Showing papers on "Parametric statistics published in 2017"

On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests

Abstract: Nonparametric two-sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being designed and analyzed, both for the unidimensional and the multivariate setting. In this short survey, we focus on test statistics that involve the Wasserstein distance. Using an entropic smoothing of the Wasserstein distance, we connect these to very different tests including multivariate methods involving energy statistics and kernel based maximum mean discrepancy and univariate methods like the Kolmogorov–Smirnov test, probability or quantile (PP/QQ) plots and receiver operating characteristic or ordinal dominance (ROC/ODC) curves. Some observations are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two-sample testing’s classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others.

Active Disturbance Rejection Adaptive Control of Hydraulic Servo Systems

Abstract: This paper presents an active disturbance rejection adaptive control scheme via full state feedback for motion control of hydraulic servo systems subjected to both parametric uncertainties and uncertain nonlinearities. The proposed controller is derived by effectively integrating adaptive control with extended state observer via backstepping method. The adaptive law is synthesized to handle parametric uncertainties and the remaining uncertainties are estimated by the extended state observer and then compensated in a feedforward way. The unique features of the proposed controller are that not only the matched uncertainties but also unmatched uncertainties are estimated by constructing two extended state observers, and the parameter adaptation law is driven by both tracking errors and state estimation errors. Since the majority of parametric uncertainties can be reduced by the parameter adaptation, the task of the extended state observer is much alleviated. Consequently, high-gain feedback is avoided and improved tracking performance can be expected. The proposed controller theoretically achieves an asymptotic tracking performance in the presence of parametric uncertainties and constant disturbances. In addition, prescribed transient tracking performance and final tracking accuracy can also be guaranteed when existing time-variant uncertain nonlinearities. Comparative experimental results are obtained to verify the high tracking performance nature of the proposed control strategy.

Adaptive sliding mode control for finite-time stability of quad-rotor UAVs with parametric uncertainties.

Abstract: Adaptive control methods are developed for stability and tracking control of flight systems in the presence of parametric uncertainties. This paper offers a design technique of adaptive sliding mode control (ASMC) for finite-time stabilization of unmanned aerial vehicle (UAV) systems with parametric uncertainties. Applying the Lyapunov stability concept and finite-time convergence idea, the recommended control method guarantees that the states of the quad-rotor UAV are converged to the origin with a finite-time convergence rate. Furthermore, an adaptive-tuning scheme is advised to guesstimate the unknown parameters of the quad-rotor UAV at any moment. Finally, simulation results are presented to exhibit the helpfulness of the offered technique compared to the previous methods.

Grey Wolf Optimizer Algorithm-Based Tuning of Fuzzy Control Systems With Reduced Parametric Sensitivity

Abstract: This paper proposes an innovative tuning approach for fuzzy control systems (CSs) with a reduced parametric sensitivity using the Grey Wolf Optimizer (GWO) algorithm. The CSs consist of servo system processes controlled by Takagi–Sugeno–Kang proportional-integral fuzzy controllers (TSK PI-FCs). The process models have second-order dynamics with an integral component, variable parameters, a saturation, and dead-zone static nonlinearity. The sensitivity analysis employs output sensitivity functions of the sensitivity models defined with respect to the parametric variations of the processes. The GWO algorithm is used in solving the optimization problems, where the objective functions include the output sensitivity functions. GWO's motivation is based on its low-computational cost. The tuning approach is validated in an experimental case study of a position control for a laboratory nonlinear servo system, and TSK PI-FCs with a reduced process small time constant sensitivity are offered.

Distributed Robust Finite-Time Secondary Voltage and Frequency Control of Islanded Microgrids

Abstract: This paper presents a distributed, robust, finite-time secondary control for both voltage and frequency restoration of an islanded microgrid with droop-controlled inverter-based distributed generators (DGs). The distributed cooperative secondary control is fully distributed (i.e., uses only the information of neighboring DGs that can communicate with one another through a sparse communication network). In contrast to existing distributed methods that require a detailed model of the system (such as line impedances, loads, other DG units parameters, and even the microgrid configuration, which are practically unknown), the proposed protocols are synthesized by considering the unmodeled dynamics, unknown disturbances, and uncertainties in their models. The other novel idea in this paper is that the consensus-based distributed controllers restore the islanded microgrid's voltage magnitudes and frequency to their reference values for all DGs within finite time, irrespective of parametric uncertainties, unmodeled dynamics, and disturbances, while providing accurate real-power sharing. Moreover, the proposed method considers the coupling between the frequency and voltage of the islanded microgrid. Unlike conventional distributed controllers, the proposed approach quickly reaches consensus and exhibits a more accurate robust performance. Finally, we verify the proposed control strategy's performance using the MATLAB/SimPowerSystems toolbox.

