# Presentation: 679073

## Probability and Statisticsfor Ensemble Forecasting

- Tom Hamill (NOAA/ESRL, Boulder)
- and
- Jim Hansen (Naval Research Lab, Monterey)

- (borrows heavily from Dan Wilks’
*Statistical Methods in the Atmospheric Sciences text*)

## Probability and statistics: (1) inherently confusing, or(2) a formal way of bamboozling and waffling?

- “Doctors say that Nordberg
- has a 50/50 chance of living,
- though there's only a 10
- percent chance of that.”

## Probability and statistics

**Probability**: a formalism for expressing uncertainty quantitatively.
**Statistics**: the science pertaining to the collection, analysis, interpretation or explanation, and presentation of data.
**Goal**: get you comfortable with the terminology the other instructors will use.

## Part 1: Probability

- photo courtesy of Lenny Smith, Oxford U. and London School of Economics

- Weather is uncertain, so we use the language of uncertainty

## Probability religions

**The Frequentists**: probability is the long-run expected frequency of occurrence.
**The Bayesians**: probability is a degree of believability.

- “*A Frequentist is a person whose long-run ambition is to be wrong 5% of the time*.”
- “*A Bayesian is one who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule*.”

- Should this matter to you?
- Luckily, no...we all worship the same mathematical deity.

## Laws of the Probabilistic Deity: the “Axioms of Probability”

- S

- No precip (E1)

- Precip (E2)

- Liquid

- Frozen and Liquid

- Frozen

- 0.0  ≤  Pr(E1)  ≤  1.0
- Pr(S) = 1.0
- Pr(E1) + Pr(E2) = 1.0

- S is the “sample space.” E1 and E2 are
- “mutually exclusive” and “collectively exhaustive”
- events that fill the sample space.

## Union of Events

- S

- No precip (E1)

- Precip (E2)

- Liquid

- Frozen and Liquid

- Frozen

- Pr(Liquid) and Pr(Frozen) = Pr(Liquid) + Pr(Frozen) - Pr(Frozen and Liquid)

## Conditional Probability

- E1

- E2

- S

## Conditional Probability

- E1

- E2

- S

- E1 and E2

- Pr(E1 | E2) = Pr(E1 given that E2 has occurred)
- 	        = Pr(E1 and E2) / Pr(E2)

- narrow the playing field ... consider only the subset where
- E2 has occurred

## Example: “loaded gun” sounding

- unconditional probability
- of a tornado is small; most
- likely it will be impossible
- to break through the
- capping inversion.

- P(tornado | thunderstorm) = 0.35

- if penetrative convection
- does happen, the large
- instability and shear increase
- the probability that the
- thunderstorm will produce a tornado.

## Independence

- E1 and E2 are independent if and only if
- 	Pr (E1 and E2) = Pr (E1) x Pr (E2)

- Probability of two sixes =
- 1/6 x 1/6 = 1/36

## Discrete vs. Continuous Probability 

**Discrete**:  probability assigned
-     to limited number of
-     possible outcomes
**Continuous**: unlimited number of outcomes.

- P(T=60.0) not meaningful, though
- P(58 ≤ T ≤ 62) is.
- probability density
- expressed relative likelihood
- of being *near* a particular value;
- and probability density follows
- other probability axioms, e.g.,

## Discrete “parametric” probability distributions: the binomial distribution

*X* is random variable
*x* is a specific number
*N* is the number of trials
*p* is the event probability

## Discrete “parametric” probability distributions: the binomial distribution

*X* is random variable
*x* is a specific number
*N* is the number of trials
*p* is the event probability

- Note: you might
- forecast 50%
- probability of rain,
- and rain may
- happen 9 / 10
- times.  That can
- happen, though
- it’s unlikely.

