[Paper Review] Predicting hurricane numbers from Sea Surface Temperature: closed form expressions for the mean, variance and standard error of the number of hurricanes

Stephen Jewson|ArXiv.org|Jan 15, 2007

Tropical and Extratropical Cyclones Research5 references3 citations

TL;DR

This paper derives closed-form analytical expressions for the mean, variance, and standard error of hurricane counts predicted from sea surface temperature (SST) using probabilistic models. It presents mathematically tractable solutions for linear-normal, linear-Poisson, and exponential-Poisson relationships between SST and hurricane numbers, enabling efficient uncertainty quantification in seasonal hurricane forecasting without simulation.

ABSTRACT

One way to predict hurricane numbers would be to predict sea surface temperature, and then predict hurricane numbers as a function of the predicted sea surface temperature. For certain parametric models for sea surface temperature and the relationship between sea surface temperature and hurricane numbers, closed-form solutions exist for the mean and the variance of the number of predicted hurricanes, and for the standard error on the mean. We derive a number of such expressions.

Motivation & Objective

• To develop analytically tractable models for predicting hurricane numbers from sea surface temperature (SST) forecasts.
• To quantify the uncertainty in predicted hurricane counts by deriving closed-form expressions for mean, variance, and standard error.
• To evaluate the impact of SST forecast uncertainty on hurricane count predictions across different statistical models.
• To enable efficient probabilistic forecasting of basin and landfalling hurricane numbers by combining SST-to-basin and basin-to-landfall relationships.
• To provide a framework for sensitivity analysis of hurricane predictions to SST mean and variance changes.

Proposed method

• Assumes SST follows a normal distribution with known mean and variance, enabling analytical propagation of uncertainty.
• Derives closed-form expressions for the mean and variance of basin hurricane counts under three models: linear-normal, linear-Poisson, and exponential-Poisson relationships with SST.
• Uses the law of total variance and moment-generating functions to propagate uncertainty from SST to hurricane counts.
• Combines SST-to-basin hurricane models with basin-to-landfall hurricane models via linear-Poisson relationships to predict landfalling counts.
• Derives standard error expressions for the predicted mean hurricane count by propagating variances and covariances of model parameters.
• Applies the delta method and moment-based approximations to derive analytical expressions for variance and standard error in hierarchical prediction chains.

Experimental results

Research questions

• RQ1 What closed-form expressions exist for the mean and variance of predicted basin hurricane counts given a probabilistic SST forecast?
• RQ2 How does the choice of SST-hurricane relationship (linear-normal, linear-Poisson, exponential-Poisson) affect the analytical tractability and uncertainty structure of hurricane predictions?
• RQ3 What is the analytical expression for the standard error of the predicted mean number of hurricanes when SST is uncertain?
• RQ4 How can the prediction of landfalling hurricane numbers be analytically derived by combining SST-to-basin and basin-to-landfall models?
• RQ5 What are the sensitivity relationships between the mean and variance of predicted hurricane counts and the mean and variance of the SST forecast?

Key findings

• For the linear-Poisson model, the mean number of basin hurricanes is μ_b = α + βμ_s, and the variance is σ²_b = β²σ²_s + μ_b.
• For the exponential-Poisson model, the mean basin hurricane count is μ_b = exp(α + β(μ_s + βσ²_s/2)), and the variance is σ²_b = μ²_b(exp(β²σ²_s) - 1) + μ_b.
• The mean number of landfalling hurricanes is μ_l = α′ + β′μ_b, with variance σ²_l = β′²σ²_b + μ_l, when using a linear-Poisson model from basin to landfall.
• The standard error of the predicted mean landfalling hurricane count is derived by combining the variances and covariances of parameters across the SST → basin → landfall chain.
• The variance of the predicted mean basin hurricane count under the linear-Poisson model is var(μ_b) = var(α) + μ²_s var(β) + β² var(μ_s) + 2μ_s cov(α,β).
• The full analytical framework allows for uncertainty quantification in hurricane forecasts without relying on Monte Carlo simulation, significantly improving computational efficiency.