1 Fundamental concepts
1.1 Definition and scope
Time series analysis studies data collected in temporal order, where each observation is linked to a specific point or interval in time. Unlike unordered data sets, time series emphasize sequence, dependence, and change. The field addresses both descriptive questions, such as how a variable has evolved, and predictive questions, such as how it may behave in the future.
The scope of time series analysis ranges from simple graphical inspection to sophisticated probabilistic modeling. It includes methods for detecting structure, comparing patterns across time, and extracting signals from noisy measurements. Because time-indexed data appear in many domains, the subject has become a core part of statistics, data science, and applied forecasting.
1.2 Time-ordered observations
A time series consists of observations arranged in chronological order, often at regular intervals such as minutes, days, or months. The time spacing may also be irregular, depending on the source of the data. In either case, the arrangement matters because nearby observations are frequently related.
This temporal dependence distinguishes time series from cross-sectional data. The value at one time point may reflect earlier values, external influences, or latent processes that unfold gradually. As a result, the ordering of observations is a defining feature rather than a mere indexing device.
1.3 Deterministic and stochastic components
Many time series are described as a combination of deterministic and stochastic elements. Deterministic components include fixed patterns such as trends or repeating seasonal effects. Stochastic components represent randomness, unpredictable shocks, or measurement error.
Separating these parts helps analysts understand what is systematic and what is uncertain. In practice, a time series may contain a smooth long-term rise, regular periodic changes, and irregular fluctuations around that structure. Modeling often focuses on representing each component as clearly as possible.
1.4 Goals of time series analysis
The main goals of time series analysis are description, inference, and forecasting. Description involves identifying visible patterns and summarizing how the series behaves over time. Inference concerns estimating relationships and testing hypotheses about the underlying process.
Forecasting aims to use past observations to predict future values. Beyond prediction, time series methods may also support monitoring, anomaly detection, control, and decision-making. These tasks are especially important when the timing of events carries substantive meaning.
2 Characteristics of time series data
2.1 Trend
A trend is a persistent upward or downward movement in a series over a long period. It may reflect growth, decline, technological change, demographic shifts, or other broad influences. Trends can be linear, curved, or piecewise, depending on the context.
Identifying a trend is useful because it can mask shorter-term behavior. Analysts often remove or model the trend separately so that other patterns become easier to study. In forecasting, a trend may be extrapolated cautiously when supported by the data.
2.2 Seasonality
Seasonality refers to regular patterns that repeat at a fixed period, such as daily, weekly, monthly, or yearly cycles. These patterns are usually tied to calendar effects, weather, routines, or institutional schedules. Seasonal variation tends to appear with predictable timing and similar magnitude from one cycle to the next.
Recognizing seasonality helps improve interpretation and forecast accuracy. For example, sales, energy demand, and temperature often show recurring seasonal structure. If unaccounted for, seasonality can distort estimates of trend and short-term change.
2.3 Cyclic behavior
Cyclic behavior consists of wave-like movements that recur, but not necessarily with a fixed period. Cycles are often associated with business activity, natural processes, or feedback mechanisms. Compared with seasonality, cycles are usually longer and less regular.
Because their timing is less exact, cycles are harder to isolate and predict. They may overlap with trend or seasonal effects, making decomposition more challenging. Analysts often use smoothing or model-based approaches to reveal cyclic structure.
2.4 Irregular variation
Irregular variation is the residual fluctuation left after systematic patterns have been considered. It includes random disturbances, unexpected shocks, and small-scale noise. Some irregularity is natural in nearly all observed series.
This component can arise from measurement limitations, sudden events, or unmodeled influences. While it may appear unstructured, it still matters because it affects uncertainty and forecast precision. A good model aims to explain as much regular behavior as possible without overfitting the noise.
2.5 Autocorrelation
Autocorrelation measures the relationship between a series and past versions of itself. It is one of the central ideas in time series analysis because observations are often not independent. Positive autocorrelation suggests that nearby values tend to resemble one another, while negative autocorrelation indicates alternating movement.
Autocorrelation helps reveal hidden structure and guides model choice. Strong dependence at short lags can indicate persistence, whereas specific lag patterns may suggest seasonality or periodic effects.
2.5.1 Lag dependence
Lag dependence describes how the current value relates to earlier values separated by one or more time steps. The lag is the delay between observations. Different lags can show different strengths of association.
Studying lag dependence allows analysts to identify the memory of a process. If a series depends heavily on recent history, short-lag relationships will dominate. In many models, lag structure is the basis for prediction.
