Transcript Comparison of Eigenpatterns 1,2 for 0 (Top) with Eigenpatterns 1,2

Data Mining Using Eigenpattern Analysis in Simulations
and Observed Data
Woodblock Print, from “Thirty-Six Views of Mt. Fuji”, by K. Hokusai, ca. 1830
John B. Rundle
Department of Physics and Colorado Center for Chaos & Complexity
University of Colorado, Boulder, CO
Presented at the GEM/ACES Workshop Maui, HI
July 30, 2001
Activity Correlation Operators
Let y(xi,t) be the number of earthquakes per unit time at location xi
and time t.
Now center the time series (remove mean and standard deviation)
y(xi,t)
 z(xi,t) … where z(xi,t) is the centered time series.
Define two correlation operators, a static correlation operator C(xi,xj )
and a rate correlation operator K(xi,xj):
C(xi,xj ) =

z(xi,t) z(xj,t) dt
K(xi,xj ) = (2) 2

{z(xi,t)/t} {z(xj,t)/t} dt
Static
Rate
Diagonalize the Correlation Operators
C(xi,xj ) and K(xi,xj ) are both symmetric, square, and postive definite
matrix operators. We can therefore apply singular value decomposition
to find the eigenvectors and eigenvalues:
C(xi,xj ) =  2 T
K(xi,xj ) =  2 T
where
T
denotes the transpose.
 is a matrix of static eigenpatterns n(xi)
 is a diagonal matrix of eigenprobabilities i2
 is a matrix of rate eigenpatterns n(xi)
 is a diagonal matrix of eigenfrequencies i2
Comparison of Eigenpatterns 1,2 for   0 (Top)
with Eigenpatterns 1,2 for  = 0 (Bottom)
Positively correlated: (red - red) & (blue - blue). Negatively correlated: (red - blue).
Uncorrelated: (red - green) & (blue - green).
JBR et al, Phys. Rev. E., v 61, 2000, & AGU Monograph “GeoComplexity & the Physics of Earthquakes”
Patterns of Earthquakes in Southern California
Earthquakes in southern California have been systematically recorded
since 1932. The rate at which these events occur can be used to define
activity time series in 10 km x 10 km spatial boxes that can be used to
find the spatial patterns.
Figures courtesy KF Tiampo
Below is a map of the first
PCA mode, which we call the
“Hazard Mode”. Red areas
tend to be active or inactive at
the same time.
Above is a map of the
relative intensity of seismic
activity in southern
California, 1932-1999. This
can be considered to be a
seismic “hazard map”.
Above is a map of the second
PCA mode, which we call the
“Landers Mode”. Red areas tend
to be inactive when blue areas
are active & vice versa. All sites
in a blue or red area tend to be
active (or inactive) at the same
time.
Comparison of Log Likelihoods for PDPC from 500 random catalogs
of seismic activity in Southern California with occurrence of future events (M >
5) with Log Likelihoods of hazard map & actual catalog via PDPC.
Example of a
PDPC arising from
a catalog that has
been randomized
in space and time.
Actual catalog: PDPC
for 1978-Dec 31, 1991
Histogram: Log Likelihoods for
500 random catalogs.
RSV: Use hazard map as
predictor.
Actual PDPC: Plot at left
Earthquake Forecasting via the Mathematics of Quantum Mechanics
Pattern techniques suggest a new approach to forecasting earthquakes. The idea
is to view the patterns in the context of PHASE DYNAMICAL SYSTEMS, whose
mathematics can be mapped into the mathematics of QUANTUM MECHANICS.
See JB Rundle et al. (2000); KF Tiampo et al. (2000)
Using this new technique, one can compute
the Phase Dynamical Probability Change
(PDPC) anomalies that develop during the
years 1988-1999.
Our retrospective studies indicate that colored
anomalies can be regarded as indicating high
probability for current and future major
earthquakes (M > 6) over the period ~ 19992009, and have considerable forecast skill.
In the PDPC method,
intensity of seismic
activity is mapped to a
“wave function (x,t)”.
Intensity of seismic activity, 1932-1999
Testing the Forecast
One way to test the forecast
for events from 2000-2010 is
to plot all events with M > 4.0
that have occurred since Jan
1, 2000, superposed upon the
colored forecast anomalies.
These events are the small