The Frontier Fields lens modelling comparison project

Abstract: Gravitational lensing by clusters of galaxies offers a powerful probe of their structure and mass distribution. Several research groups have developed techniques independently to achieve this goal. While these methods have all provided remarkably high-precision mass maps, particularly with exquisite imaging data from the Hubble Space Telescope (HST), the reconstructions themselves have never been directly compared. In this paper, we present for the first time a detailed comparison of methodologies for fidelity, accuracy and precision. For this collaborative exercise, the lens modelling community was provided simulated cluster images that mimic the depth and resolution of the ongoing HST Frontier Fields. The results of the submitted reconstructions with the un-blinded true mass profile of these two clusters are presented here. Parametric, free-form and hybrid techniques have been deployed by the participating groups and we detail the strengths and trade-offs in accuracy and systematics that arise for each methodology. We note in conclusion that several properties of the lensing clusters are recovered equally well by most of the lensing techniques compared in this study. For example, the reconstruction of azimuthally averaged density and mass profiles by both parametric and free-form methods matches the input models at the level of ∼10 per cent. Parametric techniques are generally better at recovering the 2D maps of the convergence and of the magnification. For the best-performing algorithms, the accuracy in the magnification estimate is ∼10 per cent at μ_(true) = 3 and it degrades to ∼30 per cent at μ_(true) ∼ 10.

A Hierarchical Split-Based Approach for Parametric Thresholding of SAR Images: Flood Inundation as a Test Case

Abstract: Parametric thresholding algorithms applied to synthetic aperture radar (SAR) imagery typically require the estimation of two distribution functions, i.e., one representing the target class and one its background. They are eventually used for selecting the threshold that allows binarizing the image in an optimal way. In this context, one of the main difficulties in parameterizing these functions originates from the fact that the target class often represents only a small fraction of the image. Under such circumstances, the histogram of the image values is often not obviously bimodal and it becomes difficult, if not impossible, to accurately parameterize distribution functions. Here we introduce a hierarchical split-based approach that searches for tiles of variable size allowing the parameterization of the distributions of two classes. The method is integrated into a flood-mapping algorithm in order to evaluate its capacity for parameterizing distribution functions attributed to floodwater and changes caused by floods. We analyzed a data set acquired during a flood event along the Severn River (U.K.) in 2007. It is composed of moderate (ENVISAT-WS) and high (TerraSAR-X)-resolution SAR images. The obtained classification accuracies as well as the similarity of performance levels to a benchmark obtained with an established method based on the manual selection of tiles indicate the validity of the new method.

Robust Statistical Methods for Empirical Software Engineering

Abstract: There have been many changes in statistical theory in the past 30 years, including increased evidence that non-robust methods may fail to detect important results. The statistical advice available to software engineering researchers needs to be updated to address these issues. This paper aims both to explain the new results in the area of robust analysis methods and to provide a large-scale worked example of the new methods. We summarise the results of analyses of the Type 1 error efficiency and power of standard parametric and non-parametric statistical tests when applied to non-normal data sets. We identify parametric and non-parametric methods that are robust to non-normality. We present an analysis of a large-scale software engineering experiment to illustrate their use. We illustrate the use of kernel density plots, and parametric and non-parametric methods using four different software engineering data sets. We explain why the methods are necessary and the rationale for selecting a specific analysis. We suggest using kernel density plots rather than box plots to visualise data distributions. For parametric analysis, we recommend trimmed means, which can support reliable tests of the differences between the central location of two or more samples. When the distribution of the data differs among groups, or we have ordinal scale data, we recommend non-parametric methods such as Cliff's ź or a robust rank-based ANOVA-like method.

