Lester Ingber Research Projects
https://www.ingber.com/ingber_projects.html
https://www.ingber.com/ingber_projects.txt
__________________________________________________________________
"This above all; to thine own self be true."
William Shakespeare (Polonius' advice to his son Laertes), "Hamlet"
(1601)
"... great life, if you don't weaken"
Irving Klock, Farmer, Herkimer County, NY (1956)
"Stop! All that can be said has been said!."
Eugene Wigner, communication to Lester Ingber (1965)
After my conference talk, when he took me aside and asked why nuclear
forces were so hard to calculate in detail, I replied that the nuclear
force was a cancellation of kinetic and potential energies, and
furthermore the potential energy itself, at many energies, was a
cancellation of repulsive and attractive forces from different
particle-exchange contributions. I then started to give him some
further details, whereupon he put his hand up in air and said the
above.
"Thinking about thinking can make you crazy."
Richard Feynman, Caltech, communication to Lester Ingber (1972)
[http://feynmanlectures.info/ -> Stories -> Lester Ingber]
__________________________________________________________________
ingber_projects.txt
Lester Ingber Research (LIR) develops projects in areas of expertise
documented in the ingber.com InterNet archive. Terms of use,
downloading policies, and consulting/contracting are discussed in the
file
ingber_terms.txt
Projects and interests are described in
ingber_projects_brief.pdf
__________________________________________________________________
2008
For 40+ years I have replied to queries on various aspects of files,
papers, and codes, now many of which are on my website
https://www.ingber.com/ (mirrored on
http://alumni.caltech.edu/~ingber/), especially on: my Adaptive
Simulated Annealing (ASA) and path-integral PATHINT & PATHTREE papers
and codes; my Karate books and correspondence; my Statistical Mechanics
modeling of Financial Markets (SMFM), Neocortical Interactions (SMNI)
and Combat (SMC); and on some earlier work in theoretical nuclear and
elementary-particle physics. I welcome correspondence from people
interested in these disciplines.
My more recent projects, e.g., Ideas by Statistical Mechanics (ISM) and
Real Options for Project Schedules (ROPS), span many disciplines and
businesses. I especially welcome correspondence from people interested
in these projects.
Climbing up from zero in several disciplines has given me a strong
appreciation of various stages of development of projects and people,
and of the role of interdisciplinary synergies often required to forge
successful projects. These experiences have taught me to value/evaluate
the person more than any labels defining chosen disciplines.
I also welcome correspondence from people interested in simply
registering their comments, although I may not have time to pursue any
lengthy correspondence.
__________________________________________________________________
CONTENTS
CURRENT PROJECTS
Quantum path-integral qPATHTREE and qPATHINT algorithms
Statistical-Mechanics-of-Neocortical-Interactions (SMNI)
Ideas by Statistical Mechanics (ISM)
Statistical Mechanics of Financial Markets (SMFM)
Generic Risk Management
Statistical Mechanical Tools
PREVIOUS PROJECTS
EEG Analyses
Optimization of Trading
Optimization of Combat Analyses
Short Introduction To Canonical Momenta
Gaussian-Markovian Modeling
SOME ISSUES IN MATHEMATICAL MODELING
Risk Control of Mathematical Models such as PATHTREE
Interdisciplinary Reviews of Applications of Mathematical Physics
How I Think
__________________________________________________________________
CURRENT PROJECTS
__________________________________________________________________
Jul 16
Quantum path-integral qPATHTREE and qPATHINT algorithms
I am Principal Investigator, for the National Science Foundation
(NSF.gov) resource The Extreme Science and Engineering Discovery
Environment (XSEDE.org), "Quantum path-integral qPATHTREE and qPATHINT
algorithms" project. If you are interested in working on this project,
see
lir_computational_physics_group.html
Some current papers are
>smni18_quantumCaEEG.pdf
%A L. Ingber
%T Quantum calcium-ion interactions with EEG
%J Sci
%V 1
%N 7
%P 1-20
%D 2018
%O URL https://www.ingber.com/smni18_quantumCaEEG.pdf and
https://doi.org/10.3390/sci1010007
ABSTRACT: Background: Previous papers have developed a
statistical mechanics of neocortical interactions (SMNI)
fit to short-term memory and EEG data. Adaptive Simulated
Annealing (ASA) has been developed to perform fits to such
nonlinear stochastic systems. An N-dimensional
path-integral algorithm for quantum systems, qPATHINT, has
been developed from classical PATHINT. Both fold
short-time propagators (distributions or wave functions)
over long times. Previous papers applied qPATHINT to two
systems, in neocortical interactions and financial
options. Objective: In this paper the quantum
path-integral for Calcium ions is used to derive a
closed-form analytic solution at arbitrary time that is
used to calculate interactions with classical-physics SMNI
interactions among scales. Using fits of this SMNI model
to EEG data, including these effects, will help determine
if this is a reasonable approach. Method: Methods of
mathematical-physics for optimization and for path
integrals in classical and quantum spaces are used for
this project. Studies using supercomputer resources tested
various dimensions for their scaling limits. In this paper
the quantum path-integral is used to derive a closed-form
analytic solution at arbitrary time that is used to
calculate interactions with classical-physics SMNI
interactions among scales. Results: The
mathematical-physics and computer parts of the study are
successful, in that there is modest improvement of
cost/objective functions used to fit EEG data using these
models. Conclusion: This project points to directions for
more detailed calculations using more EEG data and
qPATHINT at each time slice to propagate quantum calcium
waves, synchronized with PATHINT propagation of classical
SMNI.
path17_quantum_options_shocks.pdf
%A L. Ingber
%J The Open Cybernetics Systemics Journal
%V 11
%P 3-18
%D 2017
%O URL https://www.ingber.com/path17_qpathint.pdf
Links path17_qpathint.pdf with smni17_qpathint.pdf and
markets17_qpathint.pdf
ABSTRACT: A path-integral algorithm, PATHINT, used previously
for several systems, has been generalized from 1 dimension to N
dimensions, and from classical to quantum systems, qPATHINT.
Pilot quantum applications have been made to neocortical
interactions and financial options in 1 dimension. Future work
is planned to extend qPATHINT to multiple dimensions in these
systems, and to extend calculations to multiple scales of
interaction between classical events and expectations over
quantum processes. Each of the two systems considered contribute
insight into applications of qPATHINT to the other system,
leading to new algorithms presenting time-dependent propagation
of interacting quantum and classical scales.
path17_quantum_options_shocks.pdf
%A L. Ingber
%T Options on quantum money: Quantum path-integral with serial
shocks
%J International Journal of Innovative Research in Information
Security
%V 4
%N 2
%P 7-13
%D 2017
%O URL https://www.ingber.com/path17_quantum_options_shocks.pdf
Links path17_quantum_options_shocks.pdf with and
markets17_quantum_options_shocks.pdf
ABSTRACT: The author previously developed a numerical
multivariate path-integral algorithm, PATHINT, which has been
applied to several classical physics systems, including
statistical mechanics of neocortical interactions, options in
financial markets, and other nonlinear systems including chaotic
systems. A new quantum version, qPATHINT, has the ability to
take into account nonlinear and time-dependent modifications of
an evolving system. qPATHINT is shown to be useful to study some
aspects of serial changes to systems. Applications to options on
quantum money and blockchains in financial markets are
discussed.
path17_quantum_pathint_shocks.pdf
%A L. Ingber
%T Evolution of regenerative Ca-ion wave-packet in
neuronal-firing fields: Quantum path-integral with serial shocks
%J International Journal of Innovative Research in Information
Security
%V 4
%N 2
%P 14-22
%D 2017
%O URL https://www.ingber.com/path17_quantum_pathint_shocks.pdf
Links path17_quantum_pathint_shocks.pdf with
smni17_quantum_pathint_shocks.pdf and
markets17_quantum_pathint_shocks.pdf
ABSTRACT: The author previously developed a numerical
multivariate path-integral algorithm, PATHINT, which has been
applied to several classical physics systems, including
statistical mechanics of neocortical interactions, options in
financial markets, and other nonlinear systems including chaotic
systems. A new quantum version, qPATHINT, has the ability to
take into account nonlinear and time-dependent modifications of
an evolving system. qPATHINT is shown to be useful to study some
aspects of serial changes to systems. Applications ranging from
regenerative $\mathrm{Ca}^{2+}$ waves in neuroscience to options
on blockchains in financial markets are discussed.
