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  2. 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. 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. 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. 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. 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
  8. 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. 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. 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. 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. 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. 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. 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. 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. 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. 17.  A graph comparing the calculated value of data is shown
  18. 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
  20. 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. 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. 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. 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. 24. REFERENCES William Lowrie 2007. Fundamentals of Geophysics  William Menke 1984. Geophysical data analysis: Discrete inverse theory  Wikipedia  Ernesto(2006) 
  25. 25. THANK YOU