Structural Adaptive Smoothing in Diffusion Tensor Imaging: The R Package dti

Diffusion weighted imaging has become and will certainly continue to be an important tool in medical research and diagnostics. Data obtained with diffusion weighted imaging are characterized by a high noise level. Thus, estimation of quantities like anisotropy indices or the main diffusion direction may be significantly compromised by noise in clinical or neuroscience applications. Here, we present a new package dti for R, which provides functions for the analysis of diffusion weighted data within the diffusion tensor model. This includes smoothing by a recently proposed structural adaptive smoothing procedure based on the propagation-separation approach in the context of the widely used diffusion tensor model. We extend the procedure and show, how a correction for Rician bias can be incorporated. We use a heteroscedastic nonlinear regression model to estimate the diffusion tensor. The smoothing procedure naturally adapts to different structures of different size and thus avoids oversmoothing edges and fine structures. We illustrate the usage and capabilities of the package through some examples.

Adaptive Smoothing of Digital Images: The R Package adimpro

Digital imaging has become omnipresent in the past years with a bulk of applications ranging from medical imaging to photography. When pushing the limits of resolution and sensitivity noise has ever been a major issue. However, commonly used non-adaptive filters can do noise reduction at the cost of a reduced effective spatial resolution only. Here we present a new package adimpro for R, which implements the propagationseparation approach by (Polzehl and Spokoiny 2006) for smoothing digital images. This method naturally adapts to different structures of different size in the image and thus avoids oversmoothing edges and fine structures. We extend the method for imaging data with spatial correlation. Furthermore we show how the estimation of the dependence between variance and mean value can be included. We illustrate the use of the package through some examples.

Varying coefficient GARCH versus local constant volatility modeling. Comparison of the predictive power

GARCH models are widely used in financial econometrics. However, we show by mean of a simple simulation example that the GARCH approach may lead to a serious model misspecification if the assumption of stationarity is violated. In particular, the well known integrated GARCH effect can be explained by nonstationarity of the time series. We then introduce a more general class of GARCH models with time varying coefficients and present an adaptive procedure which can estimate the GARCH coefficients as a function of time. We also discuss a simpler semiparametric model in which the beta-parameter is fixed. Finally we compare the performance of the parametric, time varying nonparametric and semiparametric GARCH(1,1) models and the locally constant model from Polzehl and Spokoiny (2002) by means of simulated and real data sets using different forecasting criteria. Our results indicate that the simple locally constant model outperforms the other models in almost all cases. The GARCH(1,1) model also demonstrates a relatively good forecasting performance as far as the short term forecasting horizon is considered. However, its application to long term forecasting seems questionable because of possible misspecification of the model parameters.varying coefficient GARCH, adaptive weights

When did the 2001 recession really start?

The paper develops a non-parametric, non-stationary framework for business-cycle dating based on an innovative statistical methodology known as Adaptive Weights Smoothing (AWS). The methodology is used both for the study of the individual macroeconomic time series relevant to the dating of the business cycle as well as for the estimation of their joint dynamic. Since the business cycle is defined as the common dynamic of some set of macroeconomic indicators, its estimation depends fundamentally on the group of series monitored. We apply our dating approach to two sets of US economic indicators including the monthly series of industrial production, nonfarm payroll employment, real income, wholesale-retail trade and gross domestic product (GDP). We find evidence of a change in the methodology of the NBER’s Business-Cycle Dating Committee an extended set of five monthly macroeconomic indicators replaced in the dating of the last recession the set of indicators emphasized by the NBER’s Business- Cycle Dating Committee in recent decades. This change seems to seriously affect the continuity in the outcome of the dating of business cycle. Had the dating been done on the traditional set of indicators, the last recession would have lasted one year and a half longer. We find that, independent of the set of coincident indicators monitored, the last economic contraction began in November 2000, four months before the date of the NBER’s Business-Cycle Dating Committee.business cycle, non-parametric smoothing, non-stationarity

