Weather Forecasting using ARIMA & SARIMA Model

Stationarity & Differencing

Learning Outcome

Recall

Before this topic you already learned

• Time Series Components​

• Trend

• Seasonality

• Noise

• Basic Time Series Plots

Imagine measuring temperature in a room

Machine learning models prefer stable data patterns.

If patterns keep changing

Model becomes unreliable.

Therefore we need:

Stationary Time Series

What is Stationarity?

A time series is stationary if its statistical properties do not change over time

Key properties:

• Constant Mean

• Constant Variance

• Constant Autocovariance

Why Stationarity Matters

Most forecasting models assume stationarity

Examples:

AR

MA

ARMA

ARIMA

AutoRegressive

Moving Average

AR + MA Combined

Integrated ARMA

Without stationarity

Forecasts become unreliable

Types of Stationarity

Two main types:

Strict Stationarity

Weak (Covariance) Stationarity

Strict Stationarity

Distribution remains identical at every time point

Rarely used in real-world modeling

Too restrictive for practical applications

Mean

Variance

Skewness

All statistical moments

Weak Stationarity

Three Conditions:

Used in most time series modeling

How to Check Stationarity

Three ways:

Visual Inspection

Plot the time series.

Check for:

Trend

Changing variance

Seasonality

Upward or downward movement over time

Spreads or narrows across time periods

Repeating patterns at regular intervals

If pattern shifts → Non-stationary

Summary Statistics Method

Divide data into segments.

Compare statistical measures across different time periods

Compare Mean

Calculate average for each segment

Compare Variance

Measure spread for each segment

If values change significantly → Non-stationary

Statistical Test (ADF Test)

Augmented Dickey-Fuller

Most common test for stationarity

Null Hypothesis

Alternative Hypothesis

Series is non-stationarya

Series is stationary

H₀

H₁

Decision rule:

p-value < 0.05

Stationary

What is Differencing?

Removes Trend

Subtracts previous value from current value to achieve stationarity

Differencing removes trend from time series

Transformation Process

Used to convert non-stationary → stationary

First-Order Differencing

Removes Linear Trend

Subtracts consecutive observations to eliminate linear trends

Yt′=Yt−Yt−1Y'_t = Y_t - Y_{t-1}Yt′​=Yt​−Yt−1

Second-Order Differencing

• Used when trend still exists

Removes Quadratic Trend

Applied when first-order differencing is insufficient

Yt′′=Yt′−Yt−1′Y''_t = Y'_t - Y'_{t-1}Yt′′​=Yt′​−Yt−1′​

Seasonal Differencing

Removes Seasonal Patterns

Eliminates repeating seasonal cycles from time series data

Y_t - Y_{t-s}

Monthly Data Example

s = 12       for annual seasonality

Before vs After Differencing

Summary

Which method is used to remove trend from a time series?

Which method is used to remove trend from a time series?