circles at right.
Note that our method should
really only forecast events with
M > 6.0
Space-Time Patterns in Complex
Multi-Scale Earthquake Fault Systems
Since much of the dynamics is not accessible to direct observations, we
must focus on learning about the system through analysis of the
observable patterns
Space-time patterns in the system are mathematical expressions of the
strong statistical correlations between various parts of the system
The system state vector characterizes the current state of the system -- it
has an amplitude and a phase angle
Mapping Earthquake Dynamics into the
Mathematics of Quantum Mechanics
(or “Phase Dynamics”)
(JB Rundle et al., Phys. Rev. E, v61, 2416, 2000)
This new technique can be regarded as a novel datamining method
Quantum Mechanical systems are strongly correlated systems (QM is a nonlocal theory)
The mathematics of QM describe systems with periodic and quasiperiodic observables,
as well as hidden variables
Relative probabilities are well-defined quantities in QM
Normalized system state vectors are actually “WAVE FUNCTIONS” that describe
earthquake probability amplitudes
An Earthquake Forecast ?
Using our technique, we
can compute the PDPC
anomalies that develop
during the years 1988-1999.
Our retrospective studies
indicate that these
anomalies can be regarded
as forecasts for major
earthquakes (M > 6) over
the period ~ 1999-2009
Earthquake Fault System Dynamics are Strongly
Correlated in Space and Time and Lead to Patterns
Data from Last Tuesday
PDPC Forecast for ~ 1999-2010
Summary & Future Directions
The methods described here can be used to understand many classes of driven
threshold systems
Network dynamics are determined importantly by the network connectivity as
well as the details of the nonlinear threshold process
Meanfield threshold systems appear to have locally ergodic behavior
Space-time patterns of observable failures (earthquakes) can be used to
understand many facets of the underlying, unobservable dynamics
(physical state variables)
Boolean Correlation Operators and Space-Time Patterns
We can define a set of basis patterns of earthquake activity using Boolean
correlation operators. To do so, we need to define a Boolean activity time
series: y(xi,t)
As a first step, we coarse grain the domain in space and time…i.e., we divide
the region of interest up into N boxes (say, ~10 km on a side) and time into
a series of Q short intervals (say, 8 hours).
If an earthquake occurs in a spatial box centered at (xi,t), we give a value
y(xi,t) = 1 ;
y(xi,t) = 0
Otherwise.
We therefore have a set of N time series, all Q elements long:
y(xi,t) = 0,0,0,1,0,0,0,0,0,0,1,0,0,0,0… etc.
Boolean Activity Eigenpatterns from Simulations
Here we show Static or Activity Eigenpatterns from the simulation…these constitute
one possible basis set for all possible space-time patterns displayed by the
system
Key to Correlation Patterns:
Red sites are positively correlated with red (and blue with blue)
Red sites are negatively correlated with blue
Red sites & Blue sites are uncorrelated with green
The Activity Eigenpatterns are RELATIVE PROBABILITY AMPLITUDES.
( JBR et al, Phys. Rev. E., v 61, 2000, & AGU Monograph “GeoComplexity & the Physics of
Earthquakes” )
Summary & Future Directions
Numerical simulations (“Third Leg of Science”) are now being used to understand many
classes of driven threshold systems (systems with many scales of length and time)
Network dynamics of these complex systems are determined importantly by the network
connectivity as well as the details of the nonlinear threshold process
Meanfield threshold systems have dynamics that demonstrate first and second order
(phase) transitions.
Threshold systems are capable of universal computation such as that which occurs in
the human brain
Space-time patterns of observable failures (earthquakes) can be used to understand
many facets of the underlying, unobservable dynamics (physical state variables)