Experiment 1a ¶ For this process, we will exclusively use the GEVD for the data likelihood.
Under these assumptions we will try a few different models for fitting the GEVD method.
We will try the following:
Null Model (M 0 M_0 M 0 IID . M 1 M_1 M 1 GEVD methodHypothesis .
The log-likelihood loss function is very similar for both approaches.
However, the PP formulation has an extra “regularization” term which may improve or worsen the parameter estimation.
Analysis .
We do a number of analysis to assess the efficacy of the methods.
Some standard methods include:
Table 1: Table of methods
Experiment Model Name 1a M 0 M_0 M 0 GEVD 1a M 1 M_1 M 1 PP -GEVD 1b M 0 M_0 M 0 GPD 1b M 1 M_1 M 1 PP -GPD 1b M 1 M_1 M 1 PP -GPD -GEVD
Null Model - Baseline GEVD ¶ Summary Formulation .
The null model, M 0 M_0 M 0 IID
D = { y n } n = 1 N \mathcal{D} = \left\{ y_n \right\}_{n=1}^N D = { y n } n = 1 N We can write the joint distribution as
p ( y 1 : N , θ ) = p ( θ ) ∏ n = 1 N p ( y n ∣ θ ) p(y_{1:N},\boldsymbol{\theta}) = p(\boldsymbol{\theta})\prod_{n=1}^N p(y_n|\boldsymbol{\theta}) p ( y 1 : N , θ ) = p ( θ ) n = 1 ∏ N p ( y n ∣ θ ) where p ( θ ) p(\boldsymbol{\theta}) p ( θ ) p ( y n ∣ θ ) p(y_n|\boldsymbol{\theta}) p ( y n ∣ θ ) GEVD .
Data Likelihood : p ( y n ∣ θ ) = f GEVD ( y n ; θ ) \begin{aligned}
\text{Data Likelihood}: && &&
p(y_n|\boldsymbol{\theta}) &= \boldsymbol{f}_{\text{GEVD}}(y_n;\boldsymbol{\theta})
\end{aligned} Data Likelihood : p ( y n ∣ θ ) = f GEVD ( y n ; θ ) So, to find the best parameters, we can maximize the log-likelihood of the joint distribution which is given as
log p ( y 1 : N , θ ) = ∑ n = 1 N log f GEVD ( y n ; θ ) + log p ( θ ) \log p(y_{1:N},\theta) =
\sum_{n=1}^N\log \boldsymbol{f}_{\text{GEVD}}(y_n;\boldsymbol{\theta})
+ \log p(\boldsymbol{\theta}) log p ( y 1 : N , θ ) = n = 1 ∑ N log f GEVD ( y n ; θ ) + log p ( θ ) We can plug in the equations for the GEVD to obtain the full expressions
L GEVD ( θ ) : = − N log σ − ( 1 + 1 / κ ) ∑ n = 1 N log [ 1 + κ z n ] + − ∑ n = 1 N [ 1 + κ z n ] + − 1 / κ \boldsymbol{L}_\text{GEVD}(\boldsymbol{\theta})
:=
- N \log \sigma -
(1+1/\kappa)\sum_{n=1}^N
\log \left[ 1 + \kappa z_n\right]_+
-
\sum_{n=1}^N
\left[ 1 + \kappa z_n\right]_+^{-1/\kappa} L GEVD ( θ ) := − N log σ − ( 1 + 1/ κ ) n = 1 ∑ N log [ 1 + κ z n ] + − n = 1 ∑ N [ 1 + κ z n ] + − 1/ κ Results :
Model 1 - PP GEVD ¶ Summary Formulation . M 1 M_1 M 1 PP formulation to calculate the parameters of the GEVD .
