| Abdullah Al Mahmud | www.statmania.info |

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 |

Four Components

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

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

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

\(T_t =\) Trend

\(S_t =\) Seasonal

\(C_t =\) Cyclic

\(R_t =\) Random/irregular

**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).

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

Year | Production |
---|---|

2001 | 11 |

2002 | 9 |

2003 | 10 |

2004 | 16 |

2005 | 12 |

2001 | 7 |

2002 | 8 |

2003 | 6 |

2004 | 15 |

2005 | 3 |

Year | Production |
---|---|

2001 | 11 |

2002 | 9 |

2003 | 10 |

2004 | 16 |

2005 | 12 |

2001 | 7 |

2002 | 8 |

2003 | 6 |

2004 | 15 |

2005 | 3 |

**Steps**

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

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 |