Multivariate Interpolation of Wind Field Based on GPR Mia Feng - PowerPoint PPT Presentation

Multivariate Interpolation of Wind Field Based on GPR Mia Feng January 24, 2018

Incompatible interpolation operator gives rise to representa- tive error, which is a big challenge for improving the accu- racy of numerical weather prediction. The multi-kernel in- terpolation method based on Gaussian Process Regression we proposed pave a new way to make use of multi variables to infer the weather process.

Reference C. Rassemussen, "Gaussian processes for machine learning.” 2004.

Outline • GPR • Multivariate Interpolation • Issues

GPR and kernel tricks – Gaussian Process and GPR Gaussian Process GP is a collection of random variables, any finite number of which have a joint Gaussian distribution. � � x , x ′ �� f ( x ) ∼ GP m ( x ) , k (1) For any X , the collection of x in GP , it follows a jointly Gaussian distribution. For GP , knowing mean function and covariance function means knowing everything. GPR infers the predictive value based on the mean function and covariance function

GPR – Start from Bayesian linear regression f ( x ) = xw T y = f ( x ) + ε, (2) � 0 , σ 2 � ε ∼ N n Put a prior distribution on w : w ∼ N ( 0 , Σ p ) , the posterior of y ∗ at input vector x ∗ is � p ( y ∗ | x ∗ , X , y ) = p ( y ∗ | x ∗ , w ) p ( w | X , y ) d w (3) � � 1 n x ∗ T A − 1 Xy , x T ∗ A − 1 x ∗ = N σ 2 which implies it is a GRP model, where A = σ − 2 n XX T +Σ − 1 p , with a control function p ( y | X , w ) . = � n p ( y | X , w ) i = 1 p ( y i | x i , w ) (4) � X T w , σ 2 � = N n I

GPR – kernel tricks ⋆ To deal with the nonlinear problem, import mapping operator φ ( · ) Idea Change the BASIS SPACE Why is it useful? Change the measuring distance – the similarity of two input vectors x , x ′ , which implies the similarity of targets

kernel function p ( y ∗ | x ∗ , X , y ) (5) n φ ( x ∗ ) T A − 1 φ ( X ) y , φ ( x ∗ ) T A − 1 x ∗ � � 1 = N σ 2 � − 1 y , � φ T � K + σ 2 p ( y ∗ | x ∗ , X , y ) = N ∗ Σ p Φ n I (6) � − 1 Φ T Σ p φ ∗ � K + σ 2 φ T ∗ Σ p φ ∗ − φ T � ∗ Σ p Φ n I where K = φ ( X ) T Σ p φ ( X ) , Φ = φ ( X ) Suppose k ( x , x ′ ) = φ ( x ) Σ p φ ( x ′ ) � − 1 y , � K + σ 2 � p ( y ∗ | x ∗ , X , y ) = N K ( x ∗ , X ) n I (7) � − 1 K ( x ∗ , X ) � K + σ 2 K ( x ∗ , x ∗ ) − K ( x ∗ , X ) � n I

kernel function k ( · , · ) is called kernel function , it is a covariance function, which defines the similarity The predictive value of y ∗ at input vector x ∗ , � − 1 y K ( X , X ) + σ 2 � (8) y ∗ = K ( x ∗ , X ) n I with a control function what solves the unknown parameter θ , which are σ 2 n , Σ p here. log p ( y | X , θ ) = − 1 2 y T K − 1 y − 1 2 log | K | − n 2 log 2 π (9)

popular kernel function kenel formula advantages � � − r 2 SE exp smooth 2 l 2 √ √ � � � � 3 r 3 r Matérn 1 + exp − rough l l − 1 2 t T Λ − 2 t cos � � 2 π t T �� Gabor exp edge extraction p I Periodic u ( x ) = (cos ( x ) , sin ( x )) periodical space Generating kernel from old kernel k ( x , x ′ ) = k 1 ( x , x ′ ) + k 2 ( x , x ′ ) k ( x , x ′ ) = k 1 ( x , x ′ ) ∗ k 2 ( x , x ′ ) (10) k ( x , x ′ ) = α k 1 ( x , x ′ )

