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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

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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)

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Things You Need To Know First

In a nutshell…

Time Series

Stochastic Processes

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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

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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

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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

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• 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-

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Fig. 3: Example of a nonstationary process (trend)

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Things You Need To Know First

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Fig. 4: Example of a non-stationary process (seasonal)

Section 2

Looking Into Metocean Time S iSeries

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Some Metocean Time Series

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Fig. 5: Time Series of Maximum Wind Speed (taken at 10-minute intervals)

Time interval

Some Metocean Time Series

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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

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Fig. 7: Time Series of Current Speed (taken at 10-minute intervals)

Time interval

is 61.0 m

Section 3

Analyzing Time Series

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Analysis of Time Series

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Section 4

Analysis on Time Domain

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Time Domain Analyses of Time Series

Time Interval Time Interval

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Time Interval

Time Domain Analyses of Time Series

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Fig. 8: Autocorrelations of the Max Wind Speed Time Series

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Time Domain Analyses of Time Series

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Fig. 9: Autocorrelations of the Max Wind Speed Time Series (taking First Difference)

Time Domain Analyses of Time Series

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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

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Descriptive Statistics of Time Series

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Section 5

Analysis on Frequency Domain

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Frequency Domain Analyses of Time Series

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Fig. 11: Spectrum of Maximum Wind Speed

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Frequency Domain Analyses of Time Series

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Fig. 12: Spectrum of Maximum Wave Height

Frequency Domain Analyses of Time Series

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Fig. 13: Cross spectrum of Maximum Wind Speed and Maximum Wave Height

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Section 6

Forecasting of Time Series

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Forecasting of Time Series

Mixed Autoregressive – Integrated MovingAverage Processes of order (p, d, q)

or

qtqttptptt aaazzz −−−− −−−+++= θθφφ ...~...~~1111

aBzB )(~)( θφ =

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tt aBzB )()( θφ =

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Forecasting of Time Series

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Model DescriptionModel Type

Model ID MaxWind Model_1 Simple

Model fit of Maximum Wind Speed Time Series

Forecasting of Time Series

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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

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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

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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≥

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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=

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frequency:

tfc Δ=

21

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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

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Smoothing function – typically done toeliminate noise from densities.

End of Presentation

Thank you very much

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