# 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

$$Y_t = T_t + S_t + C_t + R_t$$

• $$C_t$$ 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

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