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I was learning from Elements of statistics p.109 under the topic LINEAR DISCRIMINANT ANALYSIS

and I saw the function below for linear discriminant function

$\delta_k = X^T\sum^{-1}\mu_k -\frac{1}{2}\mu_k^T\mu_k+ \log\pi_k$

where

$\pi_k = N_k/N$ where $N_k$ is the number of class-k observations;

$\mu_k$ = $\sum_{g_i = k} x_i/N_k$

$\sum = \sum_{k=1}^k\sum_{g_i = k}(x_i - \mu_k)(x_i - \mu_k)^T/(N - K)$

Please I want know the parameters for this function (I am thinking is $\mu_k$ ) and how come it has (K-1) x (p + 1) parameters

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1 Answer 1

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Below is the answer quoted from Xiaozhou's note.

For LDA, (p+1) parameters are needed to construct the discriminant function in (2). For a problem with K classes, we would only need (K-1) such discriminant functions by arbitrarily choosing one class to be the base class (subtracting the base class likelihood from all other classes). Hence, the total number of estimated parameters for LDA is (K-1)(p+1).

  • Mean: p
  • Class prior: 1

function (2): $$ \delta_k(\mathbf{x})=\mathbf{x}^T \boldsymbol{\Sigma}^{-1} \mu_k-\frac{1}{2} \mu_k^T \boldsymbol{\Sigma}^{-1} \mu_k+\log \pi_k $$

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