Principles of Extrapolation

Interpolation vs. Extrapolation

• Interpolation is filling in data points between the data that you already have;– for example - drawing a line (fitting a curve) from the first data point

you have to the last allows you to estimate data points between those two extremes (or between any data points that you have). ie. 'filling in between'

• Extrapolation is filling in data points beyond the data that you have (extending the data):– for example fitting a curve to the data that you have using an

equation, then extending that line beyond the first and last points enables you to estimate values (or extrapolate them) beyond the measured data.

Examples

• Interpolation – estimating values within a series e.g. 1.2.3.x.y.6.7

(x = 4, y = 5)

• Extrapolation – estimating values outside a known series e.g. 2 4 6

x y (x = 8, y = 10)

Interpolation

.

A method of constructing a function that crosses through a discrete set of known data points.

Interpolation

• Estimation of intermediate values between precise data points. The most common method is:

• Although there is one and only one nth-order polynomial that fits n+1 points, there are a variety of mathematical formats in which this polynomial can be expressed:– The Newton polynomial– The Lagrange polynomial

nnxaxaxaaxf 2

210)(

Steps in the Extrapolation Method

• Step 1: Analyze past behavior and development for recurring themes and underlying trends.

• Step 2: Summarize each theme and trend, taking into account the other themes and trends.

• Step 3: Project the modified themes and trends into the future by extrapolation.

• Extrapolation techniques have a higher probability of success in

• Short time horizons

• Large areas

Trend Extrapolation

• There are many techniques used to project past data into the future. These tend to be powerful forecasting techniques that are sometimes subject to unforeseen events.

• Some Common Types of Trends• Trends are often shown graphically (as line graphs) with the

level of a dependent variable on the y-axis and the time period on the x-axis. There are different "levels" of trends:

• constant• linear• exponential

Extrapolation

Practical Application

Extrapolation in math

• Extrapolation in math is the process of finding a value beyond a set of given values. You most often have to use extrapolation when you have to find values in a sequence, or when making graphs.

• When you use extrapolation, you look for the relationship between the given values. Look at the following sequences. What's the relationship between the values?

• 2, 4, 6, 8… • 5, 10, 15, 20…

What is Extrapolation?

• Extrapolation is the process of taking data values at points x1, ..., xn, and approximating a value outside the range of the given points.

• This is most commonly experienced when an incoming signal is sampled periodically and that data is used to approximate the next data point.

– For example, weather predictions take historic data and extrapolate a future weather pattern.

Interpolation & Extrapolation

(xi,yi)

Find an analytic function f(x) that passes through given N points exactly.

Polynomial Interpolation

• Use polynomial of degree N-1 to fit exactly with N data points (xi,yi), i =1, 2, …, N.

• The coefficient ci is determined by a system of linear equations

2 10 1 2 1( ) N

Nf x c c x c x c x

Examle

• Given: (0.3, 0.7), (0.5, 0.6), (0.8, 0.4), (1.2, 0.2), (1.6, -0.1) and wish to approximate the value at 2.0, then we can do nothing which may give us a good approximation.

• If, however, we are told that this data is linear, then we may find the least-squares fitting line (y(x) = -0.60830 x + 0.89531), then we may approximate the value at x = 2 by evaluating this function: y(2) = -0.60830 2 + 0.89531 = -0.32130.⋅

The interpolating polynomial• The data, the interpolating polynomial (blue), and the least-squares

line (red) are shown in

• The appropriateness of the extrapolating estimator should be apparent.

Error

• The error associated with extrapolation • If it is assumed that the data is normally distributed, it can be shown

that the error is quite easy to calculate, and for examples, when the data is known to be linear, then the error of extrapolation only increases quadratically as you move away from the average of the x values and the corresponding coefficient is significantly smaller than that for using an interpolating polynomial with only two points.

– In fact, it can be shown that any extrapolation using an interpolating linear function has no statistical significance. It is like using a single point to estimate a mean: you cannot say anything about the error associated with your estimator.

Newton’s Divided-Difference Interpolating Polynomials

Linear Interpolation/• Is the simplest form of interpolation, connecting two data

points with a straight line.

• f1(x) designates that this is a first-order interpolating polynomial.

)()()(

)()(

)()()()(

00

0101

0

01

0

01

xxxx

xfxfxfxf

xx

xfxf

xx

xfxf

Linear-interpolation formula

Slope and a finite divided difference approximation to 1st derivative

Lagrange Interpolating Polynomials

• The Lagrange interpolating polynomial is simply a reformulation of the Newton’s polynomial that avoids the computation of divided differences:

n

ijj ji

ji

n

iiin

xx

xxxL

xfxLxf

0

0

)(

)()()(

Lagrange Polynomial

Li (X) is Lagrange coefficient polynomials

Lagrange Polynomial Example

Spline Interpolation

• There are cases where polynomials can lead to erroneous results because of round off error and overshoot.

• Alternative approach is to apply lower-order polynomials to subsets of data points. Such connecting polynomials are called spline functions.

Mathlab

• http://www.mathworks.com/help/curvefit/fnxtr.html