Components of Time Series

Components of Time Series Model (Additive and Multiplicative Models)

To better understand, analyze, and forecast a time series, it is often helpful to think of the observed data as being the result of the combined influence of several underlying, unobservable factors or components. These components represent different types of variations or patterns present in the series over time. Decomposing a time series into these components helps in identifying the driving forces behind the data's movement and allows for targeted analysis and modeling of each part.

Traditionally, time series are considered to be composed of four main components:

1. Secular Trend (T or $T_t$): Represents the long-term, underlying direction or persistent movement of the time series over a considerable period.
2. Seasonal Variation (S or $S_t$): Represents patterns that repeat at fixed and known intervals within a standard period, typically a year or less, linked to calendar effects.
3. Cyclical Variation (C or $C_t$): Represents longer-term, wave-like oscillations around the trend that are not of a fixed period, often associated with broader economic or business cycles.
4. Irregular or Random Variation (I or $I_t$): Represents the unsystematic, unpredictable, and random fluctuations that remain after the other components have been removed. It is the residual variation caused by unforeseen events or random disturbances.

These four components are assumed to interact in a specific way to produce the actual observed values of the time series ($Y_t$). The two primary models describing this interaction are the Additive and Multiplicative models.

1. Additive Model

In the Additive Model, it is assumed that the observed value of the time series ($Y_t$) at any point in time $t$ is the sum of the values of the four components at that time.

where:

• $Y_t$ is the observed value of the time series at time $t$.
• $T_t$ is the value of the Trend component at time $t$.
• $S_t$ is the value of the Seasonal component at time $t$.
• $C_t$ is the value of the Cyclical component at time $t$.
• $I_t$ is the value of the Irregular component at time $t$.

Appropriateness: This model is generally suitable when the magnitude or amplitude of the seasonal, cyclical, and irregular fluctuations does not change significantly as the level of the trend changes. In other words, the variations are relatively constant in absolute terms. For example, if monthly sales typically fluctuate by around $\textsf{₹}50,000$ (above or below the trend) during a particular season, regardless of whether the annual total sales are $\textsf{₹}5$ Lakhs or $\textsf{₹}50$ Lakhs, an additive model might be appropriate. In this model, $S_t$, $C_t$, and $I_t$ represent deviations from the trend value $T_t$ in the original units of the series.

Visual Clue: On a time plot, if the seasonal swings have roughly the same vertical size across the entire series, regardless of the trend level, the additive model is likely suitable.

2. Multiplicative Model

In the Multiplicative Model, it is assumed that the observed value of the time series ($Y_t$) at any point in time $t$ is the product of the values of the four components at that time.

where the components are defined as above.

Appropriateness: This model is generally more appropriate when the magnitude or amplitude of the seasonal and cyclical variations is proportional to the level of the trend. In other words, the variations are relatively constant in percentage terms. For example, if monthly sales in December are consistently $20\%$ higher than the average sales for that year, regardless of whether the annual total sales trend is low or high, a multiplicative model is more suitable. In this model, $S_t$, $C_t$, and $I_t$ are typically expressed as ratios or indices fluctuating around 1 (e.g., a seasonal factor of 1.20 for December means sales are $20\%$ above the trend for that month).

Visual Clue: On a time plot, if the seasonal swings tend to get larger as the trend increases (heteroskedasticity with respect to level), the multiplicative model is likely suitable. This is common in economic and business series like sales, production, or prices, where fluctuations often grow with the overall scale of the activity.

Transformation to Additive: A significant advantage of the multiplicative model is that it can be easily transformed into an additive model by taking the natural logarithm of both sides:

This transforms the multiplicative relationship into a linear (additive) relationship on the logarithmic scale, allowing the use of methods developed for additive models. Analysis is performed on the log-transformed data, and results (like forecasts) are then back-transformed using the exponential function.

The choice between the additive and multiplicative models is crucial because it affects the methods used for decomposition and forecasting. While these are the two basic models, more complex time series can sometimes require hybrid models or more advanced modeling techniques.

Secular Trend (T): Definition and Properties

Definition

The Secular Trend, often simply referred to as the Trend Component ($T_t$), is the fundamental underlying movement of a time series that represents its smooth, long-term direction or tendency. It describes the general pattern of growth, decline, or stability of the series over an extended period, abstracting from the shorter-term fluctuations like seasonality, cycles, and irregular variations.

The trend reflects the aggregate influence of gradual, persistent, and long-acting forces that drive the variable's behaviour over a significant duration. Examples of such forces in economic contexts include:

• Population changes (growth or decline).
• Technological advancements and innovation.
• Changes in income levels and standards of living.
• Long-term shifts in consumer tastes and preferences.
• Capital formation and investment.
• Evolution of infrastructure and communication networks.
• Long-term government policies (e.g., trade liberalisation, environmental regulations).

The trend is what remains after the predictable periodic fluctuations and random noise have been removed or averaged out.

Properties

The key characteristics of the Secular Trend component are:

• Long-Term Nature: Trend describes movement over a considerable period, often spanning several years or even decades. It captures the overall path of the series over its entire observed history.
• Smoothness: Trend is generally a smooth, continuous movement without abrupt or sharp changes. While structural breaks (sudden shifts in the trend's level or slope) can occur due to major events (like economic crises, policy changes), the trend component itself aims to represent the underlying smooth progression.
• Direction: The trend can be upward (indicating growth, e.g., GDP, population), downward (indicating decline, e.g., sales of an obsolete product), or horizontal/stable (indicating no significant long-term change).
• Form/Shape: The trend line or curve can take various forms. Common types include:
  • Linear Trend: The series increases or decreases by a roughly constant absolute amount per unit of time (e.g., $T_t = a + bt$).
  • Non-Linear Trend: The rate of increase or decrease is not constant. Examples include:
    • Quadratic Trend ($T_t = a + bt + ct^2$)
    • Exponential Trend ($T_t = a \times b^t$ or $T_t = a \times e^{bt}$), often seen in growth processes where change is proportional to the current level (which appears as a linear trend on a logarithmic scale).
    • Logarithmic Trend ($\log Y_t = a + bt$ or $Y_t = a + b \log t$)
    • Logistic or S-shaped curves, often used for life cycles of products or technologies.
• Persistence: Once established, a trend tends to continue for a significant period, driven by long-acting forces. However, it is not necessarily irreversible and can eventually slow down, flatten, or change direction.

Estimating the trend is a crucial step in time series analysis. It allows analysts to understand the fundamental direction of the series, remove its influence to study other components, and use the estimated trend as a basis for long-term forecasts.

Seasonal Variation (S): Definition and Examples

Definition

Seasonal Variation ($S_t$) refers to the patterns or fluctuations in a time series that are regular, predictable, and repeat with a fixed period, typically within a year. These variations are directly or indirectly attributable to the calendar and are often tied to the natural seasons or specific dates and periods within the year.

The defining characteristic is the strict periodicity: the pattern repeats exactly the same way every period. The 'period' of seasonality is the number of observations within one full cycle of the seasonal pattern. For example, for monthly data exhibiting yearly seasonality, the period is 12. For quarterly data with yearly seasonality, the period is 4. For daily data with weekly seasonality, the period is 7.

Causes

Seasonal variations are caused by factors that occur regularly at specific times within the repeating period. Common causes include:

• Climate and Weather Conditions: These directly influence activities like agriculture (planting/harvesting seasons), tourism (peak/off-peak seasons), retail (seasonal clothing, heating/cooling needs), and energy consumption.
• Social Customs and Traditions: Cultural events, holidays, and festivals drive predictable patterns in spending (e.g., pre-festival shopping), travel, and consumption of specific goods.
• Administrative and Institutional Practices: School calendars (affecting transport, retail, and leisure), financial year cycles (impacting reporting and spending patterns), or tax deadlines can create regular patterns.
• Fixed Holidays: Events like Christmas, Diwali, Eid, New Year, etc., occurring at roughly the same time each year, significantly impact economic activity around those dates.

Properties

Key properties that define Seasonal Variation are:

• Regularity: The pattern occurs consistently at predictable times within the fixed period.
• Periodicity: The pattern repeats over a fixed, known interval (e.g., every 12 months, every 4 quarters, every 7 days). The length of this period is constant.
• Fixed Shape: The general shape of the seasonal pattern (e.g., peaks in summer, dips in winter for ice cream sales) is consistent from one period to the next, although its magnitude might change (especially in multiplicative models).
• Short-Term: The full cycle of seasonal variation is typically one year or less. Longer cycles that are not of a fixed period are classified as cyclical variation.