path16_quantum_path.pdf
%A L. Ingber
%T Path-integral quantum PATHTREE algorithm
%J International Journal of Innovative Research in Information
Security
%V 3
%N 5
%P 1-15
%D 2016
%O URL https://www.ingber.com/path16_quantum_path.pdf
Links path16_quantum_path.pdf with smni16_quantum_path.pdf and
markets16_quantum_path.pdf
ABSTRACT: The author previously developed a generalization of a
binomial tree algorithm, PATHTREE, to develop options pricing
for multiplicative-noise models possessing quite generally time
dependent and nonlinear means and variances. This code is
generalized here for complex variable spaces, to produce
qPATHTREE useful for quantum systems. As highlighted in this
paper, a quantum version, qPATHTREE, has the ability to take
into account time dependent modifications of an evolving system.
qPATHTREE is shown to be useful to study some aspects of serial
changes to systems. Similarly, another path-integral code,
PATHINT, used for several previous systems is being developed
into qPATHINT. An example is given for a free particle, and it
is explained when an n-tree generalization of qPATHTREE beyond
the binomial tree is required for such systems, similar to code
developed for qPATHINT. Potential applications in neuroscience
and financial markets are discussed.
Statistical-Mechanics-of-Neocortical-Interactions (SMNI)
October 2016
FREE WILL
Free Will in the context of my current project, using
Statistical Mechanics of Neocortical Interactions (SMNI) under
The Extreme Science and Engineering Discovery Environment
(XSEDE.org)
From Section 3, L. Ingber, "Evolution of regenerative Ca-ion
wave-packet in neuronal-firing fields: Quantum path-integral
with serial shocks," Report 2017:QPIS, Lester Ingber Research,
Ashland, OR, (2017). [ URL
https://www.ingber.com/path17_quantum_pathint_shocks.pdf ]
In addition to the intrinsic interest of researching short-term
memory (STM) and multiple scales of neocortical interactions via
electroencephalography (EEG) data, there is interest in
researching possible quantum influences on highly synchronous
neuronal firings relevant to STM to understand possible
connections to "Free Will" (FW).
As pointed out in some recent papers (Ingber, 2016a,b), if
neuroscience ever establishes experimental feedback from
quantum-level processes of tripartite synaptic interactions with
large-scale synchronous neuronal firings, that are now
recognized as being highly correlated with STM and states of
attention, then FW may yet be established using the quantum
"Free Will Theorem" (FWT) (Conway and Kochen, 2006, 2009).
Basically, this means that a Ca2+ quantum wave-packet may
generate a state proven to have not previously existed. In the
context of the basic premise of this paper, this state may be
influential in a large-scale pattern of synchronous neuronal
firings, thereby rendering this pattern as a truly new pattern
not having previously existed. The FWT shows that this pattern,
considered as a measurement of the Ca2+ quantum wave-packet,
surprisingly is correctly identified as itself being a new
decision not solely based on previous (deterministic) decisions,
even under reasonably stochastic experimental conditions.
Only recently has the core Statistical Mechanics of Neocortical
Interactions (SMNI) hypothesis since circa 1980 (Ingber, 1981,
1982, 1983), that highly synchronous patterns of neuronal
firings in fact process high-level information, been verified
experimentally (Asher, 2012; Salazar et al., 2012).
Clearly, even in the above context, for most people most of the
time, internal and external events affecting neural
probabilistic patterns of attention give rise to quite practical
reasonable FW. However, there also may be some Science that
establishes a truly precise FW.
REFERENCES
Asher, 2012. Asher, J., "Brain's code for visual working memory
deciphered in monkeys NIH-funded study," NIH Press Release, NIH,
Bethesda, MD (2012). URL
http://www.nimh.nih.gov/news/science-news/2012/in-sync-
brain-waves-hold-memory-of-objects-just-seen.shtml.
Conway & Kochen, 2006. Conway, J. & Kochen, S., "The free will
theorem," arXiv:quant- ph/0604079 [quant-ph], pp. 1-31,
Princeton U, Princeton, NJ (2006).
Conway & Kochen, 2009. Conway, J. & Kochen, S., "The strong free
will theorem," Notices of the American Mathematical Society
56(2), pp. 226-232 (2009).
Ingber, 1981. Ingber, L., "Towards a unified brain theory,"
Journal Social Biological Structures 4, pp. 211-224 (1981). URL
https://www.ingber.com/smni81_unified.pdf.
Ingber, 1982. Ingber, L., "Statistical mechanics of neocortical
interactions. I. Basic formulation," Physica D 5, pp. 83-107
(1982). URL https://www.ingber.com/smni82_basic.pdf.
Ingber, 1983. Ingber, L., "Statistical mechanics of neocortical
interactions. Dynamics of synaptic modification," Physical
Review A 28, pp. 395-416 (1983). URL
https://www.ingber.com/smni83_dynamics.pdf.
Ingber, 2016a. Ingber, L., "Path-integral quantum PATHTREE and
PATHINT algorithms," International Journal of Innovative
Research in Information Security 3(5), pp. 1-15 (2016a). URL
https://www.ingber.com/path16_quantum_path.pdf and
https://dx.doi.org/10.17632/xspkr8rvks.1.
Ingber, 2016b. Ingber, L., "Statistical mechanics of neocortical
interactions: Large-scale EEG influences on molecular
processes," Journal of Theoretical Biology 395, pp. 144-152
(2016b). URL
https://www.ingber.com/smni16_large-scale_molecular.pdf and
https://dx.doi.org/10.1016/j.jtbi.2016.02.003.
Salazar et al, 2012. Salazar, R.F., Dotson, N.M., Bressler, S.L.
& Gray, C.M., "Content-specific fronto-parietal synchronization
during visual working memory," Science 338(6110), pp. 1097-1100
(2012). URL https://dx.doi.org/10.1126/science.1224000.
July 2009
smni09_nonlin_column_eeg.pdf
%A L. Ingber
%T Statistical mechanics of neocortical interactions:
Nonlinear columnar electroencephalography
%J NeuroQuantology Journal
%P 500-529
%D 2009
%O URL https://www.ingber.com/smni09_nonlin_column_eeg.pdf
and
http://www.neuroquantology.com/journal/index.php/nq/articl
e/view/365/385
ABSTRACT: Columnar firings of neocortex, modeled by a
statistical mechanics of neocortical interactions (SMNI), are
investigated for conditions of oscillatory processing at
frequencies consistent with observed electroencephalography
(EEG). A strong inference is drawn that physiological states of
columnar activity receptive to selective attention support
oscillatory processing in observed frequency ranges. Direct
calculations of the Euler-Lagrange (EL) equations which are
derived from functional variation of the SMNI probability
distribution, giving most likely states of the system, are
performed for three prototypical Cases, dominate excitatory
columnar firings, dominate inhibitory columnar firings, and
in-between balanced columnar firings, with and without a
Centering mechanism (CM) (based on observed changes in
stochastic background of presynaptic interactions) which pulls
more stable states into the physical firings ranges. Only states
with the CM exhibit robust support for these oscillatory states.
These calculations are repeated for the visual neocortex, which
has twice as many neurons/minicolumn as other neocortical
regions. These calculations argue that robust columnar support
for common EEG activity requires the same columnar presynaptic
parameter necessary for ideal short-term memory (STM). It is
demonstrated at this columnar scale, that both shifts in local
columnar presynaptic background as well as local or global
regional oscillatory interactions can effect or be affected by
attractors that have detailed experimental support to be
considered states of STM. Including the CM with other proposed
mechanisms for columnar-glial interactions and for
glial-presynaptic background interactions, a path for future
investigations is outlined to test for quantum interactions,
enhanced by magnetic fields from columnar EEG, that directly
support cerebral STM and computation by controlling presynaptic
noise. This interplay can provide mechanisms for information
processing and computation in mammalian neocortex.