## Continuous parametric probability distributions: the “Normal” distribution

- Also called “Gaussian” or ‘the bell-shaped curve”
*f *(*x*)* *is the “probability density”
** is the “mean”
** is the
-    “standard deviation”

*** **= 60.0, **** = 4.0*

## Continuous parametric probability distributions: the “Normal” distribution

- Also called “Gaussian” or ‘the bell-shaped curve”
*f *(*x*)* *is the “probability density”
*    *function, or PDF
** is the “mean”
** is the
-    “standard deviation”

*** **= 60.0, **** = 4.0*

- this ensemble might
- be a random sample
- from a smooth
- distribution like this

## The gamma distribution

** = shape parameter
** = scale parameter

** = 1.0, **= 4.0

** = 3.0 **= 4.0

## “Empirical” probability distributions

- distribution derived from the data itself

## Statistics

**    ****Definition**: “the science pertaining to the collection, analysis, interpretation or explanation, and presentation of data.”

## Measures of “location”

- T = [50, 51, 53, 54, 54,  57, 59, 63, 65, 66, 84]      		(*n*=11)
- Measure the centrality of this data set in some fashion.
- Mean (also called average, or 1st moment); minimizes RMS error:
- Median: central value of the sample, here = 57. Less affected by the 84 “outlier.”  Minimizes mean absolute error.

## Measures of spread

- T = [50, 51, 53, 54, 54,  57, 59, 63, 65, 66, 84]
- Standard Deviation of sample:
- 	(variance is the square of this)
- IQR (Interquartile Range) = q0.75 - q0.25 = 65 - 53 = 12
- 	where q0.75 is the 75th percentile (quantile) of the distribution and q0.25 is the 25th percentile.

## Measures of association

- Tf   = [50, 51, 53, 54, 54,  57, 59, 63, 65, 66, 84]
- To  = [54, 53, 51, 57, 55,  53, 66, 63, 66, 71, 87_x0018__x0013_]

## Measures of association

- Pearson (ordinary) correlation:

## Correlation, mean, standard deviation

*r* = 0.953

## Regression

- Find the equation that
- minimizes the squared
- difference between
- forecasts and observations.

- Methods like this used to
- statistically adjust weather
- forecasts.

## Connection between ensemble forecasts and PDFs

- Fokker-Planck equation to describe evolution of forecast PDF

- errors due to chaos

- errors due to the model

- (there is a theory behind ensemble forecasting!)

- (in reality, we
- never can
- get the pdf
- shown on
- day 1 from
- Fokker-Planck)

## Connection, continued

- In ensemble forecasting (ideally), we sample the
- initial pdf, and...

## Connection, continued

- In ensemble forecasting (ideally), we sample the initial
- PDF, and evolve each initial condition forward with the
- forecast model(s) to randomly sample the day-1 PDF

## Connection, continued

- We might even go so far as to “fit” a PDF to the ensemble
- data to estimate probability density for other events.

## Questions?

## Baye’s Rule

- combine 2 right-hand sides and rearrange

## Law of total probability

- E1

- E2

- E3

- A and
-      E1

- A and
-    E2

- A and
- E3

- Overall “unconditional” probability can be computed summing /
- integrating the weighted conditional probabilities

- S

## Law of total probability: driving example

- Not snowy

- Snowy

- Fender-
- Bender

- P(Fender-Bender) = P(Fender-Bender| Not Snowy) P(Not Snowy)  +
-                                  P(Fender-Bender| Snowy) P(Snowy)

-                      = 0.01                     x   0.75              +
-                         0.10                     x   0.25
- 	      = 0.0325

## Law of total probability: driving example

- Not snowy

- Snowy

- Fender-
- Bender

- P(Fender-Bender) = P(Fender-Bender| Not Snowy) P(Not Snowy)  +
-                                  P(Fender-Bender| Snowy) P(Snowy)

-                      = 0.01                     x   0.75              +
-                         0.10                     x   0.25
- 	      = 0.0325  (I’m an excellent driver)

## Cumulative Distribution Function (CDF)

- F(t) = Pr {T ≤ t}
- where T is the random variable, t is some specified threshold.