Examples

Seasonal patterns are evident in many real-world time series:

• Higher sales of air conditioners and cold drinks during summer months (e.g., April to June in India).
• Increased demand for heating oil and winter clothing during colder months.
• Peak agricultural production during harvest seasons.
• Significant increase in retail sales during the festive season (e.g., October-December in India due to Diwali, Christmas).
• Higher volume of online shopping on weekends compared to weekdays.
• Quarterly peaks in revenue for companies with seasonal businesses, or due to quarterly reporting cycles.
• Increased tourist arrivals during specific holiday seasons or pleasant weather periods.

Identifying and quantifying seasonal variation is important for accurate forecasting, as it allows us to predict the regular fluctuations. Furthermore, removing seasonality (seasonal adjustment) is often done for macroeconomic time series (like GDP, Industrial Production Index) to reveal the underlying trend and cyclical movements, which are considered more indicative of the overall state of the economy.

Cyclical Variation (C): Definition and Examples

Definition

Cyclical Variation ($C_t$) refers to the oscillatory movements in a time series that represent fluctuations around the trend but do not occur at fixed or regular intervals. These cycles are typically of longer duration than seasonal variations, usually spanning several years, and their length and intensity (amplitude) can vary from one cycle to the next.

Cyclical fluctuations are often associated with the broader patterns of economic or business activity, commonly known as business cycles. A business cycle involves phases of expansion (prosperity), peak (boom), contraction (recession), and trough (depression or slowdown), which repeat over time but without a strict, predictable periodicity.

Key Distinction from Seasonality

It is crucial to differentiate cyclical variation from seasonal variation, as both involve repeating patterns:

• Period: Seasonal variation repeats over a fixed and known period (e.g., 1 year, 1 week, 1 day). Cyclical variation repeats over a variable and generally unknown period, usually longer than one year (e.g., 3-10 years or more).
• Cause: Seasonality is caused by factors tied to the calendar and natural seasons (weather, holidays, social customs). Cyclical variation is typically driven by more complex and often macroeconomic forces like changes in aggregate demand, investment patterns, interest rates, credit availability, consumer and business confidence, and inventory levels.
• Predictability: Seasonal patterns are highly predictable once identified. Cyclical patterns are much more difficult to predict accurately due to their irregular duration and amplitude. Forecasting the turning points (peaks and troughs) of cycles is a major challenge in economic forecasting.
• Analysis Focus: Seasonal variation is often explicitly removed from a series (seasonal adjustment) to better reveal the underlying trend and cycle. Cycles are usually analyzed in conjunction with the trend (often called the 'Trend-Cycle' component) due to the difficulty in separating them precisely, especially for shorter time series.

Properties

The main characteristics of Cyclical Variation include:

• Oscillatory Nature: The component exhibits recurring upward and downward swings around the long-term trend.
• Medium to Long-Term Duration: A complete cycle (from peak to peak or trough to trough) typically lasts for more than one year, often ranging from 3 to 10 years or even longer.
• Variable Periodicity and Amplitude: The length of one cycle and the magnitude of the swing from peak to trough (or vice versa) are not fixed. They can vary significantly from one cycle to the next, making the component irregular and challenging to model deterministically.
• Associated with Economic/Business Cycles: Cyclical fluctuations in many time series (like GDP, industrial production, employment, interest rates) tend to move together, reflecting the overall state of the economy or industry.

Examples

Cyclical variation is prominent in many macroeconomic and industry-specific time series:

• Phases of the Business Cycle: The most classic example is the cyclical pattern observed in GDP, unemployment rates, industrial production, and consumer spending, reflecting periods of economic expansion and contraction (recession).
• Real Estate or Housing Market Cycles: Periods of boom (rapid price increase and construction) followed by busts (price stagnation or decline and reduced construction) often span several years and are cyclical.
• Inventory Cycles: Businesses adjust inventory levels in response to expected demand, leading to cycles of accumulation and depletion that can last for a few years.
• Investment Cycles: Large-scale investments in plant and machinery or infrastructure tend to occur in waves, contributing to cyclical patterns in capital goods industries.
• Long-Term Cycles in Agricultural Commodity Prices: While influenced by weather (seasonal/irregular), prices of some commodities can also show multi-year cycles influenced by planting decisions based on previous years' prices.