RISE-Based Adaptive Control of Hydraulic Systems With Asymptotic Tracking

Abstract: Parametric uncertainty associated with unmodeled disturbance always exist in physical hydraulic systems, and complicate the advanced nonlinear controller design. In this paper, an adaptive compensation with a robust integral of the sign of the error (RISE) feedback is developed for high precise tracking control of hydraulic motion system. To handle both payload and hydraulic unknown parameters in one controller, a chain of integrator nonlinear system model is first derived, and an adaptive RISE controller is then proposed, in which adaptive law is synthesized to handle parametric uncertainty and RISE robust term to attenuate unmodeled disturbance. The major feature of the proposed controller is that it can theoretically guarantee asymptotic tracking performance with a continuous control input, in the presence of various parametric uncertainties and unmodeled disturbances such as unconsidered dynamics as well as external disturbances via Lyapunov analysis. However, the proposed controller takes the acceleration as a system state, which usually suffers heavy noise pollution and thus cannot be utilized directly in actual control. To solve this practical issue, in this paper, a tracking differentiator is employed to extract high-quality acceleration signal and to make the proposed controller feasible execution. The effectiveness of the proposed nonlinear controller is demonstrated via comparative experimental results.

Adaptive Backstepping Control of Spacecraft Rendezvous and Proximity Operations With Input Saturation and Full-State Constraint

Abstract: This paper presents a six-degree-of-freedom relative motion control method for autonomous spacecraft rendezvous and proximity operations subject to input saturation, full-state constraint, kinematic coupling, parametric uncertainty, and matched and mismatched disturbances. Relative rotational and relative translational controllers are developed separately based on a unified adaptive backstepping technique. Both element-wise and norm-wise adaptive estimation techniques are used for handling parametric uncertainties, kinematic couplings, and matched and mismatched disturbances, where the bounds of disturbances are unknown. Two auxiliary design systems are employed to deal with input saturation in the relative rotational and relative translational control designs, and the stability of the saturated control solution is verified. Full-state constraint of the relative pose motion is handled by using barrier Lyapunov functions while achieving a satisfactory control performance. All signals in the closed-loop system are guaranteed to be uniformly ultimately bounded, and the relative motion states are all restricted within the known constraints. Compared with the previous control designs of spacecraft rendezvous and proximity operations, the proposed control strategy in this paper can simultaneously deal with input saturation, full-state constraint, kinematic coupling, parametric uncertainty, and matched and mismatched disturbances. Experimental simulation results validate the performance and robustness improvement of the proposed control strategy.

Off-policy evaluation for slate recommendation

Abstract: This paper studies the evaluation of policies that recommend an ordered set of items (e.g., a ranking) based on some context---a common scenario in web search, ads, and recommendation. We build on techniques from combinatorial bandits to introduce a new practical estimator that uses logged data to estimate a policy's performance. A thorough empirical evaluation on real-world data reveals that our estimator is accurate in a variety of settings, including as a subroutine in a learning-to-rank task, where it achieves competitive performance. We derive conditions under which our estimator is unbiased---these conditions are weaker than prior heuristics for slate evaluation---and experimentally demonstrate a smaller bias than parametric approaches, even when these conditions are violated. Finally, our theory and experiments also show exponential savings in the amount of required data compared with general unbiased estimators.

Noisy Networks for Exploration

Abstract: We introduce NoisyNet, a deep reinforcement learning agent with parametric noise added to its weights, and show that the induced stochasticity of the agent's policy can be used to aid efficient exploration. The parameters of the noise are learned with gradient descent along with the remaining network weights. NoisyNet is straightforward to implement and adds little computational overhead. We find that replacing the conventional exploration heuristics for A3C, DQN and dueling agents (entropy reward and $\epsilon$-greedy respectively) with NoisyNet yields substantially higher scores for a wide range of Atari games, in some cases advancing the agent from sub to super-human performance.

