Fitting a Polynomial to Data

• Interpolation involves constructing and then evaluating an interpolating function.
• interpolant, $y=F(x)$, determined by requiring that it pass through the known data $(x_i,y_i)$.
• In its most general form, interpolation involves determining the coefficients $a_1,a_2,\ldots,a_n$
• in the linear combination of n basis functions, $\Phi(x)$, that constitute the interpolant
  $$F(x)=a_1\Phi_1(x)+a_2\Phi_2(x)+\ldots+a_n\Phi_n(x)$$
  
  
  • such that $F(x)=y_i$ for $i=1,\ldots,n$. The basis function may be polynomial
    $$F(x)=a_1+a_2x+a_3x^2+\ldots+a_nx^{n-1}$$

  • or trigonometric
    $$F(x)=a_1+a_2e^{ix}+a_3e^{i2x}+\ldots+a_ne^{i(n-1)x}$$

  • or some other suitable set of functions.

Polynomials are often used for interpolation because they are easy to evaluate and easy to manipulate analytically.

Table 7.1: Fitting a polynomial to data.

x	f(x)
3.2	22.0
2.7	17.8
1.0	14.2
4.8	38.3
5.6	51.7

• Suppose that we have a data set.
• First, we need to select the points that determine our polynomial.
• The maximum degree of the polynomial is always one less than the number of points.
• Suppose we choose the first four points. If the cubic is $\boxed{ax^3 + bx^2 + cx + d}$,

• We can write four equations involving the unknown coefficients $a, b, c$, and $d$;
  
  

• Solving these equations gives
  

• and our polynomial is
  $$\boxed{-0.5275x^3+6.4952x^2-16.1177x+24.3499}$$

• At $x=3.0$, the estimated value is 20.212.
• if we want a new polynomial that is also made to fit at the point $(5.6,51.7)$ ?
• or if we want to see what difference it would make to use a quadratic instead of a cubic?
• Example py-file: Polynomial Interpolation. Gaussian elimination & back substitution. No pivoting. myGEshow_interpolation.py

Figure 7.1: Polynomial Interpolation.

Table 7.2: Interpolation of gasoline prices.

year	price
1986	133.5
1988	132.2
1990	138.7
1992	141.5
1994	137.6
1996	144.2

• Another example;
• Use the polynomial order 5, why?
  $$P=a_1+a_2y+a_3y^2+a_4y^3+a_5y^4+a_6y^5$$

• Make a guess about the prices of gasoline at year of 1991 (2011).

• Example py-file: Interpolation of gasoline prices. Gaussian elimination & back substitution. No pivoting. myGEshow_gasoline.py
  • Now, try with the shifted dates (Xdata=Xdata-np.mean(Xdata)).
  • Make the necessary corrections for the array A.
  • What differs in the plot and why?

Figure 7.2: Polynomial Interpolation - Gasoline Case.