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The Japan Meteorological Agency (JMA) provides tertile probabilistic forecasts for 3monthaveraged sea surface temperature, surface temperature and precipitation over the global. The ordered probit model is used to calibrate tertile probabilistic forecasts using 30year hindcasts (1991 – 2020). The thresholds of tertile are determined so that the climatological chance of occurrence for each category is 33.3 % for the hindcast period from 1991 to 2020. The specification of the numerical prediction model is described on the model outline page.
The ordered probit model is an extension of the binary probit model that can be used in cases where there are multiple and ranked discrete dependent variables. Considering the simple case where the dependent variable Y takes the values 0, 1 and 2 as in the binary probit model, we 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 tertile (Y=0), the middle tertile (Y=1), and the upper tertile (Y=2) probabilities for 3monthaveraged model outputs. In the case of precipitation, the predictor X is the ensemble mean 1/4 power transformed precipitation to get better fit to the Gaussian distribution, otherwise the predictor X is a simple ensemble mean. In order to determine the three parameters of the statistical model, β, k1 and k2 by the maximun likelihood estimation method using the 30year (1991 – 2020) hindcast data. The skill of the tailored probabilistic forecasts is crossvalidated, and shown in the verification pages ( maps of BSS and ROC area scores, and reliability diagrams and ROC curves).
The probabilistic forecast maps over global or Asia region are shown. For example, Fig. 1 shows most likely Categories of surface temperature (left), precipitation (middle) and sea surface temperature (right) in the Asian region for March–April–May 2022. On the maps of surface temperature, the colors range from light to dark red is for above normal (i.e., upper third of probabilities), grey is for normal (i.e., middle third of probabilities) and from light blue to purple is for below normal (i.e., lower third of probabilities). On the maps for precipitation, the colors range from light green to blue is for above normal, grey is for normal and from dark yellow to red is for below normal. Please click any areas of your interest on the maps to display the tertile probabilities (Fig. 2). The verifications are also available from links in the forecast page at station points, where the calibrated hindcasts were verified over the 9 nearrest grids. Please note that due to a limited size of hindcast cases, the verification scores are used for guidance of forecast performance.
Figure 1 Most likely Categories of surface temperature (left), precipitation (middle) and sea surface temperature (right) in Asia region for March–April–May 2022.

Figure 2 The probability for each category in Northwestern Pacific (140E, 25N). 
