#
Kernel Two-Sample and Independence Tests for Nonstationary Random Processes^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. mmd and hsic for Nonstationary Random Processes

#### 3.1. Notation and Assumptions

#### 3.2. mmd for Nonstationary Random Processes

#### 3.3. hsic for Nonstationary Random Processes

#### 3.4. Maximising the Test Power

## 4. Experimental Results on Synthetic Data

#### 4.1. Homogeneity Tests with mmd

#### 4.1.1. Setup

- Mean shift:${\mu}_{\mathbf{X}}\left(t\right)=t$ and ${\mu}_{\mathbf{Y}}\left(t\right)=t+{\delta}_{\mu}{t}^{3}$. The basis coefficients are sampled as ${{\xi}_{\mathbf{X}}}_{i,1},\phantom{\rule{0.166667em}{0ex}}{{\xi}_{\mathbf{Y}}}_{i,1}\sim \mathcal{N}(0,10)$ and ${{\xi}_{\mathbf{X}}}_{i,2},\phantom{\rule{0.166667em}{0ex}}{{\xi}_{\mathbf{Y}}}_{i,2}\sim \mathcal{N}(0,5)$, and the additive noises are sampled as ${{\u03f5}_{\mathbf{X}}}_{i,t},\phantom{\rule{0.166667em}{0ex}}{{\u03f5}_{\mathbf{Y}}}_{i,t}\sim \mathcal{N}(0,0.25)$.
- Variance shift: We take ${\mu}_{\mathbf{X}}\left(t\right)={\mu}_{\mathbf{Y}}\left(t\right)=0$, and introduce a shift in variance in the first basis function coefficients via ${{\xi}_{\mathbf{X}}}_{i,1}\sim \mathcal{N}(0,10)$ and ${{\xi}_{\mathbf{Y}}}_{i,1}\sim \mathcal{N}(0,10+{\delta}_{\sigma})$. The second coefficients are sampled as ${{\xi}_{\mathbf{X}}}_{i,2},{{\xi}_{\mathbf{Y}}}_{i,2}\sim \mathcal{N}(0,5)$, and the noises as ${{\u03f5}_{\mathbf{X}}}_{i,t},\phantom{\rule{0.166667em}{0ex}}{{\u03f5}_{\mathbf{Y}}}_{i,t}\sim \mathcal{N}(0,0.25)$.

#### 4.1.2. Baseline Results without Test Power Optimisation

#### 4.1.3. Results of the Optimised Test

#### 4.2. Independence Tests with hsic

#### 4.2.1. Setup

- Linear dependence:$\mathbf{X}$ is generated as in (9) with ${\mu}_{\mathbf{X}}\left(t\right)=t$, basis coefficients ${{\xi}_{\mathbf{X}}}_{i,1}\sim \mathcal{N}(0,10)$, ${{\xi}_{\mathbf{X}}}_{i,2}\sim \mathcal{N}(0,5)$, and noise ${{\u03f5}_{\mathbf{X}}}_{i,t}\sim \mathcal{N}(0,0.25)$. The samples of the second process are $\mathbf{Y}={\{{x}_{i,1}+{\u03f5}_{i}\}}_{i=1}^{m}$ where ${\u03f5}_{i}\sim \mathcal{N}(0,1)$, as in [18].
- Dependence through a shared coefficient:$\mathbf{X}$ and $\mathbf{Y}$ are generated as in (9) with ${\mu}_{\mathbf{X}}\left(t\right)={\mu}_{\mathbf{Y}}\left(t\right)=t$ and independently sampled ${{\xi}_{\mathbf{X}}}_{i,1}$, ${{\xi}_{\mathbf{Y}}}_{i,1}$, ${{\u03f5}_{\mathbf{X}}}_{i,t}$, ${{\u03f5}_{\mathbf{Y}}}_{i,t}$ as in the mean shift experiments of Section 4.1, but where the stochastic processes now share the second basis function coefficient: ${{\xi}_{\mathbf{X}}}_{i,2}={{\xi}_{\mathbf{Y}}}_{i,2}$.
- Dependence through rotation: We start by generating independent ${\mathbf{X}}^{\left(0\right)}$ and ${\mathbf{Y}}^{\left(0\right)}$ as in (9) with ${\mu}_{\mathbf{X}}\left(t\right)={\mu}_{\mathbf{Y}}\left(t\right)=t$ and ${{\u03f5}_{\mathbf{X}}}_{i,t},{{\u03f5}_{\mathbf{Y}}}_{i,t}\sim \mathcal{N}(0,0.25)$, but with ${{\xi}_{\mathbf{X}}}_{i,k}$ and ${{\xi}_{\mathbf{Y}}}_{i,k}$ drawn from: (i) student-t, (ii) uniform, or (iii) exponential distributions [28] (Table 3). We next multiply $({\mathbf{X}}^{\left(0\right)},{\mathbf{Y}}^{\left(0\right)})$ by a $2\times 2$ rotation matrix $R\left(\theta \right)$ with $\theta \in [0,\pi /4]$ to generate new rotated samples $(\mathbf{X},\mathbf{Y})$, which we then test for independence. Clearly, for $\theta =0$ our samples $(\mathbf{X},\mathbf{Y})$ are independent and as $\theta $ is increased their dependence becomes easier to detect (see [7] (Section 4) and Figure A3 for implementation details).