2.5.2 Partial autocorrelation
Partial autocorrelation measures the association between values at a given lag after removing the influence of shorter lags. It isolates the direct contribution of a specific lag rather than the combined effect of intervening values. This makes it especially helpful in model identification.
Partial autocorrelation is often used alongside autocorrelation plots. Together, these tools help distinguish between autoregressive and moving average behavior. They also assist in deciding how many lag terms to include.
3 Data preparation
3.1 Missing data treatment
Missing values are common in time series because of recording failures, sensor outages, or incomplete reporting. Treating them carefully is important because gaps can disrupt estimation and distort temporal patterns. The appropriate method depends on the amount, location, and cause of the missingness.
Common approaches include interpolation, imputation, carrying forward recent values, or using model-based methods. In some cases, analysts may omit incomplete segments if the loss is small. However, ignoring missing data entirely can lead to biased results or reduced accuracy.
3.2 Smoothing and filtering
Smoothing reduces short-term noise to reveal broader patterns. Simple methods include moving averages and weighted averages, while more advanced filters may separate low-frequency structure from high-frequency variation. Filtering is widely used for exploratory analysis and preprocessing.
These techniques are helpful when raw data are too volatile for direct interpretation. They can make trends and cycles more visible, but they may also blur sharp changes. For that reason, smoothing is usually applied with care and with attention to the purpose of the analysis.
3.3 Scaling and transformation
Scaling and transformation adjust the numerical form of a series to improve comparability, stabilize variance, or better satisfy model assumptions. Common transformations include logarithms, square roots, and standardization. Such changes can make patterns more uniform and estimation more reliable.
Transformations are especially useful when the variability changes with the level of the series. They may also help reduce skewness or make relationships more nearly linear. In practice, the choice of transformation is often guided by data behavior and modeling goals.
3.3.1 Differencing
Differencing replaces each observation with the change from a previous one. It is frequently used to reduce trend and make a series more stable over time. First differences compare consecutive values, while higher-order differences compare values farther apart.
This method is central in many forecasting models because it can convert a nonstationary series into a more manageable form. Differencing highlights short-term movement rather than absolute level. Excessive differencing, however, may remove useful information.
3.3.2 Detrending
Detrending removes or accounts for a long-term pattern so that shorter-term dynamics can be examined more clearly. The trend may be estimated through regression, smoothing, or decomposition and then subtracted from the original series. The result is a residual series centered around the underlying baseline.
Detrending is useful when the trend is strong enough to obscure other features. It can improve the interpretability of remaining fluctuations and support further statistical analysis. In some contexts, detrending is preferable to differencing because it preserves more of the original structure.
3.4 Resampling and aggregation
Resampling changes the time resolution of a series by summarizing observations over new intervals. Aggregation may convert daily data into monthly totals, or minute-level data into hourly averages. This process can reduce noise and align multiple series to a common scale.
Resampling is often used when the original sampling frequency is too fine for the analysis. It may also be necessary when combining data from different sources. The choice of aggregation level affects the patterns that remain visible, so it should reflect the question being studied.
4 Time series decomposition
4.1 Additive models
In an additive decomposition, a series is represented as the sum of separate components, typically trend, seasonality, and irregular error. This structure is suitable when the size of the seasonal effect remains relatively constant over time. It provides a clear and intuitive separation of behavior.
Additive models are common in exploratory analysis and forecasting. They are easy to interpret and can be fitted with relatively simple methods. Their main limitation is that they may not capture changes in amplitude that depend on the level of the series.
4.2 Multiplicative models
A multiplicative decomposition expresses a series as the product of components. This form is useful when seasonal variation grows or shrinks in proportion to the overall level. It is often applied when fluctuations become larger as the series increases.
Multiplicative models can be converted to additive form by using a logarithmic transformation in many cases. This makes them easier to analyze with standard tools. They are particularly helpful for positive-valued data with scale-dependent variability.
4.3 Classical decomposition
Classical decomposition separates a series into trend, seasonal, and residual components using moving averages and averaging techniques. It is one of the oldest and most intuitive approaches in time series analysis. The method is often used for regular, evenly spaced data.
Although classical decomposition is simple, it relies on assumptions about the stability of seasonal patterns and the smoothness of trend. It is mainly useful as an exploratory tool. More flexible methods may be preferred when the data contain irregularities or changing seasonal behavior.