Analyzing cyclical components helps in understanding the underlying dynamics of business and economic activity, although predicting their precise turning points remains one of the most difficult aspects of time series forecasting.

Irregular/Random Variation (I): Definition and Nature

Definition

The Irregular Variation ($I_t$), also often referred to as the random component, residual component, or noise, constitutes the part of a time series that remains after the systematic components – Trend ($T_t$), Seasonal Variation ($S_t$), and Cyclical Variation ($C_t$) – have been identified and removed. It represents the unpredictable, erratic, and non-repeating fluctuations in the data.

Conceptually, the irregular component captures everything in the time series that cannot be explained by the long-term trend, the fixed seasonal patterns, or the broader business cycles. It is the residual, unsystematic variability.

Causes

Irregular variations are caused by a multitude of unpredictable and often one-off events that impact the time series. These causes can be broadly categorised as:

• Unforeseen Events: Events that are difficult or impossible to anticipate, such as natural disasters (e.g., floods, earthquakes, tsunamis), major accidents, sudden political upheavals (e.g., coups, unexpected policy announcements), or international crises (e.g., wars, pandemics like COVID-19).
• Strikes and Lockouts: Labour disputes can cause temporary disruptions in production or service delivery.
• Unusual Weather Conditions: Extreme or unexpected weather (e.g., unseasonal cold snap, unusually heavy rainfall) can temporarily affect activities like retail sales, agriculture, or construction.
• Measurement Errors: Inaccuracies in data collection or recording can introduce random noise.
• Purely Random or Chance Occurrences: There is an inherent level of randomness in many real-world processes that cannot be explained by systematic factors.

Major unforeseen events that cause significant, lasting impacts on the level or direction of the series are sometimes treated as 'outliers' or 'structural breaks' rather than purely irregular variations, especially if they alter the trend or seasonal patterns going forward.

Nature and Properties

The key nature and properties of the Irregular Component are:

• Unpredictability: By definition, irregular variations cannot be predicted in advance based on past patterns or any systematic model.
• Randomness: In well-modelled time series, the irregular component is assumed to be random, typically with a mean of zero (in the additive model) or an average around 1 (in the multiplicative model), and a constant variance. Ideally, the irregular component should resemble 'white noise', meaning its values are uncorrelated with each other across time.
• Short-Term Impact: The effects of irregular events are generally assumed to be transient or temporary. Unlike trends or cycles, they don't represent a sustained force driving the series, although a severe irregular event can sometimes trigger changes in the other components.
• Residual: The irregular component is what is left in the data after the systematic patterns ($T, S, C$) have been accounted for. In time series modelling, a key diagnostic step is to examine the residuals of the fitted model to ensure they behave like a random, irregular series, indicating that the systematic components have been adequately captured.
• Error Term: In statistical modeling, the irregular component is often treated as the error term in the model, representing the unexplained variation.

While we cannot forecast the irregular component itself (as it's unpredictable), understanding its typical magnitude (e.g., its standard deviation) is important for determining the confidence or prediction intervals around forecasts made using the systematic components. The size of the irregular component gives an indication of the inherent volatility and unpredictability of the series.

Decomposition of Time Series

Concept

Time Series Decomposition is a statistical technique used to break down an observed time series into its underlying components: Trend ($T_t$), Seasonal Variation ($S_t$), Cyclical Variation ($C_t$), and Irregular Variation ($I_t$). The goal is to separate the combined fluctuations in the raw data into these constituent parts to understand their individual behaviour and contribution to the overall series.

Decomposition is typically based on either the additive ($Y_t = T_t + S_t + C_t + I_t$) or multiplicative ($Y_t = T_t \times S_t \times C_t \times I_t$) model, chosen based on the observed relationship between the magnitude of fluctuations and the level of the series.

