Time Series Analysis Dashboard

Why do we difference time series data?

Differencing is a technique used to make a non-stationary time series stationary by removing trends and seasonality:

• First Differencing: Removes trend by taking the difference between consecutive observations. The formula is:
  \[ \nabla Y_t = Y_t - Y_{t-1} \]
  This helps stabilize the mean by removing linear trends.
• Seasonal Differencing: Removes seasonality by taking the difference between an observation and the same observation from the previous season. The formula is:
  \[ \nabla_s Y_t = Y_t - Y_{t-s} \]
  where \(s\) is the seasonal period (e.g., 4 for quarterly data, 12 for monthly data).
• Log Returns: For financial time series, taking logarithmic returns can help stabilize variance:
  \[ r_t = \log(Y_t) - \log(Y_{t-1}) \approx \frac{Y_t - Y_{t-1}}{Y_{t-1}} \]

Stationarity is important because most time series forecasting methods assume that the data is stationary.

What is stationarity?

A time series is stationary if its statistical properties (mean, variance, autocorrelation) do not change over time. Most time series models assume stationarity, so we often need to transform non-stationary data.

Augmented Dickey-Fuller (ADF) Test

The ADF test checks for a unit root in the time series. The presence of a unit root indicates non-stationarity.

Null Hypothesis (H₀): The time series has a unit root (non-stationary)

Alternative Hypothesis (H₁): The time series does not have a unit root (stationary)

If the p-value is less than the significance level (typically 0.05), we reject the null hypothesis and conclude that the time series is stationary.

ADF test regression model:

The test statistic is the t-statistic for the \(\gamma\) coefficient. If \(\gamma\) is significantly less than zero, the series is stationary.

ARIMA Models

ARIMA (AutoRegressive Integrated Moving Average) models are used for forecasting time series data. They combine three components:

• AR (AutoRegressive): Uses the relationship between an observation and a number of lagged observations.
  \[ \text{AR}(p): y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \cdots + \phi_p y_{t-p} + \epsilon_t \]
• I (Integrated): Represents differencing of observations to make the time series stationary.
  \[ \text{I}(d): (1 - B)^d y_t \]
  where B is the backshift operator: \(B y_t = y_{t-1}\)
• MA (Moving Average): Uses the dependency between an observation and residual errors from previous observations.
  \[ \text{MA}(q): y_t = c + \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + \cdots + \theta_q \epsilon_{t-q} \]

SARIMA Models

SARIMA (Seasonal ARIMA) models add seasonal components to ARIMA models.

A SARIMA model is denoted as SARIMA(p,d,q)(P,D,Q)s, where:

• (p,d,q): Non-seasonal components
• (P,D,Q): Seasonal components
• s: The number of observations per season

The general form of a SARIMA model is:

Where:

• \(\phi_p(B)\): Non-seasonal AR component
• \((1-B)^d\): Non-seasonal differencing
• \(\theta_q(B)\): Non-seasonal MA component
• \(\Phi_P(B^s)\): Seasonal AR component
• \((1-B^s)^D\): Seasonal differencing
• \(\Theta_Q(B^s)\): Seasonal MA component

Model Evaluation Metrics

To compare time series forecasting models, we use several metrics:

Root Mean Square Error (RMSE): Measures the square root of the average squared differences between predicted and actual values.

Mean Absolute Error (MAE): Measures the average absolute differences between predicted and actual values.

Mean Absolute Percentage Error (MAPE): Measures the average percentage difference between predicted and actual values.

Lower values of these metrics indicate better model performance.

About the Models

• ARIMA: Good for non-seasonal data with trends.
• SARIMA: Adds seasonal components to ARIMA, making it suitable for seasonal data.
• Holt-Winters: Uses exponential smoothing to capture level, trend, and seasonality.