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Data-Driven Docume
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Supporting Information for "Magnetically Actuated Reconfigurable Metamaterials as Conformal Electromagnetic Filters
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Primary care attitudes towards the prescription of acetazolamide for altitude illness
Stem Lesson: Building a chair K-2
Potential change in the future spatial distribution of submerged macrophyte species and species richness: the role of today's lake type and strength of compounded environmental change
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UHI in Fortaleza and trends on screen-level air temperature and humidity
Mode Test By GMM and Excess Mass Methods
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\label{sec:methods}
\label{sec:methods-gmm}
GMM (Gaussian mixture modeling) method maximizes the likelihood of the data set using EM (expectation-maximization) method.
1. Assume that data has unimodal distribution: x ∼ N(μ, σ2). Calculate μ and σ2
2. Assume that data has bimodal distribution: x ∼ N(μ1, μ2, σ12, σ22, p)
Initial guess: μ1 = μ − σ, μ2 = μ + σ, σ12 = σ22 = σ2, p = 0.5
n= number of observations
θ = (μ1, μ2, σ1, σ2, p) z = (z1, ..., zn) categorical vector, zi = 1, 2
x = (x1, ..., xn) observations, (xi|zi = 1)∼N(μ1, σ12), (xi|zi = 2)∼N(μ2, σ22)
E-step P(z1)=p, P(z2)=1 − p
Marginal likelihood: L(θ; x; z)=P(x, z|θ)=$\prod\limits_{i=1}^n P(Z_i=z_i)f(x_i|\mu_{j}, \sigma^2_{j})$
Q(θ|θ(t))=Ez|x, θ(t)(logL(θ; x; z))
$T^{(t)}_{j,i}=P(Z_i=j|X_i=x_i,\theta^{(t)})=\frac{P(z_{j})f(x_i|\mu^{(t)}_{j}, \sigma^{2(t)}_{j})}{p^{(t)} f(x_i|\mu^{(t)}_{1}, \sigma^{2(t)}_{1})+(1-p^{(t)})f(x_i|\mu^{(t)}_{2}, \sigma^{2(t)}_{2})}$
$Q(\mathbf{\theta}|\mathbf{\theta^{(t)}})=E_{\textbf{z}|\textbf{x},\mathbf{\theta^{(t)}}}(\log L(\mathbf{\theta};\textbf{x};\textbf{z})) = \sum\limits_{i=1}^n E[( \log L(\mathbf{\theta};x_{i};z_{i})] =$
$= \sum\limits_{i=1}^n \sum\limits_{j=1}^2 T^{(t)}_{j,i}[\log P(z_{j}) -\frac{1}{2}\log(2\pi) - \frac{1}{2}\log\sigma^{2}_{j} - \frac{(x_{i}-\mu_{j})^2}{2\sigma^{2}_{j}}]$
M-step θ(t + 1) = argmaxQ(θ|θ(t))
$\hat{p}^{(t+1)} = \frac{1}{n} \sum\limits_{i=1}^n T^{(t)}_{1,i}$, $\mu^{(t+1)}_{1} = \frac{\sum\limits_{i=1}^n T^{(t)}_{1,i}x_i}{\sum\limits_{i=1}^n T^{(t)}_{1,i}}$, $\sigma^{2(t+1)}_{1} = \frac{\sum\limits_{i=1}^n T^{(t)}_{1,i}(x_i-\mu^{(t+1)}_{1})^2}{\sum\limits_{i=1}^n T^{(t)}_{1,i}}$
Continue iterations t until |logL(t + 1) − logL(t)|<10−3
Conclusion about data is made based on 3 tests. H0 distribution is unimodal, H1 distribution is bimodal:
1. LRT (Likelihood ratio test) −2lnλ = 2[lnLbimodal − lnLunimodal]∼χ2 (LRT is the main test among all 3 tests for making conclusion about bimodality of data. The bigger −2lnλ is, the more we are convinced that distribution is bimodal).
2. (Bandwidth test) $D = \frac {|\mu_1 - \mu_2|}{(\sigma^2_1+\sigma^2_2)/2)^{0.5}}$ (D(distance)>2 is necessary for a clear separation of 2 peaks).
3. (Kurtosis test) kurtosis < 0 should be negative for a bimodal distribution.
In some hard cases D and kurtosis fail to detect bimodality. That is why our main test is LRT. For example on the next 2 plots distributions are bimodal, however on 1 plot D<2 (it is hard to distinguish 2 peaks) and on the 2 plot kurtosis is positive and that corresponds to unimodal distribution (it happens because distribution is biased):
Example document
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Investigation X
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