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Outline
- Introduction
- EOFVAR-X model
- Real experiments
- Conclusions and discussions
Statistical downscaling by EOFVAR-X models Jiang, Ci-Ren (Institute - - PowerPoint PPT Presentation
Statistical downscaling by EOFVAR-X models Jiang, Ci-Ren (Institute - - PowerPoint PPT Presentation
Statistical downscaling by EOFVAR-X models Jiang, Ci-Ren (Institute of Statistical Science, Academia Sinica) Chen, Lu-Hung* (Institute of Statistics, NCHU) July 17, 2017 Outline Introduction EOFVAR-X model Real experiments
SLIDE 2
SLIDE 3
Introduction
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Statistical downscaling
- Station-wise multiple linear model (and its variants)
- Bayesian spatial(-temporal) models
- Neural-networks
- Support vector regression models
- etc.
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Station-wise multiple linear model
Denote Yt = [Yt(s1),...,Yt(sm)]′ by the observations from m stations and Xt = [Xt(˜ s1),...,Xt(˜ sn)]′ by the model outputs on n grid points, a station-wise multiple linear model considers Yt(si) = X′
tβ
β β i + εt
- Independence assumptions between:
- 1. difgerent stations Yt(si) and Yt(sj)
- 2. difgerent observational time points Yt(si) and Yt′(si)
- But shouldn’t a meteorological variable be spatially and
temporally correlated?
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Principal regression by Benestad et al. (2015)
- Apply principal components analysis to both observations Yt
and model outputs Xt. That is, let Yt =
K
∑
k=1
γt,kφk and Xt =
L
∑
ℓ=1
χt,ℓψℓ. (1)
- The spatial relationships of Yt and Xt are retained in the
eigenvectors φk’s and ψℓ’s, respectively.
- By the orthonormal property of φk’s, the pc scores γt,k’s are
now uncorrelated with each other, and it is safe to consider γt,k = bk,0 +
L
∑
ℓ=1
bk,ℓχt,k + et (2) for each k = 1,2,...,K.
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Principal regression by Benestad et al. (2015)
- Denote ˆ
bk,0, ˆ bk,1,..., ˆ bk,L by the estimation of bk,0,bk,1,...,bk,L, the downscaling of Xt can be accomplished by ˆ Yt =
K
∑
k=1
ˆ γt,kφk with ˆ γt,k = ˆ bk,0 + ∑L
ℓ=1 ˆ
bk,ℓχt,k.
- Empirical studies in Benestad et al. (2015) suggest that PCR
generally performs better and is more robust to station-wise MLR with K = 4 and L = 4.
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EOFVAR-X model
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Hypothesis
- Are temporal relationships also useful for statistical
downscaling?
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EOFVAR-X model
- The pc scores γ
γ γt = [γt,1,γt,2,...,γt,K]′ and χ χ χt = [χt,1,χt,2,...,χt,L]′ are temporally correlated due to the natures of Yt and Xt.
- Thus, they can be treated as vector of time series.
- For simplicity, we adapt the easiest model for analyzing vector
- f time series, vector autoregressive model:
γt,k = αk,0 +
K
∑
κ=1
αk,κ,1γt−1,κ + ···+
K
∑
κ=1
αk,κ,pγt−p,κ +
L
∑
ℓ=1
ρk,ℓ,0χt,ℓ +
L
∑
ℓ=1
ρk,ℓ,1χt−1,ℓ + ···+
L
∑
ℓ=1
ρk,ℓ,qχt−q,ℓ + ut,k (3)
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EOFVAR-X model
- The lags p and q in equation (3) are tuning parameters and
can be selected by model selection, e.g. cross-validation, AIC, BIC, etc.
- Equation (2) by Benestad et al. (2015) is a special case of
equation (3) with αk,κ,1 = αk,κ,2 = ··· = αk,κ,p = 0 and ρk,ℓ,1 = ρk,ℓ,2 = ··· = ρk,ℓ,q = 0 for all k, κ, and ℓ.
- Thus our hypothesis can be verifjed by hypothesis testing or
model selection.
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Implementation details
- 1. Impute possible missing data in Yt by DINEOF (Beckers and
Rixen, 2003, Data Interpolating Empirical Orthogonal Functions) (provided by the R package sinkr).
- 2. Apply PCA to both Xt and Yt as described in equation (1).
- 3. Estimate the coeffjcients in equation (3) (by the R package
MTS)
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Real experiments
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Data sets
Daily mean temperatures of the following two countries are used:
- 1. German
- 254 stations from ECAD, 1979-2016.
- NCEP-DOE Reanalysis 2 (lon 5◦ −16◦, lat 47◦ −56◦)
- Rolling cross-validation from 2001/1/1.
- 2. Taiwan
- 7 stations from ASOS, 2000-2016.
- NCEP-DOE Reanalysis 2 (lon 118◦ −123◦, lat 21◦ −26◦)
- Rolling cross-validation from 2011/1/1.
As suggested by Benestad et al. (2015), we set K = L = 4 in both experiments.
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Rolling cross-validation
Figure 1: Figure from https://robjhyndman.com/hyndsight/tscv/
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Results
German Taiwan RMSE Correlation RMSE Correlation PCR 1.39±0.53 0.83±0.11 1.44±0.75 0.80±0.37 EOFVAR-X 1.14±0.49 0.88±0.08 1.00±0.57 0.83±0.33
Table 1: Cross-validated RMSE and Pearson correlation coeffjcients (mean±sd)
The source codes and notebooks for the experiments are available at the project’s github page1.
1https://github.com/chenlu-hung/eofvarx
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Conclusions and discussions
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Conclusions and discussions
- 1. We developed an EOFVAR-X model to incorporate both the
spatial and temporal relationships in a meteorological variable.
- 2. Our experimental results on two difgerent countries suggest
that spatial and temporal relationships are both useful for statistical downscaling.
- 3. Some future directions:
- More sophisticated time series models can be considered, e.g.,
ARIMA, GARCH, etc.
- Simultaneous downscaling of multiple meteorological variables.
- Nonlinear models (e.g. deep learning).
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Thank you!
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