Solving Dynamic Discrete Choice Models Using Smoothing and Sieve Methods

We propose to combine smoothing, simulations and sieve approximations to solve for either the integrated or expected value function in a general class of dynamic discrete choice (DDC) models. We use importance sampling to approximate the Bellman operators defining the two functions. The random Bellman operators, and therefore also the corresponding solutions, are generally non-smooth which is undesirable. To circumvent this issue, we introduce a smoothed version of the random Bellman operator and solve for the corresponding smoothed value function using sieve methods. We show that one can avoid using sieves by generalizing and adapting the `self-approximating' method of Rust (1997) to our setting. We provide an asymptotic theory for the approximate solutions and show that they converge with root-N-rate, where $N$ is number of Monte Carlo draws, towards Gaussian processes. We examine their performance in practice through a set of numerical experiments and find that both methods perform well with the sieve method being particularly attractive in terms of computational speed and accuracy

Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples

Agriculture spray wind velocity measurements and predictions

During the spraying seasons of 2014 and 2015, wind velocity and solar radiation (2014 only) were collected at a one meter height above the ground to simulate conditions affecting droplets near a ground-based spray boom. This instrumentation was placed in a cross pattern with sensors at the four cardinal directions (north, south, east, and west) with a fifth sensor in the center (2015 only). Data were collected at 10 Hz to measure the turbulent properties of the wind near the ground. Measurements of wind velocity profiles moving from upwind sensors to downwind sensors were used to evaluate correlation between the wind measurements. Two periods in which wind direction, on average, was collinear with multiple sensors were investigated. The first period contained five hours of data in which the average wind speed was 3.6 m/s (8 mi/h), while the second period contained 1.5 hours of data with an average wind speed of 1.5 m/s (3.4 mi/h). For the five hour dataset, correlation coefficients of 0.29 and 0.27 were found for wind direction and wind speed measured at two sensors respectively. This value fell when the five hours were broken up into multiple one minute periods. The correlation coefficients rose from less than 0.03 to greater than 0.14 once a lag term was introduced to the data. These results were not observed in the 1.5 hour dataset. Over the 1.5 hour period, the correlation coefficients were found to be less than 0.03. The introduction of a lag term had no clear effect. The entirety of the datasets that were collected in 2014 and 2015 were investigated to see under what conditions large wind change events were more likely to occur. The datasets suggest that low wind speeds lead to higher probability of large wind changes. As solar radiation increased so did the probability of large changes in wind. As a tolerance on the wind shift was tightened, the probability of wind changes became uniform. In models that predict spray drift, a popular method to simulate turbulent wind conditions in which the droplet is entrained, is to update the current wind velocities with a random process to achieve new wind velocities. This type of process is known as a random walk. The random walk hypothesis was tested using data collected at 10 Hz, and the average of the collected data to simulate data recorded at 0.5 s, 1 s, 5 s, 10 s, 30 s, 1 min, 5 min, and 10 min. For all tests below five minute averages, the test rejected the hypothesis that wind velocity updates can be independent of previous measurements at greater than 95% confidence. Indicating that updates to the current wind velocity is dependent on previous velocities. To help reduce the chances of spray drift, prediction models were developed and tested to predict wind direction 30 seconds into the future utilizing current and past measurements. The models tested included a kernel filter that is used for prediction of wind speeds for wind turbines, an autoregressive process (AR), a full ARIMA process, and a hybrid model that includes ideas from ARIMA and Taylor series expansions. The listed models were tested against a “No Model” model in which the predicted value was simply the current observed value. Models were trained over a one hour dataset and tested over a four hour data set. The AR and hybrid models lowered the RMS error value by 9% over the “No Model” model. The AR and hybrid models were outside of a 20 degree tolerance about 12% of the time. The correlation values between an upwind and downwind sensors indicate that little correlation exists. Along with the predictive models yielding limited results indicate that the wind changes rather randomly. However, results from testing the time series against the random walk hypothesis indicate that wind’s random fluctuations are correlated with one another, but these correlations are not seen using linear correlations. Further effort is needed to better model the wind process

The Surface Laplacian Technique in EEG: Theory and Methods

This paper reviews the method of surface Laplacian differentiation to study EEG. We focus on topics that are helpful for a clear understanding of the underlying concepts and its efficient implementation, which is especially important for EEG researchers unfamiliar with the technique. The popular methods of finite difference and splines are reviewed in detail. The former has the advantage of simplicity and low computational cost, but its estimates are prone to a variety of errors due to discretization. The latter eliminates all issues related to discretization and incorporates a regularization mechanism to reduce spatial noise, but at the cost of increasing mathematical and computational complexity. These and several others issues deserving further development are highlighted, some of which we address to the extent possible. Here we develop a set of discrete approximations for Laplacian estimates at peripheral electrodes and a possible solution to the problem of multiple-frame regularization. We also provide the mathematical details of finite difference approximations that are missing in the literature, and discuss the problem of computational performance, which is particularly important in the context of EEG splines where data sets can be very large. Along this line, the matrix representation of the surface Laplacian operator is carefully discussed and some figures are given illustrating the advantages of this approach. In the final remarks, we briefly sketch a possible way to incorporate finite-size electrodes into Laplacian estimates that could guide further developments.Comment: 43 pages, 8 figure

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest