Probabilistic downscaling: multimodel probabilistic prediction methods based on Bayes theorem
BAYES APPROACH WITH REGRESSION LIKELIHOOD
where posterior probability of y;
- prior - a priory knowledge about probability of y
- likelihood -
- total probability of x.
Assumption: x and y are normally distributed (need to test)
Our goal is to estimate these m and s.
and find posterior Probability Distribution Function (PDF):
Posterior Probability Distribution Function
If PDF is Gaussian, it is reasonable to assume linear relationships between x and y and link them with regression, particularly:
Derivative of function under exponent in respect to y gives 0 in the top of PDF
A posteriory PDF
yt – series of predictand - a target variable anomaly at the target station (T2m, T850, precipitation, etc.) - observations
xt – series of predictor – single model ensemble mean – series of anomaly of some quantity (grid-point value, area mean, PC, etc.) of Z500, wind component, etc.
t = 1…T
T – training period length (years)
Time (year) f – time (year) of forecast.
yf – target variable forecast value xf – model forecast value (ensemble mean)
xef – model ensemble members in year f ; N – ensemble size.
M – number of models (m = 1….M)
Forecast of predictand mean value (deterministic forecast)
Assumption for each single model participated in MME:
a and b are estimated by Least Squares Estimation using observations and forecasts (ensemble means) from the training period. Due to anomalies a=0 (we write it just not to forget).
Forecast value of y by a single model:
Estimation of uncertainty
Regression error variance
Errors in estimates of regression parameters a and b (although we deal with anomalies and a=0, error in a estimate is not zero):
Uncertainty of forecast model ensemble mean
Reminder: we have to estimate PDF parameters:
Let’s take climatology as an a priory knowledge (prior forecast consideration)
So, we can estimate parameters mp and sp from climatology
(since we deal with anomalies, mp = 0 so)
(since we deal with anomalies, a=0)
Let’s write once again and analyze the final equations for anomalies: Forecast PDF mean:
If (which we get when regression is poor – small b and large se) m ® 0
– to climatological (for anomalies) mean.
If (which we get when regression is significant – large b and small large b and small se) m ® my – to model forecast anomaly.
If is comparable with , m is between my and mp, closer to that which variance is smaller.
From it follows that forecast variance is smaller than the smallest of and
Use of Total Probability Formula (see previous lecture)
We may use climatology as a first prior and then add model by model using posterior probability from each step as a prior probability for the next step.
So, we have a recurrent formula. For each m-step (addition of m-model (1<=m<=M), with climatology being considered as a 0-model) we do