2.2 Time series patterns and their identification

Time series patterns are crucial in business forecasting. They include trends, seasonal fluctuations, cycles, and random variations. Understanding these patterns helps predict future values and make informed decisions.

Identifying time series patterns involves visual inspection, time series plots, and advanced techniques like seasonal and autocorrelation plots. These methods reveal underlying structures in data, enabling more accurate forecasts and better business planning.

Trend Patterns

Linear and Exponential Trends

• Linear trend represents a constant rate of change over time in a time series
  • Characterized by a straight line when plotted on a graph
  • Equation: $Y_t = b_0 + b_1t$, where $Y_t$ is the value at time t, $b_0$ is the y-intercept, and $b_1$ is the slope
  • Commonly observed in steady economic growth or population increase
• Exponential trend shows a constant percentage rate of change over time
  • Appears as a curved line that steepens over time when plotted
  • Equation: $Y_t = b_0 * (1 + r)^t$, where r is the growth rate
  • Often seen in compound interest calculations or rapid technological adoption rates

Recurring Patterns

Seasonal and Cyclic Patterns

• Seasonal patterns occur at fixed intervals within a year
  • Influenced by factors like weather, holidays, or cultural events
  • Repeat consistently every year (retail sales spike during holidays)
  • Can be additive (constant amplitude) or multiplicative (increasing amplitude over time)
• Cyclic patterns fluctuate over longer periods, typically more than one year
  • Not tied to calendar-based events
  • Often associated with economic or business cycles (boom and bust cycles)
  • Duration and amplitude can vary between cycles

Random Variations and Their Impact

• Random variations represent unpredictable fluctuations in a time series
  • Also known as noise or irregular component
  • Cannot be explained by trends, seasonal, or cyclic patterns
  • Caused by unexpected events or measurement errors (natural disasters, political changes)
• Impact of random variations on forecasting
  • Can obscure underlying patterns, making prediction more challenging
  • Necessitate the use of statistical techniques to separate signal from noise
  • Importance of considering confidence intervals in forecasts to account for randomness

Identification Techniques

Visual Inspection and Time Series Plots

• Visual inspection involves examining raw data or simple plots to identify patterns
  • Requires experience and intuition to interpret effectively
  • Can reveal obvious trends, seasonality, or outliers
• Time series plots graph data points against time on the x-axis
  • Provide a clear visual representation of the overall pattern
  • Help identify trends, cycles, and potential outliers
  • Can be enhanced with smoothing techniques (moving averages) to highlight underlying patterns

Advanced Plotting Techniques

• Seasonal plots overlay data from different years to highlight recurring patterns
  • Useful for identifying consistent seasonal effects
  • Can reveal changes in seasonal patterns over time
  • Help distinguish between additive and multiplicative seasonality
• Autocorrelation plots measure the correlation between a time series and its lagged values
  • Also known as correlogram
  • Reveal periodic patterns and the strength of time dependencies
  • Assist in identifying appropriate models for forecasting (ARIMA models)
  • Equation for autocorrelation: $r_k = \frac{\sum_{t=k+1}^n (y_t - \bar{y})(y_{t-k} - \bar{y})}{\sum_{t=1}^n (y_t - \bar{y})^2}$, where $r_k$ is the autocorrelation at lag k