the focus is mainly on the temporal dimension, and similar non-stationary in the spatial dimension is not really discussed

Minor Comments:
- From line 18 onwards, it is not very obvious how the discussion on anisotropic kernels constructed from weighted Sobolev spaces lead to the subsequent discussion on non-stationary kernels. Perhaps this was the key to motivating the choices of eigenfunctions and eigenvalues later on, so this section could benefit from more clarity.

Summary of Recommendation:
- Accept

From: ~Ransalu_Senanayake1
comment: Brief Summary:
-	The paper presents a kernel that is for temporal phenomena. It does this by using a Mercer series defined only on the half-plane.
General Comments:
- I like the idea of a kernel just defined on the half-plane and it is definitely relevant to representing temporal data
- The kernel’s construction and accompanying discourse was clear and well defined
-	It is mentioned that you use Laguerre polynomials instead of Chebyshev (line46) but it is not explicitly clear why Laguerre?
-	The comment on safely computing the kernel are welcome. The “safe computation” section refers to asymptotic expansions in both limiting directions. I have use previously the asymptotic form in my work and noted that the u -> \infty case (NIST online 10.30.1) is better approximated by the exponentially improved Hankel’s expansion (NIST online 10.40.1) and may be useful

Strengths:
-	A good explanation of the effect of the free parameters of the systems
-	Numerical considerations for computation of the kernel

Weaknesses:
-	Additional experiments on the predictive capabilities of the kernel as opposed to just a standard kernel would be welcome (this is not really a weakness as it is clear the focus is on the analysis of the kernel parameters)
-	A comparison or discussion alongside other available non-stationary kernels in the literature would be good too (e.g. Non-stationary Spectral Kernels, NIPS 2017)

Summary of Recommendation:
-	Accept
title: NIPS 2018 Workshop Spatiotemporal Paper14 AnonReviewer1 (appeared as an official comment)

From: NIPS.cc/2018/Workshop/Spatiotemporal/Paper14/AnonReviewer1
relevance: 5: very high
impact: 4: high
novelty: 4: high