We have a 2D space
A = { [ 0 , T ] × [ y , ∞ ) } , t ∈ T ⊆ R + y ∈ Y ⊆ R \begin{aligned}
A &= \left\{[0,T]\times[y,\infty) \right\},
&& &&
t\in\mathcal{T}\subseteq\mathbb{R}^+
&&
y\in\mathcal{Y}\subseteq\mathbb{R}
\end{aligned} A = { [ 0 , T ] × [ y , ∞ ) } , t ∈ T ⊆ R + y ∈ Y ⊆ R The joint hazard function is factored into a temporal component and a marks component
λ ( y , t ) = λ ( t ) λ ( y ) = f GEVD ( y ; θ ) \lambda(y,t) = \lambda(t)\lambda(y) = \boldsymbol{f}_{\text{GEVD}}(y;\boldsymbol{\theta}) λ ( y , t ) = λ ( t ) λ ( y ) = f GEVD ( y ; θ ) Note, we assume the ground intensity term equal 1, λ g ( t ) = 1 \lambda_g(t)=1 λ g ( t ) = 1 SPP ); a special case of the Homogeneous Poisson process (HPP ).
The cumulative hazard function can be written as
Λ ( A ) = ∫ 0 T ∫ y ∞ f GEVD ( y ; θ ) d y d τ = ∫ 0 T S GEVD ( y ; θ ) d τ \Lambda(A) = \int_0^T\int_y^\infty \boldsymbol{f}_{\text{GEVD}}(y;\boldsymbol{\theta}) dyd\tau
= \int_0^T\boldsymbol{S}_{\text{GEVD}}(y;\boldsymbol{\theta})d\tau Λ ( A ) = ∫ 0 T ∫ y ∞ f GEVD ( y ; θ ) d y d τ = ∫ 0 T S GEVD ( y ; θ ) d τ We used the relationship of the SF , the CDF , and the PDF .
We approximate this integral using a simple Riemann sum
∫ 0 T S GEVD ( y 0 ; θ ) ≈ ∑ n = 1 N T S GEVD ( y 0 ; θ ) = N T S GEVD ( y 0 ; θ ) \int_0^T\boldsymbol{S}_\text{GEVD}(y_0;\boldsymbol{\theta})
\approx
\sum_{n=1}^{N_T}
\boldsymbol{S}_\text{GEVD}(y_0;\boldsymbol{\theta})
=
N_T\boldsymbol{S}_\text{GEVD}(y_0;\boldsymbol{\theta}) ∫ 0 T S GEVD ( y 0 ; θ ) ≈ n = 1 ∑ N T S GEVD ( y 0 ; θ ) = N T S GEVD ( y 0 ; θ ) where N T N_T N T
log p ( y 1 : N , θ ) = ∑ n = 1 N T log f GEVD ( y n ∣ θ ) + log p ( θ ) − N T S GEVD ( y 0 ; θ ) \log p(y_{1:N},\theta) =
\sum_{n=1}^{N_T}
\log \boldsymbol{f}_{\text{GEVD}}(y_n|\boldsymbol{\theta})
+
\log p(\boldsymbol{\theta})
-
N_T\boldsymbol{S}_\text{GEVD}(y_0;\boldsymbol{\theta}) log p ( y 1 : N , θ ) = n = 1 ∑ N T log f GEVD ( y n ∣ θ ) + log p ( θ ) − N T S GEVD ( y 0 ; θ ) The first term is the log-likelihood of the specific events at the specific location, ( t n , y n ) (t_n,y_n) ( t n , y n ) ( 0 , T ] (0,T] ( 0 , T ] GEVD into this expression to obtain
L PP-GEVD ( θ ) : = L GEVD ( θ ) − N T [ 1 + κ z n ] + − 1 / κ \begin{aligned}
\boldsymbol{L}_\text{PP-GEVD}(\boldsymbol{\theta})
&:=
\boldsymbol{L}_\text{GEVD}(\boldsymbol{\theta})
-
N_T
\left[ 1 + \kappa z_n\right]_+^{-1/\kappa}
\end{aligned} L PP-GEVD ( θ ) := L GEVD ( θ ) − N T [ 1 + κ z n ] + − 1/ κ where L GEVD \boldsymbol{L}_\text{GEVD} L GEVD GEVD PDF (equation (15)
GPD Connections¶ We can also estimate the parameters of the Poisson-GPD using some relationships between the GEVD and the GPD .
