238 lines
7.8 KiB
Markdown
238 lines
7.8 KiB
Markdown
# Sample statistics
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## Sample statistics
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How to estimate sample median, mode and FWHM?
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. . .
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- \only<3>\strike{Binning data $\hence$ depends wildly on bin-width}
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. . .
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- Alternative solutions
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- Robust estimators
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- Kernel density estimation
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## Sample median
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\Begin{block}{Algorithm}
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::: incremental
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1. Sample points
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2. Sort sample in ascending order
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3.
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Take middle element if odd
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Take average of two middle elements if even
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:::
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\End{block}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}[remember picture, >=Stealth]
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% placeholder
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\draw [ultra thick, transparent] (-0.35,0.7) -- (-0.35,-0.7);
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% line
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\draw <1-> [line width=3, ->, cyclamen] (-5,0) -- (5,0);
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\node <1-> [right] at (5,0) {$x$};
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% points
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\draw <1-> [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1);
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% nodes
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\node <2-> [below] at (-4.7,-0.1) {1};
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\node <2-> [below] at (-4.1,-0.1) {2};
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\node <2-> [below] at (-3.4,-0.1) {3};
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\node <2-> [below] at (-2.4,-0.1) {4};
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\node <2-> [below] at (-0.7,-0.1) {5};
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\node <2-> [below] at ( 0 ,-0.1) {6};
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\node <2-> [below] at ( 1.2,-0.1) {7};
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\node <2-> [below] at ( 2.1,-0.1) {8};
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\node <2-> [below] at ( 2.8,-0.1) {9};
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\node <2-> [below] at ( 4.1,-0.1) {10};
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\draw <3-> [ultra thick] (-0.35,0.7) -- (-0.35,-0.7);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## Sample mode
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**Half Sample Mode** [@robertson74]
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\Begin{block}{Algorithm}
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::: incremental
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1. Sample points
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2. Find the smallest interval containing half points
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3. Repeat on the new interval (iterative)
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4. If less than four points, take average of the closest two
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:::
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\End{block}
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\centering
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\setbeamercovered{}
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\begin{tikzpicture}[remember picture, >=Stealth]
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% line
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\draw <1-> [line width=3, ->, cyclamen] (-5,0) -- (5,0);
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\node <1-> [right] at (5,0) {$x$};
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% points
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\draw <1-> [yellow!50!black, fill=yellow] (-4.6,-0.1) rectangle (-4.8,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-4,-0.1) rectangle (-4.2,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-3.3,-0.1) rectangle (-3.5,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-2.3,-0.1) rectangle (-2.5,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-0.6,-0.1) rectangle (-0.8,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (-0.1,-0.1) rectangle (0.1,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (1.1,-0.1) rectangle (1.3,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (2,-0.1) rectangle (2.2,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (2.7,-0.1) rectangle (2.9,0.1);
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\draw <1-> [yellow!50!black, fill=yellow] (4,-0.1) rectangle (4.2,0.1);
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% nodes
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\node <1-> at (-1,-0.3) (1a) {};
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\node <1-> at (3.1,0.3) (1b) {};
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\node <1-> at (0.9,-0.3) (2a) {};
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\node <1-> at (1.8,-0.3) (3a) {};
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\node <1-> at (2.45,-0.7) (f1) {};
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\node <1-> at (2.45,0.7) (f2) {};
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% algorithm
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\draw <2-> [gray, fill=gray, opacity=0.5] (1a) rectangle (1b);
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\draw <3-> [gray, fill=gray, opacity=0.6] (2a) rectangle (1b);
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\draw <4-> [cyclamen, thick] (3a) rectangle (1b);
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\draw <5-> [ultra thick] (f1) -- (f2);
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\end{tikzpicture}
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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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\setbeamercovered{transparent}
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. . .
