Fish 507; Lecture 23

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Interpolation Methods

• Fish 507 Lecture 23

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What is Interpolation?

• There is a function f(x) that is very expensive
  (or difficult) to evaluate. We wish to compute
  f(x) for many values of x as cheaply as
  possible.
• We have data points (x,yg(x)) but do not know
  the function g. We wish to compute g(x) for
  values of x other than those for which the data
  points are available.

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Consider the Runge Function
Given these seven data points, how do we
determine the values of y for other values for
x?
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Linear Interpolation-I

• Approximate f(x) between xi and xi1 by a
  straight line connecting xi and xi1, i.e.
• or

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Linear Interpolation-II
Linear interpolation is very accurate in this
case, but it requires knowing between which
values of xi the x we are interested in lies
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Linear Interpolation-III

• R includes the function approx which can be used
  to perform linear interpolation
• approx(x, y, xout)
• x,y the vector of (x,y) pairs on which to base
  the calculations.
• xout a vector of values of x for which values
  of f(x) are required.

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Polynomial Interpolation-I

• There are many ways to approximate a function
  using a polynomial. The simplest is to
  approximate the function by
• Note that
• this approximation is exact for xx1, x2, x3,
  etc. and
• a single equation is used to approximate the
  entire function.

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Polynomial Interpolation-II
The polynomial passes though all seven
points but.
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Polynomial Interpolation-III
A better approximation can be obtained in this
case by applying polynomial interpolation with
N3 (i.e. quadratic functions)
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Rational Interpolation-I

• Rather than approximating a function by a simple
  polynomial, we can approximate it by the ratio of
  polynomials, i.e.
• This approach can deal with poles in the function
  to be approximated (unfortunately even when they
  are not there).

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Rational Interpolation-II
Rational approximation performs better than
simple polynomial interpolation for the Runge
function (as expected?)
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More Examples
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Splines-I

• Cubic splines are piecewise cubic polynomials
  that approximate a function such that
• The zeroth, first and second derivatives are
  continuous
• the approximating function matches the points on
  which it is based and
• the second derivatives of the approximating
  function match the second derivatives of the
  function.

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Splines II

• Define a set of points (called knots) these
  are the points for which data are available
• The cubic spline is defined by the equation
• This equation has four parameters for each pair
  of knots. The values for these parameters are
  defined from the function values at the knots and
  the (numerical) second derivatives at the knots.

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Splines III
The cubic spline is able to satisfactorily mimic
the underlying shape almost perfectly
y3 lt- spline(x,y) lines(y3,col19,lwd3)
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Splines -IV

• Using a Spline for interpolation requires that
  the values for the parameters be computed.
• In order to predict the value of y for some x, it
  is necessary to apply a search algorithm (e.g.
  bisection) to find the knots between which x lies.

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Splines V

• R includes the function spline
• spline(x, y, n, periodic, xmin, xmax )
• Arguments
• x, y the data (required)
• n number of output points
• periodic is the function periodic?
• xmin / xmax range to predict over.
• Output
• List with components x and y that provide the
  results of the cubic spline fitting.

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Splines VI
Spline vs linear interpolation based on 8, 12 and
16 points
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Interpolation and Extrapolation-I

• Polynomial and rational approximation are methods
  for interpolation. In principle, they can be used
  to extrapolate beyond the range of the xs for
  which data are available. This should, however,
  be done with considerable caution.
• Consider the following selectivity ogive.

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Interpolation and Extrapolation-II
3-point polynomial interpolation. Extrapolation
beyond age0 and age25 is based on the results
for the first and last intervals