**LECTURE 6 **

**Probabilistic downscaling: multimodel probabilistic prediction methods based on Bayes theorem**

**Vladimir Kryjov**

BAYES APPROACH WITH REGRESSION LIKELIHOOD

BAYES FORMULA

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:

where

Derivative of function under exponent in respect to y gives 0 in the top of PDF

A posteriory PDF

with parameters

**Practical implementation **

**Notations: **

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.

*y _{f}* – target variable forecast value xf – model forecast value (ensemble mean)

*x*_{ef }– model ensemble members in year f ;* N* – ensemble size.

*M *– number of models (*m = 1….M*)

SINGLE MODEL

*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:

So,

Reminder: we have to estimate PDF parameters:

Let’s take climatology as an a priory knowledge (prior forecast consideration)

So, we can estimate parameters *m _{p}* and

*s*from climatology

_{p }(since we deal with anomalies, *m _{p}* = 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 *s _{e}*)

*m*® 0

– to climatological (for anomalies) mean.

If (which we get when regression is significant – large b and small large *b* and small *s _{e}*)

*m*®

*m*– to model forecast anomaly.

_{y}If is comparable with , *m* is between *m _{y}* and

*m*, closer to that which variance is smaller.

_{p}From it follows that forecast variance is smaller than the smallest of and

MULTIMODEL COMBINATION

PARALLEL

Use of Total Probability Formula (see previous lecture)

SERIAL

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.

Steps:

1.

2.

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

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