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Can we reliably forecast individual 3G usage data? An analysis using mathematical simulation of time series algorithms Cosmo Zheng

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Background Fluctuations in daily demand for bandwidth make ordinary usage pricing inefficient Solution: Time- dependent pricing to persuade users to defer usage http://scenic.princeton.edu/tube/overview.html

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Our Problem Users must be informed of expected future prices, to assess the costs of deferring usage We need a reliable way to predict future usage based on past data http://scenic.princeton.edu/tube/tech nology.html

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The Algorithms Nonlinear regression – generate a fitted function of the form D + A*sin(2πt/24) + B*sin(2πt/12) + C*sin(2πt/6) Use fitted function to extrapolate

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Algorithms (cont.) Time series decomposition – isolate trend, seasonal, and residual components Extend trend and seasonal components into the future

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Algorithms (cont.) Exponential smoothing – generate {S t } based on a weighted average of previous data Simplest form is S 1 = X 0, S t = αX t-1 + (1-α)S t-1 for t>1, where α is a smoothing factor

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The Data Use simulated datasets, representing usage each hour over 5 days {X t } for 1 <= t <= 120 First 4 days are historical data (training set), 5 th day is the test set

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Algorithm 1: Regression

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Regression (cont.) R 2 = 0.424

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Algorithm 2: Decomposition

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Decomposition (cont.) R 2 = 0.693

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Algorithm 3: Smoothing

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Smoothing (cont.) R 2 = 0.516

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Additional Trials Trial #RegressionDecompositionSmoothing 164.146.256.4 27647.461.1 365.553.953.4 461.748.946.8 558.843.153.3 668.943.551.3 759.145.440.8 859.656.658.6 975.656.459.2 1052.846.954.1 Average64.2148.8353.5 Trial #RegressionDecompositionSmoothing 10.4240.6930.516 20.3740.7210.455 30.3880.5770.543 40.530.6010.593 50.3830.6870.527 60.3820.640.682 70.5150.7220.783 80.4570.4590.389 90.5060.6120.719 100.4680.5070.348 Average0.44270.62190.5555 Sum of absolute error R2R2

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Conclusions Time series decomposition provided most accurate prediction of future usage, followed by exponential smoothing, then regression Possible explanation: usage pattern is strongly cyclic; repeats itself on a daily basis Suggestion: investigate further into better means of isolating seasonal data; some more sophisticated algorithms exist (ARIMA, stochastic volatility models).

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