A Comparison of Parametric and Non-Parametric Methods Applied to a Likert Scale

Abstract: A trenchant and passionate dispute over the use of parametric versus non-parametric methods for the analysis of Likert scale ordinal data has raged for the past eight decades. The answer is not a simple “yes” or “no” but is related to hypotheses, objectives, risks, and paradigms. In this paper, we took a pragmatic approach. We applied both types of methods to the analysis of actual Likert data on responses from different professional subgroups of European pharmacists regarding competencies for practice. Results obtained show that with “large” (>15) numbers of responses and similar (but clearly not normal) distributions from different subgroups, parametric and non-parametric analyses give in almost all cases the same significant or non-significant results for inter-subgroup comparisons. Parametric methods were more discriminant in the cases of non-similar conclusions. Considering that the largest differences in opinions occurred in the upper part of the 4-point Likert scale (ranks 3 “very important” and 4 “essential”), a “score analysis” based on this part of the data was undertaken. This transformation of the ordinal Likert data into binary scores produced a graphical representation that was visually easier to understand as differences were accentuated. In conclusion, in this case of Likert ordinal data with high response rates, restraining the analysis to non-parametric methods leads to a loss of information. The addition of parametric methods, graphical analysis, analysis of subsets, and transformation of data leads to more in-depth analyses.

Active disturbance rejection adaptive control of uncertain nonlinear systems: theory and application

Abstract: This paper proposes an active disturbance rejection adaptive controller for tracking control of a class of uncertain nonlinear systems with consideration of both parametric uncertainties and uncertain nonlinearities by effectively integrating adaptive control with extended state observer via backstepping method. Parametric uncertainties are handled by the synthesized adaptive law and the remaining uncertainties are estimated by extended state observer and then compensated in a feedforward way. Moreover, both matched uncertainties and unmatched uncertainties can be estimated by constructing an extended state observer for each channel of the considered nonlinear plant. Since parametric uncertainties can be reduced by parameter adaptation, the learning burden of extended state observer is much reduced. Consequently, high-gain feedback is avoided and improved tracking performance can be expected. The proposed controller theoretically guarantees a prescribed transient tracking performance and final tracking accuracy in general while achieving asymptotic tracking when the uncertain nonlinearities are not time-variant. The motion control of a motor-driven robot manipulator is investigated as an application example with some suitable modifications and improvements, and comparative simulation results are obtained to verify the high tracking performance nature of the proposed control strategy.

Comparing Inference Approaches for RD Designs: A Reexamination of the Effect of Head Start on Child Mortality

Abstract: The regression discontinuity (RD) design is a popular quasi-experimental design for causal inference and policy evaluation. The most common inference approaches in RD designs employ “flexible” parametric and nonparametric local polynomial methods, which rely on extrapolation and large-sample approximations of conditional expectations using observations somewhat near the cutoff that determines treatment assignment. An alternative inference approach employs the idea of local randomization, where the very few units closest to the cutoff are regarded as randomly assigned to treatment and finite-sample exact inference methods are used. In this paper, we contrast these approaches empirically by re-analyzing the influential findings of Ludwig and Miller (2007), who studied the effect of Head Start assistance on child mortality employing parametric RD methods. We first review methods based on approximations of conditional expectations, which are relatively well developed in the literature, and then present new methods based on randomization inference. In particular, we extend the local randomization framework to allow for parametric adjustments of the potential outcomes; our extended framework substantially relaxes strong assumptions in prior literature and better resembles other RD inference methods. We compare all these methods formally, focusing on both estimands and inference properties. In addition, we develop new approaches for randomization-based sensitivity analysis specifically tailored to RD designs. Applying all these methods to the Head Start data, we find that the original RD treatment effect reported in the literature is quite stable and robust, an empirical finding that enhances the credibility of the original result. All the empirical methods we discuss are readily available in general purpose software in R and Stata; we also provide the dataset and software code needed to replicate all our results.

Generative adversarial network-based postfilter for statistical parametric speech synthesis

Abstract: We propose a postfilter based on a generative adversarial network (GAN) to compensate for the differences between natural speech and speech synthesized by statistical parametric speech synthesis. In particular, we focus on the differences caused by over-smoothing, which makes the sounds muffled. Over-smoothing occurs in the time and frequency directions and is highly correlated in both directions, and conventional methods based on heuristics are too limited to cover all the factors (e.g., global variance was designed only to recover the dynamic range). To solve this problem, we focus on “spectral texture”, i.e., the details of the time-frequency representation, and propose a learning-based postfilter that captures the structures directly from the data. To estimate the true distribution, we utilize a GAN composed of a generator and a discriminator. This optimizes the generator to produce samples imitating the dataset according to the adversarial discriminator. This adversarial process encourages the generator to fit the true data distribution, i.e., to generate realistic spectral texture. Objective evaluation of experimental results shows that the GAN-based postfilter can compensate for detailed spectral structures including modulation spectrum, and subjective evaluation shows that its generated speech is comparable to natural speech.

