9.2 Identifying seasonal patterns in ACF and PACF
3 min read•july 22, 2024
Seasonal patterns in time series data repeat at fixed intervals, like yearly temperature changes. ACF and PACF plots help spot these patterns by showing significant spikes at lags that are multiples of the seasonal period.
To identify seasonal components, look for spikes at regular intervals in ACF and PACF plots. The lag of the first significant spike indicates the seasonal period. This helps distinguish between short-term fluctuations and long-term, recurring patterns in the data.
Identifying Seasonal Patterns in ACF and PACF
Seasonal patterns in ACF and PACF
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- ACF and PACF plots help identify seasonal patterns in time series data which are regular, repeating patterns occurring at fixed intervals (temperature data showing yearly )
- shows significant spikes at lags that are multiples of the seasonal period (lags 12, 24, 36 for monthly data with annual seasonality) with spikes gradually decreasing in magnitude as lag increases
- PACF plot shows significant spikes at lags that are multiples of the seasonal period with spikes cutting off abruptly after the seasonal lag (significant spike at lag 12 followed by insignificant spikes at lags 24, 36 for monthly data with annual seasonality)
Determining seasonal periods
- Seasonal period is the number of time steps between each repetition of the seasonal pattern (12 for monthly data with annual seasonality, 4 for quarterly data with annual seasonality)
- In ACF plots, the lag at which the first significant spike appears represents the seasonal period
- In PACF plots, the lag at which a significant spike appears followed by a cutoff in subsequent lags represents the seasonal period
Order of seasonal AR and MA terms
- Seasonal autoregressive (SAR) terms are identified using the PACF plot
- The order of the SAR term is determined by the number of significant spikes at lags that are multiples of the seasonal period
- A significant spike at lag 12 in the PACF plot indicates an SAR term of order 1, denoted as SAR(1)
- Seasonal moving average (SMA) terms are identified using the ACF plot
- The order of the SMA term is determined by the number of significant spikes at lags that are multiples of the seasonal period
- Significant spikes at lags 12 and 24 in the ACF plot indicate an SMA term of order 2, denoted as SMA(2)
Seasonal vs non-seasonal components
- Non-seasonal components are represented by significant spikes at short lags (lags 1, 2, 3) in both ACF and PACF plots indicating the presence of short-term dependencies in the time series data (daily stock price fluctuations)
- Seasonal components are represented by significant spikes at lags that are multiples of the seasonal period in both ACF and PACF plots indicating the presence of long-term, recurring patterns in the time series data (monthly retail sales data showing holiday seasonality)
- To distinguish between seasonal and non-seasonal components, analyze the ACF and PACF plots simultaneously
- Identify significant spikes at short lags (non-seasonal) and at lags that are multiples of the seasonal period (seasonal)
- Consider the magnitude and pattern of the spikes to determine the relative importance of seasonal and non-seasonal components (strong seasonal spikes with weak non-seasonal spikes suggest a predominantly seasonal time series)
Key Terms to Review (13)
Acf plot: An acf plot, or autocorrelation function plot, is a graphical representation that shows the correlation of a time series with its own past values over various time lags. This plot is essential in identifying the presence of patterns such as seasonality and trends in the data, which helps in understanding the underlying structure of a time series. It serves as a key tool in diagnosing the characteristics of a time series and guides model selection for forecasting.
Additive seasonality: Additive seasonality refers to a situation in a time series where seasonal variations are constant and can be added directly to the trend component. This means that the seasonal effect does not change in magnitude or scale with the level of the data, allowing for straightforward modeling and interpretation of seasonal patterns. In this context, understanding how these seasonal fluctuations interact with other components like trend and noise is crucial for accurate forecasting.
Autocorrelation: Autocorrelation is a statistical measure that assesses the relationship between a variable's current value and its past values over time. It helps in identifying patterns and dependencies in time series data, which is crucial for understanding trends, cycles, and seasonality within the dataset.
Ets model: The ets model, which stands for Error, Trend, and Seasonality, is a forecasting method that captures the underlying patterns in time series data. It is particularly useful for handling data with seasonal fluctuations, allowing for accurate predictions by decomposing the series into its components. By using exponential smoothing, the ets model provides a flexible approach to account for various types of trends and seasonal effects.
Lagged correlation: Lagged correlation is a statistical measure that assesses the relationship between a time series and a lagged version of itself over time. It helps in identifying whether past values of a series influence its current or future values, which is crucial for recognizing patterns and dependencies in time series data, especially in determining seasonal effects.
Lagged variable: A lagged variable is a previous time period's value of a variable used in a time series analysis to help understand its impact on the current value. By including lagged variables, analysts can capture the influence of past events or observations on present outcomes, which is crucial when identifying patterns such as seasonality and trends in data. This concept is especially relevant in examining autocorrelation and partial autocorrelation functions, where lagged variables reveal the relationship between current and past values over time.
Multiplicative seasonality: Multiplicative seasonality occurs when the seasonal fluctuations in a time series are proportional to the level of the series itself. This means that during high-demand periods, the fluctuations are larger, and during low-demand periods, they are smaller, suggesting that seasonal effects amplify as the underlying level of the series changes. Recognizing this pattern is crucial for accurate forecasting and modeling, especially when utilizing specific statistical approaches that accommodate seasonal variations.
Null hypothesis: The null hypothesis is a statement in statistical testing that asserts there is no effect or no difference between groups, essentially serving as a default position. It provides a baseline for comparison, allowing researchers to determine if observed data significantly deviates from this assumption, which can be crucial when assessing relationships and patterns in data over time.
Partial autocorrelation: Partial autocorrelation is a statistical measure that quantifies the correlation between a time series and its own lagged values, while controlling for the effects of intermediate lags. This allows for a clearer understanding of the direct relationship between an observation and its previous values, making it useful in identifying the order of autoregressive models. By examining the partial autocorrelation function (PACF), analysts can discern patterns, assess model suitability, and evaluate the presence of seasonality in time series data.
SARIMA: SARIMA stands for Seasonal Autoregressive Integrated Moving Average, a sophisticated statistical model used for forecasting time series data that exhibit both seasonal patterns and trends. This model extends the ARIMA framework by incorporating seasonal components, allowing it to effectively capture and predict complex seasonal fluctuations in data, making it a popular choice in various fields such as economics, meteorology, and hydrology.
Seasonal decomposition: Seasonal decomposition is a statistical technique used to break down a time series into its underlying components: trend, seasonal, and residual components. This process allows for better understanding and analysis of data by isolating seasonal patterns and trends that may not be immediately apparent in the raw data.
Seasonality: Seasonality refers to periodic fluctuations in time series data that occur at regular intervals, often influenced by seasonal factors like weather, holidays, or economic cycles. These patterns help in identifying trends and making predictions by accounting for variations that repeat over specific timeframes.
Time series plot: A time series plot is a graphical representation of data points in a time-ordered sequence, allowing viewers to visualize trends, seasonal patterns, and potential anomalies over time. This type of plot helps in analyzing how data points change at different time intervals, making it essential for understanding the underlying patterns and behaviors in time series data.