analog forecasting errors from a dynamical systems point ... · imt-a & riken feb. 10, 2020 paul...

Analog forecasting errors from a dynamical systems point of view

Paul Platzer (feb. 10, 1993)IMT-A & RIKEN workshop :

Statistical Modeling and Machine Learningin Meteorology and Oceanography, feb. 10, 2020

IMT-A & RIKENFeb. 10, 2020

Paul Platzer2/20

Analog forecasting errors from a dynamical systems point of view

Paul Platzer (feb. 10, 1993)IMT-A & RIKEN workshop :

Statistical Modeling and Machine Learningin Meteorology and Oceanography, feb. 10, 2020

Marc Schoenauer,Naonori Ueda,Said Ouala,Maha Mdini…→ bridge the gap betweendata-driven andprocess-driven→ « 5th science »

Pierre Tandeo,Yicun ZhenAnDA


Paul Platzer3/20

Two analog maps correspond to close points in phase space(Courtesy of D. Farandaand P. Yiou)

What is an analog ?

→ Introduced by Lorenzin 1969 for atmospheric predictability


Paul Platzer4/20

● ak0 = analogs of x0 : « close » to x0

● akt successors ( time t ) → estimate xt

Analog forecasting : the idea

(Lguensat et al. 2017)

0 t

a10

aK-10aK0

a1t

aK-1taKt


Paul Platzer5/20

Analog forecasting : cheap ML


Paul Platzer6/20

Analog forecasting : in practice● Large database called « catalog » (reanalysis, model output) ● Select K nearest neighbors of x0 (fast thanks to ML libraries)● Assign weights to the analogs (discard bad analogs)● Apply a given analog forecasting operator

• • •


Paul Platzer7/20

Analog forecasting operators

0

0

0

0 0

0 t

t

t

t

t

t

x0a10

a20

x0a10

a20

x0 a10

a20

x0

a20

a10

x0

a20

a10

x0

a20

a10

µLCa1t

a2t

µLIa1t

a2tµLLa1t

a2t

µLCa2t

a1t

µLI

a2t

a1t

µLL

a2t

a1t

(figure from Lguensat et al. 2017)


Paul Platzer8/20

Dynamical systems● Hypotheses :

● Distance between successor and future state :


Paul Platzer9/20

Dynamical systems● Here → looking at local error

→ similar to Nicolis et al. (2009)● Zhao and Giannakis (2016)

→ looking at global dynamical properties


Paul Platzer10/20

Dynamical systems : mean error● Hypotheses :

● Mean error of analog forecasting operators :


Paul Platzer11/20


● Mean error of analog forecasting :

→ experiments on L96 with δ=0

(figure from Lguensat et al. 2017)


Paul Platzer12/20


● Mean error of analog forecasting :→experiments on L63 with δ=0

– LC empirical error-- LC error estimation– LI empirical error-- LI error estimation

(figure from Platzer et al. 2019)


Paul Platzer13/20


● Mean error of analog forecasting :→experiments on L63 with δ=0

– LC -- LI-- LI+mean correction

-·-· LI+local correction(figure from Platzer et al. 2019)


Paul Platzer14/20

Relation between operators● Locally-linear :

residuals

→

● Locally-constant, locally-incremental :


Paul Platzer15/20

The locally-linear and the Jacobian


Paul Platzer16/20

The locally-linear and the Jacobian

Mean error of LL forecast

Quality of Jacobian estimation using LL


Paul Platzer17/20

What about the covariance ?


Paul Platzer18/20

What about the variance ?

LC (Lguensat estimation)LC (empirical from several catalogs)LI (Lguensait estimation)LI (empirical from several catalogs)

→ Experiments on L63


Paul Platzer19/20

What about the variance ?(logscale)

occu

ren c

es


Paul Platzer20/20

Conclusion● Analog forecasting = empirical, simple ML● Using dynamical systems helps

→ interpretation→ error estimation→ data-based + model-based hybrid method ?→ finer tuning ?

● Combine analogs and NN ? (analogs=first guess)


Paul Platzer21/20

Thank you !Bibliography :• Lguensat, R., Tandeo, P., Ailliot, P., Pulido, M., & Fablet, R. (2017). The analog data assimilation. Monthly Weather Review, 145(10), 4093-4107.• Lorenz, E. N. (1969). Atmospheric predictability as revealed by naturally occurring analogues. Journal of the Atmospheric sciences, 26(4), 636-646.• Nicolis, C., Perdigao, R. A., & Vannitsem, S. (2009). Dynamics of prediction errors under the combined effect of initial condition and model errors. Journal of the atmospheric sciences, 66(3), 766-778.• Platzer, P., Yiou, P., Tandeo, P., Naveau, P., & Filipot, J. F. (2019, October). Predicting Analog Forecasting Errors using Dynamical Systems.• Tippett, M. K., & DelSole, T. (2013). Constructed analogs and linear regression. Monthly Weather Review, 141(7), 2519-2525.• Zhao, Z., & Giannakis, D. (2016). Analog forecasting with dynamics-adapted kernels. Nonlinearity, 29(9), 2888.

Diapo 1Diapo 2Diapo 3Diapo 4Diapo 5Diapo 6Diapo 7Diapo 8Diapo 9Diapo 10Diapo 11Diapo 12Diapo 13Diapo 14Diapo 15Diapo 16Diapo 17Diapo 18Diapo 19Diapo 20Diapo 21

analog forecasting errors from a dynamical systems point ... · imt-a & riken feb. 10, 2020 paul...

Documents