Time Series Analysis

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(
    ts, seasonal_periods=30, trend='add', seasonal='add', initialization_method='estimated'
)
exp_result = exp_smooth.fit()

# Model diagnostics
fig = exp_result.plot_diagnostics(figsize=(12, 8))
plt.tight_layout()
plt.show()

# Custom moving average analysis
window_sizes = [7, 30, 90]
fig, ax = plt.subplots(figsize=(12, 5))

ax.plot(ts.index, ts.values, label='Original', alpha=0.7)

for window in window_sizes:
    ma = ts.rolling(window=window).mean()
    ax.plot(ma.index, ma.values, label=f'MA({window})')

ax.set_title('Moving Averages')
ax.legend()
ax.grid(True, alpha=0.3)
plt.show()

# Seasonal subseries plot
fig, axes = plt.subplots(2, 2, figsize=(12, 8))
for i, month in enumerate(range(1, 5)):
    month_data = ts[ts.index.month == month]
    axes[i // 2, i % 2].plot(month_data.values)
    axes[i // 2, i % 2].set_title(f'Month {month} Pattern')

plt.tight_layout()
plt.show()

# Forecast accuracy metrics
def calculate_forecast_metrics(actual, predicted):
    mae = np.mean(np.abs(actual - predicted))
    rmse = np.sqrt(np.mean((actual - predicted) ** 2))
    mape = np.mean(np.abs((actual - predicted) / actual)) * 100
    return {'MAE': mae, 'RMSE': rmse, 'MAPE': mape}

metrics = calculate_forecast_metrics(ts[-30:], forecast_mean[:30])
print(f"\nForecast Metrics:\n{metrics}")

# Additional analysis techniques

# Step 10: Seasonal subseries plots
fig, axes = plt.subplots(2, 2, figsize=(12, 8))
for i, season in enumerate([1, 2, 3, 4]):
    seasonal_ts = ts[ts.index.month % 4 == season % 4]
    axes[i // 2, i % 2].plot(seasonal_ts.values)
    axes[i // 2, i % 2].set_title(f'Season {season}')
plt.tight_layout()
plt.show()

# Step 11: Granger causality (for multiple series)
from statsmodels.tsa.stattools import grangercausalitytests

# Create another series for testing
ts2 = ts.shift(1).fillna(method='bfill')

try:
    print("\nGranger Causality Test:")
    print(f"Test whether ts2 Granger-causes ts:")
    gc_result = grangercausalitytests(np.column_stack([ts.values, ts2.values]), maxlag=3)
except Exception as e:
    print(f"Granger causality not performed: {str(e)[:50]}")

# Step 12: Autocorrelation and partial autocorrelation analysis
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

acf_values = acf(ts.dropna(), nlags=20)
pacf_values = pacf(ts.dropna(), nlags=20)

# Step 13: Seasonal strength
def seasonal_strength(series, seasonal_period=30):
    seasonal = seasonal_decompose(series, model='additive', period=seasonal_period)
    var_residual = np.var(seasonal.resid.dropna())
    var_seasonal = np.var(seasonal.seasonal)
    return 1 - (var_residual / (var_residual + var_seasonal)) if (var_residual + var_seasonal) > 0 else 0

ss = seasonal_strength(ts)
print(f"\nSeasonal Strength: {ss:.3f}")

# Step 14: Forecasting with uncertainty
fig, ax = plt.subplots(figsize=(12, 5))
ax.plot(ts.index[-60:], ts.values[-60:], label='Historical', linewidth=2)

# Multiple horizon forecasts
for steps_ahead in [10, 20, 30]:
    try:
        fc = arima_result.get_forecast(steps=steps_ahead)
        fc_mean = fc.predicted_mean
        ax.plot(pd.date_range(ts.index[-1], periods=steps_ahead+1)[1:],
               fc_mean.values, marker='o', label=f'Forecast (+{steps_ahead})')
    except:
        pass

ax.set_title('Multi-step Ahead Forecasts')
ax.set_xlabel('Date')
ax.set_ylabel('Value')
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

# Step 15: Model comparison summary
print("\nTime Series Analysis Complete!")
print(f"Original series length: {len(ts)}")
print(f"Trend strength: {1 - np.var(decomposition.resid.dropna()) / np.var((ts - ts.mean()).dropna()):.3f}")
print(f"Seasonal strength: {ss:.3f}")

Stationarity

• Stationary: Mean, variance, autocorrelation constant over time
• Non-stationary: Trend or seasonal patterns present
• Solution: Differencing, log transformation, or detrending

Model Selection

• ARIMA: Good for univariate forecasting
• SARIMA: Includes seasonal components
• Exponential Smoothing: Simpler, good for trends
• Prophet: Handles holidays and changepoints

Evaluation Metrics

• MAE: Mean Absolute Error
• RMSE: Root Mean Squared Error
• MAPE: Mean Absolute Percentage Error

Deliverables

• Decomposition analysis charts
• Stationarity test results
• ACF/​PACF plots
• Fitted models with diagnostics
• Forecast with confidence intervals
• Accuracy metrics comparison