Here’s a step-by-step explanation of the main time series models and how they work:​

1. Understanding Time Series Data

Time series data consists of observations recorded sequentially over time. It typically has:

• Trend: Long-term movement or direction in the data.
• Seasonality: Regular pattern or cycle in the data (e.g., monthly sales spikes).
• Noise: Random variability or irregular fluctuations.

2. Exploratory Data Analysis (EDA)

Before modeling, perform EDA to understand the time series characteristics:

• Plot the Data: Visualize the time series to identify trends, seasonality, and outliers.
• Decompose the Series: Use methods like STL (Seasonal and Trend decomposition using LOESS) to separate the time series into trend, seasonal, and residual components.

3. Stationarity

Many time series models assume the data is stationary, meaning its statistical properties (mean, variance) do not change over time. To check for stationarity:

• Plot and Test: Use visual plots and statistical tests like the Augmented Dickey-Fuller (ADF) test.
• Transformations: Apply transformations like differencing or logarithmic scaling to achieve stationarity if necessary.

4. Time Series Models

ARIMA (AutoRegressive Integrated Moving Average)

ARIMA is a popular model for univariate time series forecasting. It combines:

• AR (AutoRegressive) Term: Relates the current value to previous values.

  X_t = φ₁ X_t-1 + φ₂ X_t-2 + … + φ_p X_t-p + ε_t

• I (Integrated) Term: Differencing the data to make it stationary.

  Δ_d X_t = X_t - X_t-d

• MA (Moving Average) Term: Relates the current value to past forecast errors.

  X_t = ε_t + θ₁ε_t-1 + … + θ_qε_t-q

Steps to Apply ARIMA:

1. Identify: Determine the order of AR, I, and MA terms using methods like ACF (AutoCorrelation Function) and PACF (Partial AutoCorrelation Function) plots.
2. Fit the Model: Estimate the parameters and fit the ARIMA model to the data.
3. Diagnose: Check residuals to ensure no patterns remain.
4. Forecast: Use the model to make future predictions.

SARIMA (Seasonal ARIMA)

SARIMA extends ARIMA to handle seasonality.

• Seasonal Terms: Include seasonal AR, I, and MA terms.

  SARIMA(p, d, q) (P, D, Q)_s

Steps to Apply SARIMA:

1. Identify Seasonal Parameters: Use seasonal plots and ACF/PACF plots.
2. Fit the Model: Estimate seasonal and non-seasonal parameters.
3. Diagnose: Check residuals.
4. Forecast: Predict future values.

Exponential Smoothing Methods

These methods give more weight to recent observations:

• Simple Exponential Smoothing: For data without trend or seasonality.

  ^𝑋ᵗ⁺¹ = α Xₜ + (1 - α) ^𝑋ᵗ

  Where:
  • ^ X_t+1 is the forecast for the next time period.
  • α is the smoothing constant (0 < α < 1).
  • X_t is the actual value at time t.
  • ^ X_t is the forecast value at time t.

• Holt’s Linear Trend Model: For data with a trend.

  X^t+1 = α X_t + (1 - α)(^X^t + ^T^t)

• Holt-Winters Seasonal Model: For data with trend and seasonality.

  Xᵗ⁺¹ = (α Xₜ + (1 - α)(𝑋ᵗ + 𝑇ᵗ)) + 𝑆ᵗ⁻ˢ

  Where:
  • ^S_t-s is the seasonal component.

Steps to Apply Exponential Smoothing:

1. Select the Model: Based on the presence of trend and seasonality.
2. Estimate Parameters: Tune smoothing parameters ($\mathrm{<span\; style="line-height:1.5em;"><span\; style="font-family:arial,helvetica,sans-serif;">\alpha </span></span>}$\alpha$\alpha$, $\mathrm{<span\; style="line-height:1.5em;"><span\; style="font-family:arial,helvetica,sans-serif;">\beta </span></span>}$\beta$\beta$, $\mathrm{<span\; style="line-height:1.5em;"><span\; style="font-family:arial,helvetica,sans-serif;">\gamma </span></span>}$\gamma$\gamma$).
3. Fit the Model: Apply the model to the data.
4. Forecast: Make predictions.

State Space Models

These models represent the time series using latent variables:

• Kalman Filter: A recursive algorithm for estimating the state of a linear dynamic system.
• Dynamic Linear Models (DLMs): Generalize the Kalman filter to handle different types of time series.

Steps to Apply State Space Models:

1. Specify the Model: Define the state equations and observation equations.
2. Estimate Parameters: Use algorithms like the Kalman filter.
3. Fit the Model: Apply the state space model to the data.
4. Forecast: Generate predictions.

5. Model Evaluation

• Split the Data: Divide the data into training and test sets.
• Validation: Use metrics like Mean Absolute Error (MAE), Mean Squared Error (MSE), or Mean Absolute Percentage Error (MAPE) to evaluate model performance.
• Cross-Validation: Perform time series cross-validation if necessary to ensure robustness.

6. Deploy the Model

Once a satisfactory model is identified and validated:

• Generate Forecasts: Produce future predictions.
• Update Regularly: Retrain the model periodically as new data becomes available.