#### 4.2.2. Baseline Results without Test Power Optimisation

#### 4.2.3. Results of the Optimised Test

## 5. Application to a Socioeconomic Dataset

## 6. Discussion and Conclusions

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Appendix A

#### Appendix A.1. Results for Realisations with Varying Number of Time Points, T

**Figure A1.**Results of mmd-based homogeneity test with $T=[5,10,25,50,100]$: Percentage of rejected ${H}_{0}$ for mean shift (

**left**) and variance shift (

**right**) for sample sizes $m=n=100$ and T discrete time points in $d=1$ dimensions.

**Figure A2.**Results of the hsic-based independence test: Test power for linear dependence (

**left**) and dependence through shared coefficient (

**right**) as sample size is varied for various numbers of time points $T=[5,10,25,50,100]$.

#### Appendix A.2. Test Power Maximisation

**Table A1.**Linear search spaces for bandwidth $\sigma $ in mmd mean (

**left**) and variance (

**right**) shift experiments.

${\mathbf{\delta}}_{\mathbf{\mu}}$ | 0–2 | 2.25–3 | 3.25–5 | 5.5–8 | ${\mathbf{\delta}}_{\mathbf{\sigma}}$ | 0–4 | 5–14 | 15–32 |

Step Size = 0.25 | Step Size = 0.5 | Step Size = 1 | ||||||

search space for $\sigma $ | 1 | 6 | 11 | 16 | search space for $\sigma $ | 10 | 20 | 30 |

3 | 8 | 13 | 18 | 12 | 22 | 32 | ||

5 | 10 | 15 | 20 | 14 | 24 | 34 | ||

7 | 12 | 17 | 22 | 16 | 26 | 36 | ||

9 | 14 | 19 | 24 | 18 | 28 | 38 | ||

11 | 16 | 21 | 26 | 20 | 30 | 40 | ||

13 | 18 | 23 | 28 | 22 | 32 | 42 | ||

15 | 20 | 25 | 30 | 24 | 34 | 44 | ||

17 | 22 | 27 | 32 | 26 | 36 | 46 | ||

19 | 24 | 29 | 34 | 28 | 38 | 48 | ||

21 | 26 | 31 | 36 | 30 | 40 | 50 |

#### Appendix A.3. Distribution Specifications for Basis Function Coefficients in Rotation Mixing

**Table A2.**Specifications of distributions for the rotation mixing. They are a subset of the distributions in [28] (Table 3), and $\mathbf{Z}$ is a proxy for both $\mathbf{X}$ and $\mathbf{Y}$.

Distribution | Fourier Basis Function Coefficients | |
---|---|---|

${\mathbf{\xi}}_{\mathbf{Z}\mathbf{i}\mathbf{1}}$ | ${\mathbf{\xi}}_{\mathbf{Z}\mathbf{i}\mathbf{2}}$ | |

Exponential | $\lambda =1.5$ | $\lambda =3$ |

Student-t | $\nu =3$ | $\nu =5$ |

Uniform | $\mathcal{U}[-10,10]$ | $\mathcal{U}[-5,5]$ |

**Figure A3.**Illustration of $\mathbf{X}$ and $\mathbf{Y}$ with (

**i**) student-t, (

**ii**) uniform, and (

**iii**) exponential basis function coefficients being mixed by different rotation angles $\theta $, ordered clockwise by increasing $\theta $.

