We study a change point model based on a stochastic partial differential equation (SPDE) corresponding to the heat equation governed by the weighted Laplacian $\Delta_\vartheta = \nabla\vartheta\nabla$, where $\vartheta=\vartheta(x)$ is a space-dependent diffusivity. As a basic problem the domain $(0,1)$ is considered with a piecewise constant diffusivity with a jump at an unknown point $\tau$. Based on local measurements of the solution in space with resolution $\delta$ over a finite time horizon, we construct a simultaneous M-estimator for the diffusivity values and the change point. The change point estimator converges at rate $\delta$, while the diffusivity constants can be recovered with convergence rate $\delta^{3/2}$. Moreover, when the diffusivity parameters are known and the jump height vanishes with the spatial resolution tending to zero, we derive a limit theorem for the change point estimator and identify the limiting distribution. For the mathematical analysis, a precise understanding of the SPDE with discontinuous $\vartheta$, tight concentration bounds for quadratic functionals in the solution, and a generalisation of classical M-estimators are developed.

We consider standard gradient descent, gradient flow and conjugate gradients as iterative algorithms for minimising a penalised ridge criterion in linear regression. While it is well known that conjugate gradients exhibit fast numerical convergence, the statistical properties of their iterates are more difficult to assess due to inherent non-linearities and dependencies. On the other hand, standard gradient flow is a linear method with well-known regularising properties when stopped early. By an explicit non-standard error decomposition we are able to bound the prediction error for conjugate gradient iterates by a corresponding prediction error of gradient flow at transformed iteration indices. This way, the risk along the entire regularisation path of conjugate gradient iterations can be compared to that for regularisation paths of standard linear methods like gradient flow and ridge regression. In particular, the oracle conjugate gradient iterate shares the optimality properties of the gradient flow and ridge regression oracles up to a constant factor. Numerical examples show the similarity of the regularisation paths in practice.

We consider estimators obtained by iterates of the conjugate gradient (CG) algorithm applied to the normal equation of prototypical statistical inverse problems. Stopping the CG algorithm early induces regularisation, and optimal convergence rates of prediction and reconstruction error are established in wide generality for an ideal oracle stopping time. Based on this insight, a fully data-driven early stopping rule τ is constructed, which also attains optimal rates, provided the error in estimating the noise level is not dominant. The error analysis of CG under statistical noise is subtle due to its nonlinear dependence on the observations. We provide an explicit error decomposition and identify two terms in the prediction error, which share important properties of classical bias and variance terms. Together with a continuous interpolation between CG iterates, this paves the way for a comprehensive error analysis of early stopping. In particular, a general oracle-type inequality is proved for the prediction error at τ . For bounding the reconstruction error, a more refined probabilistic analysis, based on concentration of self-normalised Gaussian processes, is developed. The methodology also provides some new insights into early stopping for CG in deterministic inverse problems. A numerical study for standard examples shows good results in practice for early stopping at τ .

We estimate nonparametrically the spatially varying diffusivity of a stochastic heat equation from observations perturbed by additional noise. To that end, we employ a two-step localization procedure, more precisely, we combine local state estimates into a locally linear regression approach. Our analysis relies on quantitative Trotter–Kato type approximation results for the heat semigroup that are of independent interest. The presence of observational noise leads to non-standard scaling behaviour of the model. Numerical simulations illustrate the results.

We study the rank of the instantaneous or spot covariance matrix $\Sigma_X(t)$ of a multidimensional continuous semi-martingale $X(t)$. Given high-frequency observations $X(i/n)$, $i=0,\ldots,n$, we test the null hypothesis $rank(\Sigma_X(t))\le r$ for all $t$ against local alternatives where the average $(r+1)$st eigenvalue is larger than some signal detection rate $v_n$. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of $\Sigma_X(t)$. We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates $v_n$. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.

We consider the estimation of a non-linear reaction term in the stochastic heat or more generally in a semi-linear stochastic partial differential equation (SPDE). Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. Our main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. The estimation method is extended to local observations in time and space, which allows for non-parametric estimation of a reaction intensity varying in time and space. Furthermore, discrete observations in time and space can be handled. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincar\'e inequality and regularity results for SPDEs in $L^p$-interpolation spaces.