This paper demonstrates by explicit calculations that short-term
memory (STM) and EEG can indeed be correlated. At least
according to some reviewers, this seems not to have been
demonstrated previously. This paper shows that the previous SMNI
models which calculate many features measured as STM also
support EEG at columnar scales. To put this into some
perspective, many neuroscientists believe that global regional
activity supports EEG wave-like oscillatory observations, by
solving wave equations with hemisphere boundary conditions with
spherical eigenfunctions that detail the frequencies of EEG. In
this columnar study, wave-type equations are derived via
nonlinear EL equations from SMNI probability distributions, and
these are explicitly numerically solved to demonstrate that
observed EEG frequencies are supported under the same SMNI
conditions that support STM.
The next study in smni10_multiple_scales.pdf includes definitive
calculations using PATHINT to evolve multivariate probability
distributions of firing states.
2010
smni10_multiple_scales.pdf
%A L. Ingber
%A P.L. Nunez
%T Neocortical Dynamics at Multiple Scales: EEG Standing
Waves, Statistical Mechanics, and Physical Analogs
%J Mathematical Biosciences
%D 2011
%O URL https://www.ingber.com/smni10_multiple_scales.pdf
ABSTRACT: The dynamic behavior of scalp potentials (EEG) is
apparently due to some combination of global and local processes
with important top-down and bottom-up interactions across
spatial scales. In treating global mechanisms, we stress the
importance of myelinated axon propagation delays and periodic
boundary conditions in the cortical-white matter system, which
is topologically close to a spherical shell. By contrast, the
proposed local mechanisms are multiscale interactions between
cortical columns via short-ranged non-myelinated fibers. A
mechanical model consisting of a stretched string with attached
nonlinear springs demonstrates the general idea. The string
produces standing waves analogous to large-scale coherence EEG
observed in some brain states. The attached springs are
analogous to the smaller (mesoscopic) scale columnar dynamics.
Generally, we expect string displacement and EEG at all scales
to result from both global and local phenomena. A statistical
mechanics of neocortical interactions (SMNI) calculates
oscillatory behavior consistent with typical EEG, within
columns, between neighboring columns via short-ranged
non-myelinated fibers, across cortical regions via myelinated
fibers, and also derive a string equation consistent with the
global EEG model.
This paper includes PATHINT evolution of probability
distributions of columnar activity with explicit oscillatory
firings, and integration of such mesoscopic processes with
global brain EEG activity.
May 11
The SMNI project has been developed in a series of papers since
1981. Current work includes neuron-astrocyte interactions,
bringing more scales of interaction into SMNI STM. A project
that will be further developed into two invited papers is
described in a report:
smni11_stm_scales.pdf
%A L. Ingber
%T Columnar electromagnetic influences on short-term
memory at multiple scales
%R Report Ingber:2011:CEMI
%I Lester Ingber Research
%C Ashland, OR
%D 2011
%O URL https://www.ingber.com/smni11_stm_scales.pdf
Nov 11
A full review of SMNI, including its mathematical outline, in
the context of STM is in:
https://www.ingber.com/smni11_stm_scales.pdf
%A L. Ingber
%T Columnar EEG magnetic influences on molecular
development of short-term memory
%B Short-Term Memory: New Research
%E G. Kalivas
%E S.F. Petralia
%D 2012
%P 37-72
%I Nova
%C Hauppauge, NY
%O Invited Paper. URL
https://www.ingber.com/smni11_stm_scales.pdf
Jun 12
More research on a "smoking gun" for explicit top-down
neocortical mechanisms that directly drive bottom-up processes
that describe memory, attention, etc., is given in several
papers, e.g.:
https://www.ingber.com/smni12_vectpot.pdf
%A L. Ingber
%T Influence of macrocolumnar EEG on Ca waves
%J Current Progress Journal
%D 2012
%V 1
%N 1
%P 4-8
%D 2012
%O URL https://www.ingber.com/smni12_vectpot.pdf
smni14_eeg_ca.pdf
%A L. Ingber
%A Pappalepore
%A R.R. Stesiak
%T %T Electroencephalographic field influence on calcium
momentum waves
%J Journal of Theoretical Biology
%D 2014
%O URL https://www.ingber.com/smni14_eeg_ca.pdf and
https://dx.doi.org/10.1016/j.jtbi.2013.11.002
smni15_calc_conscious.pdf
%A L. Ingber
%T Calculating consciousness correlates at multiple scales
of neocortical interactions
%B Horizons in Neuroscience Research
%I Nova
%C Hauppauge, NY
%D 2015
%O URL https://www.ingber.com/smni15_calc_conscious.pdf
Feb 16
smni16_large-scale_molecular.pdf
%A L. Ingber
%T Statistical mechanics of neocortical interactions:
Large-scale EEG influences on molecular processes
%J Journal of Theoretical Biology
%D 2016
%O URL
https://www.ingber.com/smni16_large-scale_molecular.pdf
ABSTRACT: Calculations further support the premise that
large-scale synchronous firings of neurons may affect molecular
processes. The context is scalp electroencephalography (EEG)
during short-term memory (STM) tasks. The mechanism considered
is $\mathbf{\Pi} = \mathbf{p} + q \mathbf{A}$ (SI units)
coupling, where $\mathbf{p}$ is the momenta of free
$\mathrm{Ca}^{2+}$ waves $q$ the charge of $\mathrm{Ca}^{2+}$ in
units of the electron charge, and $\mathbf{A}$ the magnetic
vector potential of current $\mathbf{I}$ from neuronal
minicolumnar firings considered as wires, giving rise to EEG.
Data has processed using multiple graphs to identify sections of
data to which spline-Laplacian transformations are applied, to
fit the statistical mechanics of neocortical interactions (SMNI)
model to EEG data, sensitive to synaptic interactions subject to
modification by $\mathrm{Ca}^{2+}$ waves.
__________________________________________________________________
Ideas by Statistical Mechanics (ISM)
Jul 06
This project is described in a report:
smni06_ism.pdf
%A L. Ingber
%T Ideas by statistical mechanics (ISM)
%R Report 2006:ISM
%I Lester Ingber Research
%C Ashland, OR
%D 2006
%O URL https://www.ingber.com/smni06_ism.pdf
A short version appears as "AI and Ideas by Statistical
Mechanics (ISM)" in Encyclopedia of Artificial Intelligence, pp.
58-64 (2008), and details in this paper appear in "Ideas by
Statistical Mechanics (ISM)" the Journal of Integrated Systems
Design and Process Science, Vol. 11, No. 3, pp. 22-43 (2007),
Special Issue: Biologically Inspired Computing.
Links from smni06_ism.pdf to asa06_ism.pdf, combat06_ism.pdf,
markets06_ism.pdf, and path06_ism.pdf.
ABSTRACT: Ideas by Statistical Mechanics (ISM) is a generic program to
model evolution and propagation of ideas/patterns throughout
populations subjected to endogenous and exogenous interactions. The
program is based on the author's work in Statistical Mechanics of
Neocortical Interactions (SMNI), and uses the author's Adaptive
Simulated Annealing (ASA) code for optimizations of training sets, as
well as for importance-sampling to apply the author's copula financial
risk-management codes, Trading in Risk Dimensions (TRD), for
assessments of risk and uncertainty. This product can be used for
decision support for projects ranging from diplomatic, information,
military, and economic (DIME) factors of propagation/evolution of
ideas, to commercial sales, trading indicators across sectors of
financial markets, advertising and political campaigns, etc.
It seems appropriate to base an approach for propagation of ideas on
the only system so far demonstrated to develop and nurture ideas, i.e.,
the neocortical brain. A statistical mechanical model of neocortical
interactions, developed by the author and tested successfully in
describing short-term memory and EEG indicators, is the proposed model.
