Downloads accelerometer data for Buoy PR206 (located just off the coast of Puerto Rico at a latitude of 18° 28.46' N and a longitude of 66° 5.94' W)

Fits the significant wave height to a Weibull distribution by using two methods: maximum likelihood estimation and moment matching

Plots the fitted distributions on top of a histogram of the experimental data

$\mathrm{restart}\:$

$\mathrm{with}\left(\mathrm{plots}\right)\:\mathrm{with}\left(\mathrm{Statistics}\right)\:\mathrm{with}\left(\mathrm{Optimization}\right)\:$

$\mathrm{url}≔''http://gyre.umeoce.maine.edu/data/gomoos/buoy/php/view\_csv\_file.php?ncfile=/data/gomoos/buoy/archive/PR206/realtime/PR206.waves.triaxys.realtime.nc''\:$

$\mathrm{data}≔\mathrm{ImportMatrix}\left(\mathrm{url}\right)$

$\mathrm{sigWaveHeight}≔\mathrm{RemoveNonNumeric}\left(\mathrm{data}\left[..\,2\right]\right)\:$

$\mathrm{n}≔\mathrm{numelems}\left(\mathrm{sigWaveHeight}\right)$

$n≔8273$

$\mathrm{numBins}≔40\:$

$\mathrm{p1}≔\mathrm{Histogram}\left(\mathrm{sigWaveHeight}\,\mathrm{bincount}\=\mathrm{numBins}\,\mathrm{color}\=\mathrm{COLOR}\left(\mathrm{RGB}\,\frac{150}{255}\,\frac{40}{255}\,\frac{27}{255}\right)\,\mathrm{thickness}\=0\,\mathrm{style}\=\mathrm{patchnogrid}\,\mathrm{transparency}\=0.5\,\mathrm{background}\=\mathrm{ColorTools}:-\mathrm{Color}\left(''RGB''\,\left[\frac{236}{255}\,\frac{240}{255}\,\frac{241}{255}\right]\right)\,\mathrm{axis}\=\left[\mathrm{gridlines}\=\left[10\,\mathrm{color}\=\mathrm{RGB}\left(1\,1\,1\right)\right]\right]\,\mathrm{axesfont}\=\left[\mathrm{Arial}\right]\,\mathrm{labels}\=\left[''Significant\; Wave\; Height\; (m)''\,''Distribution''\right]\,\mathrm{labeldirections}\=\left[\mathrm{horizontal}\,\mathrm{vertical}\right]\,\mathrm{labelfont}\=\left[\mathrm{Arial}\right]\,\mathrm{size}\=\left[800\,500\right]\right)\:$

$\mathrm{display}\left(\mathrm{p1}\right)$

$\mathrm{P}≔\mathrm{unapply}\left(\mathrm{ProbabilityDensityFunction}\left(\mathrm{Weibull}\left(\mathrm{alpha}\,\mathrm{beta}\right)\,\mathrm{t}\right)\,\mathrm{t}\,\mathrm{alpha}\,\mathrm{beta}\right)$

$P≔\left(t\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\right)\mapsto \left\{\begin{array}{cc}0& t<0\\ \frac{\mathrm{\beta }\cdot t^{-1+\mathrm{\beta }}\cdot {\&ExponentialE;}^{-{\left(\frac{t}{\mathrm{\alpha }}\right)}^{\mathrm{\beta }}}}{{\mathrm{\alpha }}^{\mathrm{\beta }}}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{maxLike}:=\left(\mathrm{\alpha }\&comma;\mathrm{\beta }\right)\to \mathrm{add}\left(\ln \left(\mathrm{P}\left({\mathrm{sigWaveHeight}}_{\mathrm{i}}\&comma;\mathrm{\alpha }\&comma;\mathrm{\beta }\right)\right)\&comma;\mathrm{i}\&equals;1..\mathrm{n}\right)\&colon;$

Warning, (in maxLike) `i` is implicitly declared local

$\mathrm{resultsMLE}≔\mathrm{Maximize}\left(\mathrm{maxLike}\left(\mathrm{\alpha }\&comma;\mathrm{\beta }\right)\&comma;\mathrm{\&alpha;}\&equals;0.01..5\&comma;\mathrm{\&beta;}\&equals;0.01..5\right)$

$\mathrm{resultsMLE}≔\left[\mathrm{-4839.27465547207612}\&comma;\left[\mathrm{\alpha }=1.34452570778020\&comma;\mathrm{\beta }=2.83250005643047\right]\right]$

$\mathrm{p2}≔\mathrm{plot}\left(\mathrm{eval}\left(\mathrm{P}\left(\mathrm{t}\&comma;\mathrm{alpha}\&comma;\mathrm{beta}\right)\&comma;\mathrm{resultsMLE}\left[2\right]\right) \&comma;\mathrm{t}\&equals;\min \left(\mathrm{sigWaveHeight}\right)..\max \left(\mathrm{sigWaveHeight}\right)\&comma;\mathrm{color}\&equals;\mathrm{black}\&comma;\mathrm{legend}\&equals;''Maximum\; Likelihood\; Estimation''\&comma;\mathrm{thickness}\&equals;3\&comma;\mathrm{legendstyle}\&equals;\left[\mathrm{font}\&equals;\left[\mathrm{Arial}\right]\right]\right)\&colon;$

$\mathrm{resultsMM}:=\mathrm{fsolve}\left(\left[\mathrm{Moment}\left(\mathrm{sigWaveHeight}\&comma;1\right)\&equals;\mathrm{Moment}\left(\mathrm{Weibull}\left(\mathrm{\&alpha;}\&comma;\mathrm{\&beta;}\right)\&comma;1\right)\&comma;\mathrm{Moment}\left(\mathrm{sigWaveHeight}\&comma;2\right)\&equals;\mathrm{Moment}\left(\mathrm{Weibull}\left(\mathrm{\&alpha;}\&comma;\mathrm{\&beta;}\right)\&comma;2\right)\right]\&comma;\left\{\mathrm{\&alpha;}\&equals;1\&comma;\mathrm{\&beta;}\&equals;1\right\}\right)$

$\mathrm{resultsMM}≔\left\{\mathrm{\alpha }=1.342796022\&comma;\mathrm{\beta }=2.958762705\right\}$

$\mathrm{p3}≔\mathrm{plot}\left(\mathrm{eval}\left(\mathrm{P}\left(\mathrm{t}\&comma;\mathrm{alpha}\&comma;\mathrm{beta}\right)\&comma;\mathrm{resultsMM}\right) \&comma;\mathrm{t}\&equals;\min \left(\mathrm{sigWaveHeight}\right)..\max \left(\mathrm{sigWaveHeight}\right)\&comma;\mathrm{color}\&equals;\mathrm{RGB}\left(\frac{150}{255}\&comma;\frac{40}{255}\&comma;\frac{27}{255}\right)\&comma;\mathrm{thickness}\&equals;3\&comma;\mathrm{legend}\&equals;''Moment\; Matching''\&comma;\mathrm{legendstyle}\&equals;\left[\mathrm{font}\&equals;\left[\mathrm{Arial}\right]\right]\right)\&colon;$

$\mathrm{display}\left(\mathrm{p1}\&comma;\mathrm{p2}\&comma;\mathrm{p3}\&comma;\mathrm{size}\&equals;\left[800\&comma;500\right]\right)$