#### Appendix A.4. SDG Dataset

## References

- Christakis, N.A.; Fowler, J.H. The spread of obesity in a large social network over 32 years. N. Engl. J. Med.
**2007**, 357, 370–379. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Barabási, A.L.; Gulbahce, N.; Loscalzo, J. Network medicine: A network-based approach to human disease. Nat. Rev. Genet.
**2011**, 12, 56–68. [Google Scholar] [CrossRef] [Green Version] - Bond, R. Complex networks: Network healing after loss. Nat. Hum. Behav.
**2017**, 1, 1–2. [Google Scholar] [CrossRef] - Battiston, S.; Mandel, A.; Monasterolo, I.; Schütze, F.; Visentin, G. A climate stress-test of the financial system. Nat. Clim. Chang.
**2017**, 7, 283–288. [Google Scholar] [CrossRef] - Muandet, K.; Fukumizu, K.; Sriperumbudur, B.; Schölkopf, B. Kernel mean embedding of distributions: A review and beyond. Found. Trends Mach. Learn.
**2017**, 10, 1–141. [Google Scholar] [CrossRef] - Gretton, A.; Borgwardt, K.; Rasch, M.; Schölkopf, B.; Smola, A.J. A kernel method for the two-sample-problem. arXiv
**2008**, arXiv:0805.2368. [Google Scholar] - Gretton, A.; Fukumizu, K.; Teo, C.H.; Song, L.; Schölkopf, B.; Smola, A.J. A kernel statistical test of independence. NIPS
**2008**, 20, 585–592. [Google Scholar] - Besserve, M.; Logothetis, N.K.; Schölkopf, B. Statistical analysis of coupled time series with Kernel Cross-Spectral Density operators. In Advances in Neural Information Processing Systems 26; Curran Associates, Inc.: Red Hook, NY, USA, 2013; pp. 2535–2543. [Google Scholar]
- Chwialkowski, K.; Sejdinovic, D.; Gretton, A. A wild bootstrap for degenerate kernel tests. In Advances in Neural Information Processing Systems; Curran Associates, Inc.: Red Hook, NY, USA, 2014; pp. 3608–3616. [Google Scholar]
- Davis, R.A.; Matsui, M.; Mikosch, T.; Wan, P. Applications of distance correlation to time series. Bernoulli
**2018**, 24, 3087–3116. [Google Scholar] [CrossRef] [Green Version] - Székely, G.J.; Rizzo, M.L.; Bakirov, N.K. Measuring and testing dependence by correlation of distances. Ann. Stat.
**2007**, 35, 2769–2794. [Google Scholar] [CrossRef] - Horváth, L.; Kokoszka, P.; Reeder, R. Estimation of the mean of functional time series and a two-sample problem. J. R. Stat. Soc. Ser. B
**2012**, 75, 103–122. [Google Scholar] [CrossRef] [Green Version] - Fremdt, S.; Steinbach, J.G.; Horváth, L.; Kokoszka, P. Testing the Equality of Covariance Operators in Functional Samples. Scand. J. Stat.
**2012**, 40, 138–152. [Google Scholar] [CrossRef] [Green Version] - Panaretos, V.M.; Kraus, D.; Maddocks, J.H. Second-Order Comparison of Gaussian Random Functions and the Geometry of DNA Minicircles. J. Am. Stat. Assoc.
**2010**, 105, 670–682. [Google Scholar] [CrossRef] [Green Version] - Pomann, G.M.; Staicu, A.M.; Ghosh, S. A two-sample distribution-free test for functional data with application to a diffusion tensor imaging study of multiple sclerosis. J. R. Stat. Soc. Ser. C
**2016**, 65, 395–414. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wynne, G.; Duncan, A.B. A kernel two-sample test for functional data. arXiv
**2020**, arXiv:2008.11095. [Google Scholar] - Górecki, T.; Krzyśko, M.; Wołyński, W. Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data. Artif. Intell. Rev.
**2018**, 53, 475–499. [Google Scholar] [CrossRef] [Green Version] - Zhang, Q.; Filippi, S.; Gretton, A.; Sejdinovic, D. Large-scale kernel methods for independence testing. Stat. Comput.
**2018**, 28, 113–130. [Google Scholar] [CrossRef] [Green Version] - Sriperumbudur, B.K.; Gretton, A.; Fukumizu, K.; Schölkopf, B.; Lanckriet, G.R. Hilbert space embeddings and metrics on probability measures. J. Mach. Learn. Res.
**2010**, 11, 1517–1561. [Google Scholar] - Sriperumbudur, B.K.; Fukumizu, K.; Lanckriet, G.R. Universality, Characteristic Kernels and RKHS Embedding of Measures. J. Mach. Learn. Res.
**2011**, 12, 2389–2410. [Google Scholar] - Gretton, A.; Borgwardt, K.M.; Rasch, M.J.; Schölkopf, B.; Smola, A.J. A kernel two-sample test. J. Mach. Learn. Res.
**2012**, 13, 723–773. [Google Scholar] - Gretton, A.; Fukumizu, K.; Harchaoui, Z.; Sriperumbudur, B.K. A fast, consistent kernel two-sample test. NIPS
**2009**, 23, 673–681. [Google Scholar] - Song, L.; Smola, A.J.; Gretton, A.; Bedo, J.; Borgwardt, K. Feature selection via dependence maximization. J. Mach. Learn. Res.
**2012**, 13, 1393–1434. [Google Scholar] - Ramdas, A.; Reddi, S.J.; Póczos, B.; Singh, A.; Wasserman, L. On the decreasing power of kernel and distance based nonparametric hypothesis tests in high dimensions. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, Austin, TX, USA, 25–30 January 2015. [Google Scholar]
- Reddi, S.; Ramdas, A.; Póczos, B.; Singh, A.; Wasserman, L. On the high dimensional power of a linear-time two sample test under mean-shift alternatives. Artif. Intell. Stat.
**2015**, 38, 772–780. [Google Scholar] - Serfling, R.J. Approximation Theorems of Mathematical Statistics; Wiley Series in Probability and Mathematical Statistics; Wiley: New York, NY, USA, 2002. [Google Scholar]
- Sutherland, D.J.; Tung, H.Y.; Strathmann, H.; De, S.; Ramdas, A.; Smola, A.J.; Gretton, A. Generative models and model criticism via optimized maximum mean discrepancy. arXiv
**2016**, arXiv:1611.04488. [Google Scholar] - Gretton, A.; Herbrich, R.; Smola, A.J.; Bousquet, O.; Schölkopf, B. Kernel methods for measuring independence. J. Mach. Learn. Res.
**2005**, 6, 2075–2129. [Google Scholar] - World Bank. World Bank Country and Lending Groups. 2020. Available online: https://datahelpdesk.worldbank.org/knowledgebase/articles/906519-world-bank-country-and-lending-groups (accessed on 28 January 2020).
- World Bank. Sustainable Development Goals. 2020. Available online: https://datacatalog.worldbank.org/dataset/sustainable-development-goals (accessed on 28 January 2020).

