Transcript Statistical Forecasting 1

Statistical Forecasting [Part 1]
Dr Mark Cresswell
69EG6517 – Impacts & Models of Climate Change
Lecture Topics
• What is statistical forecasting?
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Simple linear regression
Multiple linear regression
Analog forecasting
Model Output Statistics (MOS) forecasts
PCA and EOF
Canonical Correlation Analysis
What is Statistical Forecasting?
In nature, observed phenomena are intrinsically linked to
each other by physical processes. Such processes are
referred to as the causality or causal links.
In statistical forecasting, we can exploit this causality
mathematically by replicating the pattern of change observed
for a particular set of conditions. The physical processes
represent the forcing and the observed pattern (of weather!)
is the direct result.
Different sets of conditions (forcing) will give rise to replicable
and specific patterns of weather
What is Statistical Forecasting?
Example: Unusually warm sea-surface temperature (SST) conditions
in the Indian Ocean is usually associated with a greater than normal
frequency (and magnitude) of tropical cyclones. The forcing here is
the increased SSTs whilst the observed pattern is enhanced
cyclogenesis.
Cyclone Eline
(Feb 2000)
What is Statistical Forecasting?
Model: Since we know there is a causal link (enhanced energy flux,
more evaporation over the ocean, greater convection etc) we can
determine a statistical relationship from historical observations
Simple Linear Regression
Model: The previous example illustrated how a specific forcing can
be seen to alter future weather conditions. We can summarise this
relationship mathematically in a regression equation
Regression
equation
This type of
Relationship
Is known as
empirical
Simple Linear Regression
Model: The regression model informs us of the dependence one
variable has on another. Usually, we will select variables that are
correlated with one another
Simple Linear Regression
We must be careful however when using relationships based purely
on a correlation as association is not causation
In children, shoe size may be strongly correlated with reading skills.
This does not mean that children who learn to read new words sprout
longer feet !
The simple calculation of one variable from another based on a
regression equation is known as the method of least squares.
Normally we can insert a line of best fit through a scatter-plot of X
and Y data pairs. The line that makes the smallest r.m.s (root mean
square) error in predicting Y from X is the regression line
Simple Linear Regression
The regression line is often referred to as the least squares line. We
can use the slope and intercept characteristics of a least squares line
to derive constants, m (slope) and b (intercept) that can be used in
our linear regression model equation:
y  mx  b
The intercept is the height of the least squares line when X is zero.
The slope is the rate at which Y increases per unit increase in X
Thus, for any given value of our X variable (and values for m and b
which we calculate from observations) we can estimate a value for Y
Y variable
Simple Linear Regression
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intercept
X variable
Multiple Linear Regression
Often, we will not look at the contribution of a single variable
in isolation – but instead a number of predictors will be
included in a forecast
Different predictors may be causally related to the same
weather phenomena.
If K is the number of predictor variables then:
ŷ = b0 + b1x1 + b2x2 + ····· + bkxk
Multiple Linear Regression
Linear regression models provide a “fit” for our estimate of y
for a number of observations of x
A straight-line fit (simple linear regression) will not go through
all points…but a multivariate regression line will be curved
thus allowing a better fit and a more accurate estimate of y
for a given value of x
Multivariate models are often used on weather prediction to
estimate future change based on historical observations of a
trend.
Multiple Linear Regression
Multiple Linear Regression
Multiple Linear Regression
Analog Forecasting
Not all objective statistical forecast procedures are based on
regression
Some methods were in use prior to the advent of fairly
accurate NWP forecasts (12-48hr range). One such method
is analog forecasting
Analog forecasting is still in use for long-range (seasonal)
forecasting – although the climatological database it uses is
deemed to be too short for AF forecasts to be competitive in
ordinary short-range weather forecasting
Analog Forecasting
The idea underlying AF is to search the archives of
climatological synoptic data for maps closely resembling
current observations, and assume that the future evolution of
the atmosphere will be similar to the flows that followed the
historical analogs
The method is intuitive and gains from the value provided
from experienced weather forecasters
AF is limited however as the atmosphere apparently does not
exactly repeat itself – so matches can only be approximate
Model Output Statistics (MOS)
The MOS approach is a preferred method of incorporating
NWP forecast information into statistical weather forecasts
The MOS approach has the capacity to include directly into
the regression equations the influences of specific
characteristics of different NWP models at different time
projections in the future
To develop MOS forecast equations it is necessary to have a
developmental data set composed of historical records of the
predictand, together with archived records of the forecasts
produced by the NWP model for the same days on which the
predictand was observed
PCA and EOF
Sometimes we might need to compare sets of variables
against patterns of change – and synthesise them
It might be the case that environmental change (shifts in
weather patterns) are due to more than one variable. In order
to determine the spatial limits of their influence (in a
geographical sense) we can use a spatially dependent
correlation scheme – called Principal Components Analysis
(PCA). The technique allows data reduction
PCA as a technique, became popular following papers by
Lorenz in the mid 1950s – who called the technique
Empirical Orthogonal Function (EOF) analysis. Both names
refer to the same set of procedures
PCA and EOF
The purpose of PCA is to reduce a data set containing a
large number of variables to a new data set containing far
fewer new variables – but which nevertheless represent a
large fraction of the variability contained in the original data
Following PCA analysis the method provides a number of
principal components – which constitute a compact
representation of the original data.
PCA can yield substantial insights into both the spatial and
temporal variations exhibited by the field or fields being
analysed
Canonical Correlation Analysis
CCA is a statistical technique that identifies a sequence of
pairs of patterns in two multivariate data sets – and
constructs sets of transformed variables by projecting the
original data onto these patterns
The patterns are chosen such that the new variables defined
by projection of the two data sets onto these patterns exhibit
maximum correlation
CCA is an extension of multiple regression models. It is often
applied to fields – such as SST or heights of pressure.