Extrapolation

EXTRAPOLATION

ALKESH GOYAL 2nd Year, GPT IIT ROORKEE 10411003

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.

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

ADVANTAGESExtrapolation Techniques have- Computational simplicity

Transparent methodology

  low data requirements

May work for 1.Large areas 2.Short time horizons 3.Slow grow areas

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.

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

GENERAL SHAPE OF VARIOUS CURVE FITTING FAMILIES

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

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

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

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-

2.Non linear fitting function which cannot be transformed into a linear fitting function.

example-

This type of function can be Fit by Levenberg–Marquardt algorithm (LMA)

/( ) toN t N e ln / ln oN t N

cos( ) sin( )y a bX b aX

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.

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.

( , ) ( , )i i if x f x J ( , ) /i iJ f x

iJ

The above first-order approximation of    gives

To obtained the minimum of S(β),  gradient of S with respect to δ will be zero. It gives

where I is Identity matrix

J is jacobian matrix

This is a set of linear equation which can be solved for .

2

1

( ) ( , )m

i i ii

S y f x J

( ) [ ( )]T TJ J I J y f

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)

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.

A graph comparing the calculated value of data is shown

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

NHC ‘S FORECAST MODEL OF HURRICANE ERNESTO NEAR FLORIDA

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

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.

APPLICATION IN GEOPHYSICS

Extrapolation is used in geophysical forward and inverse modelling.

Example- Temperature 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.

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 therefore it has to be x extrapolate to obtain in o order to find the depth.

cx

2 / idepth V t

it

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)

THANK YOU