## 3.6 Classical decomposition

Model time series $$\{x_t\}$$ as a combination of

1. trend ($$m_t$$)
2. seasonal component ($$s_t$$)
3. remainder ($$e_t$$)

$$x_t = m_t + s_t + e_t$$

### 3.6.1 1. The trend ($$m_t$$)

We need a way to extract the so-called signal. One common method is via “linear filters”

$m_t = \sum_{i=-\infty}^{\infty} \lambda_i x_{t+1}$

For example, a moving average

$m_t = \sum_{i=-a}^{a} \frac{1}{2a + 1} x_{t+i}$

If $$a = 1$$, then

$m_t = \frac{1}{3}(x_{t-1} + x_t + x_{t+1})$

### 3.6.2 Example of linear filtering

Here is a time series.

A linear filter with $$a=3$$ closely tracks the data.

As we increase the length of data that is averaged from 1 on each side ($$a=3$$) to 4 on each side ($$a=9$$), the trend line is smoother.

When we increase up to 13 points on each side ($$a=27$$), the trend line is very smooth.

### 3.6.3 2. Seasonal effect ($$s_t$$)

Once we have an estimate of the trend $$m_t$$, we can estimate $$s_t$$ simply by subtraction:

$s_t = x_t - m_t$

This is the seasonal effect ($$s_t$$), assuming $$\lambda = 1/9$$, but, $$s_t$$ includes the remainder $$e_t$$ as well. Instead we can estimate the mean seasonal effect ($$s_t$$).

seas_2 <- decompose(xx)$seasonal par(mai = c(0.9, 0.9, 0.1, 0.1), omi = c(0, 0, 0, 0)) plot.ts(seas_2, las = 1, ylab = "") ### 3.6.4 3. Remainder ($$e_t$$) Now we can estimate $$e_t$$ via subtraction: $e_t = x_t - m_t - s_t$ ee <- decompose(xx)$random
par(mai = c(0.9, 0.9, 0.1, 0.1), omi = c(0, 0, 0, 0))
plot.ts(ee, las = 1, ylab = "")