The graph is straightening out to a straight line in a slightly downwards direction· The graph is slowly starting to curve back upwards· The graph is starting to curve back upwards, more and more so as you move to the right
It is a very popular curve fitting algorithm used in many software applications.
It takes abt 50 iterations to obtain these value.
Blue line indicates calculated value and red dots are actual value.
Significant track errors still occur on occasion, as seen in this Ernesto (2006) early forecast. The NHC official forecast is light blue, while the storm's actual track is the white line over Florida
2.
INTRODUCTION
Extrapolation is the process of constructing new data points outside the
range of known points.
It is the statistical technique of inferring about the future based on known
facts and observation, such as estimating the size of a population a few
years from now on the basis of current population size and its rate of
growth.
It is similar to the process of Interpolation, which constructs new points
between known points.
But the results of extrapolations are often less meaningful, and are
subject to greater uncertainty.
3.
ASSUMPTIONS
Use of aggregate data, generally across time (population,
employment, etc.)
Future movement of the data series is determined by past
patterns embedded in the series
The essential information about the future of the data
series is contained in the history of the series
Past trends will continue into the future
4.
ADVANTAGES
Extrapolation Techniques have Computational simplicity
Transparent methodology
low data requirements
May work for
1.Large areas
2.Short time horizons
3.Slow grow areas
5.
DISADVANTAGES
Does not account for underlying causes / structural
conditions
Current trend often does not continue
Excludes any external considerations
Lots of possible way to extend the line.
6.
EXTRAPOLATION
TECHNIQUES
The basic procedure for Extrapolation Techniques are:
1. Acquire population data for past years
2. Plot data to determine the best fitting curve
3. Extend the curve into the future
7.
GENERAL SHAPE OF VARIOUS CURVE FITTING
FAMILIES
8.
METHODS OF FINDING BEST-FIT CURVE
Best-Fit curve can be obtained by three methods
Least-square curve fit.
Smooth curve fit
Non linear curve fit
9.
LEAST SQUARE CURVE FIT
Minimizes the square of error between the original
data and the value predicted by the equation.
The five least square fits are
1.Linear
2.Polynomial
3.Exponential
4.Logarithmic
5.Power
10.
SMOOTH CURVE FIT
These curve fit do not generate an equation for
resulting curve. These is because there is no single
equation that can be used to represent the curve.
Three smooth fit are
1.Weighted
2.Cubic spline
3.interpolate
11.
NON LINEAR CURVE FIT
It is the most general method of curve fit.
There are two types of non linear function 1.Non-linear fitting function which can be transformed
into a linear fitting function.
example
N (t ) N o e t /
ln N t / ln N o
2.Non linear fitting function which cannot be
transformed into a linear fitting function.
example-
y a cos(bX ) b sin(aX )
This type of function can be Fit by
Levenberg–Marquardt algorithm (LMA)
12.
LEVENBERG–MARQUARDT ALGORITHM
(LMA)
It provides a numerical solution to the problem of
minimizing a Non linear function, over a space of
parameters of the function. These minimization
problems arise especially in curve fitting.
It is also used for solving Non linear Geophysical
Inverse problems.
No data restrictions associated with this algorithm.
It is an Iterative method starts with the initial guess for
the unknown parameters.
It can fail if the initial guesses of the fitting parameters
are too far away from the desired solution.
13.
Given a set of m empirical pairs of independent and
dependent variables, (xi,yi), optimize the
parameters β of the model curve f(x,β) so that the sum
of the squares of the deviations becomes minimum.
Provide the initial guess for the parameter vector β.
like βT=(1,1,...,1).
In each iteration step, the parameter vector, β, is
replaced by a new estimate, β + δ. To determine δ, the
functions are approximated by their linearizations.
where
is the gradient of f w.r.t.
f ( xi , ) f ( xi , ) J i
J i f ( xi , ) /
Ji
14.
The above first-order approximation of
m
S ( ) yi f ( xi , ) J i
gives
2
i 1
To obtained the minimum of S(β), gradient of S with
respect to δ will be zero. It gives
( J T J I ) J T [ y f ( )]
where I is Identity matrix
J is jacobian matrix
This is a set of linear equation which can be solved
for .
15.
EXAMPLE
From the following vapor pressure data of methanol at
different temperature obtain the value of A,B,C by
using Levenberg–Marquardt algorithm (LMA) for
Antoine vapour pressure correlation Psat = eA-B/(T+C)
16.
After solving by LMA we obtain the value of
A = 26.9843
B = 5780.219
C = 36.06304
Putting these parameter values back into the Antoine
equation gives the table shown below.
17.
A graph comparing the calculated value of data is
shown
18.
APPLICATION OF EXTRAPOLATION
Forecasting :weather predictions take historic data
and extrapolate to obtain a future weather pattern.
Hurricane tracking chart is prepared by the help of
extrapolation
The position of the hurricane’s center is plotted on a
map every 6 hours, and future positions are
predicted by continuing the track
19.
NHC ‘S FORECAST MODEL OF HURRICANE
ERNESTO NEAR FLORIDA
20.
SEA LAKE AND OVERLAND SURGES FROM
HURRICANES (SLOSH) MODEL
It developed for use in areas of the Gulf of Mexico and near
the United States' East coast
21.
Extrapolation is used to find the properties of
system near 0 k temperature because it is
impossible to attain a temperature near 0 k in
laboratory.
22.
APPLICATION IN GEOPHYSICS
Extrapolation is used in geophysical forward and
inverse modelling.
ExampleTemperature increases linearly with depth in the
earth that is the temperature T is related to depth Z
by T(Z) = aZ + b where a and b are numerical
constants. By measuring the value of temperature
at different depths we can find the value of a and b.
Then we can extrapolate it to find the temperature
at any desired depth.
23.
Extrapolation is used for determining depth of the interface
in Refraction seismology.
The time travel curve for direct , reflected and refracted
ray is shown in figure.
The doubly refracted rays
a
are only recorded at
d
distances greater than the c
critical distance
t
xc
therefore it has to be
x
extrapolate to obtain in o
ti
order to find the depth.
depth V2 / ti
24.
REFERENCES
William Lowrie 2007. Fundamentals of Geophysics
William Menke 1984. Geophysical data analysis:
Discrete inverse theory
Wikipedia
http://en.wikipedia.org/wiki/Hurricane Ernesto(2006)
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