From 79df2aa7d7f8d95762d93c083a3a4d61aff329a8 Mon Sep 17 00:00:00 2001 From: German Aragon Date: Tue, 4 Jun 2024 08:36:17 +0200 Subject: [PATCH] Update flooding.md fix mathematical expressions --- wikiIA/sheets/flooding.md | 16 ++++------------ 1 file changed, 4 insertions(+), 12 deletions(-) diff --git a/wikiIA/sheets/flooding.md b/wikiIA/sheets/flooding.md index 08b0994..4e0e327 100644 --- a/wikiIA/sheets/flooding.md +++ b/wikiIA/sheets/flooding.md @@ -29,9 +29,7 @@ However, in spite of the computational advantages of reduced-complexity process ## Problem statement In order to overcome these limitations, machine learning is a great tool. However, the predictability problem implies that a spatial output (millions of points) needs to be predicted based on a reduced set of parameters describing the coastal forcings of the storm (wave height, period, direction and still water level). -$$ -Y = F(X) \text{, where } Y \text{ is the spatial map and } X \text{ represents Hs, Tp, Dir, SWL.} -$$ +$Y = F(X) \text{, where } Y \text{ is the spatial map and } X \text{ represents Hs, Tp, Dir, SWL.}$
@@ -82,9 +80,7 @@ The numerical simulation of the 100 cases yields a training dataset consisting o ### 4. Training the statistical model The statistical model consists of projecting the training predictand dataset (N=45000xM=100) into a reduced subset using principal component analysis (PCA). -$$ -Y_{N,M}=U_{N,N} \Delta_{N,M} V_{M,M} -$$ +$Y_{N,M}=U_{N,N} \Delta_{N,M} V_{M,M}$
@@ -96,15 +92,11 @@ $$ Then, a given flood map (Y) can be solved as a linear summation of fixed EOFs (U) multiplied by storm-dependent latent variables (PCs) -$$ -Y_{j}=\sum_{i=1}^N \alpha_{ji}U_j -$$ +$Y_{j}=\sum_{i=1}^N \alpha_{ji}U_j$ Then, the spatial characteristics of the problem is represented by invariang EOFs and the inference problem is restricted to finding the storm-dependent latent variables that weights every EOF as a function of the storm characteristics which much more doable than the initial problem. The latent variables are inferred using Gaussian processes: -$$ -\alpha_{ji}=GP(\underbrace{X}_{Hs, Tp, Dir, SWL}) -$$ +$\alpha_{ji}=GP(\underbrace{X}_{Hs, Tp, Dir, SWL})$ A balanced solution in terms of accuracy is obtained when choosing the EOFs that represent 90% of the total variance: