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

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. Algorithm 1: Regression

9. Regression (cont.) R 2 = 0.424

10. Algorithm 2: Decomposition

11. Decomposition (cont.) R 2 = 0.693

12. Algorithm 3: Smoothing

13. Smoothing (cont.) R 2 = 0.516

14. 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

15. 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).