12 research outputs found
Linear system identification using stable spline kernels and PLQ penalties
The classical approach to linear system identification is given by parametric
Prediction Error Methods (PEM). In this context, model complexity is often
unknown so that a model order selection step is needed to suitably trade-off
bias and variance. Recently, a different approach to linear system
identification has been introduced, where model order determination is avoided
by using a regularized least squares framework. In particular, the penalty term
on the impulse response is defined by so called stable spline kernels. They
embed information on regularity and BIBO stability, and depend on a small
number of parameters which can be estimated from data. In this paper, we
provide new nonsmooth formulations of the stable spline estimator. In
particular, we consider linear system identification problems in a very broad
context, where regularization functionals and data misfits can come from a rich
set of piecewise linear quadratic functions. Moreover, our anal- ysis includes
polyhedral inequality constraints on the unknown impulse response. For any
formulation in this class, we show that interior point methods can be used to
solve the system identification problem, with complexity O(n3)+O(mn2) in each
iteration, where n and m are the number of impulse response coefficients and
measurements, respectively. The usefulness of the framework is illustrated via
a numerical experiment where output measurements are contaminated by outliers.Comment: 8 pages, 2 figure
Robust EM kernel-based methods for linear system identification
Recent developments in system identification have brought attention to
regularized kernel-based methods. This type of approach has been proven to
compare favorably with classic parametric methods. However, current
formulations are not robust with respect to outliers. In this paper, we
introduce a novel method to robustify kernel-based system identification
methods. To this end, we model the output measurement noise using random
variables with heavy-tailed probability density functions (pdfs), focusing on
the Laplacian and the Student's t distributions. Exploiting the representation
of these pdfs as scale mixtures of Gaussians, we cast our system identification
problem into a Gaussian process regression framework, which requires estimating
a number of hyperparameters of the data size order. To overcome this
difficulty, we design a new maximum a posteriori (MAP) estimator of the
hyperparameters, and solve the related optimization problem with a novel
iterative scheme based on the Expectation-Maximization (EM) method. In presence
of outliers, tests on simulated data and on a real system show a substantial
performance improvement compared to currently used kernel-based methods for
linear system identification.Comment: Accepted for publication in Automatic
Stage-discharge relationship in tidal channels
Author Posting. © The Author(s), 2016. This is the author's version of the work. It is posted here by permission of Association for the Sciences of Limnology and Oceanography for personal use, not for redistribution. The definitive version was published in Limnology and Oceanography: Methods 15 (2017): 394–407, doi:10.1002/lom3.10168.Long-term records of the flow of water through tidal channels are essential to constrain
the budgets of sediments and biogeochemical compounds in salt marshes. Statistical
models which relate discharge to water level allow the estimation of such records from
more easily obtained records of water stage in the channel. Here we compare four
different types of stage-discharge models, each of which captures different characteristics
of the stage-discharge relationship. We estimate and validate each of these models on a
two-month long time series of stage and discharge obtained with an Acoustic Doppler
Current Profiler in a salt marsh channel. We find that the best performance is obtained by
models that account for the nonlinear and time-varying nature of the stage-discharge
relationship. Good performance can also be obtained from a simplified version of these
models, which captures nonlinearity and nonstationarity without the complexity of the
fully nonlinear or time-varying models.This research was supported by the National Science Foundation (awards OCE1354251,
OCE1354494, and OCE1238212).2018-04-2
Signals of nonlinear, multiscale and stochastic processes in coastal landscapes
Salt marshes are some of the most productive and valuable landscapes on earth, but they are vulnerable to the effects of sea-level rise, erosion and eutrophication. These processes act on a wide range of temporal and spatial scales, which complicate assessments of the health and stability of marsh ecosystems. High-frequency monitoring using in situ sensors captures the complete range of these dynamics, but extracting meaningful physical and ecological information from these signals requires process-based models coupled with statistical techniques. I develop and apply such methods to study two coastal landscapes, a coastal pine forest on the Eastern Shore of Virginia and a mesotidal salt marsh complex in the Plum Island Estuary, Massachusetts.
Observations from groundwater wells in the Virginia pine forest indicate that storms are the dominant controls on the hydrology of the forest and that tidal influence is nonexistent. This forest exhibits a distinct spatial pattern in age structure in which young trees do not grow at low elevations. This pattern can be explained by a model that includes the interaction of sea-level rise, storms and the age-dependent variation in tree stress response, which predicts that the long-term evolution of the boundary is an ecological ratchet. Stresses due to sea-level rise slowly push the boundary at which young trees can survive upslope. Powerful storms then kill the mature, persistent forest at low elevations, which quickly pushes the forest boundary up to the regeneration boundary.
