88 lines
1.3 KiB
Markdown
88 lines
1.3 KiB
Markdown
# Sample parameters estimation
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## Sample parameters estimation
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Once the points are sampled,
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how to estimate their median, mode and FWHM?
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. . .
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- Binning data $\quad \longrightarrow \quad$ result depending on bin-width
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. . .
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- Alternative solutions
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## Sample median
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$$
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m = Q \left( \frac{1}{2} \right)
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$$
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. . .
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- Sort points in ascending order
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. . .
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- Middle element if odd
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- Average of the two central elements if even
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## Sample mode
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Most probable value
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. . .
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HSM
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- Iteratively identify the smallest interval containing half points
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- once the sample is reduced to less than three points, take average
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## Sample FWHM
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$$
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\text{FWHM} = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}
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$$
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. . .
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KDE
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- empirical PDF construction:
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$$
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f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
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G \left( \frac{x-x_i}{\varepsilon} \right)
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$$
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The parameter $\varepsilon$ controls the strenght of the smoothing
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## Sample FWHM
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Silverman's rule of thumb:
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$$
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f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
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G \left( \frac{x-x_i}{\varepsilon} \right)
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\with
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\varepsilon = 0.63 \, S_N
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\left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}
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$$
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with:
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- $S_N$ is the sample stdev
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- $d$ number of dimensions ($d = 1$)
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. . .
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\vspace{10pt}
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Numerical root finding (Brent)
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