Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods

Blanchard, Gilles; Zadorozhnyi, Oleksandr

doi:10.3150/18-bej1095

Cited by 5 publications

(6 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However,

τ

‐mixing holds for some simple AR

false(1 false)

models that are not

α

‐mixing. Examples of

τ

‐mixing processes can be found in Blanchard and Zadorozhnyi (2019). Other types of mixing coefficients equivalent to

τ

‐mixing when the random elements are almost surely bounded are the

θ_{\infty}

‐mixing coefficients (Dedecker et al ., 2007, Definition 2.3) or the

τ_{\infty}

‐mixing coefficients (Olivier, 2010, p. 492).…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…Their rate of convergence is similar, except for the

\log false(N false) \log {false(T false)}^{1 false/ 2 γ}

factor, which appears as

\sqrt{\log false(N false)}

in their case. The reason for our different rates of convergence is that we rely on concentration inequalities for weakly dependent time series in Hilbert spaces (Blanchard and Zadorozhnyi, 2019), which are weaker and require stronger assumptions than their scalar counterparts (Merlevède et al ., 2011). Having said that, whether these rates are optimal or whether the assumptions here are the weakest possible ones remain open questions.…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…With this notation, notice that

{bold-italicb}_{i} = {sans-serifP}_{i} {bold-italicB}_{N}

. To get uniform bounds for the loadings, we need some technical assumptions, which rely on the concept of

τ

‐mixing: for a sequence of random elements

false{Y_{t} false} \subset H_{0}

such that

{(‖‖, Y_{t})}_{H_{0}} \leq c

almost surely, where

H_{0}

is a real separable Hilbert space with norm

{(‖‖, \cdot)}_{H_{0}}

, we define the

τ

‐mixing coefficients (Blanchard and Zadorozhnyi, 2019, Example 2.3)

\begin{matrix} alignleft \\ align-1 τ (h) & align-2 : = \sup \{𝔼 [Z (φ (Y_{t + h}) - 𝔼 φ (Y_{t + h}))] | t \geq 1, Z \in L^{1} (Ω, ℳ_{t}^{Y}, ℙ), \\ align-1 & align-2 𝔼 |Z| \leq 1, φ \in Λ_{1} (H_{0}, c)\}, \end{matrix}

where

ℳ_{t}^{Y}

is the

σ

‐algebra generated by

…”

Section: Theoretical Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Factor models for high‐dimensional functional time series II: Estimation and forecasting

Tavakoli

Nisol

Hallin

2023

Journal Time Series Analysis

View full text Add to dashboard Cite

This article is the second one in a set of two laying the theoretical foundations for a high‐dimensional functional factor model approach in the analysis of large cross‐sections (panels) of functional time series (FTS). Part I establishes a representation result by which, under mild assumptions on the covariance operator of the cross‐section, any FTS admits a canonical representation as the sum of a common and an idiosyncratic component; common components are driven by a finite and typically small number of scalar factors loaded via functional loadings, while idiosyncratic components are only mildly cross‐correlated. Building on that representation result, Part II is dealing with the identification of the number of factors, their estimation, the estimation of their loadings and the common components, and the resulting forecasts. We provide a family of information criteria for identifying the number of factors, and prove their consistency. We provide average error bounds for the estimators of the factors, loadings, and common components; our results encompass the scalar case, for which they reproduce and extend, under weaker conditions, well‐established similar results. Under slightly stronger assumptions, we also provide uniform bounds for the estimators of factors, loadings, and common components, thus extending existing scalar results. Our consistency results in the asymptotic regime where the number of series and the number of time points diverge thus extend to the functional context the ‘blessing of dimensionality’ that explains the success of factor models in the analysis of high‐dimensional (scalar) time series. We provide numerical illustrations that corroborate the convergence rates predicted by the theory, and provide a finer understanding of the interplay between and for estimation purposes. We conclude with an application to forecasting mortality curves, where our approach outperforms existing methods.

show abstract

“…However,

τ

‐mixing holds for some simple AR

false(1 false)

models that are not

α

‐mixing. Examples of

τ

‐mixing processes can be found in Blanchard and Zadorozhnyi (2019). Other types of mixing coefficients equivalent to

