analistica/slides/sections/4.md

Sample parameters estimation

Sample parameters estimation

Once the points are sampled,
how to estimate their median, mode and FWHM?

. . .

• Binning data \hence result depending on bin-width

. . .

• Alternative solutions

Sample median

  m = Q \left( \frac{1}{2} \right)

. . .

• Sort points in ascending order

. . .

• Middle element if odd
• Average of the two central elements if even

Sample mode

Most probable value

. . .

HSM

• Iteratively identify the smallest interval containing half points
• Once the sample is reduced to less than three points, take average

Sample FWHM

  \text{FWHM} = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}

. . .

KDE

• empirical PDF construction:

  f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
  G \left( \frac{x-x_i}{\varepsilon} \right)

The parameter \varepsilon controls the strength of the smoothing

Sample FWHM

Silverman's rule of thumb:

  f_\varepsilon(x) = \frac{1}{N\varepsilon} \sum_{i = 1}^N
  G \left( \frac{x-x_i}{\varepsilon} \right)
  \with
  \varepsilon = 0.88 \, S_N
  \left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}

with:

• S_N is the sample standard deviation
• d is number of dimensions (d = 1)

. . .

\vspace{10pt}

Numerical root finding (Brent)

Sample FWHM