As a concrete setting where stochastic partial differential equations (SPDEs) are able to model real phenomena, we propose a stochastic Meinhardt model for cell repolarisation and study how parameter estimation techniques developed for simple linear SPDE models apply in this situation. We establish the existence of mild SPDE solutions and we investigate the impact of the driving noise process on pattern formation in the solution. We then pursue estimation of the diffusion term and show asymptotic normality for our estimator as the space resolution becomes finer. The finite sample performance is investigated for synthetic and real data.

The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the parametric rate even in the nonparametric setup with a spatially varying coefficient. A numerical example illustrates the main results.

Given a sample of a Poisson point process with intensity λf(x,y)=n1(f(x)≤y), we study recovery of the boundary function f from a nonparametric Bayes perspective. Because of the irregularity of this model, the analysis is non-standard. We establish a general result for the posterior contraction rate with respect to the L1-norm based on entropy and one-sided small probability bounds. From this, specific posterior contraction results are derived for Gaussian process priors and priors based on random wavelet series.

We analyse the reconstruction error of principal component analysis (PCA) and prove nonasymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. The bounds reveal that the excess risk differs significantly from usually considered subspace distances based on canonical angles. Our approach relies on the analysis of empirical spectral projectors combined with concentration inequalities for weighted empirical covariance operators and empirical eigenvalues.

Given data from a Poisson point process with intensity $(x,y) \mapsto n \mathbf{1}(f(x)\leq y),$ frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone and piecewise constant support boundaries and adapts to H\"older smooth boundaries with smoothness index at most one. We then derive a non-standard Bernstein-von Mises result for a compound Poisson process prior and a function space with increasing parameter dimension. As an intermediate result the limiting shape of the posterior for random histogram type priors is obtained. In both settings, it is shown that the marginal posterior of the functional $\vartheta =\int f$ performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, $(1-\alpha)$-credible sets are also asymptotic $(1-\alpha)$-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions indicating that credible sets only have frequentist coverage for priors that are specifically constructed to match properties of the underlying true function.

The recent explosion of data that is routinely collected has led scientists to contemplate more and more sophisticated structural assumptions. Understanding how to harness and exploit such structure is key to improving the prediction accuracy of various statistical procedures. The ultimate goal of this line of research is to develop a set of tools that leverage underlying complex structures to pool information across observations and ultimately improve statistical accuracy as well as computational efficiency of the deployed methods. The workshop focused on recent developments in regression and matrix estimation under various complex constraints such as physical, computational, privacy, sparsity or robustness. Optimal-transport based techniques for geometric data analysis were also a main topic of the workshop.

Consider a Poisson point process with unknown support boundary curve g, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form ( )integral Phi (g (x)) dx. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Holder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on Phi which are satisfied for Phi (u) = vertical bar u vertical bar(p), p >= 1. As an application, we consider the problem of estimating the L-p-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.

We propose a new estimator for the spot covariance matrix of a multi-dimensional continuous semi-martingale log asset price process which is subject to noise and non-synchronous observations. The estimator is constructed based on a local average of block-wise parametric spectral covariance estimates. The latter originate from a local method of moments (LMM) which recently has been introduced by Bibinger et al. (2014). We extend the LMM estimator to allow for autocorrelated noise and propose a method to adaptively infer the autocorrelations from the data. We prove the consistency and asymptotic normality of the proposed spot covariance estimator. Based on extensive simulations we provide empirical guidance on the optimal implementation of the estimator and apply it to high-frequency data of a cross-section of NASDAQ blue chip stocks. Employing the estimator to estimate spot covariances, correlations and betas in normal but also extreme-event periods yields novel insights into intraday covariance and correlation dynamics. We show that intraday (co-)variations (i) follow underlying periodicity patterns, (ii) reveal substantial intraday variability associated with (co-)variation risk, (iii) are strongly serially correlated, and (iv) can increase strongly and nearly instantaneously if new information arrives.

We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension $D$. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level $\widehat m\in\{1,\ldots,D\}$ only based on the knowledge of the first $\widehat m$ singular values and vectors. We analyse in detail whether sequential {\it early stopping} rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.

We present upper bounds for the Wasserstein distance of order $p$ between the marginals of L\'evy processes, including Gaussian approximations for jumps of infinite activity. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of L\'evy processes. Connections to other metrics like Zolotarev and Toscani-Fourier distances are established. The theory is illustrated by concrete examples and an application to statistical lower bounds.