1. Curve Fitting
Filling in the gaps

2. Copyright G. A. Tagliarini, PhD
Basic Problem
Given sample points x1 < x2 < …< xn and sample data values y1, y2,…, yn corresponding to the xi, for 1 ≤ i ≤ n
Find a function f such that y1 = f(x1), y2 = f(x2), …, yn = f(xn)
Typically an approximate match is sought
Additional design constraints, such as continuity or differentiability may apply
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3. Least Squares Approximation
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Estimating Values
Interpolation is estimating the value of the data generating function for x ε [x1, xn] and x ≠ xi.
Extrapolation is estimating the value of the data generating function for x outside the interval [x1, xn].
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5. Recall from Analytic Geometry
(x2, y2)
(x1, y1)
(x2, y1)
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6. Piece-wise Linear Interpolation
(x2, y2)
(xn, yn)
(x1, y1)
(x3, y3)
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7. Polynomial Interpolation
A polynomial p(x) is a function of the form p(x) = anxn + an-1xn-1+…+a1x+a0
If n is the largest power of x for which an ≠ 0, then p(x) is of degree n
The n+1 coefficients ai are the undetermined values
Assume n+1 samples to find a polynomial of degree n that passes through the sample points…Does one exist? How to find it?
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8. Polynomial Approximation
Polynomials are universal approximators for continuous functions on closed intervals
Weierstrass Approximation Theorem
Given F(x) continuous on [a, b] and any e>0 no matter how small
There exists a polynomial f(x) such that |F(x)-f(x)|<e for all x in [a, b]
This is an existence theorem.
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9. Polynomial Interpolation Process
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10. Re-writing as a Matrix Equation
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11. Conditions for a Solution
Given the linear system y = X a from the previous slide
X-1 exists so that we can find a = X-1 y if the xi are all distinct
This condition was met by our initial assumptions
For many curve fitting problems (e.g., regression) this assumption is unmet
X rapidly becomes ill-conditioned as n increases
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12. Example: Polynomial Fit
Find a cubic passing through (-1, -11), (0, -5), (1, -5), and (2, 1)
Graph System
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13. Example: Polynomial Fit (continued)
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14. Example: Polynomial Fit (continued)
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15. Copyright G. A. Tagliarini, PhD
Lagrange Polynomials
Expansion using the natural basis 1, x, x2, …, xn,… may give rise to numerical difficulties from searching for the inverse of an ill-conditioned matrix
An alternative basis may provide a better representation
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16. Lagrange Polynomial Definition
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17. Example Lagrange Polynomials
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18. Example Lagrange Polynomials
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19. An Orthogonality Property
Qk(xj) = 1 if k=j
Qk(xj) = 0 if k≠j
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20. The Approximation Formula
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21. Plotting the Lagrange Polynomials and Their Sum
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22. Lagrange Example Finished
For the data points (-1, -11), (0, -5), (1, -5), and (2, 1)
The polynomial f is given by f(x) = -11Q1(x)+ -5Q2(x) + -5Q3(x) + 1Q4(x) = 2x3 – 3x2 + 1x – 5 which by now should look familiar! (Can you show this?)
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23. Copyright G. A. Tagliarini, PhD
Linear Correlation
Data may be represented in a scattergram
Positive correlation results when the dependent variable’s (y) values increase as the independent variable’s (x) values increase.
Negative correlation results when the dependent variable’s (y) values decrease as the independent variable’s (x) values increase.
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Sample Scattergrams
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Notation
X and Y represent the distributions of independent and dependent variable values
Critical values of r depend upon the significance level a of the test (typically, a=0.05 or a=0.01), the type of test (one tail or two tail), and the number of data pairs n
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26. Pearson’s r: The Coefficient of Correlation
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Prediction Equation
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Cubic Splines
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