Experimental investigation of parametric cell-to-cell variation and correlation based on 1100 commercial lithium-ion cells

Abstract: In order to prevent battery modules from inhomogeneity during operation, the integrated battery cells should be matched. Therefore, the cell-to-cell variations of relevant cell parameters due to manufacturing tolerances need to be quantified. Regarding cell matching, three points are lacking in current literature: first, reported parameter analyses have either based their statistic analysis on a too small number of cells. Or, second, few different cell parameters have been considered – hence, it was not possible to discuss which parameter would be beneficial to be used for classification. Third, potentially sensible correlations between different cell parameters have not been determined and discussed adequately. This paper provides a unique combination of analysing multiple different cell parameters and discussing their correlation – both based on a large number of examined cells: we investigate the parametric cell-to-cell variation and correlation of 1100 commercial production fresh LiFePO4-graphite cells which originate from two batches. The cell parameters are experimentally determined by conducting DC check-up (CU)- and AC EIS-measurements under monitored temperature and relaxation conditions for all 1100 cells. The data is statistically analysed for 15 different parameters: different discharge capacities, different DC and AC impedances, the mass and mean temperature of the cells. The determined relative variation of capacity and impedance is small: 0.28% and 0.72% respectively, which corresponds to a variation ratio of 1:2.2. The variation of the cell impedance allows no conclusion about the variation of the cell capacity. From the results, we derive four major implications concerning recommended characterisation parameters for the development and modelling of battery modules as well as for the quality control during cell production. Our experimentally determined parametric variation values and drawn conclusions are valuable for model fittings and battery pack analyses which have up to now been based on assumptions about cell-to-cell variations. The data set for all 1100 cells and 15 parameters is provided as supplementary material [1] .

Neural Network Learning Adaptive Robust Control of an Industrial Linear Motor-Driven Stage With Disturbance Rejection Ability

Abstract: In this paper, a neural network learning adaptive robust controller (NNLARC) is synthesized for an industrial linear motor stage to achieve good tracking performance and excellent disturbance rejection ability. The NNLARC scheme contains parametric adaption part, robust feedback part, and radial basis function (RBF) neural network (NN) part in a parallel structure. The adaptive part and the robust part are designed based on the system dynamics to meet the challenge of parametric variations and uncertain random disturbances. It must be noted that in actual industrial machining situations, precision motion equipment is always disturbed by unknown factors, which usually cannot be described by mathematical models but affect the tracking accuracy significantly. Therefore, the RBF NN part is employed to further approximate and compensate the complicated disturbances with high reconstructing accuracy and fast training rate. The stability of the proposed NNLARC strategy is analyzed and proved through the Lyapunov theorem. Comparative experiments under various external disturbances such as completely unknown disturbance added by polyfoam are conducted on an industrial linear motor stage. The experimental results consistently validate that the proposed NNLARC control strategy can excellently meet the challenge of complicated disturbance in practical applications. The proposed scheme also provides a guidance for control strategy synthesis with both good tracking performance and disturbance rejection.

Adaptive Finite-Time Stabilization of a Class of Uncertain Nonlinear Systems via Logic-Based Switchings

Abstract: In this paper, global adaptive finite-time stabilization is investigated by logic-based switching control for a class of uncertain nonlinear systems with the powers of positive odd rational numbers Parametric uncertainties entering the state equations nonlinearly can be fast time-varying or jumping at unknown time instants, and the control coefficient appearing in the control channel can be unknown The bounds of the parametric uncertainties and the unknown control coefficient are not required to know a priori Our proposed controller is a switching-type one, in which a nonlinear controller with two parameters to be tuned is first designed by adding a power integrator, and then a switching mechanism is proposed to tune the parameters online to finite-time stabilize the system An example is provided to demonstrate the effectiveness of the proposed result

A Neural Parametric Singing Synthesizer Modeling Timbre and Expression from Natural Songs

Abstract: We recently presented a new model for singing synthesis based on a modified version of the WaveNet architecture. Instead of modeling raw waveform, we model features produced by a parametric vocoder that separates the influence of pitch and timbre. This allows conveniently modifying pitch to match any target melody, facilitates training on more modest dataset sizes, and significantly reduces training and generation times. Nonetheless, compared to modeling waveform directly, ways of effectively handling higher-dimensional outputs, multiple feature streams and regularization become more important with our approach. In this work, we extend our proposed system to include additional components for predicting F0 and phonetic timings from a musical score with lyrics. These expression-related features are learned together with timbrical features from a single set of natural songs. We compare our method to existing statistical parametric, concatenative, and neural network-based approaches using quantitative metrics as well as listening tests.

