Holt-Winters Forecasting of Monthly Rail Passenger Numbers with Holiday Season Patterns

Project info

• Field of studyCivil Engineering and Transportation
• LevelDoctoral dissertation (S3)
• Method typeTime Series
• Estimated build time± 7 working days
• Toolsnumpy, scipy.optimize.minimize (implementasi manual Holt-Winters aditif)
• PriceUS$34 - US$213

Research goal

Forecast passenger count with Holt-Winters Exponential Smoothing using holiday season index and mean fare as supporting variables.

Variables

Example result

• 10.286RMSE (holdout)
• 10.738MAPE % (holdout)
• 8.739MAE (holdout)
• 0.1alpha (level)
• 0.01beta (trend)
• 0.3gamma (season)

The additive Holt-Winters model forecasts Passenger count 12 periods ahead with RMSE 10.286 and MAPE 10.74%. Best parameters: alpha 0.10, beta 0.01, gamma 0.30.

Computed from illustrative synthetic data, not from a real study.

About Holt-Winters Exponential Smoothing

Holt-Winters is three-component exponential smoothing (level, trend, seasonality) for forecasting seasonal time series. It suits monthly or quarterly data with fairly stable trend and seasonal patterns.

Steps

1. 1Set the season length m and split the data chronologically
2. 2Initialise level, trend, and seasonal indices
3. 3Find alpha, beta, gamma by minimising SSE
4. 4Compute fitted values and forecasts for the test period
5. 5Evaluate MAE and RMSE
6. 6Plot data, forecast, and the test period

Assumptions to check

• The seasonal pattern has a fixed length
• Trend and seasonal amplitude are fairly stable (additive model)
• Enough data, at least two full seasons

Reading and reporting the results

Report the smoothing parameters and test MAE and RMSE, and compare against a seasonal naive forecast. Also discuss that long-horizon forecasts become increasingly uncertain.

Example Python code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import minimize

def hw_run(y, m, a, b, g):
    n = len(y)
    l = y[:m].mean()
    t = (y[m:2 * m].mean() - y[:m].mean()) / m
    s = np.zeros(n + m)
    s[:m] = y[:m] - l
    fit = np.zeros(n)
    for i in range(n):
        fit[i] = l + t + s[i]
        l_new = a * (y[i] - s[i]) + (1 - a) * (l + t)
        t = b * (l_new - l) + (1 - b) * t
        s[i + m] = g * (y[i] - l_new) + (1 - g) * s[i]
        l = l_new
    return fit, l, t, s

rng = np.random.default_rng(11)
m, n = 12, 96
holiday_season_index = np.arange(n)
passenger_count = 100 + 0.8 * holiday_season_index + 10 * np.sin(2 * np.pi * holiday_season_index / m) + rng.normal(0, 2, n)
train, test = passenger_count[:-12], passenger_count[-12:]

sse = lambda p: ((train - hw_run(train, m, *p)[0]) ** 2).sum()
res = minimize(sse, [0.3, 0.1, 0.1], bounds=[(0.01, 1)] * 3, method="L-BFGS-B")
a, b, g = res.x
_, l, t, s = hw_run(train, m, a, b, g)
nt = len(train)
fc = np.array([l + (h + 1) * t + s[nt + h % m] for h in range(12)])
print("alpha, beta, gamma:", res.x.round(3))
print("MAE:", round(np.abs(test - fc).mean(), 3), "RMSE:", round(np.sqrt(((test - fc) ** 2).mean()), 3))

plt.plot(holiday_season_index, passenger_count, label="Data")
plt.plot(holiday_season_index[-12:], fc, "r", label="Forecast")
plt.legend()
plt.show()

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

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