Salt marshes need to accumulate sediment to replenish material lost as sea-level rises and creek banks erode. Fluxes of sediment can be monitored with simultaneous high-frequency observations of flow from acoustic Doppler current profilers and turbidity from optical backscattering sensors. I first investigate the relationship between water level and flow in marsh channels and develop predictive stage-discharge models to simplify the monitoring of fluxes. I then construct sediment budgets for eleven salt marshes in the Plum Island Estuary. The observed budgets depend strongly on the unique hydrodynamic conditions of each marsh channel. Variability in these conditions leads to the observed spatial and temporal variability in sediment fluxes from these marshes
SEMIPARAMETRIC SINGLE-INDEX MODELS FOR OPTIMAL TREATMENT REGIMENS WITH CENSORED OUTCOMES
There is a growing interest in precision medicine, where a potentially censored survival time is often the most important outcome of interest. To discover optimal treatment regimens for such an outcome, we propose a semiparametric proportional hazards model by incorporatingthe interaction between treatment and a single index of covariates through an unknown monotone link function. This model is flexible enough to allow non-linear treatment-covariate interactions and yet provides a clinically interpretable linear rule for treatment decision. We propose a sieve maximum likelihood estimation approach, under which the baseline hazard function is estimated nonparametrically and the unknown link function is estimated via monotone quadratic B-splines. We show that the resulting estimators are consistent and asymptotically normal with a covariance matrix that attains the semiparametric efficiency bound. The optimal treatment rule follows naturally as a linear combination of the maximum likelihood estimators of the model parameters. Through extensive simulation studies and anapplication to an AIDS clinical trial, we demonstrate that the treatment rule derived from the single-index model outperforms the treatment rule under the standard Cox proportionalhazards model. We extend the proposed method to transformation models so that optimal treatment rules can be applied to flexible hazards relationships. The transformation model introduces new challenges to both the estimation procedure and the asymptotic properties of the estimators. We design an estimation procedure with the EM algorithm by recognizing the transformation function as the distribution function of a corresponding missing random variable. We prove that the resulting estimators are consistent and asymptotically normal, with the covariancematrix estimated using the profile likelihood theory. We demonstrate the performance of the transformation single-index model in simulation studies. We show that the proposed treatment rule under the single-index transformation model is more effective than that under the single-index proportional hazards model in delaying the disease relapse of large-bowel carcinoma in a real data analysis. With improvements in technology, researchers are able to collect many clinical and genetic variables; not all the covariates may contribute to the prediction of the optimal treatmentrules. We apply the adaptive Lasso penalty to the log-likelihood of the proposed model and let the data automatically determine the important predictors in the optimal treatment regime.We propose a simple computational approach by quadratic approximation of the original objective function and utilization of the variable selection software package available for theproportional hazards model. We show that the proposed variable selection approach displaysthe oracle property. The performance of the variable selection procedure is demonstrated inextensive simulations and the analysis of a multi-cancer clinical trial.Doctor of Philosoph
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Novel regularization models for dynamic and discrete response data
This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University LondonRegularized regression models have gained popularity in recent years. The addition of a penalty term to the likelihood function allows parameter estimation where traditional methods fail, such as in the p » n case. The use of an l1 penalty in particular leads to simultaneous parameter estimation and variable selection, which is rather convenient in practice. Moreover, computationally efficient algorithms make these methods really attractive in many applications. This thesis is inspired by this literature and investigates the development of novel penalty functions and regression methods within this context. In particular, Chapter 2 deals with linear models for time-dependent response and explanatory variables. This is beyond the independent framework which is common to many of the developed regularized regression models. We propose to account for the time dependency in the data by explicitly adding autoregressive terms to the response variable together with an autoregressive process for the residuals. In addition, the use of a l1 penalized likelihood approach for parameter estimation leads to automatic order and variable selection and makes this method feasible for high-dimensional data. Theoretical properties of the estimators are provided and an extensive simulation study is performed. Finally, we show the application of the model on air pollution and stock market data and discuss its implementation in the R package DREGAR, which is freely available in CRAN. In Chapter 3, we develop a new penalty function. Despite all the advantages of the l1 penalty, this penalty is not differentiable at zero, and neither are the alternatives that are proposed in the literature. The only exception is the ridge penalty, which does not lead to variable selection. Motivated by this gap, and noting the advantages that a differentiable penalty can give, such as increased computational efficiency in some cases and the derivation of more accurate model selection criteria, we develop a new penalty function based on the error function. We study the theoretical properties of this function and of the estimators obtained in a regularized regression context. Finally, we perform a simulation study and we use the new penalty to analyse a diabetes and prostate cancer dataset. The new method is implemented in the R package DLASSO, that is freely available in CRAN. Finally, Chapter 4 deals with regression models for discrete response data, which is frequently collected in many application areas. In particular, we consider a discrete Weibull regression model that has recently been introduced in the literature. In this chapter, we propose the first Bayesian implementation of this model. We consider a general parametrization, where both parameters of the discrete Weibull distribution can be conditioned on the predictors, and show theoretically how, under a uniform noninformative
prior, the posterior distribution is proper with finite moments. In addition, we consider closely the case of Laplace priors for parameter shrinkage and variable selection. A simulation study and the analysis of four real datasets of medical records show the applicability of this approach to the analysis of count data. The method is implemented in the R package BDWreg, which is freely available in CRAN