The parameters of the GPD can be calculated by
λ = [ 1 + κ z 0 ] + − 1 / κ σ ∗ = σ + κ ( y 0 − μ ) κ ∗ = κ
\begin{aligned}
\lambda = [1 + \kappa z_0]_+^{-1/\kappa}
&& &&
\sigma^* &= \sigma + \kappa(y_0 - \mu)
&& &&
\kappa^* = \kappa
\end{aligned} λ = [ 1 + κ z 0 ] + − 1/ κ σ ∗ = σ + κ ( y 0 − μ ) κ ∗ = κ where z 0 = ( y 0 − μ ) / σ z_0 = (y_0 - \mu)/\sigma z 0 = ( y 0 − μ ) / σ
μ ∗ = μ + σ κ ( 1 − λ κ ) σ ∗ = σ λ κ κ ∗ = κ
\begin{aligned}
\mu^* = \mu + \frac{\sigma}{\kappa}(1 - \lambda^\kappa)
&& &&
\sigma^* = \sigma\lambda^{\kappa}
&& &&
\kappa^* = \kappa
\end{aligned} μ ∗ = μ + κ σ ( 1 − λ κ ) σ ∗ = σ λ κ κ ∗ = κ Note, in the experiment for the GEVD , we have 1 event per year, i.e., λ = 1 \lambda=1 λ = 1
μ ∗ = μ σ ∗ = σ κ ∗ = κ
\begin{aligned}
\mu^* = \mu
&& &&
\sigma^* = \sigma
&& &&
\kappa^* = \kappa
\end{aligned} μ ∗ = μ σ ∗ = σ κ ∗ = κ Return Period / Average Recurrence Interval ¶ We will also calculate the average recurrence interval (ARI ) and the return period (RP ) to calculate the 100-year event.
y = Q GEVD ( y p ; θ ) y p = 1 − 1 / T a y p = exp ( − 1 / T p ) \begin{aligned}
y = \boldsymbol{Q}_\text{GEVD}(y_p;\boldsymbol{\theta})
&& &&
y_p = 1 - 1/T_a
&& &&
y_p = \exp(- 1 / T_p)
\end{aligned} y = Q GEVD ( y p ; θ ) y p = 1 − 1/ T a y p = exp ( − 1/ T p ) where Q GEVD \boldsymbol{Q}_\text{GEVD} Q GEVD T a T_a T a T p T_p T p
Model 2 ¶ For this process, we will exclusively use the GPD model for the data likelihood.
Under these assumptions, we will try fitting different methods for the GPD method.
We will try the following:
M 0 M_0 M 0 IID . This will server as the null model for this experiment.M 1 M_1 M 1 M 2 M_2 M 2 Model 2a - GPD ¶ The null model, M 2 a M_{2_a} M 2 a IID
D = { y n } n = 1 N \mathcal{D} = \left\{ y_n \right\}_{n=1}^N D = { y n } n = 1 N We can write the joint distribution as
p ( y 1 : N , θ ) = p ( θ ) ∏ n = 1 N p ( y n ∣ θ ) p(y_{1:N},\boldsymbol{\theta}) = p(\boldsymbol{\theta})\prod_{n=1}^N p(y_n|\boldsymbol{\theta}) p ( y 1 : N , θ ) = p ( θ ) n = 1 ∏ N p ( y n ∣ θ ) where p ( θ ) p(\boldsymbol{\theta}) p ( θ ) p ( y n ∣ θ ) p(y_n|\boldsymbol{\theta}) p ( y n ∣ θ ) GPD .
Data Likelihood : p ( y n ∣ θ ) = f GPD ( y n ; θ ) \begin{aligned}
\text{Data Likelihood}: && &&
p(y_n|\boldsymbol{\theta}) &= \boldsymbol{f}_{\text{GPD}}(y_n;\boldsymbol{\theta})
\end{aligned} Data Likelihood : p ( y n ∣ θ ) = f GPD ( y n ; θ ) So, to find the best parameters, we can maximize the log-likelihood of the joint distribution which is given as
log p ( y 1 : N ∣ θ ) = ∑ n = 1 N log f GPD ( y n ; θ ) \log p(y_{1:N}|\theta) =
\sum_{n=1}^N\log \boldsymbol{f}_{\text{GPD}}(y_n;\boldsymbol{\theta}) log p ( y 1 : N ∣ θ ) = n = 1 ∑ N log f GPD ( y n ; θ ) We can plug in the equations for the GEVD to obtain the full expressions
L GPD ( θ ) : = − N log σ − ( 1 + 1 / κ ) ∑ n = 1 N log [ 1 + κ z n ] + \boldsymbol{L}_\text{GPD}(\boldsymbol{\theta})
:=
- N \log \sigma -
(1+1/\kappa)\sum_{n=1}^N
\log \left[ 1 + \kappa z_n\right]_+ L GPD ( θ ) := − N log σ − ( 1 + 1/ κ ) n = 1 ∑ N log [ 1 + κ z n ] + Model 2b - PP -GPD ¶ Summary Formulation .