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**Kernel Density Estimation**
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:::: {.columns}
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::: {.column width=50% .c}
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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
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sharpness of the empirical PDF
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:::
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::: {.column width=50%}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}
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% placeholder
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\draw [transparent] (-2.7,-0.2) rectangle (3,3.3);
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% bandwidth 1
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\node <4,5> [left] at (2.9,3) {$\epsilon = 1$};
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% points
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\draw <3-> [yellow!50!black, fill=yellow] (-1.2,-0.2) rectangle (-1,0);
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\draw <3-> [yellow!50!black, fill=yellow] (-0.1,-0.2) rectangle (0.1,0);
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\draw <3-> [yellow!50!black, fill=yellow] (0.7,-0.2) rectangle (0.9,0);
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\draw <3-> [yellow!50!black, fill=yellow] (1.3,-0.2) rectangle (1.5,0);
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% lines 1
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\draw <4,5> [cyclamen, dashed] (-1.1,0.1) -- (-1.1,1);
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\draw <4,5> [cyclamen, dashed] (0,0.1) -- (0,1);
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\draw <4,5> [cyclamen, dashed] (1.4,0.1) -- (1.4,1);
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\draw <4,5> [cyclamen, dashed] (0.8,0.1) -- (0.8,1);
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% Gaussians 1
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\draw <4,5> [domain=-2.6:0.4, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)) + 0.1});
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\draw <4,5> [domain=-1.5:1.5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-\x*\x + 0.1});
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\draw <4,5> [domain=-0.7:2.3, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
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\draw <4,5> [domain=-0.1:2.9, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x - 1.4)*(\x - 1.4)) + 0.1});
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% sum 1
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\draw <5> [domain=-2.6:2.9, smooth, variable=\x, yellow, very thick]
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plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)) +
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exp(-\x*\x) +
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exp(-(\x - 1.4)*(\x - 1.4)) +
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exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
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% bandwidth 2
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\node <6> [left] at (2.9,3) {$\epsilon = 0.5$};
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% lines 2
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\draw <6> [cyclamen, dashed] (-1.1,0.1) -- (-1.1,2);
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\draw <6> [cyclamen, dashed] (0,0.1) -- (0,2);
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\draw <6> [cyclamen, dashed] (1.4,0.1) -- (1.4,2);
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\draw <6> [cyclamen, dashed] (0.8,0.1) -- (0.8,2);
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% Gaussians 2
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\draw <6> [domain=-2.6:0.4, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)/0.25)/0.5 + 0.1});
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\draw <6> [domain=-1.5:1.5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-\x*\x/0.25)/0.5 + 0.1});
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\draw <6> [domain=-0.7:2.3, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x - 0.8)*(\x - 0.8)/0.25)/0.5 + 0.1});
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\draw <6> [domain=-0.1:2.9, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {exp(-(\x - 1.4)*(\x - 1.4)/0.25)/0.5 + 0.1});
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% sum
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\draw <6> [domain=-2.6:2.9, smooth, variable=\x, yellow, very thick]
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plot ({\x}, {exp(-(\x + 1.1)*(\x + 1.1)/0.25)/0.5 +
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exp(-\x*\x/0.25)/0.5 +
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exp(-(\x - 1.4)*(\x - 1.4)/0.25)/0.5 +
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exp(-(\x - 0.8)*(\x - 0.8)/0.25)/0.5 + 0.1});
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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:::
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::::
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## Sample FWHM
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**Silverman's rule of thumb** [@silver86]:
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$$
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\varepsilon = 0.88 \, S_N
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\left( \frac{d + 2}{4}N \right)^{-1/(d + 4)}
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$$
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where:
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- $S_N$ is the sample standard deviation
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- $d$ is number of dimensions ($d = 1$)
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. . .
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Minimization (Brent) for $\quad f_{\varepsilon_{\text{max}}}$
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Root finding (Brent-Dekker) for $\quad f_{\varepsilon}(x_{\pm}) =
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\frac{f_{\varepsilon_{\text{max}}}}{2}$
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## Sample FWHM
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## Bootstrap
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Estimating confidence interval:
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. . .
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\Begin{block}{Algorithm}
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::: incremental
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1. Sample $N$ points from PDF
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2. Sample with replacement $M$ times
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3. Compute the estimator for each new sample
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4. Compute mean and standard deviation
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:::
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\End{block}
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