commit a9870ed993ec131d39230759273f0778b9029c3e
parent d56d6a44e424f1229908669101f6dbbd0c1e67bc
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Mon, 29 May 2017 17:53:11 +0300
Edit ARMA section.
Diffstat:
| arma.org | | | 82 | ++++++++++++++++++++++++++++++++++++++++--------------------------------------- |
1 file changed, 42 insertions(+), 40 deletions(-)
diff --git a/arma.org b/arma.org
@@ -265,10 +265,10 @@ any scalar quantity can be a component (temperature, salinity, concentration of
some substance in water etc.). Equation parameters are AR and MA process
coefficients and order.
-Any ARMA process can be uniquely represented as a process moving average and
-autoregression process of general infinite order\nbsp{}cite:gurgenidze1988, and
-the parameters of the spectral representation are defined by the rule of
-division of power series (in a rational factorized form\nbsp{}cite:rozhkov1990:
+Any ARMA process can be uniquely represented as either MA or AR process of
+infinite order\nbsp{}cite:gurgenidze1988, and the parameters of the spectral
+representation are defined by the rule of division of power series (in a
+rational factorized form\nbsp{}cite:rozhkov1990):
\begin{equation*}
S(\omega) =
\frac{\Delta\sigma^2}{\pi}
@@ -281,7 +281,7 @@ zeros are located near the unit circle, then the spectral density will have
pronounced dips.
** Autoregressive (AR) process
-AR process is ARMA process with only one random impulse instead of theirs
+AR process is ARMA process with only one random impulse instead of their
weighted sum:
\begin{equation}
\zeta_{\vec i}
@@ -293,10 +293,10 @@ weighted sum:
.
\label{eq-ar-process}
\end{equation}
-The coefficients \(\Phi\) are calculated from ACF via three-dimensional
-Yule---Walker equations, which are obtained after multiplying both parts of the
-previous equation by \(\zeta_{\vec{i}-\vec{k}}\) and computing the expected value.
-Generic form of YW equations is
+The coefficients \(\Phi\) are calculated from auto-covariate function (ACF) via
+three-dimensional Yule---Walker (YW) equations, which are obtained after
+multiplying both parts of the previous equation by \(\zeta_{\vec{i}-\vec{k}}\)
+and computing the expected value. Generic form of YW equations is
\begin{equation}
\label{eq-yule-walker}
\gamma_{\vec k}
@@ -373,8 +373,8 @@ eliminated. Matrix \(\Gamma\) is block-toeplitz, positive definite and symmetric
hence the system is efficiently solved by Cholesky decomposition, which is
particularly suitable for these types of matrices.
-After solving this system of equations white noise variance is estimated from
-eqref:eq-yule-walker by plugging \(\vec k = \vec 0\):
+After solving this system of equations white noise variance is estimated
+from\nbsp{}eqref:eq-yule-walker by plugging \(\vec k = \vec 0\):
\begin{equation*}
\Var{\epsilon} =
\Var{\zeta}
@@ -383,6 +383,7 @@ eqref:eq-yule-walker by plugging \(\vec k = \vec 0\):
\Phi_{\vec j}
\text{ }\gamma_{\vec{j}}.
\end{equation*}
+
** Moving average (MA) process
MA process is ARMA process with \(\Phi\equiv0\):
\begin{equation}
@@ -428,6 +429,7 @@ equation with higher precision, however, this method does not work in three
dimensions. Using slower method does not have dramatic effect on the overall
programme performance, because the number of coefficients is small and most of
the time is spent generating wavy surface.
+
** Mixed autoregressive moving average (ARMA) process
:PROPERTIES:
:CUSTOM_ID: sec:how-to-mix-ARMA
@@ -446,7 +448,7 @@ are several approaches to "mix" AR and MA processes.
of the two overlapped processes.
For the first approach there is a formula to re-compute ACF for AR process, but
there is no such formula for the second approach. So, the best solution for now
-is to simply use AR and MA process exclusively.
+is to simply use AR and MA process exclusively for different types of waves.
** Process selection criteria for different wave profiles
:PROPERTIES:
@@ -457,14 +459,13 @@ different types of wave profiles different processes /must/ be used: standing
waves are modelled by AR process, and propagating waves by MA process. This
statement comes from practice: if one tries to use the processes the other way
round, the resulting realisation either diverges or does not correspond to real
-ocean waves. (The latter happens for non-invertible MA process, as it is always
-stationary.) So, the best way to apply ARMA model to ocean wave generation is to
+ocean waves. So, the best way to apply ARMA model to ocean wave generation is to
use AR process for standing waves and MA process for progressive waves.
The other problem is inability to automatically determine optimal number of
coefficients for three-dimensional AR and MA processes. For one-dimensional
-processes this can be achieved via iterative methods\nbsp{}cite:box1976time, but they
-diverge in three-dimensional case.
