Time Series Analysis

Overview

Time series analysis examines data points collected over time to identify patterns, trends, and seasonality for forecasting and understanding temporal dynamics.

When to Use

• Forecasting future values based on historical trends
• Detecting seasonality and cyclical patterns in data
• Analyzing trends over time in sales, stock prices, or website traffic
• Understanding autocorrelation and temporal dependencies
• Making time-based predictions with confidence intervals
• Decomposing data into trend, seasonal, and residual components

Core Components

• Trend: Long-term directional movement
• Seasonality: Repeating patterns at fixed intervals
• Cyclicity: Long-term oscillations (non-fixed periods)
• Stationarity: Constant mean, variance over time
• Autocorrelation: Correlation with past values

Key Techniques

• Decomposition: Separating trend, seasonal, residual components
• Differencing: Making data stationary
• ARIMA: AutoRegressive Integrated Moving Average models
• Exponential Smoothing: Weighted average of past values
• SARIMA: Seasonal ARIMA models

Implementation with Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from statsmodels.tsa.seasonal import seasonal_decompose
from statsmodels.tsa.stattools import adfuller, acf, pacf
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tsa.holtwinters import ExponentialSmoothing

# Create sample time series data
dates = pd.date_range('2020-01-01', periods=365, freq='D')
values = 100 + np.sin(np.arange(365) * 2*np.pi / 365) * 20 + np.random.normal(0, 5, 365)
ts = pd.Series(values, index=dates)

# Visualize time series
fig, axes = plt.subplots(2, 2, figsize=(14, 8))

axes[0, 0].plot(ts)
axes[0, 0].set_title('Original Time Series')
axes[0, 0].set_ylabel('Value')

# Decomposition
decomposition = seasonal_decompose(ts, model='additive', period=30)
axes[0, 1].plot(decomposition.trend)
axes[0, 1].set_title('Trend Component')

axes[1, 0].plot(decomposition.seasonal)
axes[1, 0].set_title('Seasonal Component')

axes[1, 1].plot(decomposition.resid)
axes[1, 1].set_title('Residual Component')

plt.tight_layout()
plt.show()

# Test for stationarity (Augmented Dickey-Fuller)
result = adfuller(ts)
print(f"ADF Test Statistic: {result[0]:.6f}")
print(f"P-value: {result[1]:.6f}")
print(f"Critical Values: {result[4]}")

if result[1] <= 0.05:
    print("Time series is stationary")
else:
    print("Time series is non-stationary - differencing needed")

# First differencing for stationarity
ts_diff = ts.diff().dropna()
result_diff = adfuller(ts_diff)
print(f"\nAfter differencing - ADF p-value: {result_diff[1]:.6f}")

# Autocorrelation and Partial Autocorrelation
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

plot_acf(ts_diff, lags=40, ax=axes[0])
axes[0].set_title('ACF')

plot_pacf(ts_diff, lags=40, ax=axes[1])
axes[1].set_title('PACF')

plt.tight_layout()
plt.show()

# ARIMA Model
arima_model = ARIMA(ts, order=(1, 1, 1))
arima_result = arima_model.fit()
print(arima_result.summary())

# Forecast
forecast_steps = 30
forecast = arima_result.get_forecast(steps=forecast_steps)
forecast_df = forecast.conf_int()
forecast_mean = forecast.predicted_mean

# Plot forecast
fig, ax = plt.subplots(figsize=(12, 5))
ax.plot(ts.index[-90:], ts[-90:], label='Historical')
ax.plot(forecast_df.index, forecast_mean, label='Forecast', color='red')
ax.fill_between(
    forecast_df.index,
    forecast_df.iloc[:, 0],
    forecast_df.iloc[:, 1],
    color='red', alpha=0.2
)
ax.set_title('ARIMA Forecast with Confidence Interval')
ax.legend()
ax.grid(True, alpha=0.3)
plt.show()

# Exponential Smoothing
exp_smooth = ExponentialSmoothing(