Andy Pack
c3ebefdd64
Affected files: .obsidian/workspace.json Food/Plans & Recipes.canvas Health/ADHD.md Health/Alexithymia.md Health/Anxiety.md Health/BWS.md Health/Worry Tree.md History/Britain.md Media/Listen.md Media/Watch.md Money/Assets/Asset.md Money/Assets/CFD.md Money/Assets/Derivative.md Money/Assets/Financial Instruments.md Money/Assets/Options.md Money/Assets/Security.md Money/Assets/Stocks.md Money/Econ.md Money/Equity.md Money/Inflation.md Money/Markets/Commodity.md Money/Markets/Markets.md Money/Markets/Types.md Money/Prices.md Money/To Learn.md Politcs/British/Prime Ministers.md Politcs/British/Tory.md Politcs/Neoliberalism/Thatcher.md Politcs/Now.md Projects/Dev Options.md STEM/AI/Neural Networks/Activation Functions.md STEM/AI/Neural Networks/MLP/Back-Propagation.md STEM/AI/Neural Networks/SLP/Least Mean Square.md STEM/AI/Pattern Matching/Markov/Markov.md STEM/Maths/Algebra.md STEM/Maths/Tensor.md STEM/Maths/Vector Operators.md Want/Clothes.md Want/House.md Want/Want.md Work/Board.canvas
61 lines
1.7 KiB
Markdown
61 lines
1.7 KiB
Markdown
---
|
||
tags:
|
||
- ai
|
||
- maths
|
||
---
|
||
[Hidden Markov Models - JWMI Github](https://jwmi.github.io/ASM/5-HMMs.pdf)
|
||
[Rabiner - A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition](https://www.cs.cmu.edu/~cga/behavior/rabiner1.pdf)
|
||
|
||
- Stochastic sequences of discrete states
|
||
- Transitions have probabilities
|
||
- Desired output not always produced the same
|
||
- Same pronunciation
|
||
|
||
|
||
|
||
$$P(X|M)=\left(\prod_{t=1}^Ta_{x_{t-1}x_t}\right)\eta_{x_T}$$
|
||
$$a_{x_0x_1}=\pi_{x_1}$$
|
||
|
||
# 1st Order
|
||
- Depends only on previous state
|
||
- Markov assumption
|
||
|
||
$$P(x_t=j|x_{t-1}=i,x_{t-2}=h,...)\approx P(x_t=j|x_{t-1}=i)$$
|
||
|
||
- Described by state-transition probabilities
|
||
|
||
$$a_{ij}=P(x_t=j|x_{t-1}=i), 1\leq i,j\leq N$$
|
||
|
||
- $\alpha$
|
||
- State transition
|
||
- For $N$ states
|
||
- $N$ by $N$ matrix of state transition probabilities
|
||
|
||
# Weather
|
||
|
||
|
||
$$A=\left\{a_{ij}\right\}=\begin{bmatrix} 0.4 & 0.3 & 0.3\\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.1 & 0.8 \end{bmatrix}$$
|
||
rain, cloud, sun across columns and down rows
|
||
$$A=\{\pi_j,a_{ij},\eta_i\}=\{P(x_t=j|x_{t-1}=i)\}$$
|
||
|
||
# Start/End
|
||
- Null states
|
||
- Entry/exit states
|
||
- Don't generate observations
|
||
|
||
|
||
$$\pi_j=P(x_1=j) \space 1 \leq j \leq N$$
|
||
- Sub $j$ because probability of kicking off into that state
|
||
$$\eta_i=P(x_T=i) \space 1 \leq i \leq N$$
|
||
- Sub $i$ because probability of finishing from that state
|
||
|
||
|
||
|
||
|
||
# State Duration
|
||
- Probability of staying in state decays exponentially
|
||
$$p(X|x_1=i,M)=(a_{ii})^{\tau-1}(1-a_{ii})$$
|
||
|
||
- Given, $a_{33}=0.8$
|
||
- $\times0.8$ repeatedly
|
||
- Stay in state |