Institut des dynamiquesde la surface terrestre

Spatio-Temporal Modeling of Wind Speed UsingEOF and Machine Learning

Fabian Guignard1, Federico Amato1,Sylvain Robert2 and Mikhail Kanevski1

1. IDYST, Faculty of Geosciences and Environment, University of Lausanne2. Swiss Re, ZurichContact: fabian.guignard@unil.ch

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Spatio-Temporal Prediction of Irregularly Spaced Data

Background:

I The interpolation problem of non-linear continuousspatio-temporal fields measured on a set of irregular points inspace, using Machine Learning (ML) is still under-investigated.

Aim:

I Introduce a framework for spatio-temporal prediction ofclimate and environmental data using ML.

I Show how spatio-temporal processes can be decomposed interms of a sum of products of temporally referenced basisfunctions, and of stochastic spatial coefficients which can thenbe spatially modelled and mapped on a regular grid.

I Discuss an application on a real world dataset consisting oftwo years of wind speed measurements at hourly frequencycollected in Switzerland with different ML models.

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Methods

Basis function representation can describe the spatial, temporaland spatio-temporal dependencies in the data.

Here we use the reduced-rank basis obtained through a principalcomponent analysis (PCA), also known as Empirical OrthogonalFunctions (EOFs), to decompose the data into fixed temporalbases and their corresponding spatial coefficients.

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Empirical Orthogonal Functions decompositionConsider the spatio-temporal observations {Z (si , tj)} at S spatiallocations {si : 1 ≤ i ≤ S} and T time-indices {tj : 1 ≤ j ≤ T}.Let Z̃ (si , tj) be the spatially centered data,

Z̃ (si , tj) := Z (si , tj)− µ̄(tj), µ̄(tj) :=1

S

S∑i=1

Z (si , tj),

where µ̄(tj) is the global spatial mean at time tj .The centred data Z̃ (si , tj) can be represented as

Z̃ (si , tj) =K∑

k=1

αk(si )φk(tj), (1)

where αk(si ) is the coefficient with respect to the k-th basisfunction φk at spatial location si , and K = min{T ,S − 1}. Thescalar coefficient αk(si ) only depends on the location; the temporalbasis function φk(tj) doesn’t depend of space.

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Machine Learning modelling

The spatial coefficients canbe considered as the targetvariable in a spatial regressionproblem, solved with MLtechniques. In this work wewill use :

I Extreme Learning Machine(ELM)

I K-Nearest Neighbours (KNN)

I Random Forest (RF)

I Support Vector Regression(SVR)

I General Regression NeuralNetworks (GRNN)

InputsSpatial features

ML Algorithm

ML m

odel

ling

of sp

atia

l coe

fficie

ntsS.-T. data

EOFsRecomposition

OutputsSpatial coefficients

S.-T. prediction

EOFsDecomposition

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Case Study: Wind speed prediction in Switzerland

Here we will present the interpolation of wind speed data from 209stations in Switzerland from 2017 to 2018 measured at 1 hoursampling period. Data were divided into train (166 stations) andtest (43).

A 13-d input space has been designed including the geographicalspace (latitude, longitude and elevation) and features derived byapplying filters and derivatives on the digital elevation model.

Wind phenomenon at such frequency and in complex mountainousregion is known to be very difficult to model.

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Case Study: wind speed prediction in Switzerland

0 2000 4000 6000 8000

0.03

0.02

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0.00EOF 1

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0.04

0.02

0.00

0.02

0.04EOF 2

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0.050

0.025

0.000

0.025

EOF 3

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0.020

0.015

0.010

0.005

EOF 1, first 41 days

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0.02

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0.02

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EOF 3, first 41 days

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EOF 2 (spatial coefficients)

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Figure 1: EOF decomposition: The first 3 components of the windspeed data EOFs decomposition. Top row : The temporally referencedbasis functions. Center row : Temporal basis function for the first 42days. Bottom row: The standardized spatial coefficients of thecorresponding EOFs. Only the first 70 components will be kept for signalreconstruction, corresponding to 95% of the variability of the originaldata.

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Example of prediction using Extreme Learning Machines

Figure 2: Example of the output obtained with ELM modelling: Toprow : Predicted spatial coefficient maps for the first three components.Bottom: Test MAE over the two years investigated.

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Example of prediction using Extreme Learning Machines

Figure 3: Example of the output obtained with ELM modelling:Left: The true time series (in black) at a testing station marked by across in the map below, and the predicted time series with the first 70EOFs components (in green). For visualization purposes, only the first 42days of the time series are shown. Right: Accuracy plot at the sametesting station.

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Comparison between different ML algorithms

Model MAE (mean ± sd. dev.)ELM 1.010± 9.5× 10−3

KNN 0.968± 2.2× 10−16

RF 1.032± 2.2× 10−3

SVR 0.960± 3.3× 10−16

GRNN 1.013± 2.2× 10−16

Table 1: Comparison among different ML models: The table showsthe mean and standard deviation of the Mean Absolute Error averagedover the entire study period computed after 20 runs of each MLalgorithm.

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Discussion and conclusions

I We introduced a framework for spatio-temporal prediction ofclimate and environmental data using ML.

I The decomposition of the spatio-temporal signal into fixedtemporal bases and stochastic spatial coefficients permits tofully reconstruct spatio-temporal fields starting from spatiallyirregularly distributed measurements.

I The spatial prediction of the stochastic coefficients can beperformed using any ML algorithm.

I A promising direction for further research is to developprocedures to quantify the propagation of uncertainty throughthe diverse steps of the proposed framework.

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Institut des dynamiquesde la surface terrestre

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