Autoregressive AR(p) Model for Forecasting Daily Sea Surface Temperature in a Tropical Gulf

Project info

• Field of studyFisheries and Marine Science
• LevelMaster's thesis (S2)
• Method typeTime Series
• Estimated build time± 5 working days
• Toolsnumpy.linalg.lstsq (implementasi manual AR); scipy; matplotlib
• PriceUS$22 - US$134

Research goal

Forecast sea surface temperature with Autoregression AR(p) using wind speed and cloud cover as supporting variables.

Variables

Example result

• 11Selected order p
• 129.051AIC
• 2.406RMSE (holdout)
• 7.123MAPE % (holdout)
• 0.767In-sample R2
• -0.005Residual lag-1 ACF

The AR(11) model forecasts Sea surface temperature 12 periods ahead with RMSE 2.406 and MAPE 7.12%. The order was chosen by AIC.

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

About Autoregression AR(p)

Autoregression AR(p) models the current value as a linear combination of its p previous values. It suits stationary time series whose values depend on their own history.

Steps

1. 1Check stationarity and difference if needed
2. 2Build the lag matrix for several values of p
3. 3Select the order p with AIC
4. 4Estimate coefficients by least squares
5. 5Check residual autocorrelation
6. 6Produce recursive forecasts and plot

Assumptions to check

• The series is stationary (constant mean and variance)
• Residuals are uncorrelated with zero mean
• Dependence over time is linear

Reading and reporting the results

Report the selected order, coefficients, and forecast error on test data. Include residual autocorrelation results to show the model is adequate.

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

rng = np.random.default_rng(12)
n = 300
e = rng.normal(0, 1, n)
sea_surface_temperature = np.zeros(n)
for i in range(2, n):
    sea_surface_temperature[i] = 0.6 * sea_surface_temperature[i - 1] - 0.3 * sea_surface_temperature[i - 2] + e[i]
wind_speed = np.arange(n)

train, test = sea_surface_temperature[:270], sea_surface_temperature[270:]
pmax = 6

def lagmat(y, p, start):
    X = np.column_stack([y[start - k - 1:len(y) - k - 1] for k in range(p)])
    return np.column_stack([np.ones(len(X)), X]), y[start:]

best = None
for p in range(1, pmax + 1):
    X, t = lagmat(train, p, pmax)
    beta = np.linalg.lstsq(X, t, rcond=None)[0]
    sse = ((t - X @ beta) ** 2).sum()
    aic = len(t) * np.log(sse / len(t)) + 2 * (p + 1)
    print("p =", p, "AIC =", round(aic, 2))
    if best is None or aic < best[0]:
        best = (aic, p, beta)
_, p, beta = best
print("Selected p:", p, "coef:", beta.round(3))

hist = list(train)
fc = []
for _ in range(len(test)):
    nxt = beta[0] + sum(beta[k + 1] * hist[-k - 1] for k in range(p))
    fc.append(nxt); hist.append(nxt)
print("Forecast RMSE:", round(np.sqrt(np.mean((test - np.array(fc)) ** 2)), 3))

plt.plot(wind_speed[250:], sea_surface_temperature[250:], label="Data")
plt.plot(wind_speed[270:], fc, "r", label="AR 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)

Frequently asked questions