4.4 STL decomposition
STL decomposition uses locally weighted regression to estimate trend and seasonality. The method is flexible and can adapt to changing patterns better than classical approaches. It is especially useful when seasonal effects evolve gradually over time.
STL is robust to outliers and can accommodate complex series with nonconstant behavior. It produces separate components that are easy to inspect and analyze. Because of its adaptability, it has become a standard decomposition technique in many applications.
5 Statistical foundations
5.1 Stationarity
Stationarity refers to a stable statistical structure over time. In a stationary series, properties such as mean, variance, and dependence patterns do not change systematically. This concept is fundamental because many methods assume that the underlying process is stable enough for reliable inference.
Stationarity does not mean a series is constant. Instead, it implies that fluctuations are governed by consistent rules. When a series is not stationary, transformations such as differencing or detrending may be applied.
5.1.1 Weak stationarity
Weak stationarity, also called covariance stationarity, requires a constant mean and variance, along with autocovariances that depend only on lag, not on time location. This is the most common stationarity assumption in applied work. It focuses on second-order properties rather than the full distribution.
Many linear time series models are built under weak stationarity. The condition makes it possible to describe dependence in a compact and mathematically tractable way. It also supports estimation of long-run relationships from observed data.
5.1.2 Strict stationarity
Strict stationarity requires the full joint distribution of the process to remain unchanged under time shifts. This is a stronger and more comprehensive condition than weak stationarity. It implies that every statistical feature of the series is time-invariant.
Strict stationarity is important in theoretical work because it provides a rigorous foundation for probabilistic analysis. In practice, it is often difficult to verify directly. Many applied methods therefore rely on weaker conditions that are easier to assess.
5.2 Random walks
A random walk is a process in which each value equals the previous value plus a random change. It is a basic model for nonstationary behavior and a useful benchmark in forecasting. Random walks often appear in finance and other contexts where changes accumulate over time.
Because shocks persist indefinitely, random walks can drift far from their starting point. Their future path is uncertain, but the increments may still be modeled as stable. This makes them central to understanding integrated processes and unit-root behavior.
5.3 White noise
White noise is a sequence of random variables with constant mean, constant variance, and no serial correlation. It represents purely unpredictable variation and often serves as an idealized error term. In a well-specified model, the residuals should resemble white noise.
White noise is useful as a baseline because it contains no exploitable temporal structure. If a transformed series behaves like white noise, little predictive information remains. Deviations from white noise often indicate that a model has not captured all relevant dynamics.
5.4 Autocovariance and autocorrelation functions
The autocovariance function measures how observations at different lags vary together. The autocorrelation function rescales this quantity to provide a standardized measure of dependence. Both functions summarize the memory structure of a series.
These functions are essential for identifying patterns, diagnosing models, and comparing alternative specifications. They help reveal whether dependence fades quickly or persists over longer intervals. In many analyses, the shape of the autocorrelation function is a primary clue about the underlying process.
6 Models and methods
6.1 Moving average models
Moving average models represent a series as a combination of current and past random shocks. The dependence arises through lagged error terms rather than lagged observations. This approach can capture short-term correlation effectively.
Such models are useful when disturbances influence the series for a limited number of periods. They are often combined with other structures to represent more complex behavior. The order of the model determines how many past shocks are included.
6.2 Autoregressive models
Autoregressive models express the current value as a function of past values and a random error. They are widely used because they directly encode persistence and feedback. A higher-order autoregressive model includes several lagged observations.
These models are especially effective when recent history strongly influences the present. They can approximate many types of temporal dependence with relatively simple equations. Their interpretation is intuitive: the series evolves partly from its own earlier states.
6.3 ARMA models
ARMA models combine autoregressive and moving average terms. This hybrid structure captures dependence both through past values and through past shocks. It is suitable for stationary series with short memory.
ARMA models are central in classical time series theory because they balance flexibility and mathematical clarity. They are often used as building blocks for more elaborate models. Their parameters describe both persistence and transient shock effects.
6.4 ARIMA models
ARIMA models extend ARMA methods by adding differencing to handle nonstationary series. The acronym refers to autoregressive, integrated, and moving average components. This framework is especially useful for data with trends that must be removed before modeling.
ARIMA remains one of the most influential families in forecasting. It can represent a broad range of patterns using a relatively compact specification. When appropriately chosen, it can provide strong out-of-sample performance.
6.4.1 Seasonal ARIMA
Seasonal ARIMA incorporates seasonal lag structures alongside the usual ARIMA terms. It is designed for data with repeating patterns at fixed intervals, such as monthly or quarterly series. Seasonal differencing may be used to stabilize recurring fluctuations.
This extension allows the model to capture both short-term and seasonal dependence. It is particularly effective when calendar-based effects play a major role. Seasonal ARIMA is common in demand forecasting and other regularly sampled applications.
6.4.2 Integrated processes
Integrated processes are time series that become stationary only after differencing. The number of differences needed is called the order of integration. Such processes often exhibit persistent movement and are closely related to unit roots.
Understanding integration is important because it affects model choice and statistical testing. Integrated behavior suggests that shocks have lasting effects on the level of the series. Differencing helps convert the process into one that can be analyzed with stationary methods.
6.5 Exponential smoothing
Exponential smoothing produces forecasts by weighting recent observations more heavily than older ones. The weights decline exponentially over time, giving the method a simple and adaptive character. It is widely used because of its effectiveness and ease of implementation.
Different versions of exponential smoothing handle trend and seasonality in various ways. The method is well suited to practical forecasting problems, especially when data patterns are stable but not too complex. It often performs well even with limited historical information.
6.5.1 Holt’s method
Holt’s method extends simple exponential smoothing by including a trend component. It estimates both level and slope, making it useful for series that increase or decrease over time. The trend may be additive or damped depending on the specification.
This method provides a straightforward way to forecast trending data without requiring elaborate model fitting. It is often chosen when a series shows gradual directional movement. Its simplicity makes it a standard tool in applied forecasting.
6.5.2 Holt-Winters method
Holt-Winters method adds seasonality to Holt’s trend-based framework. It is designed for data with both directional movement and repeating seasonal cycles. Variants exist for additive and multiplicative seasonal patterns.
The method is especially valuable in business and operations settings. It updates level, trend, and seasonality recursively as new data arrive. This adaptability makes it practical for regularly monitored series.
6.6 State-space models
State-space models describe a system through latent states that evolve over time and observed data linked to those states. They offer a flexible framework for handling noise, missing values, and time-varying structure. Many classical methods can be expressed in state-space form.
This framework is useful because it separates the hidden process from the measurement process. It also supports recursive estimation and forecasting. State-space models have become important in modern time series analysis due to their generality.
6.7 Vector autoregression
Vector autoregression models multiple time series jointly, allowing each variable to depend on its own past and the past of others. This makes the framework suitable for interconnected systems. It is frequently used when variables influence one another over time.
VAR models help study dynamic relationships, impulse responses, and multivariate forecasting. They are particularly useful when no single variable can be treated as fully exogenous. The approach extends univariate autoregression to a multivariate setting.
7 Model identification and estimation
7.1 Parameter estimation
Parameter estimation determines the numerical values that define a chosen model. Common approaches include least squares, maximum likelihood, and method-of-moments techniques. The goal is to find parameter values that align the model closely with the observed data.
Good estimation balances fit and robustness. Parameters should capture the main structure without becoming overly sensitive to noise. In time series settings, estimation can be more complex because observations are dependent rather than independent.
7.2 Model selection criteria
Model selection criteria help compare competing specifications and choose an appropriate level of complexity. A good model should fit the data well while remaining parsimonious. Selection methods are needed because overly flexible models may overfit, while overly simple ones may miss important structure.
These criteria are often used together with diagnostic checks and forecasting performance. They provide a systematic way to evaluate candidate models. No single criterion is universally best, so context matters.
7.2.1 AIC and BIC
AIC and BIC are information criteria that balance goodness of fit against model complexity. AIC tends to favor predictive performance, while BIC penalizes complexity more strongly. Both are widely used for selecting among time series models.
These measures are especially useful when comparing alternative lag lengths or component structures. They offer a convenient numerical summary of model adequacy. However, they should not replace subject-matter judgment or residual diagnostics.
7.2.2 Cross-validation
Cross-validation evaluates a model by testing it on data not used for fitting. In time series, the procedure must respect temporal order, so standard random partitioning is usually inappropriate. Rolling or expanding-window validation is often preferred.
This approach provides a practical estimate of forecasting performance. It helps reveal whether a model generalizes beyond the sample used to estimate it. Cross-validation is valuable when the main goal is prediction rather than formal inference.
7.3 Residual diagnostics
Residual diagnostics examine the errors left after fitting a model. If the model is adequate, residuals should show little remaining structure. They should resemble noise rather than retain patterns that could have been modeled.
Diagnostics are an essential step because a model with low error can still be flawed. Visual checks and formal tests both help identify issues such as autocorrelation, nonconstant variance, or non-normality. When problems appear, the model may need to be revised.
7.3.1 Serial correlation tests
Serial correlation tests check whether residuals remain correlated over time. Significant correlation suggests that the model has not captured all temporal dependence. Common tests assess whether autocorrelation remains at one or more lags.
These tests are important because residual dependence can undermine inference and forecasting. If residuals are still structured, the model is incomplete. Passing the test does not guarantee perfection, but it is a useful sign of adequacy.
7.3.2 Normality checks
Normality checks assess whether residuals are approximately normally distributed. Many classical methods assume normal errors, especially for interval estimation and hypothesis testing. Departures from normality may signal outliers, skewness, or heavy tails.
Normality is not always essential for forecasting, but it can matter for uncertainty quantification. Graphical tools and statistical tests are commonly used together. When residuals are clearly non-normal, robust or alternative modeling approaches may be considered.
8 Forecasting
8.1 One-step and multi-step prediction
One-step prediction estimates the next value in the series using information available up to the present. Multi-step prediction extends this idea to several future points. The latter typically becomes less precise as the horizon increases.
Forecasting methods differ in how they propagate uncertainty across time. Some models are better for short-range predictions, while others remain informative farther ahead. The choice depends on the pattern in the data and the purpose of the forecast.
8.2 Prediction intervals
Prediction intervals provide a range within which a future observation is expected to fall with a stated probability. They are more informative than point forecasts because they express uncertainty. Wider intervals indicate greater unpredictability or longer forecast horizons.
These intervals are central to practical decision-making. They help users assess risk and prepare for plausible outcomes beyond the single best estimate. Accurate interval construction depends on a realistic model of the error structure.
8.3 Forecast evaluation
Forecast evaluation measures how well predictions match realized values. Common metrics include absolute error, squared error, and percentage-based measures. Evaluation may be done on a holdout sample or through rolling forecasts.
Assessing forecast quality is essential because a model that fits the past may still perform poorly in the future. Different metrics emphasize different aspects of accuracy, such as bias or sensitivity to large errors. The best measure depends on the application and the cost of mistakes.
8.4 Forecast reconciliation
Forecast reconciliation adjusts forecasts from a hierarchical or grouped set of series so that they are mutually consistent. For example, totals at one level should equal the sum of components below. Reconciliation ensures coherence across related forecasts.
This topic is important in organizational and multiscale settings. Independent forecasts can conflict if generated separately. Reconciliation methods resolve these inconsistencies while preserving overall predictive quality.
9 Advanced topics
9.1 Nonlinear time series
Nonlinear time series models allow relationships that are not captured by linear equations. The effect of past values may depend on thresholds, interactions, or changing conditions. These models can represent more complicated dynamics than standard linear approaches.
Nonlinearity is relevant when the behavior of a series shifts across regimes or responds differently in different ranges. Such models may fit some data better, but they are often harder to estimate and interpret. They are valuable when linear assumptions are too restrictive.
9.2 Long memory processes
Long memory processes exhibit dependence that decays slowly over time. Unlike short-memory series, their distant past can still exert noticeable influence. This persistence may appear in autocorrelation patterns that fade gradually.
These processes are important in economics, climatology, and network data. They challenge standard modeling assumptions because short lag structures may be insufficient. Specialized methods are often needed to account for the extended dependence.
9.3 Regime switching models
Regime switching models assume that a series can move between distinct states or regimes. Each regime may have different mean behavior, variance, or dynamic rules. The switches are often latent rather than directly observed.
Such models are useful when a process alternates between stable periods, volatile periods, or other structural modes. They provide a flexible way to represent abrupt changes. Regime switching is common in systems where behavior depends on hidden conditions.
9.4 Cointegration
Cointegration refers to a long-run equilibrium relationship among nonstationary series. Even if individual series drift over time, a particular combination of them may remain stable. This concept is especially important in multivariate economic and financial analysis.
When series are cointegrated, standard differencing may remove useful long-term information. Specialized models can capture both short-term dynamics and long-run links. Cointegration helps distinguish shared trends from independent movement.
9.5 Time series in the frequency domain
The frequency domain views a time series through the periodic components that compose it. Rather than focusing on time order directly, it examines how variance is distributed across different frequencies. This perspective is useful for studying cycles and oscillatory behavior.
Frequency-domain methods complement time-domain approaches. They are especially effective when repeated patterns are easier to detect as waves than as sequential changes. The analysis often reveals hidden periodic structure.
9.5.1 Spectral density
Spectral density describes how the variability of a series is distributed across frequencies. Peaks in the spectrum indicate dominant periodic components. It is the frequency-domain analogue of autocorrelation-based summaries.
Spectral density helps identify whether a series is dominated by slow movements, fast fluctuations, or particular cycles. It is widely used in signal analysis and related fields. The concept provides a compact view of temporal structure.
9.5.2 Periodogram
The periodogram is an empirical estimate of spectral density. It is computed from observed data and provides a graph of power against frequency. Although simple to construct, it can be noisy without additional smoothing.
Periodograms are useful for exploratory analysis and for detecting prominent cycles. They often serve as a first step before more refined frequency-domain modeling. Their interpretation is most effective when combined with other tools.
10 Applications
10.1 Economics and finance
Time series methods are extensively used in economics and finance to study prices, returns, inflation, interest rates, and macroeconomic indicators. These data often show trends, cycles, volatility changes, and structural shifts. Forecasting and risk assessment are major goals in this area.
Models in this domain support portfolio analysis, trading, budgeting, and policy planning. Because observations are sequential, dependence over time is central to interpretation. Accurate modeling can improve both short-term prediction and long-run understanding.
10.2 Environmental and climate data
Environmental and climate series include temperature, precipitation, river flow, air quality, and atmospheric measurements. They frequently display seasonality, long-term changes, and irregular variability. Time series analysis helps identify patterns that may be tied to natural processes or human activity.
Such data are important for studying weather behavior, resource management, and ecological conditions. Forecasts can support planning in agriculture, water use, and disaster preparedness. The presence of noise and multiple scales often makes these applications analytically demanding.
10.3 Engineering and control systems
In engineering, time series methods are used to analyze sensors, machine signals, feedback systems, and operational performance. The aim may be monitoring, fault detection, stabilization, or control. Many systems generate high-frequency data that require efficient processing.
Time series tools help detect drift, vibration, overload, and other signals of changing behavior. They are also used in automated control loops where past output affects future input. Reliable analysis supports safety, efficiency, and maintenance planning.
10.4 Medicine and public health
Medical and public health time series include patient monitoring data, disease incidence counts, hospital admissions, and physiological signals. These series can reveal outbreaks, seasonal illness patterns, treatment responses, and changes in health status. Timely analysis may support early warning and resource allocation.
The field is valuable for both individual and population-level applications. It helps interpret changes over time and assess whether an intervention is associated with a shift in behavior. Data quality, reporting delays, and irregular sampling are common challenges.
10.5 Digital signal processing
Digital signal processing uses time series methods to analyze audio, image-related signals, communications data, and other sequences of measurements. The focus is often on filtering, compression, detection, and reconstruction. Frequency-domain analysis is particularly important in this area.
Signal-processing applications benefit from techniques that separate useful information from background noise. Time series models can improve denoising, pattern recognition, and system identification. The overlap between statistics and engineering is especially strong in this domain.
</INTERNAL_LINK_CANDIDATES> Autocorrelation function (a summary measure of dependence between values at different time lags) Autoregressive model (a model that predicts a value from its own past values) ARIMA model (a forecasting model combining autoregression, differencing, and moving average terms) AIC (an information criterion used to compare model fit and complexity) BIC (an information criterion with a stronger penalty for complexity than AIC) Cross-validation (a method for evaluating predictive performance on held-out time-ordered data) Cointegration (a long-run equilibrium relationship among nonstationary series) Classical decomposition (a method that separates trend, seasonality, and residual components) Differencing (a transformation that analyzes changes between successive observations) Exponential smoothing (a forecasting method that weights recent observations more heavily) Forecast reconciliation (adjusting related forecasts so they are mutually consistent) Holt-Winters method (an exponential smoothing method for trend and seasonality) Lag dependence (the relationship between current observations and earlier ones) Moving average model (a model that represents a series using past random shocks) Partial autocorrelation (the direct association at a given lag after removing shorter-lag effects) Random walk (a process where each value equals the previous value plus a random change) Seasonal ARIMA (an ARIMA model that incorporates repeating seasonal structure) Spectral density (the distribution of variance across frequencies) Stationarity (a property of stable statistical behavior over time) State-space model (a model with latent states that evolve over time)