高斯过程回归（GPR）

$$f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}'))$$

$$k_{RBF}(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \exp\left(-\frac{||\mathbf{x} - \mathbf{x}'||^2}{2l^2}\right)$$

$$k_{Matern}(\mathbf{x}, \mathbf{x}') = \sigma_f^2 \frac{2^{1-\nu}}{\Gamma(\nu)} \left(\frac{\sqrt{2\nu}||\mathbf{x} - \mathbf{x}'||}{l}\right)^\nu K_\nu\left(\frac{\sqrt{2\nu}||\mathbf{x} - \mathbf{x}'||}{l}\right)$$

$$\mathcal{D} = \{(\mathbf{x}_i, y_i)\}_{i=1}^{N}$$

$$f_* | \mathbf{x}_*, \mathcal{D} \sim \mathcal{N}(\mu(\mathbf{x}_*), \sigma^2(\mathbf{x}_*))$$

$$\mu(\mathbf{x}_*) = \mathbf{k}_*^T \mathbf{K}_y^{-1} \mathbf{y}$$

$$\sigma^2(\mathbf{x}_*) = k(\mathbf{x}_*, \mathbf{x}_*) - \mathbf{k}_*^T \mathbf{K}_y^{-1} \mathbf{k}_*$$

$$\log p(\mathbf{y} | \mathbf{X}) = -\frac{1}{2}\mathbf{y}^T \mathbf{K}_y^{-1}\mathbf{y} - \frac{1}{2}\log|\mathbf{K}_y| - \frac{N}{2}\log(2\pi)$$

$$v_{t+1} = f(t, h, \phi, v_t, v_{t-1}, ...)$$

$$[ \mu - 1.96\sigma, \mu + 1.96\sigma ]$$