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Related Concept Videos

Introduction to Polynomial Functions01:26

Introduction to Polynomial Functions

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Polynomial functions are fundamental elements in algebra and calculus, defined by expressions that combine variables and constants through addition, subtraction, and multiplication, with the variable raised to nonnegative integer exponents. A general polynomial function of degree n is given byWhere an ≠ 0. The term anxn is the leading term, and an is the leading coefficient, while a0 is referred to as the constant term.Characteristics and ClassificationPolynomials are categorized by their...
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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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Regression analysis is a statistical tool that describes a mathematical relationship between a dependent variable and one or more independent variables.
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In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a...
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Related Experiment Video

Updated: Mar 26, 2026

O-cresol Concentration Online Measurement Based On Near Infrared Spectroscopy Via Partial Least Square Regression
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A Note On Polynomial Regression.

D V Budescu

    Multivariate Behavioral Research
    |January 27, 2016
    PubMed
    Summary

    High-degree polynomial regression models with a normally distributed independent variable face increasing collinearity. This study reiterates existing recommendations and proposes alternative solutions for this common statistical problem.

    Area of Science:

    • Statistics
    • Econometrics
    • Data Science

    Background:

    • Collinearity, or multicollinearity, poses challenges in polynomial regression models.
    • High-degree polynomials can exacerbate predictor variable correlations, impacting model stability and interpretation.
    • Understanding these effects is crucial for accurate statistical modeling.

    Purpose of the Study:

    • To analyze the problem of collinearity in polynomial regression.
    • To demonstrate how predictor collinearity intensifies with increasing polynomial degree.
    • To offer solutions and alternative methods for managing collinearity.

    Main Methods:

    • Theoretical analysis of polynomial regression models.
    • Examination of predictor variable collinearity under standard normal distribution assumptions.

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  • Review and extension of existing recommendations for addressing collinearity.

    • Main Results:

    • Collinearity significantly increases with the degree of the polynomial regression.
    • This effect is particularly pronounced when the independent variable follows a standard normal distribution.
    • Existing strategies may not fully mitigate the issue in all cases.

    Conclusions:

    • The degree of polynomial regression is a critical factor driving collinearity.
    • Alternative approaches beyond standard recommendations may be necessary.
    • Careful model selection and diagnostic checks are essential for reliable polynomial regression analysis.