In this paper we study the kernel change-point algorithm (KCP) proposed by Arlot, Celisse and Harchaoui [4], which aims at locating an unknown number of change-points in the distribution of a sequence of independent data taking values in an arbitrary set. The change-points are selected by model selection with a penalized kernel empirical criterion. We provide a non-asymptotic result showing that, with high probability, the KCP procedure retrieves the correct number of change-points, provided that the constant in the penalty is well-chosen; in addition, KCP estimates the change-points location at the optimal rate. As a consequence, when using a characteristic kernel, KCP detects all kinds of change in the distribution (not only changes in the mean or the variance), and it is able to do so for complex structured data (not necessarily in R d ). Most of the analysis is conducted assuming that the kernel is bounded; part of the results can be extended when we only assume a finite second-order moment.MSC 2010 subject classifications: Primary 62M10; secondary 62G20.

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Interpretability of machine learning algorithm is a pressing need. Numerous methods appeared in recent years, but do they make sense in simple cases? In this paper, we present a thorough theoretical analysis of Tabular LIME. In particular, we show that the explanations provided by Tabular LIME are close to an explicit expression in the large sample limit. We leverage this knowledge when the function to explain has some nice algebraic structure (linear, multiplicative, or depending on a subset of the coordinates) and provide some interesting insights on the explanations provided in these cases. In particular, we show that Tabular LIME provides explanations that are proportional to the coefficients of the function to explain in the linear case, and provably discards coordinates unused by the function to explain in the general case.

Fractional Brownian Motions (fBm) are nonstationary and selfsimilar stochastic processes which extend ordinary Brownian motion and are of great importance for modeling processes with long-term dependencies, such as l/f-type processes. Identification of fBm amounts to estimate one single scalar parameter : the fractal dimension, related to the roughness of fBm's samples. According to the structural properties of fBm, different fractal dimension estimators can be considered. We have chosen five of them, which operate either in the frequency domain (identification of a spectral exponent via spectrum analysis), in the time domain (maximum likelihood on one hand, methods based on length measurements of fBm's samples at different observation scales on the other hand) or in a mixed time-scale domain (identification of a self-similarity parameter via the variance of wavelets coefficients). The relevance of these differents estimators is discussed and their performance is compared on simulated and real data.I -FRACTIONAL BROWNIAN MOTIONS I n a number of physical phenomena, strong long-term dependencies are involved and l/fp spectral behaviors are observed over a wide range of frequencies [ 11. A convenient modeling for processes of this kind has been proposed by Mandelbrot and Van Ness [2] and is referred to as fractional Brownian motion (fBm). By definition, fBm is a nonstationary process whose expression reads (S) .P + j ( t -s)H-l/*dB(s)); 0 < H < 1, where B ( t ) = B,/2(t) is ordinary Brownian motion. The increments of fBm are stationary and, moreover, they are statistically self-similar, which means that for any a > 0 (and with the convention BH(0) = 0), [ Btl(at)) and [ aHB&)) undergo the same probabilistic behavior.In the frequency domain, an average power spectrum density (PSD) of fBm can be defined [3] and it tums out that it is proportional to I/fp with p = 2H + 1. In the time domain, the selfsimilarity structure which underlies the definition of fBm allows to associate to it a fractal dimension D = 2 -H , which varies therefore between 1 and 2 [2]. This fractal dimension is related to the roughness of fBm's samples and it appears as a relevant parameter for identification and classification [4], motivating hence the development of efficient procedures aimed at its estimation. 0 * fomcrly with LTS-ICPI Lyon # formerly with EDFIDER'RISDM t also with G D R 134 CNRS "Trarlemenl du Signal el Images" I1 -FRACTAL DIMENSION ESTIMATORS According to the above properties, five different estimators of D have been considered, each based on some specific feature of fBm. A. Spectrum AnalysisThis exploits the fact that, because fBm has a llfp spectral behavior, its PSD is supposed to be a straight line in a log-log plot. The slope -b of this line can therefore be estimated by a least-squares fit on some log-log periodogram estimate ( Fig. 1) and D can be deduced from pas D = (5 -m/2.It is clear that such an estimator necessitates a large amount of data points for achieving a significant variance reduction. Another fundame...