M 2 b M_{2_b} M 2 b PP formulation to calculate the parameters of the GEVD .
We have a 2D space as shown in equation (6)
λ ( y , t ) = λ ( t ) λ ( y ) = f GPD ( y ; θ ) \lambda(y,t) = \lambda(t)\lambda(y) = \boldsymbol{f}_{\text{GPD}}(y;\boldsymbol{\theta}) λ ( y , t ) = λ ( t ) λ ( y ) = f GPD ( y ; θ ) Note, we assume the ground intensity term equal 1, λ g ( t ) = 1 \lambda_g(t)=1 λ g ( t ) = 1 SPP ); a special case of the Homogeneous Poisson process (HPP ).
The cumulative hazard function can be written as
Λ ( A ) = ∫ 0 T ∫ y ∞ f GPD ( y ; θ ) d y d τ = ∫ 0 T S GPD ( y ; θ ) d τ \Lambda(A) = \int_0^T\int_y^\infty \boldsymbol{f}_{\text{GPD}}(y;\boldsymbol{\theta}) dyd\tau
= \int_0^T\boldsymbol{S}_{\text{GPD}}(y;\boldsymbol{\theta})d\tau Λ ( A ) = ∫ 0 T ∫ y ∞ f GPD ( y ; θ ) d y d τ = ∫ 0 T S GPD ( y ; θ ) d τ We used the relationship of the SF , the CDF , and the PDF .
We approximate this integral using a simple Riemann sum
∫ 0 T S GPD ( y 0 ; θ ) ≈ ∑ n = 1 N T S GPD ( y 0 ; θ ) = N T S GPD ( y 0 ; θ ) \int_0^T\boldsymbol{S}_\text{GPD}(y_0;\boldsymbol{\theta})
\approx
\sum_{n=1}^{N_T}
\boldsymbol{S}_\text{GPD}(y_0;\boldsymbol{\theta})
=
N_T\boldsymbol{S}_\text{GPD}(y_0;\boldsymbol{\theta}) ∫ 0 T S GPD ( y 0 ; θ ) ≈ n = 1 ∑ N T S GPD ( y 0 ; θ ) = N T S GPD ( y 0 ; θ ) where N T N_T N T
log p ( y 1 : N , θ ) = ∑ n = 1 N T log f GPD ( y n ; θ ) + log p ( θ ) − N T S GPD ( y 0 ; θ ) \log p(y_{1:N},\theta) =
\sum_{n=1}^{N_T}\log \boldsymbol{f}_{\text{GPD}}(y_n;\boldsymbol{\theta})
+ \log p(\boldsymbol{\theta})
-
N_T\boldsymbol{S}_\text{GPD}(y_0;\boldsymbol{\theta}) log p ( y 1 : N , θ ) = n = 1 ∑ N T log f GPD ( y n ; θ ) + log p ( θ ) − N T S GPD ( y 0 ; θ ) The first term is the log-likelihood of the specific events at the specific location, ( t n , y n ) (t_n,y_n) ( t n , y n ) ( 0 , T ] (0,T] ( 0 , T ] GEVD into this expression to obtain
L PP-GPD ( θ ) : = L GPD ( θ ) − N T [ 1 + κ z 0 ] − 1 κ \begin{aligned}
\boldsymbol{L}_\text{PP-GPD}(\boldsymbol{\theta})
&:=
\boldsymbol{L}_\text{GPD}(\boldsymbol{\theta})
-
N_T
\left[ 1 + \kappa z_0\right]^{-\frac{1}{\kappa}}
\end{aligned} L PP-GPD ( θ ) := L GPD ( θ ) − N T [ 1 + κ z 0 ] − κ 1 where z 0 = ( y 0 − μ ) / σ z_0 = (y_0 - \mu)/\sigma z 0 = ( y 0 − μ ) / σ L GPD \boldsymbol{L}_\text{GPD} L GPD GPD PDF (equation (21)
Model 2c - PP -GPD -GEVD ¶ M 2 c M_{2_c} M 2 c GEVD and the GPD .
