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
https://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
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
http://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 http://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
http://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/
article/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
http://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) 1994-2018 Lester Ingber. All Rights Reserved.
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