Multi-kernel k v ( · , · ) = k m ( · , · ) + k p ( · , · ) + k g ( · , · ) + k ε ( · , · ) (11) where k m , k p , k g , k ε denotes Matérn of t = 3 2 , periodical Matérn, gabor and noise kernel, respectively. The information that they try to capture: kenel target k m depicting the roughness of weather process periodical information of vortex in typhoon region or wind belt k p extracting the vortex texture, the edge of wind belt k g k ε noise

Multivariate Multi-kernel Limitation • 1D-GPR equals the piecewise spline • The partial information of spacial distribution Tips: importing more information • space information(longitude and latitude) – key feature • the principle component of wind direction, pressure and temperature – secondary feature

Multivariate Multi-kernel ′ k vms ( · , · ) = k v ( · , · ) + k v ( · , · ) (12) ′ k vmp ( · , · ) k v ( · , · ) ∗ k v ( · , · ) = where k vms denotes the correction kernel for the weather in normal condition, k vmp denotes the correction kernel for the weather in extreme condition. k v is trained by key feature, k ′ v is trained by secondary fea- ture. ♣ Should I Explain it?

Wind field interpolation – Normal weather condition 0.08 0.07 0.08 0.08 ku kv ku kv (a) (b) (a) (b) 0.06 su sv su sv 0.06 0.06 0.06 0.05 0.04 RMSE RMSE RMSE RMSE 0.04 0.04 0.04 0.03 0.02 0.02 0.02 0.02 0.01 0 0 0 0 Region Index Region Index Region Index Region Index space series space series 0.12 0.14 0.12 0.1 ku kv ku kv (a) (b) (a) (b) su 0.12 sv 0.1 su sv 0.1 0.08 0.1 0.08 0.08 RMSE RMSE RMSE RMSE 0.08 0.06 0.06 0.06 0.06 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0 0.02 Region Index Region Index Region Index Region Index time series time series k v k vms

Wind field interpolation – Normal weather condition 25 30 k vms k vms (a) (b) 25 k v 20 RMSE Percentage RMSE Percentage k v k vms 20 15 15 k vms 10 k v 10 k v 5 5 0 0 u v u v space series time series ♣ Experiment?

Wind field interpolation – Extreme weather condition (a) (b) 24 ° N 24 ° N 22 ° N 22 ° N 20 ° N 20 ° N 18 ° N 18 ° N 16 ° N 16 ° N (a) reference field 145 ° E 150 ° E 145 ° E 150 ° E (c) (d) (b) k vmp 24 ° N 24 ° N (c) spline 22 ° N 22 ° N 20 ° N 20 ° N (d) BP 18 ° N 18 ° N 16 ° N 16 ° N 145 ° E 150 ° E 145 ° E 150 ° E Spd m/s 0 5 10 15

Wind field interpolation – Extreme weather condition (a) (b) (c) 24 ° N 24 ° N 24 ° N 22 ° N 22 ° N 22 ° N 20 ° N 20 ° N 20 ° N 18 ° N 18 ° N 18 ° N 16 ° N 16 ° N 16 ° N 145 ° E 150 ° E 145 ° E 150 ° E 145 ° E 150 ° E Speed error Speed error Speed error 0 0.04 -0.1 0 0.1 -8 -4 0

Wind field interpolation – Extreme weather condition

GPR and 3DVAR loss function 2 y T K − 1 y − 1 − 1 2 log | K | − n 2 log 2 π − 1 2 ( x a − x b ) T B − 1 ( x a − x b ) − 1 2 ( y − H ( x a )) T R − 1 ( y − H ( x a )) Dose the matrix B describes the similarity?

GPR and 3DVAR assumption y ∼ GP prior : x ∼ N ( x b , B ) x a follows a single Gaussian distribution, Whether or not? What if we suppose x a comes from a Gaussian mixture dis- tribution?

kernel functions step (1) Mining: the backgroud of your dataset (2) Draw: visualizing your data (3) Choice: appropriate kernel – stationary? periodical? linear? smooth? (4) Try

Autoregression model – may be nonsense AR,MA,ARMA etc. Taking the series itself as the only explaining variable, the idea of them is extremely simple. They are popular in sta- tionary series analysis.

Some about M.L. Any model can be powerful, even the simplest one, as long as you make a good decision, that is, choose a model that fits your data. Neither M.L. nor D.L. are the magician, you are, you are the one who teach them how to do and what to do.