Nonparametric estimation of a shot-noise process

We propose an efficient method to estimate in a nonparametric fashion the marks' density of a shot-noise process in presence of pileup from a sample of low-frequency observations. Based on a functional equation linking the marks' density to the characteristic function of the observations and its derivative, we propose a new time-efficient method using B-splines to estimate the density of the underlying γ-ray spectrum which is able to handle large datasets used in nuclear physics. A discussion on the numerical computation of the algorithm and its performances on simulated data are provided to support our findings.

[1]  Alain Destexhe,et al.  Inhibition Determines Membrane Potential Dynamics and Controls Action Potential Generation in Awake and Sleeping Cat Cortex , 2007, The Journal of Neuroscience.

[2]  A. Burkitt,et al.  Shot noise in the leaky integrate-and-fire neuron. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Odile Macchi,et al.  Estimation and detection of weak optical signals , 1972, IEEE Trans. Inf. Theory.

[4]  Richard A. Davis,et al.  Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes , 2007, Journal of Applied Probability.

[5]  Michael H. Neumann,et al.  Nonparametric estimation for L\'evy processes from low-frequency observations , 2007, 0709.2007.

[6]  M. Steiner,et al.  Random drug excess , 1990, European Journal of Drug Metabolism and Pharmacokinetics.

[7]  J. Hadamard.  Sur les problemes aux derive espartielles et leur signification physique , 1902 .

[8]  Valerie Connaughton,et al.  Analytical modeling of pulse-pileup distortion using the true pulse shape; applications to Fermi-GBM , 2012, 1211.6592.

[9]  G. Wahba,et al.  Generalized Inverses in Reproducing Kernel Spaces: An Approach to Regularization of Linear Operator Equations , 1974 .

[10]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[11]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[12]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[13]  Katsuyuki Taguchi,et al.  An analytical model of the effects of pulse pileup on the energy spectrum recorded by energy resolved photon counting x-ray detectors , 2010, Medical Imaging.

[14]  J. Florens,et al.  Série Scientifique Scientific Series 2003 s-02 Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions , 2002 .

[15]  Philippe Owezarski,et al.  Modeling Internet backbone traffic at the flow level , 2003, IEEE Trans. Signal Process..

[16]  Esko Valkeila,et al.  An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, 2nd Edition by Daryl J. Daley, David Vere‐Jones , 2008 .

[17]  Francesco Laio,et al.  Advances in shot noise modeling of daily streamflows , 2005 .

[18]  Robert Lund,et al.  Inference for Shot Noise , 2006 .

[19]  H. Künsch.  The Jackknife and the Bootstrap for General Stationary Observations , 1989 .

[20]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[21]  Peter J. Brockwell,et al.  Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations , 2012, J. Multivar. Anal..

[22]  Bernard A. Mair,et al.  Statistical Inverse Estimation in Hilbert Scales , 1996, SIAM J. Appl. Math..

[23]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[24]  J. Rice.  On generalized shot noise , 1977, Advances in Applied Probability.

[25]  G. Wahba.  Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy , 1977 .

[26]  K. Perez.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment , 2014 .

[27]  S. Wood.  Modelling and smoothing parameter estimation with multiple quadratic penalties , 2000 .

[28]  Gennady Samorodnitsky,et al.  A Class of Shot Noise Models for Financial Applications , 1996 .

[29]  Eric Moulines,et al.  Nonparametric estimation of mark's distribution of an exponential Shot-noise process , 2015, 1506.08047.

[30]  S. Lahiri.  Theoretical comparisons of block bootstrap methods , 1999 .

[31]  Hervé Cardot,et al.  Spatially Adaptive Splines for Statistical Linear Inverse Problems , 2002 .

[32]  G. Wahba.  Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind , 1973 .

[33]  Raúl E. Sequeira,et al.  Intensity estimation from shot-noise data , 1995, IEEE Trans. Signal Process..

[34]  W. Schottky.  Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern , 1918 .

[35]  H. White,et al.  Automatic Block-Length Selection for the Dependent Bootstrap , 2004 .

[36]  Raúl E. Sequeira,et al.  Blind intensity estimation from shot-noise data , 1997, IEEE Trans. Signal Process..

[37]  N. Heckman,et al.  The theory and application of penalized methods or Reproducing Kernel Hilbert Spaces made easy , 2011 .

[38]  Eric Barat,et al.  Fast Digital Filtering of Spectrometric Data for Pile-up Correction , 2015, IEEE Signal Processing Letters.

[39]  Jun Yu.  Empirical Characteristic Function Estimation and Its Applications , 2003 .

[40]  H.W. Kraner,et al.  Radiation detection and measurement , 1981, Proceedings of the IEEE.

[41]  Thomas Trigano,et al.  Pileup Correction Algorithm using an Iterated Sparse Reconstruction Method , 2015, IEEE Signal Processing Letters.

[42]  A. Holden.  Models of the stochastic activity of neurones , 1976 .