best.md

Polynomial Least Squares

Let the pairs of the dataset (xᵢ, yᵢ) contain error.
We want to be able to approximate this dataset via a polynomial.

Two things are needed to establish this polynomial:

• the order of the polynomial (linear, quadratic and so on);
• a metric/criterion measuring the viability of the polynomial.

Viability

Let E = y - (a₀ + a₁x + a₂x² + ...) be the error, y be the data vector and (a₀ + a₁x + a₂x² + ...) be the approximating polynomial.
By this metric, the more appropriate polynomial would be the one to yield E smallest in value.

Popular criteria:

• min(L1(E)) (L1-Norm of Error Vector);
• min(L2(E)) (Euclidean Norm of E);
• min(L∞(E)) (Maximum Component of E).

In Least Squares Fitting, the best fit polynomial is the one satisfying the criterion of the L2-Norm.

Line of Best Fit

Satisfying the Criterion

Equaling to zero (for the minimum/minima) should yield the following system of equations:

Matricial Form

The following linear system is equivalent to the above:

and is easily solvable.

Best Fit Polynomials

The procedure is similar for finding other best fit polynomials. The matrix form can be generalized as:

It being a positive-definite matrix, the solution is unique.