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--- kb:linear_regression [2022-03-02 02:29] – ↷ Page name changed from kb:inear_regression to kb:linear_regression jaeyoung
+++ kb:linear_regression [2024-04-30 04:03] (current) – external edit 127.0.0.1
@@ Line 88: / Line 88: @@
 In other words, $R^2$ describes how much of the variation in $y$ is captured by the regression.
+==== Predictive distribution ====
+Predictive error is the difference between the prediction and true value.
+$$ \hat{Y}(x) - a^* - b^*x = \frac{1}{n} \sum_{i = 1}^{n} \varepsilon_i \left\{ 1 + \frac{(x_i - \bar{x})(x - \bar{x})}{\sigma_x^2} \right\} $$
+The expectation is $0$, of course:
+$$ \mathbb{E}[\hat{Y}(x) - a^* - b^*x] = 0 $$
+The variance is:
+$$ \mathrm{Var}[\hat{Y}(x) - a^* - b^*x] = \mathbb{E}[(\hat{Y}(x) - a^* - b^*x)^2] = \frac{\sigma^2}{n} \left( \frac{(x - \bar{x})^2}{\sigma_x^2} \frac{n-1}{n} + 1 \right) $$
+The distribution is Gaussian if $\varepsilon_i$ are Gaussian. If it is Gaussian, then we can easily compute [[kb:confidence_interval|confidence intervals]].
 ===== Multivariate linear regression =====