polynomial regression

(noun)

a higher order form of linear regression in which the relationship between the independent variable $x$ and the dependent variable $y$ is modeled as an $n$th order polynomial

Related Terms

  • least squares
  • orthogonal

Examples of polynomial regression in the following topics:

  • Polynomial Regression

    • Polynomial regression is a higher order form of linear regression in which the relationship between the independent variable $x$ and the dependent variable $y$ is modeled as an $n$th order polynomial.
    • For this reason, polynomial regression is considered to be a special case of multiple linear regression.
    • Polynomial regression models are usually fit using the method of least-squares.
    • 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.
    • Therefore, non-parametric regression approaches such as smoothing can be useful alternatives to polynomial regression.
  • Model Assumptions

    • This trick is used, for example, in polynomial regression, which uses linear regression to fit the response variable as an arbitrary polynomial function (up to a given rank) of a predictor variable.
    • This makes linear regression an extremely powerful inference method.
    • In fact, models such as polynomial regression are often "too powerful" in that they tend to overfit the data.
    • Error will not be evenly distributed across the regression line.
    • Bayesian linear regression is a general way of handling this issue.
  • Adding and Subtracting Polynomials

    • Note that any two polynomials can be added or subtracted, regardless of the number of terms in each, or the degrees of the polynomials.
    • The resulting polynomial will have the same degree as the polynomial with the higher degree in the problem.
    • For example, one polynomial may have the term $x^2$, while the other polynomial has no like term.
    • Note that the term $5x^3$ in the first polynomial does not have a like term; neither does $7x$ in the second polynomial.
    • Notice that the answer is a polynomial of degree 3; this is also the highest degree of a polynomial in the problem.
  • Introduction to Factoring Polynomials

    • A polynomial consists of a sum of monomials.
    • However, sometimes it will be more useful to write a polynomial as a product of other polynomials with smaller degree, for example to study its zeros.
    • is a factorization of a polynomial of degree $3$ into $3$ polynomials of degree $1$.
    • The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials.
    • One way to factor polynomials is factoring by grouping.
  • The Fundamental Theorem of Algebra

    • The fundamental theorem of algebra says that every non-constant polynomial in a single variable $z$, so any polynomial of the form
    • For example, the polynomial
    • So since the property is true for all polynomials of degree $0$, it is also true for all polynomials of degree $1$.
    • And since it is true for all polynomials of degree $1$, it is also true for all polynomials of degree $2$.
    • The multiplicities of the complex roots of a nonzero polynomial with complex coefficients add to the degree of said polynomial.
  • Polynomial Inequalities

    • The best way to solve a polynomial inequality is to find its zeros.
    • The easiest way to find the zeros of a polynomial is to express it in factored form.
    • Graph of the third-degree polynomial with the equation $y=x^3+2x^2-5x-6$.
    • This polynomial has three roots.
    • Solve for the zeros of a polynomial inequality to find its solution
  • Basics of Graphing Polynomial Functions

    • A polynomial function in one real variable can be represented by a graph.
    • Polynomials appear in a wide variety of areas of mathematics and science.
    • A typical graph of a polynomial function of degree 3 is the following:
    • A polynomial of degree 6.
    • A polynomial of degree 5.
  • Dividing Polynomials

    • Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree.
    • The calculated polynomial is the quotient, and the number left over (−123) is the remainder: $x^3 - 12x^2 - 42 = (x - 3)(x^2 - 9x - 27) - 123$.
    • Polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree.This method is a generalized version of the familiar arithmetic technique called long division.It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
    • The calculated polynomial is the quotient, and the number left over (−123) is the remainder:
  • Multiplying Polynomials

    • To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.
    • So for the multiplication of a monomial with a polynomial we get the following procedure:
    • To multiply a polynomial $P(x) = M_1(x) + M_2(x) + \ldots + M_n(x)$ with a polynomial $Q(x) = N_1(x) + N_2(x) + \ldots + N_k(x)$, where both are written as a sum of monomials of distinct degrees, we get
    • Since we made sure that the product of polynomials abides the same laws as if the variables were real numbers, the evaluation of a product of two polynomials in a given point will be the same as the product of the evaluations of the polynomials:
    • So the roots of a product of polynomials are exactly the roots of its factors, i.e.
  • Multiple Regression Models

    • Multiple regression is used to find an equation that best predicts the $Y$ variable as a linear function of the multiple $X$ variables.
    • You use multiple regression when you have three or more measurement variables.
    • One use of multiple regression is prediction or estimation of an unknown $Y$ value corresponding to a set of $X$ values.
    • Multiple regression is a statistical way to try to control for this; it can answer questions like, "If sand particle size (and every other measured variable) were the same, would the regression of beetle density on wave exposure be significant?
    • As you are doing a multiple regression, there is also a null hypothesis for each $X$ variable, meaning that adding that $X$ variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.
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