ISM develops subsets of macrocolumnar activity of multivariate
stochastic descriptions of defined populations, with macrocolumns
defined by their local parameters within specific regions and with
parameterized endogenous inter-regional and exogenous external
connectivities. Parameters with a given subset of macrocolumns will be
fit using ASA to patterns representing ideas. Parameters of external
and inter-regional interactions will be determined that promote or
inhibit the spread of these ideas. Tools of financial risk management,
developed by the author to process correlated multivariate systems with
differing non-Gaussian distributions using modern copula analysis,
importance-sampled using ASA, will enable bona fide correlations and
uncertainties of success and failure to be calculated. Marginal
distributions will be evolved to determine their expected duration and
stability using algorithms developed by the author, i.e., PATHTREE and
PATHINT codes.
__________________________________________________________________
Statistical Mechanics of Financial Markets (SMFM)
Dec 03
Oct 04
Jan 06
Apr 06
Previous work, mostly published, developed two-shell recursive trading
systems. An inner-shell of Canonical Momenta Indicators (CMI) is
adaptively fit to incoming market data. A parameterized trading-rule
outer-shell uses the global optimization code Adaptive Simulated
Annealing (ASA) to fit the trading system to historical data. A simple
fitting algorithm, usually not requiring ASA, is used for the
inner-shell fit.
An additional risk-management middle-shell has been added to create a
three-shell recursive optimization/sampling/fitting algorithm.
Portfolio-level distributions of copula-transformed multivariate
distributions (with constituent markets possessing different marginal
distributions in returns space) are generated by Monte Carlo samplings.
ASA is used to importance-sample weightings of these markets. There are
many publications in the academic and commercial literature on similar
treatments of markets. The Buddha is in the details :).
TRD processes Training and Testing trading systems on historical data,
and consistently interacts with RealTime trading platforms -- all at
minute resolutions. Faster or slower resolutions can be developed using
the present structure of TRD. The code is written in vanilla C, and
runs across platforms such as XP/Cygwin, SPARC/Solaris, i386/FreeBSD,
i386/NetBSD, etc. TRD can be run as an independent executable or called
as a DLL. Some more detail is given in
markets05_trd.pdf
%A L. Ingber
%T Trading in Risk Dimensions (TRD)
%D 2005
%R Report 2005:TRD
%C Ashland, OR
%I Lester Ingber Research
%O URL https://www.ingber.com/markets05_trd.pdf
An updated shorter paper with this title is published in the
Handbook of Trading: Strategies for Navigating and Profiting
from Currency, Bond, and Stock Markets (McGraw-Hill, 2010).
To illustrate how TRD can robustly and flexibly interact with various
trading platforms, I have developed a working interface with
TradeStation and have outlined an interface with Fidelity's Wealth-Lab.
Beware Greeks Bearing Gifts: Mathematical models are only as good -- at
best -- as the data supporting them. It may not make sense applying
models based on historical data to current time. It is important to
stress-test models using random data generated in large windows around
historical data.
__________________________________________________________________
Generic Risk Management
The LIR basic concepts, architecture, and TRD code described above can
be used to further develop LIR's published approaches to projects in
other disciplines, as well as to generate other projects.
These concepts can be applied to developing tools for decision-makers
of companies and government agencies, assembling functions,
departments, processes, etc., tailored to specific client requirements,
into an overall "portfolio" from which top-level measures of
performance are developed with associated measures of risk, and with
audit trails back to the member constituents. For example, marginal
distributions can be formulated based on many kinds of sets of data. A
report describes how options (in the sense of financial options) and
risk management can be developed for project schedules:
markets07_rops.pdf
%A L. Ingber
%T Real Options for Project Schedules (ROPS)
%R Report 2007:ROPS
%I Lester Ingber Research
%C Ashland, OR
%D 2007
%O URL https://www.ingber.com/markets07_rops.pdf
An updated invited paper is published in International Journal
of Science, Technology & Management (201o)
ABSTRACT: Real Options for Project Schedules (ROPS) has three recursive
sampling/optimization shells. An outer Adaptive Simulated Annealing
(ASA) optimization shell optimizes parameters of strategic Plans
containing multiple Projects containing ordered Tasks. A middle shell
samples probability distributions of durations of Tasks. An inner shell
samples probability distributions of costs of Tasks. PATHTREE is used
to develop options on schedules.
__________________________________________________________________
Statistical Mechanical Tools
Adaptive Simulated Annealing (ASA) and path-integral techniques, e.g.,
PATHINT and PATHTREE, have been published, demonstrating their utility
in statistical mechanical studies in finance, neuroscience, combat
analyses, neuroscience, and other selected nonlinear multivariate
systems. PATHTREE has been used extensively to price financial options.
PATHTREE can be be generalized, from folding forward in time a wide
class of nonlinear stochastic 1-dimensional distributions, to
n-dimensional distributions in an n-PATHTREE code. PATHINT becomes
extremely computationally intensive with just 3 variables, but
n-PATHTREE likely can be much quicker with the same extreme accuracy
for higher dimensional systems. A matrix formulation of binomial trees
could be the focus for developing n-PATHTREE. This algorithm can be
used to fit the shape of distributions to data, providing a robust
bottom-up approach to "curve-fitting" systematics of empirical data.
These tools also are being applied to price complex projects as
financial options with alternative schedules and strategies. PATHTREE
processes real-world options, including nonlinear distributions and
time-dependent starting and stopping of sub-projects, with parameters
of shapes of distributions fit using ASA to optimize cost and duration
of sub-projects.
__________________________________________________________________
PREVIOUS PROJECTS
__________________________________________________________________
EEG Analyses
1997
High-quality EEG data has been used to perform ASA optimization of
"canonical momenta" indicators (CMI). This work demonstrates how the
theory of statistical mechanics of neocortical interactions (SMNI) can
describe individuals' macroscopic brain function as measured by raw
EEG.
The first paper giving a detailed calculation of CMI in EEG was
smni97_cmi.pdf
%A L. Ingber
%T Statistical mechanics of neocortical interactions: Canonical
momenta indicators of electroencephalography
%J Physical Review E
%V 55
%N 4
%P 4578-4593
%D 1997
%O URL https://www.ingber.com/smni97_cmi.pdf
ABSTRACT: A series of papers has developed a statistical mechanics of
neocortical interactions (SMNI), deriving aggregate behavior of
experimentally observed columns of neurons from statistical
electrical-chemical properties of synaptic interactions. While not
useful to yield insights at the single neuron level, SMNI has
demonstrated its capability in describing large-scale properties of
short-term memory and electroencephalographic (EEG) systematics. The
necessity of including nonlinear and stochastic structures in this
development has been stressed. Sets of EEG and evoked potential data
were fit, collected to investigate genetic predispositions to
alcoholism and to extract brain "signatures" of short-term memory.
Adaptive Simulated Annealing (ASA), a global optimization algorithm,
was used to perform maximum likelihood fits of Lagrangians defined by
path integrals of multivariate conditional probabilities. Canonical
momenta indicators (CMI) are thereby derived for individual's EEG data.
The CMI give better signal recognition than the raw data, and can be
used to advantage as correlates of behavioral states. These results
give strong quantitative support for an accurate intuitive picture,
portraying neocortical interactions as having common algebraic or
physics mechanisms that scale across quite disparate spatial scales and
functional or behavioral phenomena, i.e., describing interactions among
neurons, columns of neurons, and regional masses of neurons.
