# Time Series Analysis: Definition, How it Works, Purpose, and Uses

Time series analysis is a statistical technique employed to model and explore patterns in data recorded sequentially over time. It involves visualizing time series data to identify trends, seasonality, and other patterns before selecting appropriate modeling techniques. Common approaches include autoregressive, moving average, and ARIMA modeling, which aim to quantify relationships between past and current values.

The primary goal of time series analysis is fourfold: to make valuable predictions, explain the drivers behind observed changes, identify cycles and durations, and monitor for anomalies. It offers several advantages, such as the ability to detect trends, quantify relationships, handle non-stationarity, and filter out noise from the data.

This analysis finds application across various industries and sciences. Examples include sales forecasting in retail, weather prediction in meteorology, anomaly detection in manufacturing processes, and econometric modeling of national indicators in economics. The diverse range of use cases highlights the practical utility and versatility of time series analysis techniques.

## What is Time Series Analysis?

Time series analysis is a statistical technique used to analyze data recorded over time, typically at successive equally spaced points. Its objective is to study past patterns in the data to make future predictions.

Time series data consists of ordered sequences over time, with data points usually representing successive measurements made over evenly spaced intervals. Examples include daily stock prices, monthly sales figures, and yearly temperature readings.

Techniques for time series analysis fall into two main categories: time domain and frequency domain. Time domain techniques analyze data observations as they evolve, such as plotting time series on a graph to identify trends and cycles. Statistical methods like moving averages and autocorrelation are used for trend analysis and correlation between successive observations at different time lags.

#### Frequency domain techniques

Frequency domain techniques, like the Fourier transform, decompose overall variations into distinct cyclical components of different frequencies. These techniques help identify underlying patterns in the data.

Time series often exhibit long-term trends, like steadily increasing sales over several years, which trend analysis can identify and predict. They may also display short-term cycles, such as seasonal fluctuations, which seasonal adjustment techniques can remove to reveal underlying trends.

Random disturbances in time series data add variability, which smoothing techniques filter out to uncover meaningful patterns. Many time series techniques require stationarity, where statistical properties like mean and variance remain constant over time. Differencing and transformations are used to stabilize the time series.

Successive observations in time series data are usually interdependent, and autocorrelation quantifies these relationships, incorporating dependence into models.

Time series analysis finds applications in various fields, including economics, finance, geology, and meteorology. It is used for forecasting, tracking trends, detecting seasonality, understanding cyclic behavior, and recognizing relationships between variables. Overall, time series analysis is a vital statistical tool for making data-driven predictions based on historical observations taken sequentially over time.

### What are the components of Time Series Analysis?

Time series analysis comprises six main components, beginning with data preparation. This involves collecting historical data, checking for completeness and accuracy, handling missing values, outliers, and ensuring stationarity.

Exploratory data analysis follows, providing initial insights into data patterns, trends, seasonality, correlations, and anomalies using statistical techniques and plotting tools.

Model identification is the next step, where various forecasting models like AR, MA, ARMA, ARIMA, or SARIMA are tested based on observed patterns. Statistical tests help select the best-fitting model.

Once a model is identified, it’s used for forecasting future values of the time series. Forecast accuracy is evaluated using key metrics on test datasets, and additional variables may be included if needed.

Model maintenance involves monitoring the model’s performance over time, periodically re-evaluating and updating it as new observations become available to ensure continued reliability.

Lastly, the learned concepts and models find application across various domains such as demand forecasting, sales prediction, disease outbreaks, and stock market trend analysis, highlighting the broad utility of time series analysis. However, domain knowledge is crucial for constructing applicable models tailored to specific fields.

### How Does Time Series Analysis Work?

Time series analysis involves applying statistical methods to data observed and recorded over uniform time intervals to uncover patterns, build models, and generate forecasts.

It starts with data collection, gathering observations over consistent time intervals, with missing values handled using interpolation or imputation methods.

#### Exploratory analysis

Exploratory analysis follows, where the dataset is visualized with plots to reveal trends, spikes, seasonal cycles, and anomalies. Statistical summaries like mean, variance, autocorrelations, and spectral density provide additional insights.

Non-stationarity in real-world time series is addressed through mathematical transformations like smoothing, detrending, differencing, or removing seasonal/cyclical components.

Based on exploratory analysis, an appropriate model is selected, such as autoregressive models, moving average models, ARIMA models, regression models, or neural networks, considering data characteristics and domain knowledge.

Model parameters are estimated from the data, often using methods like ordinary least squares, and the fitted model is validated through residual analysis, prediction errors, and information criteria.

#### Forecasts

Forecasts are then generated, including in-sample forecasts on historical data and out-of-sample forecasts for future values, with point and interval forecasts to indicate probable values.

Ongoing maintenance involves monitoring prediction errors, detecting structural changes in data patterns, and often using ensemble forecasts for robustness.

### How are Graphs and Charts used in Time Series Analysis?

In time series analysis, graphs and charts play a crucial role in visualizing data characteristics before applying statistical models.

Time series plots, with observations on the y-axis and time on the x-axis, reveal trends, seasonal patterns, outliers, and structural breaks.

Smoothed or moving average lines overlaid on raw data highlight longer-term patterns, while transformations like logarithms and differencing reveal non-stationarity.

Scatter plots show correlations between two time series, while lag plots display autocorrelations at various lags.

Correlograms compactly visualize autocorrelations for multiple lags, and spectral plots depict the frequency composition of a series.

Charts illustrating growth rates, cyclical components, and trends aid decomposition into underlying components for economic time series.

Histograms, density plots, and normal quantile plots summarize univariate distributions of raw values, returns, or forecast errors to assess distributional assumptions.

Heatmaps and contour plots depict time-varying patterns in matrix data like volatilities, correlations, and covariances, while cluster heatmaps identify groups of related series.

Visualized raw data guides the fitting of an appropriate time series model, and scatter plots of residuals check model adequacy.

### What is the purpose of Time Series Analysis?

The primary purpose of time series analysis is to extract insights from time-dependent data, uncovering patterns, trends, and relations. It helps forecast future values based on historical observations, aiding decision-making in areas like inventory management and demand projections. By capturing seasonality, trend, and noise components, it predicts where a series is heading in upcoming periods, supporting resource allocation and production planning. Time series analysis also identifies cyclical fluctuations, assessing external factors’ impacts over time and aiding risk analysis. Models detect auto-correlations within a series, assisting in generating accurate forecasts and describing dynamic system patterns.

### What are the Types of Time Series Analysis?

Time series analysis encompasses various statistical models and techniques to uncover patterns, understand structure, and make predictions with temporal data. Here are the main categories:

#### 1. Regression Models

Regression models relate the time series to other explanatory variables like economic factors or demographics. They quantify causal effects and can capture linear or nonlinear relationships.

#### 2. Autoregressive Models

Autoregressive models predict future values using a weighted sum of past observations plus an error term. They are effective at modeling the autocorrelation present in time series data. A popular example is ARIMA models.

#### 3. Smoothing Models

Smoothing models use weighted averaging of past observations to reduce noise and highlight trends and seasonality. Examples include simple, exponential, and state space smoothing methods.

#### 4. Machine Learning Models

Machine learning techniques such as artificial neural networks, random forests, and gradient-boosting machines have been adapted to handle time series data, offering flexibility in making predictions.

#### 5. State Space Models

State space models represent time series using latent state variables that evolve over time. They allow for modeling complex dynamics through multi-component state vectors.

#### 6. Gaussian Processes

Gaussian processes provide a flexible nonparametric Bayesian approach to time series modeling. They utilize kernel functions to model correlations between time points.

#### 7. Spectral Analysis

Spectral analysis decomposes a time series into underlying cyclical components of different frequencies using Fourier and wavelet transforms. This method reveals periodicities in the data.

#### 8. Threshold Models

Threshold models allow for different data-generating processes and model parameters to apply before and after crossing threshold values. They are useful for capturing structural breaks in the data.

#### 9. Markov Models

Markov models represent transitions between possible states over time as a Markov process. The following state depends only on the current state.

#### 10. Hidden Markov Models

Hidden Markov models expand Markov models by assuming that the states are hidden. They use only the observable time series to estimate state sequences and transition probabilities.

#### 11. Archetype Analysis

Archetype analysis identifies recurrent archetypal patterns in time series using basis functions from a dictionary of archetypes. It provides a concise lower-dimensional representation of the data.

These various types of time series analysis models offer a diverse toolkit to address different temporal data problems, ranging from linear and interpretable models to highly flexible nonlinear approaches.

### What are the Advantages of Time Series Analysis?

1. Making Predictions Time series models generate forecasts for future values, aiding in planning and decision-making.
2. Understanding Drivers Provides insights into the underlying factors driving changes over time, enhancing understanding.
3. Quantifying Relationships Measures like autocorrelation and regression quantify relationships, enabling hypothesis testing.
4. Handling Nonlinearity Machine learning models capture complex nonlinear relationships in data.
5. Modelling Trends Effectively detects and characterizes trends amidst noise, supporting long-term analysis.
6. Modelling Cycles Extracts short, medium, and long-term cyclical patterns, identifying recurring periodicities.
7. Estimating Seasonality Techniques adjust for regular seasonal fluctuations, improving forecasting accuracy.
8. Filtering Noise Smoothing methods uncover stable underlying components, reducing randomness.
9. Handling Missing Data Methods estimate missing observations and interpolate gaps, leveraging temporal correlations.
10. Detecting Anomalies Identifies abnormal patterns and outliers, aiding in anomaly detection and intervention.
11. Comparing Series Allows comparison and alignment of different time series to identify shared behaviors.

Time series analysis offers a comprehensive toolkit for transforming temporal data into actionable insights, facilitating prediction, understanding relationships, and detecting patterns.

### What are the Disadvantages of Time Series Analysis?

1. Data DependenceRequires sufficient historical data for training and validation, impacting performance with short series or small samples.
2. Assumption DependenceRelies on assumptions like stationarity and normality, violations of which undermine reliability.
3. OverfittingComplex models may overfit training data, limiting generalizability without proper regularization.
4. NonstationarityHandling nonstationary data adds complexity, as models may fail if nonstationarity is undetected.
5. ExpenseRequires significant storage capacity and resources, with software and technical skills adding to costs.
6. MaintenanceModels require ongoing monitoring and re-estimation to align with changing data patterns.
7. Spurious RelationshipsCorrelations may reveal false associations unrelated to causal effects due to confounding factors.
8. Unpredictable ShocksUnexpected events can disrupt historical patterns, challenging forecasting accuracy.
9. Simplifying AssumptionsMathematical models simplify real-world phenomena, losing nuance and introducing uncertainty.
10. Temporal MisalignmentComparing time series requires aligning them temporally, which can degrade accuracy.
11. Technical ComplexityAdvanced techniques require specialized expertise, while simpler methods may suffice for basic tasks.

Time series analysis presents inherent challenges, requiring careful consideration of data limitations and thoughtful analytic design to mitigate disadvantages.

### How to use Time Series Analysis?

Time series analysis offers valuable methods for modeling sequentially collected data. Here’s a systematic approach to effectively utilize time series analysis:

1. Specify the ProblemClearly define analytical goals and questions to focus the analysis and metrics, such as forecasting or trend analysis.
2. Gather Time Series DataCollect measurements over evenly-spaced intervals covering a sufficient period, using standard time scales.
3. Clean and Prepare DataAddress anomalies and missing data through preprocessing tasks like smoothing, interpolation, and outlier removal.
4. Visualize PatternsPlot time series to observe overall patterns, trends, seasonality, structural changes, and anomalies for insights.
5. Check StationarityAssess stationarity with statistical tests and plots, transforming non-stationary data if necessary.
6. Measure DependencyQuantify autocorrelation at various lags using correlograms to inform model selection and assumptions.
7. Decompose ComponentsIsolate trend, seasonal, cyclical, and noise components through smoothing, seasonal adjustment, and spectral analysis.
8. Build ModelsIdentify candidate models based on observed patterns and theory, assessing fit and avoiding overfitting.
9. Make ForecastsUtilize fitted models to forecast future points and horizons, while calculating prediction intervals to quantify uncertainty.
10. Validate ModelsEvaluate goodness-of-fit and forecast accuracy, checking residuals for independence and normality.
11. Monitor PerformanceContinuously track forecasts against actuals, updating models for changing data dynamics.
12. Choose Parsimonious ModelsPrefer simpler models over complex ones if performance is comparable, guarding against overfitting.

By following these steps, practitioners can effectively navigate the phases of problem definition, data preparation, analysis, modeling, prediction, validation, and monitoring in time series analysis, ensuring practical application and interpretation of results.

### How is Time Series Used in Trading?

Time series analysis plays a pivotal role in financial trading strategies and algorithms, leveraging historical price data for assets like stocks, currencies, and cryptocurrencies to identify profitable trades. Here’s how time series analysis is utilized in trading:

1. Price Forecasting Time series models like ARIMA forecast future prices or price changes based on historical data, aiding in speculation on future market direction. Technical indicators, such as moving averages, generate trade signals for entry and exit points.
2. Volatility Modeling Volatility analysis, including GARCH and stochastic volatility models, assesses the variability of returns. Accurate volatility forecasts optimize position sizing and risk management strategies.
3. Cycle Analysis Cycle analysis detects periodic patterns across various timescales, aiding in trade timing for entries, exits, and profit-taking. Techniques like Fourier analysis and wavelets quantify cycles and inform trading decisions.
4. Backtesting Trading systems are rigorously tested through backtesting on historical data to evaluate performance. Statistical validation ensures observed returns meet acceptable risk thresholds, avoiding overfitting.
5. Model Validation Techniques like walk-forward analysis validate trading models by partitioning historical data into in-sample fitting and out-of-sample testing periods, ensuring robustness and reliability.
6. Correlation Analysis Correlation analysis quantifies the relationships between different instruments, allowing traders to exploit temporary divergences between correlated assets for pair trading strategies and risk reduction through hedging.
7. Feature Engineering Informative input variables are extracted from raw price data through feature engineering, including lagged prices, technical indicators, and macroeconomic data, enhancing the effectiveness of trading models.

By employing time series analysis techniques in trading, traders can make informed decisions, optimize risk management, and capitalize on profitable opportunities in financial markets.

### Is Time Series Analysis used to predict stock prices?

Yes, time series analysis is commonly utilized to predict and forecast future stock prices, employing methods like ARIMA, GARCH models, and machine learning techniques. However, accurately modeling financial time series data poses challenges due to several factors:

1. Limited Predictability: While in-sample model fit to past data is often robust, out-of-sample accuracy diminishes as the forecast horizon extends into the uncertain future. Stock prices adhere to random walk patterns according to efficient market theory, restricting predictability.
2. Structural Breaks: Unexpected news events, economic shifts, and investor behavior changes can abruptly alter established price patterns, leading to structural breaks that invalidate statistical models derived from historical data.
3. Volatility and Nonstationarity: High volatility and frequent large price spikes in stock prices make prediction challenging compared to smoother time series. Nonstationarity necessitates transformations, but distributions often remain non-normal.
4. Spurious Correlations and Overfitting: Without causative explanatory variables, models risk spurious correlations and overfitting in-sample, which doesn’t guarantee reliable out-of-sample forecasts. Simplicity helps mitigate overfitting risks.
5. Trading Frictions: Trading frictions like spreads, commissions, and slippage can erode the profitability of minor predictable price movements. Consequently, large price swings are needed for profitable trading, but these are inherently less predictable.

Despite these challenges, time series analysis remains a valuable tool for predicting stock prices, offering insights into historical patterns and trends that inform trading strategies and decision-making processes.

### Is Time Series Analysis effective in forecasting stock prices?

Yes, time series analysis is commonly utilized for forecasting stock prices, albeit with limitations to its effectiveness.

Time series models, such as ARIMA and exponential smoothing, can provide short-term forecasts by extrapolating historical patterns into the future. These methods, often applied to daily closing prices, predict the next day’s price or price change. Additionally, technical analysis indicators derived from time series data, like moving averages, aim to forecast future movements.

Machine learning algorithms, leveraging time series data as inputs, have shown promise in making profitable stock price predictions. Techniques such as LSTM neural networks can capture complex nonlinear patterns, sometimes incorporating other predictive variables like fundamentals, news sentiment, or technical indicators.

However, the efficacy of time series analysis in forecasting stock prices is constrained by the unpredictability and efficient market assumptions surrounding stock prices. While models can predict based on historical correlations continuing into the future, new events and information rapidly alter investor behavior, invalidating observed patterns. Consequently, the relevance of historical extrapolations diminishes over longer forecasting horizons.

In highly liquid stocks, arbitrage activity swiftly eliminates predictable patterns learned by models, adhering to the efficient market hypothesis. This often means consistent excess returns are challenging to achieve solely from data-driven forecasts, with spurious patterns occasionally arising from randomness rather than genuine information.

Time series models may have a predictive edge in less efficient market segments with lower trading volumes. However, even in these cases, structural changes in market dynamics can render patterns obsolete over time.

For longer-term forecasts, fundamental valuation factors and earnings expectations tend to dominate stock price movements, whereas time series analysis holds more relevance for higher-frequency price changes over short horizons of days to weeks.

### How can Time Series Analysis and Moving Average work together?

Time series analysis and moving averages complement each other to enhance understanding and forecasting of temporal data.

Moving averages serve to smooth short-term fluctuations in time series data, accentuating longer-term trends and cycles while filtering out high-frequency noise. By employing different window lengths for moving averages, various components of the data can be filtered and analyzed.

Visualizing moving averages alongside raw time series data on line charts provides a quick visual insight into trends, local cycles, and potential turning points. Simple trading indicators, such as the crossover of short and long-moving averages, can be derived from these visualizations.

Moving averages provide a quantification of local means at each point in the time series. Analyzing the difference between the series and its moving averages facilitates the decomposition of the data into short and long-term components, aiding in modeling efforts.

Autocorrelation analysis of moving averages, rather than raw observations, eliminates noise and stabilizes the time series, thereby improving the fit of autoregressive models like ARIMA that assume stationarity.

Integrating moving averages directly as input variables or components within autoregressive and machine learning models furnishes valuable smoothed explanatory features. Models trained on such inputs learn to make predictions partially based on the signals derived from moving averages.

Conducting experiments with different moving average window lengths to optimize predictive performance through validation assists in selecting the most suitable smoothing parameters tailored to a specific time series and forecast horizon.

Synthesizing multiple moving averages, rather than relying solely on a single smoothed series, enhances adaptability to shifts in the underlying data-generating process. This approach of combining signals enhances model robustness and predictive accuracy.

### Can Time Series Analysis be Used with Exponential Moving Average?

Yes, time series analysis seamlessly integrates with exponential moving averages (EMAs).

Exponential moving averages employ weighting factors that exponentially diminish with time, granting more weight to recent observations compared to simple moving averages.

EMAs serve as effective smoothing filters, refining noisy raw time series data. The resulting smoothed EMA series unveils slower-moving components such as trends and cycles, which prove invaluable in forecasting models.

Moreover, an EMA offers a flexible localized approximation of the time series. The EMA’s level and slope at any given point encapsulate the local trend and level, serving as key variables in time series models.

Utilizing the difference between the original series and its EMA filters out short-term fluctuations, enhancing volatility and variance forecasts by eliminating much of the transient noise.

EMAs are integral to exponential smoothing techniques employed in time series forecasting. Simple exponential smoothing employs an EMA with an adaptive weighting parameter, while more advanced methods extend this to trended and seasonal data, inherently incorporating EMAs into the models.

Additionally, crossovers of short and long-window EMAs generate trade signals indicative of momentum shifts, complementing statistical model forecasts with empirical rules for market entry and exit timing.

By comparing model predictions to EMA benchmarks, analysts can diagnose instances of underperformance and identify structural changes in the data necessitating model retraining. Furthermore, the EMA reference helps guard against overfitting.

In summary, combining time series analysis with exponential moving averages enriches forecasting capabilities and provides valuable insights into market dynamics.

### What are Examples of Time Series Analysis data?

Here are seven examples of time series data frequently analyzed using statistical and machine learning models:

1. Stock PricesDaily closing prices of a stock over many years form a time series, reflecting market forces. Analysts study trends, volatility, seasonality, and predictive signals, while time series models forecast future prices.
2. Sales FiguresMonthly sales totals for a company constitute a time series showcasing seasonal patterns, trends, and business cycles. Time series decomposition isolates components, and ARIMA models forecast future sales.
3. Weather RecordingsMeasurements of temperature, rainfall, or wind speed collected at regular intervals represent weather data, exhibiting seasonality and longer climate cycles. Time series methods detect patterns and predict the weather.
4. Economic IndicatorsKey metrics like GDP, unemployment, manufacturing activity, and consumer confidence tracked over years or decades reveal economic trends, recessions, and booms. Regression analysis establishes relationships between indicators.
5. Web TrafficThe daily or hourly count of website visitors forms a time series indicating popularity. Variations in trends, seasons, events, and news are analyzed, and anomaly detection identifies unusual changes.
6. Sensor ReadingsTime series data from Internet of Things sensors, such as temperature, pressure, power consumption, or vibration, is utilized to monitor equipment health, detect anomalies, and predict failures.
7. Audio SignalsSound waveform samples over time, encompassing music, speech, or noise, constitute audio time series. Analysis involves filtering, spectral analysis, and digital signal processing for compression or feature extraction.

