Here is a complete tutorial on Fourier Series with detailed explanations, illustrations, and Python code examples — designed to help you understand how Fourier terms are used in time series analysis and feature engineering, especially for seasonality modeling.

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🎼 Fourier Series for Time Series Analysis

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📘 What is a Fourier Series?

The Fourier Series represents a function (especially periodic ones) as a sum of sine and cosine functions.

For a time series \(y(t)\), the Fourier series approximation is:

\[ y(t) = \sum_{k=1}^{K} \left[ a_k \cos\left(\frac{2\pi k t}{T} \right) + b_k \sin\left(\frac{2\pi k t}{T} \right) \right] \]

• \(T\): period of the seasonal cycle (e.g., 12 for months, 365 for daily data)
• \(K\): number of harmonics (Fourier order)
• \(a_k, b_k\): coefficients for cosine and sine terms

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📌 Why Use Fourier Terms?

• Models complex and smooth seasonal patterns
• Often used in Prophet, ARIMA with exogenous variables, and neural nets
• Captures both short-term and long-term cyclic behaviors

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🧮 Fourier Series in Python

🧰 Function to Create Fourier Features

import numpy as np
import pandas as pd

def create_fourier_terms(df, time_col, period, order):
    """
    Add Fourier terms to a time series DataFrame.

    :param df: DataFrame with a datetime index
    :param time_col: Name of the datetime index or column
    :param period: Seasonality period (e.g., 365 for yearly)
    :param order: Number of harmonics
    :return: DataFrame with Fourier terms
    """
    df = df.copy()
    t = np.arange(len(df))

    for k in range(1, order + 1):
        df[f'sin_{k}'] = np.sin(2 * np.pi * k * t / period)
        df[f'cos_{k}'] = np.cos(2 * np.pi * k * t / period)

    return df

📈 Example: Simulate Seasonal Time Series

import matplotlib.pyplot as plt

# Simulate a signal with yearly seasonality
t = np.linspace(0, 365, 365)
y = np.sin(2 * np.pi * t / 365) + 0.5 * np.sin(2 * np.pi * 2 * t / 365)

plt.figure(figsize=(10, 4))
plt.plot(t, y, label='Simulated Seasonal Signal')
plt.title('Fourier Series Approximation of Seasonality')
plt.xlabel('Day of Year')
plt.ylabel('Signal')
plt.legend()
plt.grid(True)
plt.show()

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📊 Use Case: Add Fourier Terms to Real Data

# Example using airline passenger dataset
url = 'https://raw.githubusercontent.com/jbrownlee/Datasets/master/airline-passengers.csv'
df = pd.read_csv(url, parse_dates=['Month'], index_col='Month')

# Add Fourier terms for seasonal pattern (monthly data: T = 12)
df_fourier = create_fourier_terms(df, time_col='Month', period=12, order=2)

df_fourier.head()

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🤖 Fourier Features with Time Series Models

You can use Fourier features as exogenous variables (X) in time series models like:

🧠 ARIMA / SARIMAX:

from statsmodels.tsa.statespace.sarimax import SARIMAX

model = SARIMAX(
    df['Passengers'], 
    exog=df_fourier[['sin_1', 'cos_1', 'sin_2', 'cos_2']], 
    order=(1, 1, 1), 
    seasonal_order=(1, 1, 1, 12)
)
results = model.fit()
print(results.summary())

🕵️ Prophet:

from prophet import Prophet
from prophet.make_holidays import make_holidays_df

df_prophet = df.reset_index().rename(columns={"Month": "ds", "Passengers": "y"})

m = Prophet()
m.add_seasonality(name='yearly', period=365.25, fourier_order=4)
m.fit(df_prophet)

future = m.make_future_dataframe(periods=24, freq='MS')
forecast = m.predict(future)

m.plot(forecast)

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📌 Tips on Choosing Period and Order

Use Case	Period	Fourier Order
Monthly data (yearly seasonality)	12	2–6
Daily data (yearly seasonality)	365	4–10
Weekly data (annual seasonality)	52	2–8

Higher order allows capturing more complex cycles but may lead to overfitting.

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✅ Summary

Concept	Purpose
Fourier Series	Decompose signal into sine/cosine
Period	Time span of one seasonal cycle
Order	Number of harmonics
Use cases	ARIMA, Prophet, deep learning models
Benefits	Capture smooth and non-linear seasonality

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📁 Optional: Export Fourier Features

df_fourier.to_csv("data_with_fourier_features.csv")