Can we reliably forecast individual 3G usage data?

An analysis using mathematical simulation of time series algorithms

Cosmo Zheng

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

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/technology.html

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

Algorithms (cont.)

• Time series decomposition – isolate trend, seasonal, and residual components

• Extend trend and seasonal components into the future

Algorithms (cont.)

• Exponential smoothing – generate {St} based on a weighted average of previous data

• Simplest form is S1 = X0, St = αXt-1 + (1-α)St-1 for t>1, where α is a smoothing factor

The Data

• Use simulated datasets, representing usage each hour over 5 days

• {Xt} for 1 <= t <= 120• First 4 days are

historical data (training set), 5th day is the test set

Algorithm 1: Regression

Regression (cont.)

R2 = 0.424

Algorithm 2: Decomposition

Decomposition (cont.)

R2 = 0.693

Algorithm 3: Smoothing

Smoothing (cont.)

R2 = 0.516

Additional Trials

Trial # Regression Decomposition Smoothing

1 64.1 46.2 56.4

2 76 47.4 61.1

3 65.5 53.9 53.4

4 61.7 48.9 46.8

5 58.8 43.1 53.3

6 68.9 43.5 51.3

7 59.1 45.4 40.8

8 59.6 56.6 58.6

9 75.6 56.4 59.2

10 52.8 46.9 54.1

Average 64.21 48.83 53.5

Trial # Regression Decomposition Smoothing

1 0.424 0.693 0.516

2 0.374 0.721 0.455

3 0.388 0.577 0.543

4 0.53 0.601 0.593

5 0.383 0.687 0.527

6 0.382 0.64 0.682

7 0.515 0.722 0.783

8 0.457 0.459 0.389

9 0.506 0.612 0.719

10 0.468 0.507 0.348

Average 0.4427 0.6219 0.5555

Sum of absolute error R2

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).