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| < 3. Linear Regression | 3.1.1 Estimating the Coefficients > |
3.1 Simple Linear Regression
Simple linear regression lives up to its name: it is a very straightforward approach for predicting a quantitative response $Y$ on the basis of a single predictor variable $X$. It assumes that there is approximately a linear relationship between $X$ and $Y$. Mathematically, we can write this linear relationship as
\[Y \approx \beta_0 + \beta_1 X \quad (3.1)\]You might read “$\approx$” as “is approximately modeled as”. We will sometimes describe (3.1) by saying that we are regressing $Y$ on $X$ (or $Y$ onto $X$).
For example, $X$ may represent TV advertising and $Y$ may represent sales. Then we can regress sales onto TV by fitting the model
In Equation 3.1, $\beta_0$ and $\beta_1$ are two unknown constants that represent the intercept and slope terms in the linear model. Together, $\beta_0$ and $\beta_1$ are known as the model coefficients or parameters. Once we have used our training data to produce estimates $\hat{\beta}_0$ and $\hat{\beta}_1$ for the model coefficients, we can predict future sales on the basis of a particular value of TV advertising by computing
\[\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x \quad (3.3)\]where $\hat{y}$ indicates a prediction of $Y$ on the basis of $X = x$. Here we use a hat symbol, $\hat{ }$, to denote the estimated value for an unknown parameter or coefficient, or to denote the predicted value of the response.
Sub-Chapters
3.1.1 Estimating the Coefficients
3.1.2 Assessing the Accuracy of the Coefficient Estimates
3.1.3 Assessing the Accuracy of the Model
| < 3. Linear Regression | 3.1.1 Estimating the Coefficients > |