Post on 10-Mar-2015
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OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Stochastic Processes of Metocean Data
ASSOC. PROF. IR. DR. MOHD SHAHIR LIEWUNIVERSITI TEKNOLOGI PETRONAS
Presentation Fast Forward
• Let’s Get Started!
• Looking Into Metocean Time Series• Looking Into Metocean Time Series
• Analysis on Time Domain
• Analysis on Frequency Domain
• Forecasting of Time Series
• Estimation Procedures and Errors
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
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Section 1
Let’s Get Started!(Parable of the mathematician the engineer and the(Parable of the mathematician, the engineer and the
statistician)
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Things You Need To Know First
In a nutshell…
Time Series
Stochastic Processes
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Series
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Things You Need To Know First
What is a “time series”?
A set of observationsgenerated sequentially intime.
Can be discrete orcontinuous
Fig 1: Example of time series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 1: Example of time series
Things You Need To Know First
What is a “stochasticprocess”?
A statistical phenomenonthat evolves or varies intime according toprobabilistic laws
Can be stationary orFig 1: Example of time seriesnon-stationary
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 1: Example of time series
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Things You Need To Know First
Stationary Process• Average values of interest in the time
series remain constant with changes intime
Non-stationary Process
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
• Average values of interest in the timeseries vary with changes in time
Things You Need To Know First
Fig. 2: Example of a stationary processy p
Fig. 3: Example of a non-
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 3: Example of a nonstationary process (trend)
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Things You Need To Know First
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 4: Example of a non-stationary process (seasonal)
Section 2
Looking Into Metocean Time S iSeries
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
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Some Metocean Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 5: Time Series of Maximum Wind Speed (taken at 10-minute intervals)
Time interval
Some Metocean Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 6: Time Series of Maximum Wave Height (taken at 10-minute intervals)
Time interval
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Some Metocean Time Series
Info: at 2.2-metre depth above bed
Total depth
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 7: Time Series of Current Speed (taken at 10-minute intervals)
Time interval
is 61.0 m
Section 3
Analyzing Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
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Analysis of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Section 4
Analysis on Time Domain
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
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Time Domain Analyses of Time Series
Time Interval Time Interval
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Time Interval
Time Domain Analyses of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 8: Autocorrelations of the Max Wind Speed Time Series
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Time Domain Analyses of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 9: Autocorrelations of the Max Wind Speed Time Series (taking First Difference)
Time Domain Analyses of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 10: Cross-correlations of Max Wind Speed and Max Wave Height
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Descriptive Statistics of Time Series
• Some other analyses (e.g. autocovariance)require descriptive statistics of the timeseries that is being analyzedseries that is being analyzed
• For this example, the Maximum WindSpeed time series is used
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Descriptive Statistics of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
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Section 5
Analysis on Frequency Domain
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Frequency Domain Analyses of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 11: Spectrum of Maximum Wind Speed
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Frequency Domain Analyses of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 12: Spectrum of Maximum Wave Height
Frequency Domain Analyses of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Fig. 13: Cross spectrum of Maximum Wind Speed and Maximum Wave Height
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Section 6
Forecasting of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Forecasting of Time Series
Mixed Autoregressive – Integrated MovingAverage Processes of order (p, d, q)
or
qtqttptptt aaazzz −−−− −−−+++= θθφφ ...~...~~1111
aBzB )(~)( θφ =
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
tt aBzB )()( θφ =
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Forecasting of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Model DescriptionModel Type
Model ID MaxWind Model_1 Simple
Model fit of Maximum Wind Speed Time Series
Forecasting of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Model fit of Maximum Wave Height Time Series
Model DescriptionModel Type
Model ID MaxWave Model_1 ARIMA(0,1,2)
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Forecasting of Time Series
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Model fit of Current Speed Time Series
Model DescriptionModel Type
Model ID CurrentSpeed Model_1 ARIMA(0,1,14)
Section 7
Estimation Procedures and EErrors
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
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Estimation Procedures and Errors
Given an analog time history record oflength T, the record can be converted to asequence of N equally spaced samplesequence of N equally spaced samplevalues with no significant loss ofinformation if:
BTN 2≥
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
where B = cyclical frequency bandwidth ofthe data
Estimation Procedures and Errors
Assuming that the time-resolution intervalbetween the sample values is Δt:
Highest frequency that will appear insample data, called Nyquist cutofffrequency:
tNT Δ=
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
frequency:
tfc Δ=
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Estimation Procedures and Errors
Minimum frequency-resolution bandwidthavailable from the data record is given by:
Smoothing function typically done to
tNTBf e Δ
===Δ11
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY
Smoothing function – typically done toeliminate noise from densities.
End of Presentation
Thank you very much
OFFSHORE ENGINERING: ADVANCES AND SUSTAINABILITY