Simple Regression

  • Kunio Takezawa


When the data \(\{(x_{i},y_{i})\}\) (1 ≤ in) are given, a 0 and a 1 are derived by minimizing the residual sum of squares (RSS) in a procedure called a simple regression:
$$\displaystyle{ RSS =\sum _{ i=1}^{n}{(y_{ i} - a_{0} - a_{1}x_{i})}^{2} =\sum _{ i=1}^{n}e_{ i}^{2}, }$$
where \((y_{i} - a_{0} - a_{1}x_{i})\) ( = e i ) is a residual. This process yields the regression equation:
$$\displaystyle{ y =\hat{ a}_{0} +\hat{ a}_{1}x, }$$
where a 0 is the intercept and a 1 is the gradient (slope). Each data point is represented as
$$\displaystyle{ y_{i} =\hat{ a}_{0} +\hat{ a}_{1}x_{i} + e_{i}. }$$
Values such as a 0 and a 1 are called regression coefficients. The “ \(\widehat{}\) ” (hat) of \(\hat{a}_{0}\) and \(\hat{a}_{1}\) indicates that these values are estimates.


Null Hypothesis Regression Coefficient Simulation Data Probability Density Function Prediction Error 
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Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Kunio Takezawa
    • 1
    • 2
  1. 1.National Agricultural and Food Research OrganizationTsukubaJapan
  2. 2.Graduate School of Life and Environmental SciencesUniversity of TsukubaTsukubaJapan

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