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[Description] Quality of Estimation - Understanding parameter uncertainty using Fisher Information Matrix
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Quality of Estimation Worksheet
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Learning Objectives:
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1. Understand the meaning of the error covariance matrix for parameter estimates
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2. Calculate the Fisher Information Matrix (FIM) using numerical derivatives
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3. Compute standard errors and confidence intervals for model parameters
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4. Interpret parameter correlations and their implications for model identifiability
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Key Formulas:
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• Beverton-Holt model: f(S; α, k) = αS / (1 + S/k)
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• Measurement variance: σ² = SSE / (n - p)
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• Numerical derivative: ∂f/∂θ ≈ [f(θ+h) - f(θ-h)] / (2h)
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• Fisher Information Matrix: FIM = Jᵀ × J (unscaled)
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• Error Covariance Matrix: C = σ² × (Jᵀ × J)⁻¹ = σ² × FIM⁻¹
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• Standard Error: SE(θ) = √Cᵢᵢ
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• 95% CI: θ̂ ± t_{α/2, n-p} × SE(θ)
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• Parameter Correlation: ρᵢⱼ = Cᵢⱼ / (SE(θᵢ) × SE(θⱼ))
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Data from Lecture 03a:
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• Model: Beverton-Holt (M. merluccius fish stock-recruitment)
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• Optimal parameters: α = 5.75, k = 33.16
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• Sum of Squared Errors: SSE = 2809.01
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• Number of observations: n = 15
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• Number of parameters: p = 2
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Color Guide:
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Given Data (Input)
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Intermediate Calculations
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Your Input (Calculate)
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Correct Answer (✓)
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Incorrect Answer (✗)
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