τ

‐mixing when the random elements are almost surely bounded are the

θ_{\infty}

‐mixing coefficients (Dedecker et al ., 2007, Definition 2.3) or the

τ_{\infty}

‐mixing coefficients (Olivier, 2010, p. 492).…”

Section: Theoretical Resultsmentioning

confidence: 99%

“…Their rate of convergence is similar, except for the

\log false(N false) \log {false(T false)}^{1 false/ 2 γ}

factor, which appears as

\sqrt{\log false(N false)}

Section: Theoretical Resultsmentioning

confidence: 99%

“…With this notation, notice that

{bold-italicb}_{i} = {sans-serifP}_{i} {bold-italicB}_{N}

. To get uniform bounds for the loadings, we need some technical assumptions, which rely on the concept of

τ

‐mixing: for a sequence of random elements

false{Y_{t} false} \subset H_{0}

such that

{(‖‖, Y_{t})}_{H_{0}} \leq c

almost surely, where

H_{0}

is a real separable Hilbert space with norm

{(‖‖, \cdot)}_{H_{0}}

, we define the

τ

‐mixing coefficients (Blanchard and Zadorozhnyi, 2019, Example 2.3)

\begin{matrix} alignleft \\ align-1 τ (h) & align-2 : = \sup \{𝔼 [Z (φ (Y_{t + h}) - 𝔼 φ (Y_{t + h}))] | t \geq 1, Z \in L^{1} (Ω, ℳ_{t}^{Y}, ℙ), \\ align-1 & align-2 𝔼 |Z| \leq 1, φ \in Λ_{1} (H_{0}, c)\}, \end{matrix}

where

ℳ_{t}^{Y}

is the

σ

‐algebra generated by

…”

Section: Theoretical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Factor models for high‐dimensional functional time series II: Estimation and forecasting

Tavakoli

Nisol

Hallin

2023

Journal Time Series Analysis

View full text Add to dashboard Cite

show abstract

“…Mixing series however exclude many stochastic processes, as discussed in the monograph Dedecker et al (2007). Weak dependance coefficients cover a wider range of processes for which Bernstein type inequalities are proven for example in Collet et al (2002); Doukhan and Neumann (2007); Wintenberger (2010); Merlevède et al (2011); Blanchard and Zadorozhnyi (2017). Dynamic systems are examples of processes where only X 1 is random, each X t is then a deterministic fonction of X t−1 .…”

Section: State Of the Artmentioning

confidence: 99%

Exponential inequalities for nonstationary Markov chains

Alquier

Doukhan

Fan

2019

Dependence Modeling

View full text Add to dashboard Cite

Exponential inequalities are main tools in machine learning theory. To prove exponential inequalities for non i.i.d random variables allows to extend many learning techniques to these variables. Indeed, much work has been done both on inequalities and learning theory for time series, in the past 15 years. However, for the non independent case, almost all the results concern stationary time series. This excludes many important applications: for example any series with a periodic behaviour is nonstationary. In this paper, we extend the basic tools of Dedecker and Fan (2015) to nonstationary Markov chains. As an application, we provide a Bernstein-type inequality, and we deduce risk bounds for the prediction of periodic autoregressive processes with an unknown period.

show abstract

“…Recently, Blanchard and Zadorozhnyi (2019) derived a Bernstein-type inequality for Hilbert space processes for a class of mixing properties called C-mixing (Maume-Deschamps, 2006). As a special case, the authors show that under restrictive Lipschitz conditions on the feature map ϕ, this mixing property is preserved under the RKHS embedding of a so-called τ -mixing process.…”

Section: Introductionmentioning

confidence: 99%

Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence

Mollenhauer¹,

Klus²,

Schütte³

et al. 2020

Preprint

View full text Add to dashboard Cite

We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and prove several asymptotic results as well as finite sample error bounds with a detailed analysis for the Gaussian kernel. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.

show abstract

Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods

Cited by 5 publications

References 53 publications

Factor models for high‐dimensional functional time series II: Estimation and forecasting

Factor models for high‐dimensional functional time series II: Estimation and forecasting

Exponential inequalities for nonstationary Markov chains

Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence

Contact Info

Product

Resources

About