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Unformatted text preview: Canonical Momenta Indicators of Financial Markets and Neocortical EEG
Lester Ingber
Lester Ingber Research
P.O. Box 857, McLean, Virginia 22101, U.S.A.
[email protected], [email protected]
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 outofsample 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 mathematicalphysics formalism to generate indicators which are
used by AItype rulebased models of management.
1. Introduction
Over a decade ago, the author published a paper suggesting the use of newly developed methods of
multivariate nonlinear nonequilibrium calculus to approach a statistical mechanics of ﬁnancial markets (SMFM) [1].
These methods were applied to interestrate termstructure systems [2,3]. Still, for some time, the standard accepted
paradigm of ﬁnancial markets has been rooted in equilibrium processes [4]. There is a current effort by many to
examine nonlinear and nonequilibrium processes in these markets [5], and this paper reinforces this point of view.
Another paper gives some earlier 1991 results using this approach [6].
There are several issues that are clariﬁed here, by presenting calculations of a speciﬁc trading model: (A) It is
demonstrated how multivariate markets might be formulated in a nonequilibrium paradigm. (B) It is demonstrated
that numerical methods of global optimization can be used to ﬁt such SMFM models to data. (C) A variational
principle possessed by SMFM permits derivation of technical indicators, such as canonical momenta, that can be
used to describe deviations from most likely evolving states of the multivariate system. (D) These technical
indicators can be embedded in realistic trading scenarios, to test whether they can proﬁt from nonequilibrium in
markets.
Section 2 outlines the formalism used to develop the nonlinear nonequilibrium SMFM model. Section 3
describes application of SMFM to SP500 cash and future data, using Adaptive Simulated Annealing (ASA) [7] to ﬁt
the shorttime conditional probabilities developed in Section 2, and to establish trading rules by recursively
optimizing with ASA, using optimized technical indicators developed from SMFM. These calculations were brieﬂy
mentioned in another ASA paper [8]. Section 4 describes similar applications, now in progress, to correlating
customized electroencephalographic (EEG) momenta indicators to physiological and behavioral states of humans.
Section 5 is a brief conclusion.
2. SMFM Model
2.1. Random walk model
The use of Brownian motion as a model for ﬁnancial systems is generally attributed to Bachelier [9], though
he incorrectly intuited that the noise scaled linearly instead of as the square root relative to the random logprice
variable. Einstein is generally credited with using the correct mathematical description in a larger physical context
of statistical systems. However, several studies imply that changing prices of many markets do not follow a random
walk, that they may have longterm dependences in price correlations, and that they may not be efﬁcient in quickly
arbitraging new information [1012]. A random walk for returns, rate of change of prices over prices, is described
by a Langevin equation with simple additive noise η , typically representing the continual random inﬂux of
information into the market. ˙
Γ = −γ 1 + γ 2η ,
˙
Γ = d Γ/ dt ,
< η (t ) >η = 0 , < η (t ), η (t ′) >η = δ (t − t ′) ,
(1)
where γ 1 and γ 2 are constants, and Γ is the logarithm of (scaled) price. Price, although the most dramatic observable, may not be the only appropriate dependent variable or order parameter for the system of markets [13].
This possibility has also been called the “semistrong form of the efﬁcient market hypothesis” [10].
It is necessary to explore the possibilities that a given market evolves in nonequilibrium, e.g., evolving
irreversibly, as well as nonlinearly, e.g., γ 1,2 may be functions of Γ. Irreversibility, e.g., causality [14] and
nonlinearity [15], have been suggested as processes necessary to take into account in order to understand markets,
but modern methods of statistical mechanics now provide a more explicit paradigm to consistently include these
processes in bona ﬁde probability distributions. Reservations have been expressed about these earlier models at the Canonical momenta indicators of ﬁnancial markets 2 Lester Ingber time of their presentation [16].
Developments in nonlinear nonequilibrium statistical mechanics in the late 1970’s and their application to a
variety of testable physical phenomena illustrate the importance of properly treating nonlinearities and
nonequilibrium in systems where simpler analyses prototypical of linear equilibrium Brownian motion do not
sufﬁce [17].
2.2. Statistical mechanics of large systems
Aggregation problems in nonlinear nonequilibrium systems, e.g., as deﬁnes a market composed of many
traders [1], typically are “solved” (accommodated) by having new entities/languages developed at these disparate
scales in order to efﬁciently pass information back and forth [18,19]. This is quite different from the nature of quasiequilibrium quasilinear systems, where thermodynamic or cybernetic approaches are possible. These approaches
typically fail for nonequilibrium nonlinear systems.
These new methods of nonlinear statistical mechanics only recently have been applied to complex largescale
physical problems, demonstrating that observed data can be described by the use of these algebraic functional forms.
Success was gained for largescale systems in neuroscience, in a series of papers on statistical mechanics of
neocortical interactions [2030], and in nuclear physics [3133]. This methodology has been used for problems in
combat analyses [19,3437]. These methods have been suggested for ﬁnancial markets [1], applied to a term
structure model of interest rates [2,3], and to optimization of trading [6].
2.3. Statistical development
When other order parameters in addition to price are included to study markets, Eq. (1) is accordingly
generalized to a set of Langevin equations. ˙G
M = f G + gG η j , (G = 1, . . . , Λ) , ( j = 1, . . . , N ) ,
ˆj
˙G
M = dM G / d Θ ,
< η j (Θ) >η = 0 , < η j (Θ), η j ′ (Θ′) >η = δ jj ′δ (Θ − Θ′) ,
G gG
ˆj (2)
G are generally nonlinear functions of mesoscopic order parameters M , j is a microscopic index
where f and
indicating the source of ﬂuctuations, and N ≥ Λ. The Einstein convention of summing over repeated indices is
used. Vertical bars on an index, e.g., j, imply no sum is to be taken on repeated indices. Θ is used here to
emphasize that the most appropriate time scale for trading may not be real time t .
