Section CF Curve Fitting

Given two points in the plane, there is a unique line through them. Given three points in the plane, and not in a line, there is a unique parabola through them. Given four points in the plane, there is a unique polynomial, of degree 3 or less, passing through them. And so on. We can prove this result, and give a procedure for finding the polynomial with the help of Vandermonde matrices (Section VM).

Theorem IP
Interpolating Polynomial
Suppose \left \{\left .({x}_{i},\kern 1.95872pt {y}_{i})\right \vert 1 ≤ i ≤ n + 1\right \} is a set of n + 1 points in the plane where the x-coordinates are all different. Then there is a unique polynomial of degree n or less, p(x), such that p({x}_{i}) = {y}_{i}, 1 ≤ i ≤ n + 1. □

Proof Write p(x) = {a}_{0} + {a}_{1}x + {a}_{2}{x}^{2} + \mathrel{⋯} + {a}_{ n}{x}^{n}. To meet the conclusion of the theorem, we desire,

\eqalignno{ {y}_{i} & = p({x}_{i}) = {a}_{0} + {a}_{1}{x}_{i} + {a}_{2}{x}_{i}^{2} + \mathrel{⋯} + {a}_{ n}{x}_{i}^{n} &1 ≤ i ≤ n + 1 & & & & & }

This is a system of n + 1 linear equations in the n + 1 variables {a}_{0},\kern 1.95872pt {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{n}. The vector of constants in this system is the vector containing the y-coordinates of the points. More importantly, the coefficient matrix is a Vandermonde matrix (Definition VM) built from the x-coordinates {x}_{1},\kern 1.95872pt {x}_{2},\kern 1.95872pt {x}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {x}_{n+1}. Since we have required that these scalars all be different, Theorem NVM tells us that the coefficient matrix is nonsingular and Theorem NMUS says the solution for the coefficients of the polynomial exists, and is unique. As a practical matter, Theorem SNCM provides an expression for the solution. ■

Example PTFP
Polynomial through five points
Suppose we have the following 5 points in the plane and we wish to pass a degree 4 polynomial through them.

The required system of equations has a coefficient matrix that is the Vandermonde matrix where row i is successive powers of {x}_{i}

\eqalignno{ A & = \left [\array{ 1&−3& 9 &−27& 81\cr 1&−1 & 1 & −1 & 1 \cr 1& 2 & 4 & 8 & 16\cr 1& 3 & 9 & 27 & 81 \cr 1& 6 &36&216&1296 } \right ] & & }

\eqalignno{ \left [\array{ {a}_{0} \cr {a}_{1} \cr {a}_{2} \cr {a}_{3} \cr {a}_{4} } \right ] & = {A}^{−1}\left [\array{ 276\cr 16 \cr 31\cr 144 \cr 2319 } \right ] = \left [\array{ −{1\over 15}& {9\over 14} & {9\over 10} & −{1\over 2} & {1\over 42} \cr 0 & −{3\over 7} & {3\over 4} & −{1\over 3} & {1\over 84} \cr {5\over 108} & −{1\over 56} & −{1\over 4} & {17\over 72} &−{11\over 756} \cr −{1\over 54}& {1\over 21} &−{1\over 12}& {1\over 18} &− {1\over 756} \cr {1\over 540} &− {1\over 168}& {1\over 60} &−{1\over 72}& {1\over 756} } \right ]\left [\array{ 276\cr 16 \cr 31\cr 144 \cr 2319 } \right ] = \left [\array{ 3\cr −4 \cr 5\cr −2 \cr 2 } \right ] & & }

The unique polynomial passing through a set of points is known as the interpolating polynomial and it has many uses. Unfortunately, when confronted with data from an experiment the situation may not be so simple or clear cut. Read on.

Subsection DF: Data Fitting

Suppose that we have n real variables, {x}_{1},\kern 1.95872pt {x}_{2},\kern 1.95872pt {x}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {x}_{n}, that we can measure in an experiment. We believe that these variables combine, in a linear fashion, to equal another real variable, y. In other words, we have reason to believe from our understanding of the experiment, that

\eqalignno{ y & = {a}_{1}{x}_{1} + {a}_{2}{x}_{2} + {a}_{3}{x}_{3} + \mathrel{⋯} + {a}_{n}{x}_{n} & & }

where the scalars {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt {a}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{n} are not known to us, but are instead desirable. We would call this our model of the situation. Then we run the experiment m times, collecting sets of values for the variables of the experiment. For run number k we might denote these values as {y}_{k}, {x}_{k1}, {x}_{k2}, {x}_{k3}, …, {x}_{kn}. If we substitute these values into the model equation, we get m linear equations in the unknown coefficients {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt {a}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{n}. If m = n, then we have a square coefficient matrix of the system which might happen to be nonsingular and there would be a unique solution.

However, more likely m > n (the more data we collect, the greater our confidence in the results) and the resulting system is inconsistent. It may be that our model is only an approximate understanding of the relationship between the {x}_{i} and y, or our measurements are not completely accurate. Still we would like to understand the situation we are studying, and would like some best answer for {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt {a}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{n}.

Let y denote the vector with {\left [y\right ]}_{i} = {y}_{i}, 1 ≤ i ≤ m, let a denote the vector with {\left [a\right ]}_{j} = {a}_{j}, 1 ≤ j ≤ n, and let X denote the m × n matrix with {\left [X\right ]}_{ij} = {x}_{ij}, 1 ≤ i ≤ m, 1 ≤ j ≤ n. Then the model equation, evaluated with each run of the experiment, translates to Xa = y. With the presumption that this system has no solution, we can try to minimize the difference between the two side of the equation y − Xa. As a vector, it is hard to imagine what the minimum might be, so we instead minimize the square of its norm

\eqalignno{ S ={ \left (y − Xa\right )}^{t}\left (y − Xa\right ) & & }

To keep the logical flow accurate, we will define the minimizing value and then give the proof that it behaves as desired.

Definition LSS
Least Squares Solution
Given the equation Xa = y, where X is an m × n matrix of rank n, the least squares solution for a is {\left ({X}^{t}X\right )}^{−1}{X}^{t}y. △

Theorem LSMR
Least Squares Minimizes Residuals
Suppose that X is an m × n matrix of rank n. The least squares solution of Xa = y, {a}^{′} ={ \left ({X}^{t}X\right )}^{−1}{X}^{t}y, minimizes the expression

\eqalignno{ S ={ \left (y − Xa\right )}^{t}\left (y − Xa\right ) & & }

Proof We begin by finding the critical points of S. In preparation, let {X}_{j} denote column j of X, for 1 ≤ j ≤ n and compute partial derivatives with respect to {a}_{j}, 1 ≤ j ≤ n. A matrix product of the form {x}^{t}y is a sum of products, so a derivative is a sum of applications of the product rule,

\eqalignno{ {∂\over ∂{a}_{j}}S & = {∂\over ∂{a}_{j}}\left ({\left (y − Xa\right )}^{t}\left (y − Xa\right )\right ) & & \cr & ={ \mathop{∑ }}_{i=1}^{m} {∂\over ∂{a}_{j}}\left ({\left [y − Xa\right ]}_{i}\right ){\left [y − Xa\right ]}_{i} +{ \left [y − Xa\right ]}_{i} {∂\over ∂{a}_{j}}\left ({\left [y − Xa\right ]}_{i}\right ) & & \cr & = 2{ \mathop{∑ }}_{i=1}^{m} {∂\over ∂{a}_{j}}\left ({\left [y − Xa\right ]}_{i}\right ){\left [y − Xa\right ]}_{i} & & \cr & = 2{ \mathop{∑ }}_{i=1}^{m} {∂\over ∂{a}_{j}}\left ({\left [y\right ]}_{i} −{\mathop{∑ }}_{k=1}^{n}{\left [X\right ]}_{ ik}{\left [a\right ]}_{k}\right ){\left [y − Xa\right ]}_{i} & & \cr & = 2{ \mathop{∑ }}_{i=1}^{m} −{\left [X\right ]}_{ ij}{\left [y − Xa\right ]}_{i} & & \cr & = −2{\left ({X}_{j}\right )}^{t}\left (y − Xa\right ) & & \cr & & }

