chapter-04/4.3.2-computations-for-LDA.ipynb

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    "# 4.3.2 Computations for LDA"
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    "Suppose we compute the eigen-decomposition for each $\\hat{\\Sigma}_k=\\mathbf{U}_k\\mathbf{D}_k\\mathbf{U}_k^T$, then ingredients for $\\delta_k(x)$ is:\n",
    "\n",
    "- $(x-\\hat{\\mu}_k)^T\\hat{\\Sigma}_k^{-1}(x - \\hat{\\mu}_k)=\\left[\\mathbf{U}_k^T (x - \\hat{\\mu}_k)\\right]^T\n",
    "\\mathbf{D}_k^{-1}\n",
    "\\left[\\mathbf{U}_k^T(x-\\hat{\\mu}_k)\\right]\n",
    "$\n",
    "\n",
    "- $\\log |\\hat{\\Sigma}_k|=\\Sigma_l \\log d_{kl}$\n",
    "\n",
    "the LDA classifier can be implemented by the following steps:\n",
    "\n",
    "- Sphere the data w.r.t the covariance estimate $\\hat{\\Sigma}: X^{*}=\\mathbf{D}^{-1/2}\\mathbf{U}^TX$\n",
    "\n",
    "- Classify to the closest class centroid in the transformed space, modulo the effect of the class prior probabilities $\\pi_k$."
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