A follow-up study, including testing the CMI on out-of-sample data, is
in
smni98_cmi_test.pdf
%A L. Ingber
%T Statistical mechanics of neocortical interactions: Training
and testing canonical momenta indicators of EEG
%J Mathematical Computer Modelling
%V 27
%N 3
%P 33-64
%D 1998
%O URL https://www.ingber.com/smni98_cmi_test.pdf
Additional results (tables of ASA-fitted parameters and 60 files
containing 240 PostScript graphs) are contained in
smni97_eeg_cmi.tar.gz
Some background and results in lecture-plate form are given in
smni97_lecture.pdf
%A L. Ingber
%T Statistical mechanics of neocortical interactions (SMNI)
%R SMNI Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 1997
%O URL https://www.ingber.com/smni97_lecture.pdf
smni01_lecture.pdf
%A L. Ingber
%T Statistical Mechanics of Neocortical Interactions (SMNI):
Multiple Scales of Short-Term Memory and EEG Phenomena
%R SMNI Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 2001
%O URL https://www.ingber.com/smni01_lecture.pdf
smni01_lecture.pdf
%A L. Ingber
%T Statistical Mechanics of Neocortical Interactions (SMNI):
Multiple Scales of Short-Term Memory and EEG Phenomena
%R SMNI Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 2001
%O URL https://www.ingber.com/smni01_lecture.pdf
Dec 06
Some of the algorithms used in the ISM project (above) are used in
smni06_ppi.pdf
%A L. Ingber
%T Statistical mechanics of neocortical interactions: Portfolio
of physiological indicators
%R Report 2006:PPI
%I Lester Ingber Research
%C Ashland, OR
%D 2006
%O URL https://www.ingber.com/smni06_ppi.pdf
A modified is published in The Open Cybernetics Systemics
Journal, vol. 3, pp. 5-18 (2009).
__________________________________________________________________
Optimization of Trading
An approach to developing a quasi-automated trading system, based on a
theory of a statistical mechanics of financial markets (SMFM) has been
developed in several papers, e.g.,
markets96_momenta.pdf
%A L. Ingber
%T Canonical momenta indicators of financial markets and
neocortical EEG
%B Progress in Neural Information Processing
%E S.-I. Amari, L. Xu, I. King, and K.-S. Leung
%I Springer
%C New York
%P 777-784
%D 1996
%O Invited paper to the 1996 International Conference on Neural
Information Processing (ICONIP'96), Hong Kong, 24-27 September
1996. ISBN 981 3083-05-0. URL
https://www.ingber.com/markets96_momenta.pdf
Tables of data supporting this paper are given in
markets96_momenta_tbl.txt.gz
markets96_lag_cmi.c contains C-code for the Lagrangian cost
function described in /markets96_momenta.pdf to be fit to data.
Also included is code for the CMI derived from this Lagrangian.
ABSTRACT: A paradigm of statistical mechanics of financial markets
(SMFM) is fit to multivariate financial markets using Adaptive
Simulated Annealing (ASA), a global optimization algorithm, to perform
maximum likelihood fits of Lagrangians defined by path integrals of
multivariate conditional probabilities. Canonical momenta are thereby
derived and used as technical indicators in a recursive ASA
optimization process to tune trading rules. These trading rules are
then used on out-of-sample data, to demonstrate that they can profit
from the SMFM model, to illustrate that these markets are likely not
efficient. This methodology can be extended to other systems, e.g.,
electroencephalography. This approach to complex systems emphasizes the
utility of blending an intuitive and powerful mathematical-physics
formalism to generate indicators which are used by AI-type rule-based
models of management.
The calculations contained in
markets96_trading.pdf
%A L. Ingber
%T Statistical mechanics of nonlinear nonequilibrium financial
markets: Applications to optimized trading
%J Mathematical Computer Modelling
%V 23
%N 7
%P 101-121
%D 1996
%O URL https://www.ingber.com/markets96_trading.pdf
were done in 1991. The calculations in markets96_momenta.pdf and in
markets01_optim_trading.pdf
%A L. Ingber
%A R.P. Mondescu
%T Optimization of Trading Physics Models of Markets
%V 12
%N 4
%P 776-790
%D 2001
%J IEEE Trans. Neural Networks
%O Invited paper for special issue on Neural Networks in
Financial Engineering. URL
https://www.ingber.com/markets01_optim_trading.pdf
give even stronger support to the use of canonical momenta for
financial indicators.
A brief and less technical discussion of this approach and of ASA is
given in
markets96_brief.pdf
%A L. Ingber
%T Trading markets with canonical momenta and adaptive simulated
annealing
%R Report 1996:TMCMASA
%I Lester Ingber Research
%C Ashland, OR
%C McLean, VA
%D 1996
%O URL https://www.ingber.com/markets96_brief.pdf
This paper gives relatively non-technical descriptions of ASA
and canonical momenta, and their applications to markets and
EEG. The paper was solicited by AI in Finance prior to cessation
of publication.
A brief discussion and motivation for work in progress, further
developing SMFM, is given in
markets98_smfm_appl.pdf
%A L. Ingber
%T Some Applications of Statistical Mechanics of Financial
Markets
%R LIR-98-1-SASMFM
%I Lester Ingber Research
%C Chicago, IL
%D 1998
%O URL https://www.ingber.com/markets98_smfm_appl.pdf
The approach of using the Lagrangian as a cost function to fit data in
financial systems (to my knowledge, to fit any highly nonlinear
stochastic multivariate system) was first proposed in
markets84_statmech.pdf
%A L. Ingber
%T Statistical mechanics of nonlinear nonequilibrium financial
markets
%J Mathematical Modelling
%V 5
%N 6
%P 343-361
%D 1984
%O URL https://www.ingber.com/markets84_statmech.pdf
Application of SMFM to developing volatility of volatility in the
context of Eurodollar options is given in
markets99_vol.pdf
%A L. Ingber
%A J.K. Wilson
%T Volatility of volatility of financial markets
%J Mathematical Computer Modelling
%V 29
%P 39-57
%D 1998
%O URL https://www.ingber.com/markets99_vol.pdf
In
https://www.ingber.com/markets00_exp.pdf
%A L. Ingber
%A J.K. Wilson
%T Statistical mechanics of financial markets: Exponential
modifications to Black-Scholes
%J Mathematical Computer Modelling
%V 31
%N 8/9
%P 167-192
%D 2000
%O URL https://www.ingber.com/markets00_exp.pdf
and
https://www.ingber.com/markets00_highres.pdf
%A L. Ingber
%T High-resolution path-integral development of financial
options
%J Physica A
%V 283
%N 3-4
%P 529-558
%D 2000
%O URL https://www.ingber.com/markets00_highres.pdf
Both ASA and a path-integral code, PATHINT, discussed below, are used
to develop new options models:
ABSTRACT: The Black-Scholes theory of option pricing has been
considered for many years as an important but very approximate
zeroth-order description of actual market behavior. We generalize the
functional form of the diffusion of these systems and also consider
multi-factor models including stochastic volatility. We use a previous
development of a statistical mechanics of financial markets to model
these issues. Daily Eurodollar futures prices and implied volatilities
are fit to determine exponents of functional behavior of diffusions
using methods of global optimization, Adaptive Simulated Annealing
(ASA), to generate tight fits across moving time windows of Eurodollar
contracts. These short-time fitted distributions are then developed
into long-time distributions using a robust non-Monte Carlo
path-integral algorithm, PATHINT, to generate prices and derivatives
commonly used by option traders. The results of our study show that
there is only a very small change in at-the money option prices for
different probability distributions, both for the one-factor and
two-factor models. There still are significant differences in risk
parameters, partial derivatives, using more sophisticated models,
especially for out-of-the-money options.
A very quick and robust algorithm motivated by PATHINT, PATHTREE, has
been developed:
path01_pathtree.pdf
%A L. Ingber
%A C. Chen
%A R.P. Mondescu
%A D. Muzzall
%A M. Renedo
%T Probability tree algorithm for general diffusion processes
%J Physical Review E
%V 64
%N 5
%P 056702-056707
%D 2001
%O URL https://www.ingber.com/path01_pathtree.pdf
Link from path01_pathtree.pdf to markets01_pathtree.pdf.
ABSTRACT: Motivated by path-integral numerical solutions of diffusion
processes, PATHINT, we present a new tree algorithm, PATHTREE, which
permits extremely fast accurate computation of probability
distributions of a large class of general nonlinear diffusion
processes.
