Hierarchical Time Series Forecasting

Key Concepts:

1. Top-down Approach:

   • Start by forecasting at the highest level (e.g., total sales for a company).

   • Allocate the forecast to lower levels based on historical proportions or other rules.

2. Bottom-up Approach:

   • Forecast at the lowest level (e.g., sales for each store).

   • Aggregate these forecasts to get higher-level forecasts.

3. Middle-out Approach:

   • Combine top-down and bottom-up approaches.

   • Start from an intermediate level and use it to refine forecasts at higher and lower levels.

4. Reconciliation:

   • Adjust forecasts across levels to ensure coherence. For example, the sum of city-level forecasts should equal the state-level forecast.

Algorithms for HTSF:

1. Forecast at each level independently:

   • Use models like ARIMA, SARIMA, or machine learning models at each hierarchy level.

   • Reconcile forecasts afterward.

2. Reconciliation Methods:

   • OLS Reconciliation: Uses ordinary least squares to adjust forecasts.

   • MinT (Minimum Trace): Minimizes forecast errors while reconciling.

   • BU (Bottom-Up): Aggregates lower-level forecasts directly.

   • TD (Top-Down): Allocates higher-level forecasts to lower levels.

Tools for HTSF:

1. R:

   • hts package for hierarchical and grouped time series forecasting.

   • forecast package for general time series modeling.

2. Python:

   • Statsmodels: For ARIMA/SARIMAX modeling.

   • scikit-hts: For hierarchical time series forecasting.

   • Prophet: Can be adapted for hierarchical forecasting using aggregation.

3. Others:

   • Custom models using machine learning (e.g., LSTMs or XGBoost).

Example of Hierarchical Time Series Forecasting for sales data of 10 stores in 5 regions, where each region has 2 stores, and one store in each region is in an urban city.

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Scenario:

1. Hierarchy:

   • Level 1 (Total Sales): Aggregated sales for all 10 stores.

   • Level 2 (Regions): Aggregated sales for 5 regions.

   • Level 3 (Stores): Individual sales for 10 stores.

   • Urban City Classification: Indicates whether a store is in an urban city (binary feature).

2. Objective:

   • Forecast future sales for all stores.

   • Ensure that:

     • Store sales sum up to region sales.

     • Region sales sum up to total sales.

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Synthetic Data Generation:

For simplicity, we'll generate random sales data to demonstrate the hierarchy.

import pandas as pd
import numpy as np

# Generate synthetic sales data
np.random.seed(42)

# Store details
stores = [f"Store_{i+1}" for i in range(10)]
regions = [f"Region_{i//2 + 1}" for i in range(10)]  # 2 stores per region
urban_city = [True if i % 2 == 0 else False for i in range(10)]  # Urban city flag

# Generate random sales for each store
dates = pd.date_range(start="2024-01-01", periods=30)  # 30 days of data
data = {
    "Date": np.repeat(dates, len(stores)),
    "Store": stores * len(dates),
    "Region": regions * len(dates),
    "Urban_City": urban_city * len(dates),
    "Sales": np.random.randint(50, 200, len(stores) * len(dates)),
}

df = pd.DataFrame(data)

# Aggregate sales at region and total levels
region_sales = df.groupby(["Date", "Region"])["Sales"].sum().reset_index()
total_sales = df.groupby("Date")["Sales"].sum().reset_index()

# Display the datasets
print("Sample Store-Level Data:")
print(df.head())

print("\nSample Region-Level Data:")
print(region_sales.head())

print("\nSample Total Sales Data:")
print(total_sales.head())

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Forecasting Example:

1. Forecast at Store Level:

   • Use any time series model (e.g., ARIMA, SARIMA) for individual stores.

   • Example for one store:

from statsmodels.tsa.arima.model import ARIMA

# Forecast for one store (e.g., Store_1)
store_1_sales = df[df["Store"] == "Store_1"].set_index("Date")["Sales"]
model = ARIMA(store_1_sales, order=(1, 1, 1))
model_fit = model.fit()
forecast_store_1 = model_fit.forecast(steps=7)  # Forecast 7 days ahead
print(f"Forecast for Store_1: {forecast_store_1}")

1. Aggregate to Region Level:

   • Sum the forecasts for the two stores in each region.

region_forecasts = (
    df.groupby(["Date", "Region"])["Sales"]
    .sum()
    .groupby("Region")
    .apply(lambda x: x.resample("D").sum())
)

1. Aggregate to Total Sales:

   • Sum all region forecasts to get the total sales.

total_forecast = region_forecasts.sum(level="Date")
print(f"Total Sales Forecast: {total_forecast}")

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Reconciliation:

To ensure consistency:

• Use Bottom-Up, Top-Down, or MinT Reconciliation methods to adjust forecasts.

• Example using Bottom-Up:

  • Sum store-level forecasts to match region-level.

  • Sum region-level forecasts to match total sales.