**Figure 1.**Results of our mmd-based homogeneity test for nonstationary random processes: percentage of rejected ${H}_{0}$ as mean shift (

**left**) and variance shift (

**right**) are varied. Our baseline method (solid lines) is compared to [15] (dashed lines) for different sample sizes $m=n=100,200,300,500$ and $T=100$ discrete time points.

**Figure 2.**Results of homogeneity test with optimising for test power: percentage of rejected ${H}_{0}$ for mean shift (

**left**) and variance shift (

**right**) for sample sizes $m=n=100,200$ and $T=100$ discrete time points. Our optimised test power method (dotted lines) is compared to our baseline method (solid lines) and [15] (dashed lines).

**Figure 3.**Results of the hsic-based independence test: Test power for linear dependence (

**left**) and dependence through shared coefficients (

**right**) as sample size is varied for various numbers of time points. For the linear dependence, we compare our baseline results to SubCorr and SubHSIC; for the shared coefficient, we compare against two spectral approximations [18] (Section 5.1).

**Figure 4.**Results of the hsic-based independence test: Percentage of rejected ${H}_{0}$ in rotation dependence experiments for different number of discrete time points T and coefficients ${{\xi}_{\mathbf{X}}}_{i,k}$ and ${{\xi}_{\mathbf{Y}}}_{i,k}$ drawn from three distributions: (

**i**) student-t, (

**ii**) uniform, and (

**iii**) exponential (see Appendix A.3). The sample size is $m=200$. The violet dotted lines are the results of our test power maximisation.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Laumann, F.; Kügelgen, J.v.; Barahona, M.
Kernel Two-Sample and Independence Tests for Nonstationary Random Processes. *Eng. Proc.* **2021**, *5*, 31.
https://doi.org/10.3390/engproc2021005031

**AMA Style**

Laumann F, Kügelgen Jv, Barahona M.
Kernel Two-Sample and Independence Tests for Nonstationary Random Processes. *Engineering Proceedings*. 2021; 5(1):31.
https://doi.org/10.3390/engproc2021005031

**Chicago/Turabian Style**

Laumann, Felix, Julius von Kügelgen, and Mauricio Barahona.
2021. "Kernel Two-Sample and Independence Tests for Nonstationary Random Processes" *Engineering Proceedings* 5, no. 1: 31.
https://doi.org/10.3390/engproc2021005031