Some background and results in lecture-plate form are given in
markets98_lecture.pdf
%A L. Ingber
%T Statistical mechanics of financial markets (SMFM)
%R SMFM Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 1998
%O Invited talk to U of Chicago Financial Mathematics Seminar,
20 Nov 1998. URL https://www.ingber.com/markets98_lecture.pdf
markets01_lecture.pdf
%A L. Ingber
%T Statistical Mechanics of Financial Markets (SMFM):
Applications to Trading Indicators and Options
%R SMFM Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 2001
%O URL https://www.ingber.com/markets01_lecture.pdf
__________________________________________________________________
Optimization of Combat Analyses
A theory of statistical mechanics of combat (SMC) is given in
combat97_cmi.pdf
%A M. Bowman
%A L. Ingber
%T Canonical momenta of nonlinear combat
%B Proceedings of the 1997 Simulation Multi-Conference, 6-10
April 1997, Atlanta, GA
%I Society for Computer Simulation
%C San Diego, CA
%D 1997
%O URL https://www.ingber.com/combat97_cmi.pdf
ABSTRACT: The context of nonlinear combat calls for more sophisticated
measures of effectiveness. We present a set of tools that can be used
as such supplemental indicators, based on stochastic nonlinear
multivariate modeling used to benchmark Janus simulation to exercise
data from the U.S. Army National Training Center (NTC). As a prototype
study, a strong global optimization tool, adaptive simulated annealing
(ASA), is used to explicitly fit Janus data, deriving coefficients of
relative measures of effectiveness, and developing a sound intuitive
graphical decision aid, canonical momentum indicators (CMI), faithful
to the sophisticated algebraic model. We argue that these tools will
become increasingly important to aid simulation studies of the
importance of maneuver in combat in the 21st century.
Some background and results in lecture-plate form are given in
combat01_lecture.pdf
%A L. Ingber
%T Statistical Mechanics of Combat (SMC): Mathematical
Comparison of Computer Models to Exercise Data
%R SMC Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 2001
%O URL https://www.ingber.com/combat01_lecture.pdf
__________________________________________________________________
Short Introduction To Canonical Momenta
21 Oct 95
Here is an edited reply to a query on "canonical momenta," a common
feature of the above projects, that may be useful to some readers.
: Just out of curiosity. Could you briefly define "canonical momenta
: indicator?" Very few people understand the definition. If that's
: too much time, please tell me where I could find the information.
Yes, it does require some advanced physics to articulate in detail.
When dealing with truly nonlinear multivariate systems, the details go
beyond a normal PhD physics education, using some advanced calculus
developed in the late '70's.
That said, let me try to give a thumbnail sketch:
A stochastic differential equation (sde) like
x_dot = [x(t+dt) - x(t)]/dt = f(t) + g(t) n(t)
where n(t) is "white" (Wiener) noise. Here, f = f(t) and g = g(t), the
Ito representation, a favorite of economists, likely because it
requires a (really not so) "special" calculus. When f and g are not
constant, it turns out to be quite important just where in the interval
dt these are defined. When f and g are defined at the midpoint of t and
t+dt, this is the Stratonovich representation, where the standard
calculus holds.
This sde can be written as a conditional short-time distribution
p[x(t+dt) | x(t)] = (2 pi dt g**2)**-1/2 exp(- dt L)
where the "Lagrangian" L
L = (x_dot - f)**2 / (2 g**2)
(where g**2 = g * g, etc.). The long-time evolution of p is given by
the path integral, sometimes called the Chapman-Kolmogorov equation. As
finally detailed in the late '70's, in the Stratonovich representation,
L becomes the Feynman Lagrangian, and many more terms appear in L for
more than one dim when f and g are not constant; an induced Riemannian
geometry becomes explicit.
The momentum is
DL/Dx_dot = (x_dot - f)/g**2
where DL means {partial L}, etc. If we just let f be zero, we see that
L is just the "kinetic energy" in terms of "velocity" x_dot and "mass"
1/g**2. The momentum is mass "times" velocity.
In this simple one-dim example, g is just the standard deviation, but
in more than one dim, g**2 becomes the covariance matrix. It turns out
that this is the inverse-metric of the space as well, and it enters
into the calculation of the "canonical momenta." Perhaps the easiest
way to see this is to look at the third mathematically equivalent
representation, the Fokker-Planck partial differential equation (pde)
for p:
Dp/Dt = - D(f p)/Dx + 1/2 D**2(g**2 p)/(Dx)**2
This is a "Schroedinger"-type equation, and the methods developed for
statistical mechanical systems in the late 1970's are quite similar to
techniques first explored for looking at quantum gravity in 1957. The
point is that the covariance matrix g**2 enters the second partial
derivative, which "warps" x-space, similar to the effects of a
"gravitational" field.
A very simple but useful text on the physical relevance of such sde and
pdf across many physical and biological systems is
%A H. Haken
%T Synergetics
%S 3rd ed.
%I Springer
%C New York
%D 1983
There are quite a few books with the title "Synergetics," but the
"Synergetics," but the others are more specialized proceedings of
conferences; the one above is a text book.
There is a nice chapter on "The Principle of Least Action,"
illustrating how "F = ma" is derived from the Lagrangian in Volume III,
Chapter 19 of
%A R.P. Feynman
%A R.B. Leighton
%A M. Sands
%T The Feynman Lectures on Physics
%I Addison Wesley
%C Reading, MA
%D 1963
A book that shows all the gory glory of the additional complications
that must be dealt with when multivariate nonlinear systems are
considered is
%A F. Langouche
%A D. Roekaerts
%A E. Tirapegui
%T Functional Integration and Semiclassical Expansions
%I Reidel
%C Dordrecht, The Netherlands
%D 1982
My work in neuroscience since the late '70's, e.g., in the smni...
papers in my archive, first took advantage of these new mathematical
physics developments in multivariate nonequilibrium nonlinear
statistical mechanics. I then applied these techniques to markets,
nuclear physics, and combat analyses, e.g., in my markets...,
nuclear..., and combat... papers in my archive, which give more details
and references.
I developed this formalism into a practical maximum likelihood
numerical tool for fitting parameters in these kinds of systems with
the use of VFSR/ASA, e.g., illustrated with the use of some of the
asa... papers in my archive. The ASA code is in
%A L. Ingber
%T Adaptive Simulated Annealing (ASA)
%R Global optimization C-code
%I Caltech Alumni Association
%C Pasadena, CA
%D 1993
%O URL https://www.ingber.com/#ASA-CODE
Some background and results in lecture-plate form are given in
asa01_lecture.pdf
%A L. Ingber
%T Adaptive Simulated Annealing (ASA) and Path-Integral
(PATHINT) Algorithms: Generic Tools for Complex Systems
%R ASA-PATHINT Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 2001
%O URL https://www.ingber.com/asa01_lecture.pdf
The long-time evolution of such multivariate systems is accomplished
with the use of algorithms such as PATHINT and PATHTREE, e.g.,
illustrated in the path... papers in my archive. This is required, for
example, to calculate many kinds of financial instruments, e.g., bond
prices, options, derivatives, etc. The famous Black-Scholes model for
options pricing is such an example of a one-variable distribution.
People have developed numerical algorithms for each representation,
i.e., for the SDE, PDE, and the Lagrangian probability representations.
Examples of the use of PATHINT for options pricing are given in
https://www.ingber.com/markets00_exp.pdf and
https://www.ingber.com/markets00_highres.pdf and the use of PATHTREE is
given in markets01_pathtree.pdf as outlined above.
Some background and results in lecture-plate form are given in
path01_lecture.pdf
%A L. Ingber
%T Adaptive Simulated Annealing (ASA) and Path-Integral
(PATHINT) Algorithms: Generic Tools for Complex Systems
%R ASA-PATHINT Lecture Plates
%I Lester Ingber Research
%C Chicago, IL
%D 2001
%O URL https://www.ingber.com/asa01_lecture.pdf
In addition to canonical momenta used as a natural coordinate system to
study dynamically evolving multivariate systems, their value has
several explicit aspects as used in a code I developed, TRD. With
respect to my markets96_trading.pdf and markets96_momenta.pdf papers,
in addition to canonical momenta being accurate as well as intuitive
measures of "flows" to and from evolving steady states, I have made the
points:
Although only one variable, the futures SP500, was actually
traded (the code can accommodate trading on multiple markets),
note that the multivariable coupling to the cash market entered
in three important ways: (1) The SMFM fits were to the coupled
system, requiring a global optimization of all parameters in
both markets to define the time evolution of the futures market.