The parameters of the GPD can be calculated by
λ = [ 1 + κ z 0 ] + − 1 / κ σ ∗ = σ + κ ( y 0 − μ ) κ ∗ = κ
\begin{aligned}
\lambda = [1 + \kappa z_0]_+^{-1/\kappa}
&& &&
\sigma^* &= \sigma + \kappa(y_0 - \mu)
&& &&
\kappa^* = \kappa
\end{aligned} λ = [ 1 + κ z 0 ] + − 1/ κ σ ∗ = σ + κ ( y 0 − μ ) κ ∗ = κ where z 0 = ( y 0 − μ ) / σ z_0 = (y_0 - \mu)/\sigma z 0 = ( y 0 − μ ) / σ
μ ∗ = μ + σ κ ( 1 − λ κ ) σ ∗ = σ λ κ κ ∗ = κ
\begin{aligned}
\mu^* = \mu + \frac{\sigma}{\kappa}(1 - \lambda^\kappa)
&& &&
\sigma^* = \sigma\lambda^{\kappa}
&& &&
\kappa^* = \kappa
\end{aligned} μ ∗ = μ + κ σ ( 1 − λ κ ) σ ∗ = σ λ κ κ ∗ = κ Note, in the experiment for the GEVD , we have 1 event per year, i.e., λ = 1 \lambda=1 λ = 1
μ ∗ = μ σ ∗ = σ κ ∗ = κ
\begin{aligned}
\mu^* = \mu
&& &&
\sigma^* = \sigma
&& &&
\kappa^* = \kappa
\end{aligned} μ ∗ = μ σ ∗ = σ κ ∗ = κ Model 3 ¶ For this process, we will exclusively use a combination of the GPD model for the data likelihood.
Under these assumptions, we will try fitting different methods for the GPD method.
We will try the following:
M 0 M_0 M 0 IID . This will server as the null model for this experiment.M 1 M_1 M 1 M 2 M_2 M 2 Model 3a ¶ We have the hazard function
λ ( t , y ) = λ g ( t ) p ( y ∣ t ) = λ f GPD ( y ; θ ) \lambda (t,y) =
\lambda_g(t)p(y|t) =
\lambda \hspace{1mm}\boldsymbol{f}_\text{GPD}(y;\boldsymbol{\theta}) λ ( t , y ) = λ g ( t ) p ( y ∣ t ) = λ f GPD ( y ; θ ) So, we can calculate the cumulative Hazard function just on the ground intensity
Λ ( T ) = ∫ 0 T λ d τ = λ T \Lambda(\mathcal{T}) = \int_0^T\lambda d\tau = \lambda T Λ ( T ) = ∫ 0 T λ d τ = λ T Now, we can plug this into our log-likelihood function.
