forked from asadoughi/stat-learning
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy path3.Rmd
42 lines (39 loc) · 1.15 KB
/
3.Rmd
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
3
========================================================
$$
p_k(x) = \frac {\pi_k
\frac {1} {\sqrt{2 \pi} \sigma_k}
\exp(- \frac {1} {2 \sigma_k^2} (x - \mu_k)^2)
}
{\sum {
\pi_l
\frac {1} {\sqrt{2 \pi} \sigma_l}
\exp(- \frac {1} {2 \sigma_l^2} (x - \mu_l)^2)
}}
\\
\log(p_k(x)) = \frac {\log(\pi_k) +
\log(\frac {1} {\sqrt{2 \pi} \sigma_k}) +
- \frac {1} {2 \sigma_k^2} (x - \mu_k)^2
}
{\log(\sum {
\pi_l
\frac {1} {\sqrt{2 \pi} \sigma_l}
\exp(- \frac {1} {2 \sigma_l^2} (x - \mu_l)^2)
})}
\\
\log(p_k(x))
\log(\sum {
\pi_l
\frac {1} {\sqrt{2 \pi} \sigma_l}
\exp(- \frac {1} {2 \sigma_l^2} (x - \mu_l)^2)
})
= \log(\pi_k) +
\log(\frac {1} {\sqrt{2 \pi} \sigma_k}) +
- \frac {1} {2 \sigma_k^2} (x - \mu_k)^2
\\
\delta(x)
= \log(\pi_k) +
\log(\frac {1} {\sqrt{2 \pi} \sigma_k}) +
- \frac {1} {2 \sigma_k^2} (x - \mu_k)^2
$$
As you can see, $\delta(x)$ is a quadratic function of $x$.