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and t 36525 t 0 1 sin log t 0 1 sin 2t t 2t t

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Unformatted text preview: q 0.00 Precipitation (in) q q q q q q q q q q q q q q q q q q q Jan q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Mar q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q May q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Jul q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Sep q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Nov Non-Stationarity Cyclic variation Fort Collins, Colorado precipitation Orthogonal approach. First fit annual cycle to Poisson rate parameter (T = 365.25): 2πt log λ(t) = λ0 + λ1 sin T 2πt + λ2 cos T Giving ˆ (t) ≈ −3.72 + 0.22 sin 2πt log λ T 2πt − 0.85 cos T Likelihood ratio test for λ1 = λ2 = 0 (p-value ≈ 0). Non-Stationarity Cyclic variation Fort Collins, Colorado precipitation Orthogonal approach. Next fit GPD with annual cycle in scale parameter. ∗ ∗ log σ ∗(t) = σ0 + σ1 sin 2πt T ∗ + σ2 cos 2πt T Giving 2πt ˆ log σ (t) ≈ −1.24 + 0.09 sin T ∗ 2πt − 0.30 cos T ∗ Likelihood ratio test for σ1 = σ ∗2 = 0 (p-value < 10−5) Non-Stationarity Cyclic variation Fort Collins, Colorado precipitation Annual cycle in location and scale parameters of the GEV re-parameterization approach point process model with t = 1, 2, ..., and T = 365.25. µ(t) = µ0 + µ1 sin log σ (t) = σ0 + σ1 sin 2πt T 2πt T ξ (t) = ξ + µ2 cos + σ2 cos 2πt T 2πt T Non-Stationarity Cyclic variation Fort Collins, Colorado precipitation ˆ µ(t) ≈ 1.281 − 0.085 sin 2πt T log σ (t) ≈ −0.847 − 0.123 sin ˆ − 0.806 cos 2πt...
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