log p ( A ) = N log λ − λ T + ∑ n = 1 N T log f GPD ( y n ; θ ) \log p(A) =
N\log\lambda
-
\lambda T
+
\sum_{n=1}^{N_T} \log \boldsymbol{f}_\text{GPD}(y_n;\boldsymbol{\theta}) log p ( A ) = N log λ − λ T + n = 1 ∑ N T log f GPD ( y n ; θ ) Model 3b ¶ We will use some reparameterization of the lambda term, λ, to be in terms of the
λ = [ 1 + κ z 0 ] − 1 / κ z 0 = ( y 0 − μ ) / σ σ ∗ = σ + κ ( y 0 − μ ) κ ∗ = κ \begin{aligned}
\lambda &= [1 + \kappa z_0]^{-1/\kappa}
&& &&
z_0 = (y_0 - \mu) / \sigma \\
\sigma^* &=
\sigma + \kappa (y_0 - \mu) \\
\kappa^* &=
\kappa
\end{aligned} λ σ ∗ κ ∗ = [ 1 + κ z 0 ] − 1/ κ = σ + κ ( y 0 − μ ) = κ z 0 = ( y 0 − μ ) / σ We can expand this likelihood function fully to be
log p ( A ) = − N κ log [ 1 + κ z 0 ] − N T y [ 1 + κ z 0 ] − 1 / κ + ∑ n = 1 N T log f GPD ( y n ; θ ) \log p(A) =
-\frac{N}{\kappa}\log[1 + \kappa z_0]
-
N_{T_y} [1 + \kappa z_0]^{-1/\kappa}
+
\sum_{n=1}^{N_T} \log \boldsymbol{f}_\text{GPD}(y_n;\boldsymbol{\theta}) log p ( A ) = − κ N log [ 1 + κ z 0 ] − N T y [ 1 + κ z 0 ] − 1/ κ + n = 1 ∑ N T log f GPD ( y n ; θ ) where N T y N_{T_y} N T y
Model 3c ¶ λ ( t , y ) = λ g ( t ) λ g ( y ) p ( y ∣ t ) \lambda (t,y) =
\lambda_g(t)\lambda_g(y)p(y|t) λ ( t , y ) = λ g ( t ) λ g ( y ) p ( y ∣ t ) So
Ground Intensity (Time) : λ g ( t ) = 1 Ground Intensity (Marks) : λ g ( y ) = f GEVD ( y 0 ; θ ) Conditional Marks : p ( y ∣ t ) = f GPD ( y ; θ ) \begin{aligned}
\text{Ground Intensity (Time)}: && &&
\lambda_g(t) &= 1 \\
\text{Ground Intensity (Marks)}: && &&
\lambda_g(y) &= \boldsymbol{f}_\text{GEVD}(y_0;\boldsymbol{\theta}) \\
\text{Conditional Marks}: && &&
p(y|t) &= \boldsymbol{f}_\text{GPD}(y;\boldsymbol{\theta})
\end{aligned} Ground Intensity (Time) : Ground Intensity (Marks) : Conditional Marks : λ g ( t ) λ g ( y ) p ( y ∣ t ) = 1 = f GEVD ( y 0 ; θ ) = f GPD ( y ; θ ) We have already seen the intensity terms in the above section.
This is equivalent to Model M 1 b M_{1_b} M 1 b
L PP-GPD-GEVD ( θ ) : = L PP-GPD-GEVD ( θ ) + L GPD ( θ ) = ∑ n = 1 N T log f GEVD ( y n ∣ θ ) − N T S GEVD ( y 0 ; θ ) + ∑ n = 1 N T log f GPD ( y n ; θ ) \begin{aligned}
\boldsymbol{L}_\text{PP-GPD-GEVD}(\boldsymbol{\theta})
&:=
\boldsymbol{L}_\text{PP-GPD-GEVD}(\boldsymbol{\theta})
+
\boldsymbol{L}_\text{GPD}(\boldsymbol{\theta}) \\
&=
\sum_{n=1}^{N_T}
\log \boldsymbol{f}_{\text{GEVD}}(y_n|\boldsymbol{\theta})
-
N_T\boldsymbol{S}_\text{GEVD}(y_0;\boldsymbol{\theta})
+
\sum_{n=1}^{N_T} \log \boldsymbol{f}_\text{GPD}(y_n;\boldsymbol{\theta})
\end{aligned} L PP-GPD-GEVD ( θ ) := L PP-GPD-GEVD ( θ ) + L GPD ( θ ) = n = 1 ∑ N T log f GEVD ( y n ∣ θ ) − N T S GEVD ( y 0 ; θ ) + n = 1 ∑ N T log f GPD ( y n ; θ ) where L PP-GEVD ( θ ) \boldsymbol{L}_\text{PP-GEVD}(\boldsymbol{\theta}) L PP-GEVD ( θ ) (11) L GPD ( θ ) \boldsymbol{L}_\text{GPD}(\boldsymbol{\theta}) L GPD ( θ ) (21)