Understanding Patterns

Describing Time-Series Patterns

• A trend exists when there is a long-term change in the data

  • This long-term change can be non-linear or linear

• A seasonal trend refers to a trend consisting of seasonal factors

  • E.g. time of the year or the day of the week
  • Seasonality always occurs over a fixed and known time interval and frequency

• A cycle refers to a trend that doesn't have a fixed frequency

  • E.g. a business cycle may include fluctuations lasting at least 2 years
  • E.g. a seasonality are cycles repeating regularly over time

Illustrating Seasonal Plots

• A seasonality chart is similar to a simple time-series chart
• However, the data is grouped into years and plotted against individual seasons

• By plotting seasonal plots, we can determine if:

  • A seasonality trend exists in our data
  • Any outliers exist
• The following is an example of this chart:

Summarizing Autocorrelation in Time-Series Data

• Correlation measures the strength of a linear relationship between two variables

• Autocorrelation measures the strength of a linear relationship between lagged values of a variable

  • Roughly, it determines whether there are trends amongst the values of a particular variable (across an indexed time variable)
• In other words, they can be used for checking randomness in data values across time

• Typically, autocorrelation is measured for lagged values using autocorrelation coefficients

  • These are calculated similarly as simple correlation coefficients

$$r_{k} = \frac{\sum_{t=k+1}^{T} (y_{t} - \bar{y}) (y_{t-k} - \bar{y})}{\sum_{t=1}^{T} (y_{t} - \bar{y})^{2}}$$

Illustrating Lag Plots and Autocorrelation

• Lag plots illustrate $y_{t}$ plotted against $y_{t-k}$ for different $k$ time values
• Lag plots help by showing if there is any autocorrelation or not
• Autocorrelation is a measure of the linear relationship between values at a specific time value and its values at previous time values
• The following $8$ autocorrelation coefficients correspond to bars in the correlogram:

Interpreting Correlogram for Time-Series Data

• A correlogram plots the autocorrelation coefficients on the y-axis
• A correlogram plots lagged time-values $t-k$ with respect to an initial time-value $t$

• A correlogram can have the following applications:

  • Is the data random?
  • Are observations correlated with recent observations?
  • Is there a seasonality trend?
  • Is there white noise?

• Suppose $k=1$, then we may observe the correlation of:

  • October values with November values
  • Or November values with December values
  • Etc.

• Suppose $k=2$, then we may observe the correlation of:

  • September values with November values
  • Or October values with December values
  • Or November values with January values
  • Etc

• In the following chart, there are high positive correlations in the first chart

  • These correlations slowly decline with increasing lags
  • Indicating, these is a high amount of autocorrelation, especially in recent time-points
  • Which, we'll need to account for in modeling

• In the following chart, there are small correlations in the second chart

  • Thus, there aren't any time trends

Illustrating White Noise in Time-Series Charts

• Time-series values $y_{t}$ without any autocorrelation is known as white noise

• In other words, the values $y_{t}$ are randomly distributed across time

  • Thus, aren't correlated with a time variable
• The second chart in the above image is an example of white noise

References

• Wiki about Correlograms
• Reading a Correlogram Chart
• Textbook for Forecasting: Principles and Practices