wiki/scratch.page

114 lines
2.4 KiB
Plaintext
Raw Normal View History

2013-09-15 03:48:19 +00:00
---
title: Scratch Page
toc: no
format: markdown
...
# Math Stuff
On observation of $\mathcal{D}$, the *likelihood* of hypothesis
$\mathcal{R}_{\alpha}$ is
$\mathit{P}(\mathcal{D}|\mathcal{R}_{\alpha})$.
## Fingerprint Variance
Additionally, we associate a collective **RSS Variance**
$\sigma_{F_s}$ with each fingerprint, which is a weighed average of
the RSS values of each of the vector elements using $C_i$ as the
weight. It is calculated in this manner:
$$
2013-09-15 04:25:10 +00:00
\sigma_{F_s} = \frac{\sum_{i \in F_s} \sigma_i\, C_i}
2013-09-15 03:48:19 +00:00
{\sum_{i \in F_s} C_i}
$$
2013-09-15 04:26:05 +00:00
2013-09-15 04:25:10 +00:00
# Bayesian Regression
First, specify a set of probabilistic models of the data.
Let a member of this set be denoted by $\mathcal{R}_\alpha$
$\mathcal{R}_\alpha$ has a *prior* probability $P(\mathcal{H}_\alpha)$
On observation of $\mathcal{D}$, the *likelihood* of hypothesis
$\mathcal{R}_{\alpha}$ is
$\mathit{P}(\mathcal{D}|\mathcal{R}_{\alpha})$.
The *posterior* probability of $\mathcal{R}_{\alpha}$ is then given by
$\mathit{P}(\mathcal{H}_{\alpha})\mathit{P}(\mathcal{D}|\mathcal{H}_{\alpha})$
This follows from **Bayes' Theorem** which says
$$
P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}
2014-01-26 01:57:08 +00:00
$$
# Matrix Stuff
This is a column vector:
$$\vec v = \left(\begin{matrix}
1\\
3\\
7
\end{matrix}\right)$$
The *vector sum* of $\vec u$ and $\vec v$ is:
$$\vec u + \vec v =
\left(\begin{matrix}
u_1\\
\vdots\\
u_n
\end{matrix}\right) + \left(\begin{matrix}
v_1\\
\vdots\\
v_n
\end{matrix}\right) = \left(\begin{matrix}
u_1 + v_1\\
\vdots\\
u_n + v_n
\end{matrix}\right)
$$
The *scalar multiplication* of the real number $r$ and the vector $\vec v$ is:
$$
r \cdot \vec v = r \cdot \left(\begin{matrix}
v_1\\ \vdots \\ v_n
\end{matrix}\right) = \left(\begin{matrix}
rv_1 \\ \vdots \\ rv_n
\end{matrix}\right)
2014-01-26 02:55:27 +00:00
$$
This system:
$$\begin{alignedat}{4}
x & {}-{} & y & {}+{} & z & = 1\\
3x & {}+{} & & & z & = 3\\
5x & {}-{} & 2y & {}+{} & 3z & = 5
\end{alignedat}$$
reduces
$$
\left(\begin{array}{rrr|r}
1&-1&1&1\\
3&0&1&3\\
5&-2&3&5\\
\end{array}\right) \longrightarrow_{-5\rho_1+\rho_3}^{-3\rho_1+\rho_2}
\left(\begin{array}{rrr|r}
1&-1&1&1\\
0&3&-2&0\\
0&3&-2&0\\
\end{array}\right) \longrightarrow^{-\rho_2+\rho_3}
\left(\begin{array}{rrr|r}
1&-1&1&1\\
0&3&-2&0\\
0&0&0&0\\
\end{array}\right)
$$
to a one parameter solution set:
$$
\{ \begin{pmatrix}1\\ 0\\ 0\end{pmatrix} +
\begin{pmatrix}-1/3\\ 2/3\\ 1\end{pmatrix} z \mid z \in \Bbb{R} \}
2013-09-15 04:25:10 +00:00
$$