Adaptive finite‐time fault‐tolerant tracking control for a class of MIMO nonlinear systems with output constraints

Abstract: Summary In this work, we present a novel adaptive finite-time fault-tolerant control algorithm for a class of multi-input multi-output nonlinear systems with constraint requirement on the system output tracking error. Both parametric and nonparametric system uncertainties can be effectively dealt with by the proposed control scheme. The gain functions of the nonlinear systems under discussion, especially the control input gain function, can be not fully known and state-dependent. Backstepping design with a tan-type barrier Lyapunov function and a new structure of stabilizing function is presented. We show that under the proposed control scheme, finite-time convergence of the output tracking error into a small set around zero is guaranteed, while the constraint requirement on the system output tracking error will not be violated during operation. An illustrative example on a robot manipulator model is presented in the end to further demonstrate the effectiveness of the proposed control scheme. Copyright © 2016 John Wiley & Sons, Ltd.

Bridging asymptotic independence and dependence in spatial extremes using Gaussian scale mixtures

Abstract: Gaussian scale mixtures are constructed as Gaussian processes with a random variance. They have non-Gaussian marginals and can exhibit asymptotic dependence unlike Gaussian processes, which are asymptotically independent except in the case of perfect dependence. In this paper, we study the extremal dependence properties of Gaussian scale mixtures and we unify and extend general results on their joint tail decay rates in both asymptotic dependence and independence cases. Motivated by the analysis of spatial extremes, we propose flexible yet parsimonious parametric copula models that smoothly interpolate from asymptotic dependence to independence and include the Gaussian dependence as a special case. We show how these new models can be fitted to high threshold exceedances using a censored likelihood approach, and we demonstrate that they provide valuable information about tail characteristics. In particular, by borrowing strength across locations, our parametric model-based approach can also be used to provide evidence for or against either asymptotic dependence class, hence complementing information given at an exploratory stage by the widely used nonparametric or parametric estimates of the χ and χ coefficients. We demonstrate the capacity of our methodology by adequately capturing the extremal properties of wind speed data collected in the Pacific Northwest, US.

Artificial neural networks for vibration based inverse parametric identifications

Abstract: Display Omitted ANNs-solved vibration based parametric identification studies are reviewed.Factors which affect identification result are discussed.Pros and cons of ANN approaches are mentioned.Suggestions are given to potential researchers based on the discussion.Analysis with experimental results is provided to justify some point of view. Vibration behavior of any solid structure reveals certain dynamic characteristics and property parameters of that structure. Inverse problems dealing with vibration response utilize the response signals to find out input factors and/or certain structural properties. Due to certain drawbacks of traditional solutions to inverse problems, ANNs have gained a major popularity in this field. This paper reviews some earlier researches where ANNs were applied to solve different vibration-based inverse parametric identification problems. The adoption of different ANN algorithms, input-output schemes and required signal processing were denoted in considerable detail. In addition, a number of issues have been reported, including the factors that affect ANNs prediction, as well as the advantage and disadvantage of ANN approaches with respect to general inverse methods Based on the critical analysis, suggestions to potential researchers have also been provided for future scopes.

Structure Preserving Model Reduction of Parametric Hamiltonian Systems

Abstract: While reduced-order models (ROMs) have been popular for efficiently solving large systems of differential equations, the stability of reduced models over long-time integration presents challenges. We present a greedy approach for a ROM generation of parametric Hamiltonian systems that captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the linear vector space to increase the accuracy of the reduced basis. We use the error in the Hamiltonian due to model reduction as an error indicator to search the parameter space and identify the next best basis vectors. Under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters, we show that the greedy algorithm converges at an exponential rate. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplecti...