Commit fd08b92e authored by Jana Ulrich's avatar Jana Ulrich
Browse files

(try to) fix equations in README

parent 87b349f6
---
output: github_document
output:
github_document:
pandoc_args: "--webtex"
---
<!-- README.md is generated from README.Rmd. Please edit that file -->
......@@ -114,11 +116,13 @@ $$
$$
The function `gev.d.fit` provides the options:
* `theta_zero = TRUE` $\theta = 0$
* `eta2_zero = TRUE` $\eta_2 = \eta$
* `tau_zero = TRUE` $\tau = 0$
resulting in the following features for IDF-curves:
* simple scaling: using only parameters $\tilde{\mu}, \sigma_0, \xi, \eta$
* curvature for small durations: allowing $\theta \neq 0$ (default)
* multi-scaling: allowing $\eta_2 \neq \eta$
......@@ -164,8 +168,9 @@ for(i.fit in 1:length(all.fits)){
mtext('Duration [h]',1,2)
}
for(i.p in 1:length(idf.probs)){
IDF.plot(1/60*2^(seq(0,13,0.5)),gev.d.params(all.fits[[i.fit]]),probs = idf.probs[i.p],add = TRUE,legen=FALSE
,lty=i.p,cols = fit.cols[i.fit])
# plotting IDF curves for each model (different colors) and probability (different lty)
IDF.plot(1/60*2^(seq(0,13,0.5)),gev.d.params(all.fits[[i.fit]]),probs = idf.probs[i.p]
,add = TRUE,legend = FALSE,lty = i.p,cols = fit.cols[i.fit])
}
mtext(fit.labels[i.fit],3,-1.25)
}
......
......@@ -74,13 +74,13 @@ fit <- gev.d.fit(xdat = ann.max$xdat,ds = ann.max$ds,sigma0link = make.link('log
#> [1] 0
#>
#> $nllh
#> [1] 57.67781
#> [1] 60.23787
#>
#> $mle
#> [1] 7.253936e+00 2.905276e-01 9.574134e-02 1.067175e-10 8.025047e-01
#> [1] 5.682697e+00 5.816338e-01 -8.017104e-02 7.574509e-09 8.059212e-01
#>
#> $se
#> [1] 4.308783e-01 6.889894e-02 6.099439e-02 2.000065e-06 1.119207e-02
#> [1] 3.530129e-01 6.884231e-02 4.942244e-02 2.000060e-06 1.153130e-02
# checking the fit
gev.d.diag(fit,pch=1,)
```
......@@ -91,8 +91,8 @@ gev.d.diag(fit,pch=1,)
# parameter estimates
params <- gev.d.params(fit)
print(params)
#> mut sigma0 xi theta eta eta2 tau
#> 1 7.253936 1.337133 0.09574134 1.067175e-10 0.8025047 0.8025047 0
#> mut sigma0 xi theta eta eta2 tau
#> 1 5.682697 1.788959 -0.08017104 7.574509e-09 0.8059212 0.8059212 0
# plotting the probability density for a single duration
q.min <- floor(min(ann.max$xdat[ann.max$ds%in%1:2]))
......@@ -126,28 +126,57 @@ IDF.plot(durations,params,add=TRUE)
This Example depicts the different features that can be used to model
the IDF curves. Here we assume, that the block maxima of each duration
can be modeled with the GEV distribution (\(\xi\neq0\)):
\[G(z;\mu,\sigma,\xi)=\exp \left\lbrace -\left[
1+\xi \left( \frac{z-\mu}{\sigma} \right)
\right]^{-1/\xi} \right\rbrace,\]
can be modeled with the GEV distribution
(![\\xi\\neq0](https://latex.codecogs.com/png.latex?%5Cxi%5Cneq0
"\\xi\\neq0")):
![G(z;\\mu,\\sigma,\\xi)=\\exp \\left\\lbrace -\\left\[ &#10;1+\\xi
\\left( \\frac{z-\\mu}{\\sigma} \\right)&#10;\\right\]^{-1/\\xi}
\\right\\rbrace,](https://latex.codecogs.com/png.latex?G%28z%3B%5Cmu%2C%5Csigma%2C%5Cxi%29%3D%5Cexp%20%5Cleft%5Clbrace%20-%5Cleft%5B%20%0A1%2B%5Cxi%20%5Cleft%28%20%5Cfrac%7Bz-%5Cmu%7D%7B%5Csigma%7D%20%5Cright%29%0A%5Cright%5D%5E%7B-1%2F%5Cxi%7D%20%5Cright%5Crbrace%2C
"G(z;\\mu,\\sigma,\\xi)=\\exp \\left\\lbrace -\\left[
1+\\xi \\left( \\frac{z-\\mu}{\\sigma} \\right)
\\right]^{-1/\\xi} \\right\\rbrace,")
where the GEV parameters depend on duration according to:
\[
\sigma(d)=\sigma_0(d+\theta)^{-\eta_2}+\tau, \\
\mu(d) = \tilde{\mu}\cdot\sigma_0(d+\theta)^{-\eta}+\tau, \\
\xi(d) = \text{const.}
\]
The function `gev.d.fit` provides the options: \* `theta_zero = TRUE`
\(\theta = 0\) \* `eta2_zero = TRUE` \(\eta_2 = \eta\) \* `tau_zero =
TRUE` \(\tau = 0\)
resulting in the following features for IDF-curves: \* simple scaling:
using only parameters \(\tilde{\mu}, \sigma_0, \xi, \eta\) \* curvature
for small durations: allowing \(\theta \neq 0\) (default) \*
multi-scaling: allowing \(\eta_2 \neq \eta\) \* flattening for long
durations: allowing \(\tau \neq 0\).
