Ridge regression

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Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where linearly independent variables are highly correlated.[1] It has been used in many fields including econometrics, chemistry, and engineering.[2]

The theory was first introduced by Hoerl and Kennard in 1970 in their Technometrics papers “RIDGE regressions: biased estimation of nonorthogonal problems” and “RIDGE regressions: applications in nonorthogonal problems”.[3][4][1] This was the result of ten years of research into the field of ridge analysis.[5]

Ridge regression was developed as a possible solution to the imprecision of least square estimators when linear regression models have some multicollinear (highly correlated) independent variables—by creating a ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller than the least square estimators previously derived.[6][2]

Mathematical details[edit]

In standard linear regression, an column vector is to be projected onto the column space of the design matrix (typically ) whose columns are highly correlated. The ordinary least squares estimator of the coefficients by which the columns are multiplied to get the orthogonal projection is

(where is the transpose of ).

By contrast, the ridge regression estimator is

where is the identity matrix and is small. The name 'ridge' refers to the shape along the diagonal of I.


  1. ^ a b Hilt, Donald E.; Seegrist, Donald W. (1977). Ridge, a computer program for calculating ridge regression estimates. doi:10.5962/bhl.title.68934.[page needed]
  2. ^ a b Gruber, Marvin (1998). Improving Efficiency by Shrinkage: The James--Stein and Ridge Regression Estimators. CRC Press. p. 2. ISBN 978-0-8247-0156-7.
  3. ^ Hoerl, Arthur E.; Kennard, Robert W. (1970). "Ridge Regression: Biased Estimation for Nonorthogonal Problems". Technometrics. 12 (1): 55–67. doi:10.2307/1267351. JSTOR 1267351.
  4. ^ Hoerl, Arthur E.; Kennard, Robert W. (1970). "Ridge Regression: Applications to Nonorthogonal Problems". Technometrics. 12 (1): 69–82. doi:10.2307/1267352. JSTOR 1267352.
  5. ^ Beck, James Vere; Arnold, Kenneth J. (1977). Parameter Estimation in Engineering and Science. James Beck. p. 287. ISBN 978-0-471-06118-2.
  6. ^ Jolliffe, I. T. (2006). Principal Component Analysis. Springer Science & Business Media. p. 178. ISBN 978-0-387-22440-4.