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Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2 , where n is the order of the polynomial equation.
Although polynomial regression is technically a special case of multiple linear regression, the interpretation of a fitted polynomial regression model requires a somewhat different perspective. It is often difficult to interpret the individual coefficients in a polynomial regression fit, since the underlying monomials can be highly correlated.
A ninth order polynomial interpolation (exact replication of the red curve at 10 points) In the mathematical field of numerical analysis, Runge's phenomenon (German:) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points.
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.
Cubic, quartic and higher polynomials. For regression with high-order polynomials, the use of orthogonal polynomials is recommended. [15] Numerical smoothing and differentiation — this is an application of polynomial fitting. Multinomials in more than one independent variable, including surface fitting; Curve fitting with B-splines [12]
The propositions for the degree of sums and products of polynomials in the above section do not apply, if any of the polynomials involved is the zero polynomial. [ 8 ] It is convenient, however, to define the degree of the zero polynomial to be negative infinity , − ∞ , {\displaystyle -\infty ,} and to introduce the arithmetic rules [ 9 ]
Local regression or local polynomial regression, [1] also known as moving regression, [2] is a generalization of the moving average and polynomial regression. [3] Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing), both pronounced / ˈ l oʊ ɛ s / LOH-ess.
Polynomial extrapolation is typically done by means of Lagrange interpolation or using Newton's method of finite differences to create a Newton series that fits the data. The resulting polynomial may be used to extrapolate the data. High-order polynomial extrapolation must be used with due care.