Time Series Analysis: Advanced Techniques

STL (Seasonal-Trend decomposition using Loess) is a robust and versatile method for decomposing a time series into seasonal, trend, and remainder components. Unlike simple moving averages or exponential smoothing, STL can handle complex seasonal patterns and is less sensitive to outliers. The Loess smoothing process is applied iteratively to the time series to extract these components.

Example:

Imagine analyzing monthly sales data. STL can decompose this data into:

• Seasonal Component: Reflects the yearly sales cycle (e.g., higher sales during holiday seasons).
• Trend Component: Indicates the long-term growth or decline in sales.
• Remainder Component: Represents the random fluctuations or noise in the data.

Code Snippet (Python - using statsmodels):

import pandas as pd
from statsmodels.tsa.seasonal import STL

# Assuming 'sales_data' is your time series (Pandas Series)
stl = STL(sales_data, period=12) # period=12 for monthly data (yearly seasonality)
results = stl.fit()

# Accessing components:
seasonal = results.seasonal
trend = results.trend
residual = results.resid

# Plotting components (optional)
import matplotlib.pyplot as plt
results.plot()
plt.show()

Prophet, developed by Facebook, is designed for forecasting time series data with strong seasonal components and holiday effects. It's particularly useful for business time series. Prophet is a decomposable model with a trend component, a seasonality component, and a holiday component. The trend is modeled using piecewise linear or logistic growth. Seasonality can be additive or multiplicative, and holiday effects are easily incorporated.

Example:

Forecasting daily website traffic. You can include major holidays as a special effect.

Code Snippet (Python - using Prophet):

from prophet import Prophet
import pandas as pd

# Prepare the data (Prophet requires 'ds' (datetime) and 'y' (value) columns)
df = pd.DataFrame({'ds': pd.to_datetime(dates), 'y': values})

# Create a Prophet model
model = Prophet()

# Add holidays (optional)
holidays = pd.DataFrame({
  'holiday': 'US_Holiday',
  'ds': pd.to_datetime(holiday_dates),
  'lower_window': 0,
  'upper_window': 0,
})
model = Prophet(holidays=holidays)

# Fit the model
model.fit(df)

# Create a future dataframe for forecasting
future = model.make_future_dataframe(periods=365) # Forecast for next 365 days

# Make predictions
forecast = model.predict(future)

# Plot the forecast
fig1 = model.plot(forecast)
plt.show()

# Plot the components
fig2 = model.plot_components(forecast)
plt.show()

ARIMA (Autoregressive Integrated Moving Average) models can be extended to include exogenous variables (ARIMAX). These variables are external factors that can influence the time series. This allows for incorporating information beyond the historical values of the time series itself, leading to improved forecasts.

Example:

Forecasting sales, including advertising spending as an exogenous variable.

Code Snippet (Python - using statsmodels):

import pandas as pd
from statsmodels.tsa.arima.model import ARIMA

# Assuming 'sales' is your time series, 'advertising' is your exogenous variable
# and that you have a dataframe called 'df' with sales data (y) and advertising data (x)

# Define the ARIMAX model
model = ARIMA(df['y'], exog=df['x'], order=(5,1,0))
model_fit = model.fit()

# Generate forecasts
predictions = model_fit.predict(start=len(df), end=len(df)+10, exog=df['x'].iloc[len(df):len(df)+10])

State-space models provide a flexible framework for modeling time series data, allowing for the inclusion of multiple sources of variation and the ability to handle missing data. Exponential Smoothing State Space Models (ETS) are a specific type of state-space model that extends exponential smoothing methods to model level, trend, and seasonality. They are defined by the error, trend, and seasonal components (e.g., multiplicative or additive models).

Example:

Modeling the level, trend, and seasonal components of retail sales data.

Code Snippet (Python - using statsmodels):

import pandas as pd
from statsmodels.tsa.statespace.exponential_smoothing import ExponentialSmoothing

# Assuming 'sales_data' is your time series

# Fit the ETS model (e.g., multiplicative seasonality)
model = ExponentialSmoothing(sales_data, seasonal_periods=12, trend='add', seasonal='mul')
model_fit = model.fit()

# Generate forecasts
predictions = model_fit.forecast(12)  # Forecast for the next 12 periods

Anomaly detection identifies unusual patterns in time series data. Common methods include:

• Moving Average with Thresholds: Calculate a moving average and set thresholds (e.g., based on standard deviations from the moving average).
• Z-score: Calculate Z-scores for each data point and flag values exceeding a threshold (e.g., +/- 3 standard deviations).
• Statistical Process Control (SPC): Use control charts to monitor the process and detect when values fall outside control limits.

Example (Z-score):

Detecting unusual spikes in website traffic.

Code Snippet (Python):

import numpy as np
import pandas as pd

# Assuming 'traffic' is your time series
window_size = 30
rolling_mean = traffic.rolling(window=window_size).mean()
rolling_std = traffic.rolling(window=window_size).std()

z_scores = (traffic - rolling_mean) / rolling_std

threshold = 3
anomalies = z_scores[np.abs(z_scores) > threshold]