Gaussian Models

2020-03-06 本文已影响0人水豚2号

Basic Knowledge

Normal Distribution / Multivariate Gaussian
$\mathcal N(x|\mu, \sigma) = ······$
Eigenvalue Decomposition
Estimate MLE for $\mu, \sigma$
Same $\mu, \sigma$ , Gaussian Distribution has Maximum Entropy
Joint Gaussian Distribution

Interpolation

$x_{i} = f(t_{i}), t_{i} \in [0, T]$
Data points: total $D$ , observed $N$
Suppose we can fit the data, then using smoothness assumption $x_{j} = \frac{1}{2} (x_{j-1} + x_{j+1}) + \epsilon_{j}$ , we get $LX=\epsilon, X=L^{-1}\epsilon$
$X \sim N(0, \sigma^{2}(L^{T}L)^{-1})$

Noise-free Observation

Partition $X$ to $[X_{1}, X_{2}]$ , $L$ to $[L_{1}, L_{2}]$ , where $X_{2}$ are $N$ observed noise-free observations. If observations are not adjacent data points, we can adjust $X$ along with $L$
Then, $X_{1} \sim N(0, \sigma^{2}(L_{1}^{T}L_{1})^{-1}), X_{2} \sim N(0, \sigma^{2}(L_{2}^{T}L_{2})^{-1})$
Use formula, get conditional distribution, aka., $X_{1}|X_{2}$ 's distribution $f_{(D-N) * 1}$ (unknown data's distribution)
Generate $f(t)$

Noisy Observation

Observed $N$ noisy data $y_{i} = x_{i} + \eta_{i}$
Use selection matrix $A$ , such that $Y_{D*1}=A_{D*D} * X_{D*1}+\eta$
Now, $X \sim N(0, \sigma^{2}(L^{T}L)^{-1}), Y|X \sim N(AX, a^{2}I)$
Use theorem, get posterior distribution, aka., $X|Y$ 's distribution $f_{D*1}$ (whole data's distribution)
Generate $f(t)$

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