Hans Arnfinn Karlsen

Nonparametric estimation in a nonlinear cointegration type model

We derive an asymptotic theory of nonparametric estimation for an nonlinear transfer function model Z(t) = f (Xt) + Wt where {Xt} and {Zt} are observed nonstationary processes and {Wt} is a stationary process. IN econometrics this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results have wider interest. The class of nonstationary processes allowed for {Xt} is a subclass of the class of null recurrent.. Markov chains. This subclass contains the random walk model and the unit root processes. WE derive the asymptotics of an nonparametric estimate of f(z) under two alternative sets of assumptions on {Wt}: i) {Wt} is a linear process ii) {Wt} is a Markov chain satisfying some mixing conditions. The latter requires considerably more work but also holds larger promise for further developments. The finite sample properties f(x) are studied via a set of simulation experiments.

NULL RECURRENT UNIT ROOT PROCESSES

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and I²-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is I²-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made I²-null recurrent for any I² ∈ (0, 1) and can also be made null recurrent without being I²-null recurrent.

Comparing the adaptive Gaussian mixture filter with the ensemble Kalman filter on synthetic reservoir models

Over the last years, the ensemble Kalman filter (EnKF) has become a very popular tool for history matching petroleum reservoirs. EnKF is an alternative to more traditional history matching techniques as it is computationally fast and easy to implement. Instead of seeking one best model estimate, EnKF is a Monte Carlo method that represents the solution with an ensemble of state vectors. Lately, several ensemble-based methods have been proposed to improve upon the solution produced by EnKF. In this paper, we compare EnKF with one of the most recently proposed methods, the adaptive Gaussian mixture filter (AGM), on a 2D synthetic reservoir and the Punq-S3 test case. AGM was introduced to loosen up the requirement of a Gaussian prior distribution as implicitly formulated in EnKF. By combining ideas from particle filters with EnKF, AGM extends the low-rank kernel particle Kalman filter. The simulation study shows that while both methods match the historical data well, AGM is better at preserving the geostatistics of the prior distribution. Further, AGM also produces estimated fields that have a higher empirical correlation with the reference field than the corresponding fields obtained with EnKF.

Autoregressive segmentation of signal traces with applications to geological dipmeter measurements

A general method and algorithm for segmenting data traces according to changes in the shape and scale of their autoregressive frequency structure are discussed. The algorithm is based on the least-squares principle, which is utilized in several iterations and on several levels. In the underlying model, the transitions from one segment to another take place according to a Markov mechanism, but the algorithm works for any signal trace where the segmentwise characteristics occur in a repetitive manner. The method is illustrated on four-track geological dipmeter measurements from an unidentified oil-well drilling hole in the North Sea. How more precise boundaries between geological layers can be determined by using the algorithm is indicated. Extensive testing of the algorithm on simulated data is discussed. >

Large Sample Properties of the Adaptive Gaussian Mixture Filter

AbstractIn high-dimensional dynamic systems, standard Monte Carlo techniques that asymptotically reproduce the posterior distribution are computationally too expensive. Alternative sampling strategies are usually applied and among these the ensemble Kalman filter (EnKF) is perhaps the most popular. However, the EnKF suffers from severe bias if the model under consideration is far from linear. Another class of sequential Monte Carlo methods is kernel-based Gaussian mixture filters, which reduce the bias but maintain the robustness of the EnKF. Although many hybrid methods have been introduced in recent years, not many have been analyzed theoretically. Here it is shown that the recently proposed adaptive Gaussian mixture filter can be formulated in a rigorous Bayesian framework and that the algorithm can be generalized to a broader class of interpolated kernel filters. Two parameters—the bandwidth of the kernel and a weight interpolation factor—determine the filter performance. The new formulation of the fi...

Modeling and Analysis of Daily Rainfall Data

In this work we have tried to develop a statistical model for the daily rainfall measurements. Data series from Bergen the last 107 years and from Sviland, Rogaland, the last 115 years, obtained by The Norwegian Meteorological Institute, has specifically been studied. This statistical approach is based only on measured precipitation data. The high frequency of daily data may contain important information of the underlying meteorological process and we investigate possibilities to extract this information. Simple parametric modeling with components for seasonal variation is used to represent the data. A parsimonious parametric model may contribute to increased statistical power of analyses and hypothesis testing of possible changes in the meteorological process. Our parametric model with quite few parameters can used for a detailed study of the properties of trends and changes over time. Properties such as occurrence of wet days, expected amount of rain, spell lengths and extreme events are of special interest. Similar modeling of daily rainfall data has been used by others, but not for the same purposes as we suggest. We have used generalized linear models as statistical tool for fitting the data to the model. Based on simulation studies, comparisons of model estimated quantities and corresponding data quantities, the model seems to fit series of daily rainfall data very well. We also combine this modeling with techniques for statistical process control to detect changes in the rainfall process. For two particular series considered, we see indications of development in jumps between levels rather than slowly evolving trends.

Comparing the Adaptive Gaussian Mixture Filter with the Ensemble Kalman Filter

Over the last years the ensemble Kalman filter (EnKF) and related versions have become a very popular tool for reservoir characterization. The EnKF presents the history matching result and uncertainty in terms of an ensemble of models generated from a prior model and updated sequentially in time to account for the measurements. From a statistical point of view, the optimal solution to the history matching problem is the posterior distribution of the parameters in the reservoir given all the measurements. However, since the EnKF update is linear it has severe limitations when the posterior distribution to be estimated is multimodal and/or strongly skewed due to nonlinearity of the system. As standard sequential Monte Carlo (SMC) techniques are too expensive for large models. Several methods have been proposed to combine the EnKF with SMC methods. In this paper we apply, for the first time, the recently proposed Adaptive Gaussian Mixture filter (AGM), introduced by Stordal et al. 2009, on a reservoir model and compare the results with the traditional EnKF. The AGM tries to loosen up the requirement of a nearly linear/Gaussian model by combining a relaxed EnKF update with an importance weights resampling approach, thereby taking advantage of some of the higher order moments information as in standard SMC methods such as particle filter whilst keeping the robustness of EnKF. The reservoir is a 2D synthetic reservoir model with 4 producers and 1 injector. The permeability and porosity fields are estimated. Although both methods produce good history matching, the results show that the AGM better preserves the geology of the prior model. Moreover, the last updated fields with AGM are closer to the truth than the corresponding EnKF results.