(2) The canonical momenta for the futures market is in terms of
the partial derivative of the full Lagrangian; the dependency on
the cash market enters both as a function of the relative value
of the off-diagonal to diagonal terms in the metric, as well as
a contribution to the drifts and diffusions from this market.
(3) The canonical momenta of both markets were used as technical
indicators for trading the futures market.
E.g., while it is common for traders to "look" at info in markets other
than those they are actually trading, this particular kind of indicator
also has the feature of including such info in a more detailed manner.
These ideas were implemented in a realtime tick-resolution trading
system, as described in markets01_optim_trading.pdf
Similarly, in EEG work, it is possible to monitor a subset of electrode
activity while incorporating information from additional electrodes.
Papers markets84_statmech.pdf, markets91_interest.pdf and
smni91_eeg.pdf contain an Appendix giving a compact derivation of the
path-integral Lagrangian representation equivalent to the Langevin
rate-equation and Fokker-Planck/Schroedinger-type representations for
multivariate systems with nonlinear drifts and diffusions.
My experience with nonlinear systems, based on my own research into
selected topics in selected disciplines as well as my interactions with
many other experts and nonlinear systems, e.g., through my support of
ASA, is that nonlinear systems are typically non-typical. It typically
requires quite a bit of work understanding the nature of each each
system before applying or evolving models and techniques borrowed from
other disciplines.
I hope this helps.
Lester
__________________________________________________________________
Gaussian-Markovian Modeling
With just dimension, D = 1, a formula for the probability of having any
value x(t) at time t+dt given the value of x(t+dt) at time t would be:
P[x, t+dt | x, t] = A exp (- L dt)
A = (2 pi dt G^2)^D/2
L = (x_dot - F)^2 / (2 G^2)
x_dot = [x(t+dt) - x(t)] / dt
F = a + b x
G = c + d x
D = 1
where just to be specific I have picked some arbitrary forms for the
functions F and G. This requires fitting {a, b, c, d} to some data, and
regularly updating the coefficients. In general, you have to try more
complex forms, e.g.,
F = a + b x + e x^2
G = c + d x + f x^2
etc.
If we are dealing with more than one dimension, D, then G become a
matrix and f become a vector, e.g., for x and y:
F_x = a + b x + c y
F_y = a + e x + f y
G_xx = g + h x
G_yy = j + k y
G_xy = G_yx = m
etc. The functions F and G can evolve, as more experience is gained
with the system.
I have developed the above formulation into a numerical procedure for
nonlinear multivariate systems that otherwise would be described as
x_dot = F_x + G_x_1 n_1 + G_x_2 n_2
y_dot = F_y + G_y_1 n_1 + G_y_2 n_2
where the n's represent independent "noise" variables and the above
{G_xx, G_yy, G_xy, G_yx} are in terms of these {G_x_1, G_x_2, G_y_1,
G_y_2}. These equations look "simpler," but in practice they are much
less reliable to fit coefficients, to get the long-time probabilities,
etc. The gory details of properly handling nonlinear F's and G's are
not touched here, but are given in my papers and references therein.
If [G_i_k] is the matrix of the coefficients in the differential rate
equations, and [G_ij] is the covariance matrix (the inverse of the
metric matrix for this space) in the Lagrangian discussed above,
[G_ij] = [G_i_k] [G_j_k]~
where the "[.]~" means the transpose matrix. Note that if there are NxM
entries in the [G_i_k] matrix (most physical systems have M >= N),
there are only at most N(N+1)/2 independent elements in the symmetric
[G_ij] matrix that enters the Lagrangian, i.e., that must be fit to the
data.
As the number of dimensions increases, so do the number of parameters
to be fit. As the functions F and G become more nonlinear, as they will
with increasing D, and as there definitely is "noise" in the data, it
becomes increasingly important to use some powerful global optimization
code like ASA to fit the data, else the coefficients will just be
modeling the "noise."
An additional hard problem arises in forecasting P for times greater
than dt in the future. You have to do the "path integral," i.e.,
integrals over all the x's for each of the intermediate times:
P[x, T | x, t] =
int dx(t) dx(t + dt) dx(t+ 2 dt) ... dx(T - dt)
P[x, t + dt | x, t]
P[x, t + 2 dt | x, t + dt]
P[x, t + 3 dt | x, t + 2 dt]
...
P[x, T - dt | x, T - 2 dt]
P[x, T | x, T - dt]
where of course x is now a vector, so each dx means
dx(t) = dx_1(t) ... dx_N(t)
where N is the dimension of the model being developed.
Only if very simple forms for F and G are selected, can these integrals
be performed to give a closed expression; otherwise they must be
performed numerically. PATHINT and PATHTREE do this for _some_ classes
of integrals. However, even for dimension D = 2 the CPU time is large,
and for D = 3 or 4, this might be huge. If simple forms are taken for
the F's and G's, all the global optimization fits still must be
performed, but perhaps you can get by without using something like
PATHINT.
A paper on the general utility of this approach to modeling complex
systems is given in
path98_datamining.pdf
%A L. Ingber
%T Data mining and knowledge discovery via statistical mechanics
in nonlinear stochastic systems
%J Mathematical Computer Modelling
%V 27
%N 3
%P 9-31
%D 1998
%O URL https://www.ingber.com/path98_datamining.pdf
ABSTRACT: A modern calculus of multivariate nonlinear multiplicative
Gaussian-Markovian systems provides models of many complex systems
faithful to their nature, e.g., by not prematurely applying
quasi-linear approximations for the sole purpose of easing analysis. To
handle these complex algebraic constructs, sophisticated numerical
tools have been developed, e.g., methods of adaptive simulated
annealing (ASA) global optimization and of path integration (PATHINT).
In-depth application to three quite different complex systems have
yielded some insights into the benefits to be obtained by application
of these algorithms and tools, in statistical mechanical descriptions
of neocortex (short-term memory and electroencephalography), financial
markets (interest-rate and trading models), and combat analysis
(baselining simulations to exercise data).
__________________________________________________________________
SOME ISSUES IN MATHEMATICAL MODELING
__________________________________________________________________
Risk Control of Mathematical Models such as PATHTREE
Jun 10
PATHTREE was developed in response to a problem faced by a trader, who
felt that the Black-Scholes (BS) pricing model was too sensitive to the
BS distribution. His empirical solution was to distort the scale of
input underlying variables, e.g., by factors of thousands (larger than
any prices on the open market!?), and then to similarly scale back the
output. I thought that using the actual distribution of the data would
be best, even if the distribution was quite an odd-shape -- as long as
the cumulative distribution was a bona fide distribution. This meant
developing an algorithm that, while still delivering all common Greeks
used for trading, could process distributions with quite general first
and second moments, e.g., nonlinear in the dependent variables and
time, etc. The focus was to fit parameters defining such odd shapes to
strikes of the options, thereby developing a bottom-up approach to
"smiles" (non-linear curves of strikes versus variables like
volatility). (The usual top-down approach to smiles is to
interpolate/extrapolate output of BS models to the strike data.)
Just about as soon as I articulated the problem, I realized that my
past published works in several disciplines, using nonlinear
nonequilibrium multivariate Gaussian-Markovian distributions, was
relevant. That is, it turns out that even though the first two moments
of a distribution might only be accurate to order (dt)^(1/2), the
conditional probability distribution is accurate to order (dt)^(3/2).
(This requires quite a bit of heavy math to prove, as referenced in my
publications.) Accuracy to order (dt) is required to use standard
numerical binomial trees to solve the associated options equations,
whether written equivalently as differential stochastic equations,
Fokker-Planck equations, or path-integral equations, the latter being
the representation of the binomial tree algorithm. In a few minutes I
modified the binomial-tree code we were using to test this idea, and it
worked perfectly!
I then had my team thoroughly test this algorithm, by calculating and
plotting all options being used by all traders in the firm, with
respect to all variables and parameters being used at the time for
their BS codes, e.g., asset price, strike price, time to expiration,
risk free rate, cost of carry, volatility, volatility step size, number
of time steps, American and European options, and boundary values at
far distances (factors of 1/10 to 10) from actual values touched by the
present calculations, to ensure proper behavior for extreme
calculations. This produced thousand of pages of graphs, which were
divided among the team to examine for any odd behavior. We found none.