Analyzed time series may exhibit various characteristics such as stationarity or nonstationarity, cyclic behavior, seasonality, noise, intermittency, and irregularity. Any phenomenon measured sequentially over uniform time intervals qualifies as a time series, underscoring the versatility of time series modeling techniques.

### Is Time Series Analysis Hard?

Time series analysis is often perceived as challenging, but its difficulty varies depending on factors such as context and methodology.

While advanced statistical techniques like ARIMA and vector autoregression demand mathematical expertise for correct application and interpretation, simpler forecasting methods and visual analysis suffice for many scenarios.

Understanding the theory behind complex time series models can be intellectually demanding. However, modern software automates numerous calculations and diagnostics, easing the practical application provided proper caution is exercised.

Dealing with nonstationary, noisy data featuring multiple seasonal cycles can be complex. However, transforming and decomposing time series into more manageable components simplifies analysis, mitigating overall complexity.

In trading applications, additional challenges arise due to low signal-to-noise ratios, non-normal distributions, and structural breaks. Nonetheless, maintaining reasonable expectations and integrating quantitative models with human insight can make trading endeavors feasible.

### Is Time Series Analysis Quantitative?

Yes, time series analysis is fundamentally a quantitative discipline that heavily relies on statistical modeling and mathematical techniques.

The analysis of time series data involves employing quantitative methods drawn from various fields such as statistics, econometrics, signal processing, and machine learning. A solid understanding and application of statistical reasoning and mathematical concepts are essential in this field.

Quantitative techniques like autocorrelation analysis, spectral decomposition, stationarity tests, and variance stabilization transforms are commonly used in exploratory analysis. When modeling time series data, one must choose from mathematically formulated model classes such as ARIMA, state space models, or VAR processes.

Estimating time series models involves quantitative estimation methods like regression, maximum likelihood estimation, Bayesian inference, or numerical nonlinear optimization. Model adequacy is assessed through quantitative diagnostic checks on residuals, information criteria, and prediction errors.

Forecasting in time series analysis requires quantifying uncertainty using numerical metrics such as mean squared error, interval widths, empirical prediction distributions, and quantile loss. Additionally, numerical evaluation metrics are employed for model selection and comparison.

The outputs of time series analysis, including point forecasts, prediction intervals, probabilistic forecasts, and quantified model uncertainties, enable data-driven decision-making. However, interpreting these results accurately depends on a solid grasp of statistical concepts.

While domain expertise and qualitative context remain crucial for the applicability of time series analysis, the underlying techniques for processing, modeling, forecasting, and decision-making are fundamentally quantitative. Even basic visualizations like time series plots rely on quantitative axes and numerical summaries.

In summary, time series analysis is a quantitative discipline that leverages mathematical and statistical competencies to extract patterns, insights, and forecasts from data ordered over time. The quantitative orientation complements qualitative perspectives from domain knowledge, resulting in impactful time series modeling.

### What is the difference between Time Series Analysis and Cross-Sectional Analysis?

Here is an overview of the critical differences between time series and cross-sectional analyses:

#### Time Series Analysis

• Studies observations recorded sequentially over time for a single entity or process.
• Observations are temporally ordered and spaced at uniform time intervals.
• Models temporal dependencies like trends, autocorrelation, and cyclical patterns.
• Commonly used for forecasting future values.

#### Cross-Sectional Analysis

• Studies observations recorded across multiple entities or subjects at a single time.
• Observations are for different subjects rather than timed observations of one subject.
• Models relationships between variables observed on the same subject.
• Used to determine correlation and causality between variables.

#### Other Differences

• Time series data has a natural ordering that cross-sectional data lacks.
• Time series is prone to autocorrelation, while cross-sectional errors are independent.
• Time series sometimes exhibit non-stationarity, whereas cross-sectional data is stationary.
• Time series models extrapolate patterns over time, and cross-sectional models interrelate explanatory variables.
• Time series analysis aims to forecast, and cross-sectional analysis aims to infer causal relationships.
• Time series has temporal dependencies, and cross-sectional analysis studies spatial relationships.

In summary, time series analysis focuses on sequential data over time, while cross-sectional analysis studies data across multiple subjects simultaneously. Their domains, assumptions, methods, and objectives differ fundamentally due to the unique structure of temporal versus spatial data.