Pascal Bondon

Second-order statistics of complex signals

The second-order statistical properties of complex signals are usually characterized by the covariance function. However, this is not sufficient for a complete second-order description, and it is necessary to introduce another moment called the relation function. Its properties, and especially the conditions that it must satisfy, are analyzed both for stationary and nonstationary signals. This leads to a new perspective concerning the concept of complex white noise as well as the modeling of any signal as the output of a linear system driven by a white noise. Finally, this is applied to complex autoregressive signals, and it is shown that the classical prediction problem must be reformulated when the relation function is taken into consideration.

A Class of Antipersistent Processes

We introduce a class of stationary processes characterized by the behaviour of their infinite moving average parameters. We establish the asymptotic behaviour of the covariance function and the behaviour around zero of the spectral density of these processes, showing their antipersistent character. Then, we discuss the existence of an infinite autoregressive representation for this family of processes, and we present some consequences for fractional autoregressive moving average models.

Polynomial estimation of the amplitude of a signal

The problem of estimating the amplitude of a signal is addressed using higher-order statistics. The probability distribution of the noise is assumed to be unknown so that the maximum likelihood estimator cannot be calculated. The estimator is taken as a polynomial of the observation, the coefficients of which are determined so that the estimate is unbiased with minimum variance. This method generalizes the linear approach, and the estimate variance is reduced. The ease of linear-quadratic estimation is detailed, and numerical examples are presented. >

Polyspectrum modeling using linear or quadratic filters

The polyspectrum modeling problem using linear or quadratic filters is investigated. In the linear case, it is shown that, if the output pth-order cumulant function of a filter, driven by a white noise, is of finite extent, then the filter necessarily has a finite-extent impulse response. It is proved that every factorable polyspectrum with a non-Gaussian white noise can also be modeled with a quadratic filter driven by a Gaussian white noise. It is shown that, if the quadratic filter has a finite-extent impulse response, then the output pth-order cumulant function is of finite extent; and if the output pth-order cumulant function of a quadratic filter is of finite extent, then the impulse response may or may not be of finite extent. It is shown that there exist finite and infinite extent pth-order cumulant functions that are not factorable but can be modeled with quadratic filters. >

A new unscented particle filter

We present a new unscented particle filter for dynamic systems that outperforms the general particle filter and the unscented particle filter when the variance of the observation noise is small. Our algorithm uses a bank of unscented Kalman filters to refine the prediction in particle filter. The key difference with the traditional unscented particle filter is the introduction of an auxiliary model and a bank of unscented Kalman filter with this auxiliary model to generate the importance distribution in the particle filter. This structure makes efficient use of the latest observation information. Our new algorithm use fewer particles than the general particle filters and its performance outperforms them.

Estimation of autoregressive models with epsilon-skew-normal innovations

A non-Gaussian autoregressive model with epsilon-skew-normal innovations is introduced. Moments and maximum likelihood estimators of the parameters are proposed and their limit distributions are derived. Monte Carlo simulation results are analysed and the model is fitted to a real time series.

Efficiency of high-order moment estimates

Some results on the estimation of high-order moments of a real random variable are presented. For a given order, the estimator considered is the corresponding moment of the samples, and we study the relative variance of this estimator. General results on the sequence of relative variances indexed by the order are established. Finally, some examples and counterexamples are presented.

Robust estimation of SARIMA models: Application to short-term load forecasting

This paper presents a new robust method to estimate the parameters of a SARIMA model. This method uses robust autocorrelations estimates based on sample medians coupled with a robust filter cleaner which rejects deviant observations. Our procedure is compared with other robust methods via evaluation of the different robustness measures such as maximum bias, breakdown point and influence function. The asymptotic properties of our method (strong consistency and central limit theorem) are established for a gaussian AR process. We show that the method improves the French load forecasting for “normal days” and offers good robustness, easiness and fast execution.

Prediction with incomplete past of a stationary process

An explicit formula is obtained for the prediction error of a future value of a stationary process when the infinite past is altered by some missing observations with an arbitrary pattern. Then the autoregressive representation of the predictor is derived and the processes for which the missing observations in the past do not affect the prediction of a future value are characterized. Some properties for autoregressive processes and for moving average processes with finite orders are established.

Robust short-term load forecasting using projection statistics

It has been observed that the French electric load series possesses outliers and breaks. Outliers are deviant data points while breaks are lasting abrupt changes in the stochastic pattern of the series. It turns out that outliers and breaks significantly degrade the reliability and accuracy of conventional day-ahead estimation and forecasting methods. Robust methods are needed for this application. In this paper, we propose to use a robust diagnostic approach for which the identification of outliers and breaks is carried out via a robust multivariate estimation of location and covariance based on projection statistics (PS). The developed procedure consists of the following steps: (i) estimate the parameters and the order of a high order autoregressive AR(p*) by means of the PS, (ii) execute a robust filter cleaner to identify and reject the outliers, and (iii) apply a maximum-likelihood estimator defined at the Gaussian distribution that handles missing values. The performance of this method has been evaluated on the French electric demand in terms of execution time and forecasting accuracy. This approach improves the load forecasting quality for “normal days” and presents several interesting properties such as fast execution, good robustness, simplicity and easy on-line implementation. A novel multivariate approach is also proposed in order to deal with heteroscedasticity.