Purpose of Decomposition

Decomposing a time series serves several important purposes:

• Enhanced Understanding: It allows analysts to gain deeper insights into the underlying patterns driving the series. By viewing the separated components, it becomes easier to discern the long-term direction (trend), the regular repeating patterns (seasonality), and the wave-like fluctuations (cycle).
• Quantifying Components: Decomposition methods provide estimates for the values of the trend, seasonal effects, and cyclical components at each point in time. This quantitative information is valuable for descriptive analysis.
• Seasonal Adjustment: A very common application is seasonal adjustment, where the estimated seasonal component ($S_t$) is removed from the original series ($Y_t$). In an additive model, the seasonally adjusted series is $Y_t - S_t = T_t + C_t + I_t$. In a multiplicative model, it is $Y_t / S_t = T_t \times C_t \times I_t$. Seasonally adjusted series are widely used, especially for macroeconomic indicators (like CPI, WPI, IIP, GDP), as they reveal the non-seasonal trend and cyclical movements more clearly, facilitating comparisons between consecutive periods.
• Forecasting (Component-Based): While modern forecasting models often handle components implicitly (like ARIMA), traditional forecasting approaches sometimes involve forecasting each component separately (e.g., fitting a model to the trend-cycle, using average seasonal factors) and then combining the forecasts to predict the original series.
• Identifying Irregular Events: The irregular component ($I_t$) represents the unexpected fluctuations. Analysing the magnitude and timing of large irregular values can help identify specific historical events that significantly impacted the series.

General Approach (Conceptual Steps)

Various methods exist for time series decomposition (e.g., Classical Decomposition using Moving Averages, Census Method I, X-11, X-12-ARIMA, X-13ARIMA-SEATS, STL). While the specific calculations differ, the conceptual process often follows a sequence of steps, typically involving smoothing techniques like moving averages:

1. Estimate the Trend-Cycle (TC): The first step is usually to estimate the combined Trend and Cyclical components. This is often done by applying a moving average filter to the original series $Y_t$. The moving average smoothes out the short-term seasonal and irregular fluctuations, leaving behind the longer-term Trend-Cycle component $(TC)_t$. The length of the moving average should typically be equal to the period of seasonality (e.g., 12 for monthly data, 4 for quarterly data).
2. Isolate the Seasonal and Irregular (SI): Once the Trend-Cycle is estimated, it is removed from the original series to obtain the combined Seasonal and Irregular components $(SI)_t$. In an additive model, this is done by subtraction: $(SI)_t = Y_t - (TC)_t$. In a multiplicative model, it is done by division: $(SI)_t = Y_t / (TC)_t$.
3. Estimate the Seasonal Component (S): The Seasonal component is estimated by averaging the $(SI)_t$ values for each specific season across all years. For example, for monthly data, all January $(SI)$ values are averaged to get the seasonal factor for January; all February $(SI)$ values are averaged for February, and so on. These preliminary seasonal factors are then typically adjusted (e.g., by subtracting their average in the additive case, or dividing by their average in the multiplicative case) so that they sum to zero (additive) or average to one (multiplicative) over a full seasonal period. This adjusted average is the estimated Seasonal component $S_t$ for each season.
4. Isolate the Irregular Component (I): Finally, the estimated Seasonal component is removed from the $(SI)_t$ series to obtain the residual, which is the estimated Irregular component $I_t$. In an additive model: $I_t = (SI)_t - S_t$. In a multiplicative model: $I_t = (SI)_t / S_t$.
5. Estimate the Trend (T) and Cycle (C): In many decomposition methods, the Trend and Cycle are not estimated separately in the first pass. The initial step estimates the combined Trend-Cycle $(TC)$. If a separate trend is needed, a smoothing method (like moving averages again, or curve fitting) might be applied to the $(TC)_t$ series to extract the smooth trend $T_t$, leaving the residual cyclical component $C_t = (TC)_t - T_t$ (additive) or $C_t = (TC)_t / T_t$ (multiplicative).

The output of a decomposition process is a table or plot showing the original series alongside the estimated values for its $T_t$, $S_t$, $(TC)_t$ or separate $T_t$ and $C_t$, and $I_t$ components. This breakdown provides a powerful visual and statistical summary of the time series dynamics.