Fitting Wave Height Data to a Probability Distribution
Introduction
The University of Maine records real-time accelerometer data from buoys deployed in the Gulf of Maine and the Caribbean (http://gyre.umeoce.maine.edu/buoyhome.php). The data can be downloaded from their website, and includes the significant wave height recorded at regular intervals for the last few months.
This application does the following:
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
The Google Maps component below gives the location of buoy PR206.
Latitude
Longitude
Zoom Level
Map Type
roadmapsatelliteterrainhybrid
Download and Plot Significant Wave Height Data in a Histogram
restart:
withplots:withStatistics:withOptimization:
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:
data≔ImportMatrixurl
data≔ 8274 x 6 MatrixData Type: anythingStorage: rectangularOrder: Fortran_order
sigWaveHeight≔RemoveNonNumericdata..,2:
n≔numelemssigWaveHeight
n≔8273
numBins≔40:
p1≔HistogramsigWaveHeight,bincount=numBins,color=COLORRGB,150255,40255,27255,thickness=0,style=patchnogrid,transparency=0.5,background=ColorTools:-ColorRGB,236255,240255,241255,axis=gridlines=10,color=RGB1,1,1,axesfont=Arial,labels=Significant Wave Height (m),Distribution,labeldirections=horizontal,vertical,labelfont=Arial,size=800,500:
displayp1
Maximum Likelihood Estimation
P≔unapplyProbabilityDensityFunctionWeibullalpha,beta,t,t,alpha,beta
P≔t,α,β↦0t<0βt−1+βⅇ−tαβαβotherwise
maxLike:=α,β→addlnPsigWaveHeighti,α,β,i=1..n:
resultsMLE≔MaximizemaxLikeα,β,α=0.01..5,β=0.01..5
resultsMLE≔−4839.27465547207703,α=1.34452570778020,β=2.83250005643047
p2≔plotevalPt,alpha,beta,resultsMLE2 ,t=minsigWaveHeight..maxsigWaveHeight,color=black,legend=Maximum Likelihood Estimation,thickness=3,legendstyle=font=Arial:
Moment Matching
resultsMM:=fsolveMomentsigWaveHeight,1=MomentWeibullα,β,1,MomentsigWaveHeight,2=MomentWeibullα,β,2,α=1,β=1
resultsMM≔α=1.342796022,β=2.958762705
p3≔plotevalPt,alpha,beta,resultsMM ,t=minsigWaveHeight..maxsigWaveHeight,color=RGB150255,40255,27255,thickness=3,legend=Moment Matching,legendstyle=font=Arial:
Results
displayp1,p2,p3,size=800,500
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