Just to be careful, and email was sent to all traders, telling them
that if they started trading any new options or present options within
new ranges of any variables, they were to first contact me so I could
run their options through these tests which were now part of the code.
As it of course would turn out, someone did not heed the warnings. The
trader for whom the new algorithm was developed just decided that
reality was not good enough, and he persisted in using scales at
thousands of times the actual data. After a while, going along quite
well but not telling us of his mangling of the basic codes, he hit a
problem in some discontinuity of results among different strikes and he
lost a lot of money, which he attributed probably incorrectly to the
code per se. When he called in the midst of trading, after a few
minutes of reflection, I had my team immediately run his values, and we
in fact saw this small discontinuity. I immediately guessed it had to
do with the way boundaries were being enforced across the parameterized
distribution and the far-away distribution (which he was hitting with
his huge scaling of the data). This was an easy immediate fix and the
new code was delivered to him in less than half an hour after his call.
Of course I was blamed for the error, which was true, except of course
the error would never have been used if the trader has used the due
diligence he was warned to exercise.
The "moral" of this incident is that (a) "Black Swans" (unknown unknown
future events beyond known stochastic models) always are potentially
threatening, (b) risk controls must be in place to mitigate such
threats, and (c) all players must follow the discipline of such risk
controls. We had risk controls in place, but they were not followed
when the model being used was used beyond its intended context for
which it was tested within broad risk bands of 10-100 (but not 1000's).
Mathematical models are just those, and they not only depend on good
data, but also on controls to account for errors/deficiencies in the
models when they are stretched beyond their original domains. Such
abuses also were present on the first decade of this century,
leveraging the nightmare of the meltdown of the global financial
system.
This new algorithm was eventually published in:
%A L. Ingber
%A C. Chen
%A R.P. Mondescu
%A D. Muzzall
%A M. Renedo
%T Probability tree algorithm for general diffusion processes
%J Physical Review E
%V 64
%N 5
%P 056702-056707
%D 2001
%O URL https://www.ingber.com/path01_pathtree.pdf
__________________________________________________________________
Interdisciplinary Reviews of Applications of Mathematical Physics
1997
Every worthwhile project I've undertaken -- from Nuclear Physics to
Statistical Mechanics (SM) projects to Karate -- each took some years
of study to understand a basic set of problems in a given discipline.
It just takes a lot of time to get familiar with the current issues of
a discipline as viewed by experts, to understand the most important
unsolved or poorly solved issues, and to see if new approaches can be
applied to create better solutions, without being distracted by
focusing on easier different problems that do not address the important
issues. It is to be expected that a newcomer to any discipline should
be expected to do the "homework" required. I addressed real problems
requiring such studies, not only to appreciate the core problems at an
expert level, but to forge the necessary tools to solve these problems.
In the SM studies, this required creating such tools as ASA, PATHINT
and PATHTREE, and many other lesser-known algorithms.
Some tools had to be developed to accommodate particular constraints.
Similar to optimizing some complex systems, some projects require
constraints additional to simply developing analytic solutions to
scientific problems, e.g., including an institution's focus,
capabilities of team members, ability of decision makers to work with
results, etc. These projects can be quite interesting and challenging,
requiring blending mathematical science with social or personnel
objectives, forming a larger system to be considered. In such cases,
once a reasonably solid intuitive understanding is reached for an
approach, analysis should strive to not waiver from this intuition, but
to be faithful to the original understanding. Of course, often further
analysis uncovers new aspects of a system demanding attention and
modification, often requiring problems and solutions to converge only
at late stages of solution. This is different from simply giving into a
too common temptation for analysts to bend their analysis to suit the
ease of solution rather than to deliver projects that solve the
original problem.
When my Statistical Mechanics of Neocortical Interactions (SMNI)
approach first appeared circa 1978, it was not hard to understand why
many people in neuroscience trained either in medicine or abstract
mathematics could not readily understand this mathematical-physics
approach to the neocortex. Especially at that time, neural networks
were much simpler to understand, albeit conveniently glossing over
issues of relevance to real neocortex. However, even supporters of this
work with extremely solid scientific credentials did not stop a few
other reviewers from attempting to cut funding and publication of this
work. One early reviewer, a well-known neurologist claimed that the
math was fabricated as a smoke screen to cover the results claimed by
the algebraic and numerical calculations. Since then, SMNI papers have
successfully applied this math-physics to numerically detail properties
of EEG and short-term memory.
When my Statistical Mechanics of Financial Markets (SMFM) approach
first appeared circa 1982, the paper finally published in 1984 was
delayed a couple of years. For example, an editor of a premier
economics journal agreed with a reviewer (who stated he had a
graduate-student physics background) that such math as used in that
paper did not exist and could not be correct. Since then, this
math-physics has successfully been applied to derivatives, trading
systems and risk-management of portfolios. Success in trading firms
I've worked with, albeit a profitable enterprise, most certainly is not
documentation of validity of this work, and I have managed to publish
some of this work to get genuine peer review.
When my Statistical Mechanics of Combat (SMC) approach first appeared
circa 1987, in the context of studies of studies on large-scale
government simulations, several operations-research/statistician
government analysts stated a similar complaint that they could not find
any such math in their previous math or physics textbooks. One OR
professor actually sent out letters to Department Heads claiming I was
promoting bogus studies. They claimed "path integrals" were only a
theoretical abstraction without any foundation for application to
anything except quantum mechanics -- pretty crazy complaints!. After
the dust settled somewhat, I was thanked by an Asst. Secretary of the
Army for my work in promoting this work, which led to use of these
simulations in training. Upon request, I may give a URL to a file
giving more detail on this bizarre tragicomedy, including intervention
required by a Congressman.
Clearly, the overwhelming majority of other reviewers of my work, who
endorsed publication, have at the least established that these fewer
critics were uninformed. Such meritless and meretricious reviews only
bring into question the integrity of the reviewers. As I am quick to
point out, these people were in the minority of reviewers. Nevertheless
they represent a set of people that should be excluded from any serious
reviewing of just about anything, but especially any interdisciplinary
research.
In a given focused discipline, like nuclear physics, most often authors
get pretty fair reviews, albeit this process can get strained a bit
when a reviewer's own grants and contracts are threatened. In martial
arts you can at least try to defend yourself face-to-face against your
opponent -- until you get to a high enough rank that you have to deal
with their politics!. Interdisciplinary research requires much more of
reviewers than just extracting their professional opinions on their own
documented expertise in a given specialty. They have to be honest with
themselves, and often with their colleagues, on just what they know and
what they do not know. Good common-sense judgment (intuitive and
probabilistic, tempered by analysis and experience), separate from
previous expert knowledge often is required. Interdisciplinary
reviewers must be prepared to acknowledge that they are not capable of
reviewing a particular paper or project. My own experience with the
frequent failure of interdisciplinary reviewing has led me to take care
to honor this practice, and I often review papers and contracts for
several institutions.
In Science, as elsewhere, opinions issued as dogma spewed by tyrants
must be met by equal and opposite forces of reason and experimental
data.
The only way to preserve integrity is to always tell the truth and to
be true to your own intuitions and analyses, albeit this
self-discipline most certainly will not make everyone happy.
__________________________________________________________________
How I Think
26 March 2011
Every since I got my skull cracked open by a spoon during an argument
over a red truck when I was about two years old, I've had problems
holding on to chains of thought. I quickly learned to compensate by
"thinking" in overlapping patterns, so that whenever such a lapse
occurs, I just about always can quickly reconstruct my chain of
thought. At a certain age, like mine at 70, these are often described
as "senior moments," but I have had these moments all my life. I think
this has turned into a asset, making me very creative in all my
endeavors, as I uncover new patterns of information relying on such
processes more than most people do, instead of having to be led by
logic.
__________________________________________________________________
__________________________________________________________________
Lester Ingber
Copyright (c) 1994-2019 Lester Ingber. All Rights Reserved.
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