6.1 Time Series and Their Components (HW)

Basically a Time Series is variables that are collected over time. The variables are highly likely to have autocorrelation

Autocorellation: Variables are automatically dependant on each other over time, and the mere aspect of these synergies (patterns) where one will often be able to prove correlation between the variables

One approach to assessing time series is by decomposing the patterns by finding the components hereof, these are:

  1. Trend(T): if it is linear, then it can be explained by \(\hat{T}_t=\beta_0+\beta_1t\), hence we apply the linear function, hence what in statistics is \(\hat{y}=\hat{T}\) in time series

  2. Cyclical(T or C)Note, often included in practice as the trend, as it can be difficult to extinguish

  3. Seasonal(S)

  4. Irregular(random)(I) - Notice! We dont want to use this for assembling the model, as it is random

These are also called deterministic variables

The purpose of decomposing the time series data, can be either for exploration or prediction. Hence you can estimate the coefficients of the components by breaking down the data.

Although the typical purpose of time series is exploration of the data and assess if there are seasons, trends etc. and perhaps to pinpoint whether you are above or below the season/trend/cycle.

If the Y observations is the sum of the components, then we have additive model, if they are the product of the components, then it is called multiplicative model

Time series is typically an additive model, if the variance is more or less the same, it is a multiplicative model if the variance increases with time

Note, one can transform a multiplicative model to an additive model by taking the logarithm

6.1.1 Additional on trend

Quadratic trend

e.g. where we have curvature

\[\hat{T}_t=\beta_0+\beta_1t+\beta_1t^2\]

Exponential trend

e.g. exponentially growing population

\[\hat{T}_t=\beta_0*\beta_2^t\]

NOTE, one may transform this into a logistic trend instead, as continuous exponential trend is not typical


6.1.2 Additional on seasonal pattern

  • One can manually rule out seasonality by adding seasonal index (that is hard coding the expected index in the respective periods)
    • One must rule out other factors before doing this!

Seasonally adjusted data

For additive

\[Y_t - S_t = T_t + I_t\]

For multiplicative

\[\frac{Y_t}{S_t} = T_t * I_t\]

One does often take out seasonality to better compare data and also create short term forecasts.

6.1.3 Cyclical and Irregular Variations

One can often rule out (or at least smooth out) irregularities by taking the moving average