On Convergence of Epanechnikov Mean Shift

Epanechnikov Mean Shift is a simple yet empirically very effective algorithm for clustering. It localizes the centroids of data clusters via estimating modes of the probability distribution that generates the data points, using the `optimal' Epanechnikov kernel density estimator. However, since the procedure involves non-smooth kernel density functions, the convergence behavior of Epanechnikov mean shift lacks theoretical support as of this writing---most of the existing analyses are based on smooth functions and thus cannot be applied to Epanechnikov Mean Shift. In this work, we first show that the original Epanechnikov Mean Shift may indeed terminate at a non-critical point, due to the non-smoothness nature. Based on our analysis, we propose a simple remedy to fix it. The modified Epanechnikov Mean Shift is guaranteed to terminate at a local maximum of the estimated density, which corresponds to a cluster centroid, within a finite number of iterations. We also propose a way to avoid running the Mean Shift iterates from every data point, while maintaining good clustering accuracies under non-overlapping spherical Gaussian mixture models. This further pushes Epanechnikov Mean Shift to handle very large and high-dimensional data sets. Experiments show surprisingly good performance compared to the Lloyd's K-means algorithm and the EM algorithm.Comment: AAAI 201

Non-Linear Exchange Rate Dynamics in Target Zones: A Bumpy Road Towards A Honeymoon Some Evidence from the ERM, ERM2 and Selected New EU Member States

This study investigates exchange rate movements in the Exchange Rate Mechanism (ERM) of the European Monetary System (EMS) and in the Exchange Rate Mechanism II (ERM-II). On the basis of Bessec (2003), we set up a three-regime self-exciting threshold autoregressive model (SETAR) with a non-stationary central band and explicit modelling of the conditional variance. This modelling framework is employed to model daily DM-based and median currency-based bilateral exchange rates of countries participating in the original ERM and also for exchange rates of the Czech Republic, Hungary, Poland and Slovakia from 1999 to 2004. Our results confirm the presence of strong non-linearities and asymmetries in the ERM period, which, however, seem to differ across countries and diminish during the last stage of the run-up to the euro. Important non-linear adjustments are also detected for Denmark in ERM-2 and for our group of four CEE economies.target zone, ERM, non-linearity, SETAR.

Non-Linear Exchange Rate Dynamics in Target Zones: A Bumpy Road towards a Honeymoon - Some Evidence from the ERM, ERM2 and Selected New EU Member States

This study investigates exchange rate movements in the Exchange Rate Mechanism (ERM) of the European Monetary System (EMS) and in the Exchange Rate Mechanism II (ERM-II). On the basis of the variant of the target zone model proposed by Bartolini and Prati (1999) and Bessec (2003), we set up a three-regime self-exciting threshold autoregressive model (SETAR) with a non-stationary central band and explicit modelling of the conditional variance. This modelling framework is employed to model daily DM-based and median currency-based bilateral exchange rates of countries participating in the original ERM and also for exchange rates of the Czech Republic, Hungary, Poland and Slovakia from 1999 to 2004. Our results confirm the presence of strong non-linearities and asymmetries in the ERM period, which, however, seem to differ across countries and diminish during the last stage of the run-up to the euro. Important non-linear adjustments are also detected for Denmark in ERM-2 and for our group of four CEE economies.target zone, ERM, non-linearity, SETAR

Equilibrium Heterogeneous-Agent Models as Measurement Tools: some Monte Carlo Evidence

This paper discusses a series of Monte Carlo experiments designed to evaluate the empirical properties of heterogeneous-agent macroeconomic models in the presence of sampling variability. The calibration procedure leads to the welfare analysis being conducted with the wrong parameters. The ability of the calibrated model to correctly predict the long-run welfare changes induced by a set of policy experiments is assessed. The results show that, for the policy reforms with sizable welfare effects (i.e., more than 0.2%), the model always predict the right sign of the welfare effects. However, the welfare effects can be evaluated with the wrong sign, when they are small and when the sample size is fairly limited. Quantitatively, the maximum errors made in evaluating a policy change are very small for some reforms (in the order of 0.02 percentage points), but bigger for others (in the order of 0.6 p.p.). Finally, having access to better data, in terms of larger samples, does lead to substantial increases in the precision of the welfare effects estimates, though the rate of convergence can be slow.Ex-ante Policy Evaluation, Incomplete Markets, Heterogeneous Agents, Monte Carlo, Welfare

Bandwidth selection for kernel estimation in mixed multi-dimensional spaces

Kernel estimation techniques, such as mean shift, suffer from one major drawback: the kernel bandwidth selection. The bandwidth can be fixed for all the data set or can vary at each points. Automatic bandwidth selection becomes a real challenge in case of multidimensional heterogeneous features. This paper presents a solution to this problem. It is an extension of \cite{Comaniciu03a} which was based on the fundamental property of normal distributions regarding the bias of the normalized density gradient. The selection is done iteratively for each type of features, by looking for the stability of local bandwidth estimates across a predefined range of bandwidths. A pseudo balloon mean shift filtering and partitioning are introduced. The validity of the method is demonstrated in the context of color image segmentation based on a 5-dimensional space