The first partial derivatives will allow us to find critical points, while second partial derivatives will be needed to confirm that a critical point will yield a minimum. Return to the next-to-last expression for the first partial derivative of S,

\eqalignno{ {∂\over ∂{a}_{ℓ}{a}_{j}}S & = {∂\over ∂{a}_{ℓ}}2{ \mathop{∑ }}_{i=1}^{m} −{\left [X\right ]}_{ ij}{\left [y − Xa\right ]}_{i} & & \cr & = −2{ \mathop{∑ }}_{i=1}^{m} {∂\over ∂{a}_{ℓ}}{\left [X\right ]}_{ij}{\left [y − Xa\right ]}_{i} & & \cr & = −2{ \mathop{∑ }}_{i=1}^{m}{\left [X\right ]}_{ ij} {∂\over ∂{a}_{ℓ}}\left ({\left [y\right ]}_{i} −{\mathop{∑ }}_{k=1}^{n}{\left [X\right ]}_{ ik}{\left [a\right ]}_{k}\right ) & & \cr & = −2{ \mathop{∑ }}_{i=1}^{m}{\left [X\right ]}_{ ij}\left (−{\left [X\right ]}_{iℓ}\right ) & & \cr & = 2{ \mathop{∑ }}_{i=1}^{m}{\left [X\right ]}_{ ij}{\left [X\right ]}_{iℓ} & & \cr & = 2{ \mathop{∑ }}_{i=1}^{m}{\left [{X}^{t}\right ]}_{ ji}{\left [X\right ]}_{iℓ} & & \cr & = 2{\left [{X}^{t}X\right ]}_{ jℓ} & & }

For 1 ≤ j ≤ n, set {∂\over ∂{a}_{j}}S = 0. This results in the n scalar equations

\eqalignno{ {\left ({X}_{j}\right )}^{t}Xa & ={ \left ({X}_{ j}\right )}^{t}y &1 ≤ j ≤ n & & & & & }

\eqalignno{ {X}^{t}Xa = {X}^{t}y & & }

{X}^{t}X is an n × n matrix and since we have assumed that X has rank n, {X}^{t}X will also have rank n. Since {X}^{t}X is invertible, we have a critical point at

\eqalignno{ {a}^{′} ={ \left ({X}^{t}X\right )}^{−1}{X}^{t}y & & }

Is this lone critical point really a minimum? The matrix of second partial derivatives is constant, and a positive multiple of {X}^{t}X. Theorem CPSM tells us that this matrix is positive semi-definite. In an advanced course on multivariable calculus, it is shown that a minimum occurs exactly where the matrix of second partial derivatives is positive semi-definite. You may have seen this in the two-variable case, where a check on the positive semi-definiteness is disguised with a determinant of the 2 × 2 matrix of second partial derivatives. ■

Subsection EXC: Exercises

T20 Theorem IP constructs a unique polynomial through a set of n + 1 points in the plane, \left \{\left .({x}_{i},\kern 1.95872pt {y}_{i})\right \vert 1 ≤ i ≤ n + 1\right \}, where the x-coordinates are all different. Prove that the expression below is the same polynomial and include an explanation of the necessity of the hypothesis that the x-coordinates are all different.

\eqalignno{ p(x) & ={ \mathop{∑ }}_{i=1}^{n+1}{y}_{ i}{ \mathop{∏ }}_{\begin{array}{c}j=1 \\ j\mathrel{≠}i \end{array}}^{n+1} {x − {x}_{j}\over { x}_{i} − {x}_{j}} & & }

This is known as the Lagrange form of the interpolating polynomial.
Contributed by Robert Beezer