In this work we deal with the mathematical analysis and application of piecewise (or segmented) polynomial regression. Motivated by an application in neurobiology we allow the residual processes of our model to exhibit long memory, short memory or antipersistence. As a solid biological background is essential for understanding the application in this work, we start with an introduction to neuro...

Both equation discovery and regression methods aim at inducing models of numerical data. While the equation discovery methods are usually evaluated in terms of comprehensibility of the induced model, the emphasis of the regression methods evaluation is on their predictive accuracy. In this paper, we present Ciper, an efficient method for discovery of polynomial equations and empirically evaluat...

Longitudinal shape analysis often relies on the estimation of a realistic continuous growth scenario from data sparsely distributed in time. In this paper, we propose a new type of growth model parameterized by acceleration, whereas standard methods typically control the velocity. This mimics the behavior of biological tissue as a mechanical system driven by external forces. The growth trajecto...

—In this paper, we present a novel algorithm for piecewise linear regression which can learn continuous as well as discontinuous piecewise linear functions. The main idea is to repeatedly partition the data and learn a liner model in each partition. While a simple algorithm incorporating this idea does not work well, an interesting modification results in a good algorithm. The proposed algorith...

We deal with the problem of choosing a piecewise constant estimator of a regression function s mapping X into R. We consider a non Gaussian regression framework with deterministic design points, and we adopt the non asymptotic approach of model selection via penalization developed by Birgé and Massart. Given a collection of partitions of X , with possibly exponential complexity, and the corresp...

For least squares regression, Efron et al. (2004) proposed an efficient solution path algorithm, the least angle regression (LAR). They showed that a slight modification of the LAR leads to the whole LASSO solution path. Both the LAR and LASSO solution paths are piecewise linear. Recently Wu (2011) extended the LAR to generalized linear models and the quasi-likelihood method. In this work we ex...

The open-loop Threshold Model, proposed by Tong [23], is a piecewise-linear stochastic regression model useful for modeling conditionally normal response time-series data. However, in many applications, the response variable is conditionally non-normal, e.g. Poisson or binomially distributed. We generalize the open-loop Threshold Model by introducing the Generalized Threshold Model (GTM). Speci...

The open-loop Threshold Model, proposed by Tong [30], is a piecewise-linear stochastic regression model useful for modeling conditionally normal response time-series data. However, in many applications, the response variable is conditionally non-normal, e.g. Poisson or binomially distributed. We generalize the open-loop Threshold Model by introducing the Generalized Threshold Model (GTM). Speci...

Asymptotically exact and conservative confidence bands are obtained for a nonparametric regression function, using piecewise constant and piecewise linear spline estimation, respectively. Compared to the pointwise confidence interval of Huang (2003), the confidence bands are inflated by a factor proportional to {log (n)}, with the same width order as the Nadaraya-Watson bands of Härdle (1989), ...

We compute upper and lower bounds on the VC dimension and pseudo-dimension of feedforward neural networks composed of piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension and pseudo-dimension grow as WlogW, where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbo...