polynomial regression

(noun)

a higher order form of linear regression in which the relationship between the independent variable xxx and the dependent variable yyy is modeled as an nnnth 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 xxx and the dependent variable yyy is modeled as an nnnth 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.
  • Glossary

    • In regression analysis (such as linear regression) the criterion variable is the variable being predicted.
    • Polynomial regression is a form of multiple regression in which powers of a predictor variable instead of other predictor variables are used.
    • Regression means "prediction. " The regression of Y on X means the prediction of Y by X.
    • A regression coefficient is the slope of the regression line in simple regression or the partial slope in multiple regression.
    • In linear regression, the line of best fit is called the regression line.
  • Multiple Regression Models

    • Multiple regression is used to find an equation that best predicts the YYY variable as a linear function of the multiple XXX 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 YYY value corresponding to a set of XXX 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 XXX variable, meaning that adding that XXX variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.
  • Estimating and Making Inferences About the Slope

    • You use multiple regression when you have three or more measurement variables.
    • When the purpose of multiple regression is prediction, the important result is an equation containing partial regression coefficients (slopes).
    • When the purpose of multiple regression is understanding functional relationships, the important result is an equation containing standard partial regression coefficients, like this:
    • Where b1′b'_1b​1​′​​ is the standard partial regression coefficient of yyy on X1X_1X​1​​.
    • A graphical representation of a best fit line for simple linear regression.
  • Evaluating Model Utility

    • Multiple regression is beneficial in some respects, since it can show the relationships between more than just two variables; however, it should not always be taken at face value.
    • It is easy to throw a big data set at a multiple regression and get an impressive-looking output.
    • But many people are skeptical of the usefulness of multiple regression, especially for variable selection, and you should view the results with caution.
    • You should examine the linear regression of the dependent variable on each independent variable, one at a time, examine the linear regressions between each pair of independent variables, and consider what you know about the subject matter.
    • You should probably treat multiple regression as a way of suggesting patterns in your data, rather than rigorous hypothesis testing.
  • Predictions and Probabilistic Models

    • Regression models are often used to predict a response variable yyy from an explanatory variable xxx.
    • In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function, which can be described by a probability distribution.
    • Regression analysis is widely used for prediction and forecasting.
    • Performing extrapolation relies strongly on the regression assumptions.
    • Here are the required conditions for the regression model:
  • The Regression Fallacy

    • The regression fallacy fails to account for natural fluctuations and rather ascribes cause where none exists.
    • The regression (or regressive) fallacy is an informal fallacy.
    • This use of the word "regression" was coined by Sir Francis Galton in a study from 1885 called "Regression Toward Mediocrity in Hereditary Stature. " He showed that the height of children from very short or very tall parents would move towards the average.
    • Assuming athletic careers are partly based on random factors, attributing this to a "jinx" rather than regression, as some athletes reportedly believed, would be an example of committing the regression fallacy.
    • A picture of Sir Francis Galton, who coined the use of the word "regression
  • The Equation of a Line

    • In statistics, linear regression can be used to fit a predictive model to an observed data set of yyy and xxx values.
    • In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable.
    • Simple linear regression fits a straight line through the set of nnn points in such a way that makes the sum of squared residuals of the model (that is, vertical distances between the points of the data set and the fitted line) as small as possible.
    • Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.
    • If the goal is prediction, or forecasting, linear regression can be used to fit a predictive model to an observed data set of yyy and XXX values.
  • Slope and Intercept

    • In the regression line equation the constant mmm is the slope of the line and bbb is the yyy-intercept.
    • Regression analysis is the process of building a model of the relationship between variables in the form of mathematical equations.
    • A simple example is the equation for the regression line which follows:
    • The case of one explanatory variable is called simple linear regression.
    • For more than one explanatory variable, it is called multiple linear regression.
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