![&#10;\\sigma(d)=\\sigma\_0(d+\\theta)^{-\\eta\_2}+\\tau,
\\\\&#10;\\mu(d) =
\\tilde{\\mu}\\cdot\\sigma\_0(d+\\theta)^{-\\eta}+\\tau,
\\\\&#10;\\xi(d) = \\text{const.}
&#10;](https://latex.codecogs.com/png.latex?%0A%5Csigma%28d%29%3D%5Csigma_0%28d%2B%5Ctheta%29%5E%7B-%5Ceta_2%7D%2B%5Ctau%2C%20%5C%5C%0A%5Cmu%28d%29%20%3D%20%5Ctilde%7B%5Cmu%7D%5Ccdot%5Csigma_0%28d%2B%5Ctheta%29%5E%7B-%5Ceta%7D%2B%5Ctau%2C%20%20%5C%5C%0A%5Cxi%28d%29%20%3D%20%5Ctext%7Bconst.%7D%20%0A
"
\\sigma(d)=\\sigma_0(d+\\theta)^{-\\eta_2}+\\tau, \\\\
\\mu(d) = \\tilde{\\mu}\\cdot\\sigma_0(d+\\theta)^{-\\eta}+\\tau, \\\\
\\xi(d) = \\text{const.}
")
The function `gev.d.fit` provides the options:
- `theta_zero = TRUE` ![\\theta
= 0](https://latex.codecogs.com/png.latex?%5Ctheta%20%3D%200
"\\theta = 0")
- `eta2_zero = TRUE` ![\\eta\_2 =
\\eta](https://latex.codecogs.com/png.latex?%5Ceta_2%20%3D%20%5Ceta
"\\eta_2 = \\eta")
- `tau_zero = TRUE` ![\\tau
= 0](https://latex.codecogs.com/png.latex?%5Ctau%20%3D%200
"\\tau = 0")
resulting in the following features for IDF-curves:
- simple scaling: using only parameters ![\\tilde{\\mu}, \\sigma\_0,
\\xi,
\\eta](https://latex.codecogs.com/png.latex?%5Ctilde%7B%5Cmu%7D%2C%20%5Csigma_0%2C%20%5Cxi%2C%20%5Ceta
"\\tilde{\\mu}, \\sigma_0, \\xi, \\eta")
- curvature for small durations: allowing ![\\theta
\\neq 0](https://latex.codecogs.com/png.latex?%5Ctheta%20%5Cneq%200
"\\theta \\neq 0") (default)
- multi-scaling: allowing ![\\eta\_2 \\neq
\\eta](https://latex.codecogs.com/png.latex?%5Ceta_2%20%5Cneq%20%5Ceta
"\\eta_2 \\neq \\eta")
- flattening for long durations: allowing ![\\tau
\\neq 0](https://latex.codecogs.com/png.latex?%5Ctau%20%5Cneq%200
"\\tau \\neq 0").
Example:
......@@ -190,8 +219,9 @@ for(i.fit in 1:length(all.fits)){
mtext('Duration [h]',1,2)
}
for(i.p in 1:length(idf.probs)){
IDF.plot(1/60*2^(seq(0,13,0.5)),gev.d.params(all.fits[[i.fit]]),probs = idf.probs[i.p],add = TRUE,legen=FALSE
,lty=i.p,cols = fit.cols[i.fit])
# plotting IDF curves for each model (different colors) and probability (different lty)
IDF.plot(1/60*2^(seq(0,13,0.5)),gev.d.params(all.fits[[i.fit]]),probs = idf.probs[i.p]
,add = TRUE,legend = FALSE,lty = i.p,cols = fit.cols[i.fit])
}
mtext(fit.labels[i.fit],3,-1.25)
}
......
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