polynomial regression

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.

• 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 $Y$ 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.

• 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 $b'_1$ is the standard partial regression coefficient of $y$ on $X_1$.
  • 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 $y$ from an explanatory variable $x$.
  • 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 $y$ and $x$ 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 $n$ 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 $y$ and $X$ values.

• Slope and Intercept

  • In the regression line equation the constant $m$ is the slope of the line and $b$ is the $y$-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.