+processes this can be achieved via iterative methods\nbsp{}cite:box1976time, but
+they diverge in three-dimensional case.
The final problem, which is discussed in [[#sec:how-to-mix-ARMA]], is inability to
"mix" AR and MA process in three dimensions.
@@ -483,7 +484,7 @@ the first time slice of the ACF, and the rest is left to MA process.
To summarise, the only established scenario of applying ARMA model to ocean wave
generation is to use AR process for standing waves and MA process for
-propagating waves. With new formulae for 3 dimensions a single mixed ARMA
+propagating waves. With a new formulae for 3 dimensions a single mixed ARMA
process might increase model precision, which is one of the objectives of the
future research.
@@ -536,7 +537,7 @@ Find ACF via analytic method. Multiplying the formula by a decaying exponent
\sin (k_x x + k_y y) \sin (\sigma t).
\label{eq-decaying-standing-wave}
\end{equation}
-Then, apply 3D Fourier transform to both sides of the equation via symbolic
+Then, apply 3D Fourier transform to the both sides of the equation via symbolic
computation programme, fit the resulting polynomial to the following
approximation:
\begin{equation}
@@ -553,10 +554,10 @@ at \((0,0,0)\) equals to the ARMA process variance, and if one used sines the
value would be wrong.
If one tries to replicate the same formula via empirical method, the usual way
-is to adapt eqref:eq-decaying-standing-wave to match eqref:eq-standing-wave-acf.
-This can be done either by changing the phase of the sine, or by substituting
-sine with cosine to move the maximum of the function to the origin of
-coordinates.
+is to adapt\nbsp{}eqref:eq-decaying-standing-wave to
+match\nbsp{}eqref:eq-standing-wave-acf. This can be done either by changing the
+phase of the sine, or by substituting sine with cosine to move the maximum of
+the function to the origin of coordinates.
**** Propagating wave ACF.
Three-dimensional profile of plain propagating wave is given by
@@ -596,14 +597,14 @@ steps.
trigonometric identities to shift the phase.
** Evaluation
-In\nbsp{}cite:degtyarev2011modelling,degtyarev2013synoptic,boukhanovsky1997thesis AR
-model the following items are verified experimentally:
+In\nbsp{}cite:degtyarev2011modelling,degtyarev2013synoptic,boukhanovsky1997thesis
+for AR model the following items were verified experimentally:
- probability distributions of different wave characteristics (wave heights,
lengths, crests, periods, slopes, three-dimensionality),
- dispersion relation,
- retention of integral characteristics for mixed wave sea state.
-In this work both AR and MA model are verified by comparing probability
-distributions of different wave characteristics.
+In this work we repeat probability distribution tests for three-dimensional AR
+and MA model.
*** Verification of wavy surface integral characteristics
In\nbsp{}cite:rozhkov1990 the authors show that several ocean wave
@@ -694,20 +695,21 @@ Graph tails in fig.\nbsp{}[[propagating-wave-distributions]] deviate from origin
distribution for individual wave characteristics, because every wave have to be
extracted from the resulting wavy surface to measure its length, period and
height. There is no algorithm that guarantees correct extraction of all waves,
-because they may and often overlap each other. Weibull distribution right tail
-represents infrequently occurring waves, so it deviates more than left tail.
-
-Correspondence rate for standing waves (fig.\nbsp{}[[standing-wave-distributions]])
-is lower for height and length, roughly the same for surface elevation and
-higher for wave period distribution tails. Lower correspondence degree for
-length and height may be attributed to the fact that Weibull distributions were
-obtained empirically for ocean waves which are typically propagating, and
-distributions may be different for standings waves. Higher correspondence degree
-for wave periods is attributed to the fact that wave periods of standing waves
-are extracted more precisely as the waves do not move outside simulated wavy
-surface region. The same correspondence degree for wave elevation is obtained,
-because this is the characteristic of the wavy surface (and corresponding AR or
-MA process) and is not affected by the type of waves.
+because they may overlap each other. Weibull distribution right tail represents
+infrequently occurring waves, so it deviates more than left tail.
+
+Degree of correspondence for standing waves
+(fig.\nbsp{}[[standing-wave-distributions]]) is lower for height and length, is
+roughly the same for surface elevation and is higher for wave period
+distribution tails. Lower correspondence degree for length and height may be
+attributed to the fact that Weibull distributions were obtained empirically for
+ocean waves which are typically propagating, and distributions may be different
+for standings waves. Higher correspondence degree for wave periods is attributed
+to the fact that wave periods of standing waves are extracted more precisely as
+the waves do not move outside simulated wavy surface region. The same
+correspondence degree for wave elevation is obtained, because this is the
+characteristic of the wavy surface (and corresponding AR or MA process) and is
+not affected by the type of waves.
** Discussion
ARMA model, owing to its non-physical nature, does not have the notion of ocean
