Differentiated logdensity approximants

Provost, Serge B.; Ha, Hyung-Tae

doi:10.1016/j.stamet.2015.02.005

Cited by 8 publications

(13 citation statements)

References 25 publications

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“…Finally, we mention that there exists an alternative expression to the above development, in terms of Laguerre polynomials (see Provost, 2005). It has the advantage of not requiring the matrix inversion in Eq.…”

Section: The D-moment Approximationmentioning

confidence: 99%

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

Lenoir

Crucifix

2018

Nonlin. Processes Geophys.

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Abstract. We develop a general framework for the frequency analysis of irregularly sampled time series. It is based on the Lomb–Scargle periodogram, but extended to algebraic operators accounting for the presence of a polynomial trend in the model for the data, in addition to a periodic component and a background noise. Special care is devoted to the correlation between the trend and the periodic component. This new periodogram is then cast into the Welch overlapping segment averaging (WOSA) method in order to reduce its variance. We also design a test of significance for the WOSA periodogram, against the background noise. The model for the background noise is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, more general than the classical Gaussian white or red noise processes. CARMA parameters are estimated following a Bayesian framework. We provide algorithms that compute the confidence levels for the WOSA periodogram and fully take into account the uncertainty in the CARMA noise parameters. Alternatively, a theory using point estimates of CARMA parameters provides analytical confidence levels for the WOSA periodogram, which are more accurate than Markov chain Monte Carlo (MCMC) confidence levels and, below some threshold for the number of data points, less costly in computing time. We then estimate the amplitude of the periodic component with least-squares methods, and derive an approximate proportionality between the squared amplitude and the periodogram. This proportionality leads to a new extension for the periodogram: the weighted WOSA periodogram, which we recommend for most frequency analyses with irregularly sampled data. The estimated signal amplitude also permits filtering in a frequency band. Our results generalise and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. They also constitute the starting point for an extension to the continuous wavelet transform developed in a companion article (Lenoir and Crucifix, 2018). All the methods presented in this paper are available to the reader in the Python package WAVEPAL.

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Section: The D-moment Approximationmentioning

confidence: 99%

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

Lenoir

Crucifix

2018

Nonlin. Processes Geophys.

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“…On making use of a recursive relationship given in [11], which can also be deduced for instance from Theorem 3.2b.2 in [6], one can determine the m th moment of Q(X) as follows: 10) where κ * U (m − i) is as specified by (3.9). In this example, we approximate the distribution of Q 3 (X) whose support is non-negative by making use of a generalized gamma distribution.…”

Section: Quadratic Forms In Order Statistics From An Exponential Popumentioning

confidence: 99%

“…As explained in [10], an initial density approximation can be adjusted by means of a polynomial whose coefficients are such that the first d moments of the approximation coincide with those of a target distribution. For instance, on the basis of the first d moments of a positive definite quadratic form Q(X), a density approximation of the following form can be assumed for Q(X):…”

Section: Appendix a Polynomially Adjusted Density Functionsmentioning

confidence: 99%

On Approximating the Distribution of Quadratic Forms in Gamma Random Variables and Exponential Order Statistics

Mohsenipour¹,

Provost²

2013

JSTA

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This paper proposes a moment-based approximation to the distribution of quadratic forms in gamma random variables. Quadratic forms in order statistics from an exponential population are considered as well. Actually, several test statistics can be expressed in terms of the latter. The density approximants are expressible as the product of a gamma type distributed base density function and a polynomial adjustment. Several illustrative examples are provided.

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“…Silverman (1986) provides a survey of the various available methodologies while (Reid, 1988) focuses on the saddlepoint approximation. Moment-based techniques are described for instance in (Elderton & Johnson, 1969), (Solomon & Stephens, 1978) and (Provost, 2005). Efromovich (1999) presents a unified account of nonparametric approaches to density estimation.…”

Section: Introductionmentioning

confidence: 99%

A Log-Density Estimation Methodology Applicable to Massive Bivariate Data

Zareamoghaddam¹,

Provost²,

Ahmed³

2017

IJSP

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First, it is shown that a univariate bona fide density approximation can be obtained by assuming that the derivative of the logarithm of the density function under consideration is expressible as a rational function or a polynomial. Then, the density function of a bivariate continuous random vector is approximated by standardizing it and applying a polynomial adjustment to the product of the density approximants of the marginal distributions. As well, it is explained that this approach can easily be extended to the estimation of density functions. For illustrative purposes, the proposed methodology is applied to several datasets. Since this technique is solely based on sample moments, it readily lends itself to the modeling of large datasets.

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Differentiated logdensity approximants

Cited by 8 publications

References 25 publications

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

On Approximating the Distribution of Quadratic Forms in Gamma Random Variables and Exponential Order Statistics

A Log-Density Estimation Methodology Applicable to Massive Bivariate Data

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