1. GAUSSIAN PROCESS REGRESSION FORECASTING OF COMPUTER NETWORK PERFORMANCE CHARACTERISTICS 1 Departments of Computer Science and Mathematics, 2 Department of Mathematics Bucknell University, Lewisburg, PA Christina Garman 1 (‘11) & Michael Frey, Ph.D. 2 Introduction Computer network conditions (available bandwidth, latency, and loss) concern: Users with large data transfers or resource-intensive applications Network engineers monitoring the quality of their network Network researchers We have investigated Gaussian process regression for forecasting network conditions. Gaussian process regression (GPR) allows us to make predictions of continuous quantities based on “learning” from a set of training data GPR accommodates – Asynchronous data sources – Periodic data – Actively measured data – Missing data – Structural data Because of the nature of computer networks, all of these situations are likely to occur, so our forecasting framework must be able to handle them. Gaussian Process Regression A Gaussian process is an indexed set of random variables, any finite number of which have a joint Gaussian distribution and can be completely specified by a mean function and covariance function Important Features – Covariance (or kernel) function – gives a model of the data and controls the properties of the Gaussian process – Hyperparameters – adjustable parameters, “learned” or inferred from a set of training data, allowing the kernel function to provide the best description of the current data GPR can model various different trends and properties of a data set Simple covariance functions can be combined to create more complex ones In the example below 3, note the long term rising trend, seasonal variation, and some small irregularities and how GPR is able to incorporate all of them into the forecast Basic Algorithm New Formulae for Updating GPR Forecasts Computationally efficient – no new matrix inversions No need to redo whole process each time a new data point is received Variance – Two Questions 1.What is the effect of history length on prediction error? 2.How does the variance change as our forecasting point moves out in time? Both of these questions boil down to a study of the same quantity: Using the Rayleigh-Ritz theorem, we can bound the quantity that we are interested in, giving us: Or more simply: Our forecasting efforts focus on the Department of Energy’s Energy Sciences Network (ESnet) pictured above 1 Forecasts are done in MATLAB 2. We have created a framework that allows the code to be run directly in MATLAB or from a C program. Revisit this work from an information theoretic perspective Improve network performance characteristics forecasting using multivariate data Future Work Acknowledgements Department of Energy Research Assistantship MATLAB Code: Carl Edward Rasmussen and Hannes Nickisch References 1.Department of Energy, Energy Sciences Network (Esnet), http://www.es.net/pub/maps/topology.html 2.Gaussian Process Regression and Classification Toolbox version 3.0, Carl Edward Rasmussen and Hannes Nickisch, 2005-2010, http://www.gaussianprocess.org/gpml/code/matlab/doc/ 3.Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. Maximum Likelihood Estimation Forecasting