Uma vez calculados os parˆametros da distribui¸c˜ao, a probabilidade acumulada da distribui¸c˜ao Pearson III obt´em-se atrav´es de:
F (x) = 1 αΓ(β) Z x γ x − γ α β−1 e−(x−γα )dx (A.1)
Para se obter as probabilidades acumuladas recorre-se a um m´etodo anal´ıtico que parte de uma transforma¸c˜ao das s´eries de precipita¸c˜oes reais (x) at´e `as s´eries (y) de acordo com os parˆametros da sua distribui¸c˜ao de frequˆencias. Onde x ´e o valor de precipita¸c˜ao para um dado mˆes.
y = x − γ α
(A.2) A sua probabilidade acumulada ser´a:
F (y) = 1 Γ(β)
Z x−γα
0
yβ−1e−ydy (A.3)
O valor de F (y) pode obter-se de analiticamente de forma similar a Abramowitz e Stegun (1965). Assim, F (y) pode escrever-se como:
F (y) = F (χ2|ν) (A.4)
Sendo esta a distribui¸c˜ao Chi-quadrado com ν = 2β e χ2 = 2y.
Kendall e Stuart (1963) utilizam a seguinte express˜ao para aproximar uma distribui¸c˜ao normal standard quando o n´umero de registos e igual ou superior a 30, como ´e o caso do SPI: u = " χ2 ν 1/3 + 2 9ν − 1 # 9ν 2 1/2 (A.5) Assim, as s´eries y s˜ao transformadas em s´eries u que constituem aproxima¸c˜oes a uma distribui¸c˜ao normal standard. A probabilidade acumulada de F (u) obt´em-se por (Abra-
Avalia¸c˜ao da seca global num clima em mudan¸ca
mowitz e Stegun, 1965):
F (u) = 1 − f (u)(b1q + b2q2+ b3q3+ b4q4+ b5q5) + ε1(u) (A.6)
Em que:
q = 1
1 + pu (A.7)
Sendo p = 0.2316419 e u deve ser positivo tal que −u = u. Os parˆametros bi com i = 1...5
da equa¸c˜ao (A.6) s˜ao:
b1 = 0.139381530
b2 = −0.356563782
b3 = 1.781477937
b4 = −1.821255978
b5 = 1.330274429
A fun¸c˜ao f (u) ´e definida por:
f (u) = (a0+ a2u2+ a4u4+ a6u6+ a8u8+ a10u10)−1+ ε2(u) (A.8)
Em que os parˆametros a2(i−1) com i = 1...6 s˜ao:
a0 = 2.5052367 a2 = 1.2831204 a4 = 0.2264718 a6 = 0.1306469 a8 = −0.0202490 a10 = 0.0039132
Nas equa¸c˜oes (A.6) e (A.8) considera-se que ε1(u) < 2.3 × 10−4 e ε2 < 7.5 × 10−8,
respectivamente. Quando os valores de u s˜ao negativos ent˜ao:
F (−u) = 1 − F (u) (A.9)
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