Tokyo Climate Center

Ordered Probit Model

The Ordered Probit model is a fairly straight-forward extension of the binary probit model that can be used in cases where there are multiple and ranked discrete dependent variables. Consider the simple case, where the dependent variable Y takes the values 0, 1, or 2. As in the binary probit model, define an unobserved index function Y* as:

	Y* = X β + ε

and assume:

	Y = 0 if  Y* < k1,
	Y = 1 if k1 ≤ Y* < k2,
	Y = 2 if k2 ≤ Y*,

where k1 and k2 are "cut points" and k1 < k2.
Then, the conditional probabilities Pr(Y=0 | X), Pr(Y=1 | X), and Pr (Y=2 | X) can be written as

	Pr(Y=0 | X) = Pr(X β + ε < k1) = Pr(ε < - X β + k1) = F(- X β + k1),
	Pr(Y=2 | X) = Pr(X β + ε > k2) = Pr(ε > - X β + k2) = 1 - F(- X β + k2),
	Pr(Y=1 | X) = 1 - Pr(Y=0) - Pr(Y=2) = F(- X β + k2) - F(- X β + k1),

where F is the cumulative distribution function of residual ε. In the Ordered Probit model, it is assumed that the residual ε has the standard normal distribution N(0,1). Thus, F is the cumulative function of N(0,1).
The statistical model is used to calculate the lower tercile (Y=0), the middle tercile (Y=1), and the upper tercile (Y=2) probabilities for 3-month-averaged surface temperature and precipitation from the numerical model output. The predictors X for the probabilities of temperature and precipitation are the ensemble mean forecast temperature and 1/4 power transformed precipitation, respectively. In order to determine the three parameters of the statistical model, β, k1 and k2 by the maximun likelihood estimation method, the Model Output Statistics (MOS) technique based on the 30-year (1979-2008) hindcast data is used. The skill of the MOS based model is verified in cross-validation mode, and shown at the Verification page.