Varying coefficient GARCH versus local constant volatility modeling: comparison of the predictive power

GARCH models are widely used in financial econometrics. However, we show by mean of a simple simulation example that the GARCH approach may lead to a serious model misspecification if the assumption of stationarity is violated. In particular, the well known integrated GARCH effect can be explained by nonstationarity of the time series. We then introduce a more general class of GARCH models with time varying coefficients and present an adaptive procedure which can estimate the GARCH coefficients as a function of time. We also discuss a simpler semiparametric model in which the ¯ - parameter is fixed. Finally we compare the performance of the parametric, time varying nonparametric and semiparametric GARCH(1,1) models and the locally constant model from Polzehl and Spokoiny (2002) by means of simulated and real data sets using different forecasting criteria. Our results indicate that the simple locally constant model outperforms the other models in almost all cases. The GARCH(1,1) model also demonstrates a relatively good forecasting performance as far as the short term forecasting horizon is considered. However, its application to long term forecasting seems questionable because of possible misspecification of the model parameters

Structural adaptive smoothing by propagation-separation methods

Propagation-Separation stands for the main properties of a new class of adaptive smoothing methods. An assumption that a prespecified type of models allows for a good local approximation within homogeneous regions in the design (structural assumption), is utilized to both recover homogeneous regions and to efficiently estimate the regression function. Locality is defined by pairwise weights. Propagation stands for the unrestricted expansion of weights within homogeneous regions. Separations characterizes the restriction of positive weights to homogeneous regions with respect to the specified model. The procedures have remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. They are fully adaptive and dimension free. We here provide a short introduction into Propagation-Separation procedures in the context of image processing. Properties are illustrated by a series of examples

Vector adaptive weights smoothing with applications to MRI

We consider the problem of adaptive spatial smoothing for a time series of images. This type of data typically occurs in functional and dynamic Magnet Resonance Imaging (MRI). We propose a new method based on spatial smoothing with adaptively chosen weights. We show how this procedure can be used for efficient image estimation and classification in functional and dynamic MRI experiments. The performance of the procedure is illustrated by applications to simulated and real data

Adaptive weights smoothing with applications to image segmentation

We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data-points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to Magnetic Resonance Imaging

Spatially adaptive regression estimation: Propagation-separation approach

Polzehl and Spokoiny (2000) introduced the adaptive weights smoothing (AWS) procedure in the context of image denoising. The procedure has some remarkable properties like preservation of edges and contrast, and (in some sense) optimal reduction of noise. The procedure is fully adaptive and dimension free. Simulations with artificial images show that AWS is superior to classical smoothing techniques especially when the underlying image function is discontinuous and can be well approximated by a piecewise constant function. However, the latter assumption can be rather restrictive for a number of potential applications. Here we present a new method based on the ideas of propagation and separation which extends the AWS procedure to the case of an arbitrary local linear parametric structure. We also establish some important results about properties of the new `propagation-separation' procedure including rate optimality in the pointwise and global sense. The performance of the procedure is illustrated by examples for local polynomial regression and by applications to artificial and real images

Beyond the Gaussian Model in Diffusion-Weighted Imaging: The Package dti

Diffusion weighted imaging (DWI) is a magnetic resonance (MR) based method to investigate water diffusion in tissue like the human brain. Inference focuses on integral properties of the tissue microstructure. The acquired data are usually modeled using the diffusion tensor model, a three-dimensional Gaussian model for the diffusion process. Since the homogeneity assumption behind this model is not valid in large portion of the brain voxel more sophisticated approaches have been developed. This paper describes the R package dti. The package offers capabilities for the analysis of diffusion weighted MR experiments. Here, we focus on recent extensions of the package, for example models for high angular resolution diffusion weighted imaging (HARDI) data, including Q-ball imaging and tensor mixture models, and fiber tracking. We provide a detailed description of the package structure and functionality. Examples are used to guide the reader through a typical analysis using the package. Data sets and R scripts used are available as electronic supplements