Time Series

Abdullah Al Mahmud

Time Series

What is Time Series Data?

Data arranged chronologically

\(Y_t = f(t); t = t_1, t_2, t_3, \cdots, t_n\)

Example

Year Production
2001 11
2002 9
2003 10
2004 16
2005 12

Components of Time Series

Four Components

  • Trend (increase/decrease)
  • Seasonal variation
  • Cyclic variation
  • Irregular/Random variation

Uses

  • Analyze past behavior
  • Forecasting
  • Comparison by time/place
  • Segregation of components
  • Performance measure

Symbols

\(Y_t =\) Values of series at time t

\(T_t =\) Trend

\(S_t =\) Seasonal

\(C_t =\) Cyclic

\(R_t =\) Random/irregular

Models

Additive Model

\(Y_t = T_t + S_t + C_t + R_t\)

  • \(C_t\) and \(S_t\) can be \(\pm\)ve
  • \(R_t\) can also be \(\pm\)ve, but in the long run, \(\sum R_t = 0\)

Multiplicative Model

  • \(Y_t = T_t \times S_t \times C_t \times R_t\)
  • \(S_t, C_t, R_t\) refer to deviation from unit
  • \(S_t\) equals unity in 1 year, \(C_t\) in a cycle, and GM of \(R_t\) is unity (1).

Comparison of Models

  • Components in additive models are independent.
  • In multiplicative models, components are interwined.

Measuring Trend

Graphical/Free-hand Method

Year Production
2001 11
2002 9
2003 10
2004 16
2005 12
2001 7
2002 8
2003 6
2004 15
2005 3

Sem-average

Year Production
2001 11
2002 9
2003 10
2004 16
2005 12
2001 7
2002 8
2003 6
2004 15
2005 3

Steps

  1. Separate the data into two equal parts (if odd-numbered, omit middle-most)
  2. Estimate averages of each group
  3. Put these two values on the scatter plot and extend

Moving Average

Year Production 3-Yearly Moving Average
2001 412 NA
2002 438 \({412+438+446}\over{3}=432\)
2003 446 \(\frac{438+446+454}3=446\)
2004 454 457
2005 470 469
2006 483 \(\frac{470+483+490}3=481\)
2007 490 NA