Via a somewhat lengthy, albeit instructive calculation, outlined in several other papers [1,3,25], involving an
¨
intermediate derivation of a corresponding FokkerPlanck or Schrodingertype equation for the conditional
probability distribution P [ M (Θ) M (Θ0 )], the Langevin rate Eq. (2) is developed into the probability distribution
for M G at longtime macroscopic time event Θ = (u + 1)θ + Θ0 , in terms of a Stratonovich pathintegral over
mesoscopic Gaussian conditional probabilities [3840]. Here, macroscopic variables are deﬁned as the longtime
¨
limit of the evolving mesoscopic system. The corresponding Schrodingertype equation is [39,41] ∂ P /∂Θ = 1
2 ( gGG ′ P ),GG ′ − ( gG P ),G + V , gGG ′ = k T δ jk gG gG ′ , gG = f G +
ˆ j ˆk 1
2 δ jk G ′ G
g j g k ,G ′
ˆˆ , [. . .],G = ∂[. . .]/∂ M G . (3) This is properly referred to as a FokkerPlanck equation when V ≡ 0. Note that although the partial differential Eq.
(3) contains equivalent information regarding M G as in the stochastic differential Eq. (2), all references to j have
ˆj
been properly averaged over. I.e., gG in Eq. (2) is an entity with parameters in both microscopic and mesoscopic
spaces, but M is a purely mesoscopic variable, and this is more clearly reﬂected in Eq. (3).
The path integral representation is given in terms of the Lagrangian L . P [ M Θ  M Θ0 ] dM (Θ) =
S= −
k T1 Θ min ∫ d Θ′ L ,
Θ
0 ∫ . . . ∫ DM exp(−S )δ [ M (Θ0) = M0]δ [ M (Θ) = MΘ] , Canonical momenta indicators of ﬁnancial markets 3 Lester Ingber u +1 Π g1/2 Π (2π θ )−1/2 dM G ,
ρ
G DM = lim u→∞ ρ =1 G ˙
L ( M , M G , Θ) =
hG = gG − 1
2 1
2 G ˙
˙
( M − hG ) gGG ′ ( M G′ − hG ′ ) + 1
2 hG ;G + R/6 − V , g−1/2 ( g1/2 gGG ′ ),G ′ , gGG ′ = ( gGG ′ )−1 , g = det( gGG ′ ) ,
F
hG ;G = hG + ΓGF hG = g−1/2 ( g1/2 hG ),G ,
,G Γ F ≡ g LF [ JK , L ] = g LF ( g JL , K + g KL , J − g JK , L ) ,
JK
R = g JL R JL = g JL g JK R FJKL ,
R FJKL = 1
2 MN
MN
( g FK , JL − g JK , FL − g FL , JK + g JL , FK ) + g MN (Γ FK Γ JL − Γ FL Γ JK ) . (4) Mesoscopic variables have been deﬁned as M G in the Langevin and FokkerPlanck representations, in terms of their
development from the microscopic system labeled by j . The Riemannian curvature term R arises from nonlinear
gGG ′ , which is a bona ﬁde metric of this parameter space [39].
2.4. Algebraic complexity yields simple intuitive results
It must be emphasized that the output need not be conﬁned to complex algebraic forms or tables of numbers.
Because L possesses a variational principle, sets of contour graphs, at different longtime epochs of the pathintegral
of P over its variables at all intermediate times, give a visually intuitive and accurate decisionaid to view the
dynamic evolution of the scenario. For example, this Lagrangian approach permits a quantitative assessment of
concepts usually only loosely deﬁned. “Momentum” = ΠG =
“Mass” gGG ′
“Force” = ∂L
,
∂(∂ M G /∂Θ) ∂2 L
=
,
∂(∂ M G /∂Θ)∂(∂ M G ′ /∂Θ) ∂L
,
∂MG ∂L
∂
∂L
−
,
(5)
G
∂M
∂Θ ∂(∂ M G /∂Θ)
where M G are the variables and L is the Lagrangian. These physical entities provide another form of intuitive, but
“F = ma”: δ L = 0 = quantitatively precise, presentation of these analyses. For example, daily newspapers use this terminology to discuss
the movement of security prices. Here, we will use the canonical momenta as indicators to develop trading rules.
2.5. Fitting parameters
The shorttime pathintegral Lagrangian of a Λdimensional system can be developed into a scalar “dynamic
cost function,” C , in terms of parameters, e.g., generically represented as C (α ),
˜ C (α ) = L ∆Θ +
˜ Λ
2 ln(2π ∆Θ) − 1
2 ln g , (6) which can be used with the ASA algorithm [7], originally called Very Fast Simulated Reannealing (VFSR) [42], to
ﬁnd the (statistically) best ﬁt of parameters. The cost function for a given system is obtained by the product of P ’s
over all data epochs, i.e., a sum of C ’s is obtained. Then, since we essentially are performing a maximum likelihood
ﬁt, the cost functions obtained from somewhat different theories or data can provide a relative statistical measure of
their likelihood, e.g., P 12 ∼ exp(C 2 − C 1 ).
If there are competing mathematical forms, then it is advantageous to utilize the pathintegral to calculate the
longtime evolution of P [19,35]. Experience has demonstrated that the longtime correlations derived from theory, Canonical momenta indicators of ﬁnancial markets 4 Lester Ingber measured against the observed data, is a viable and expedient way of rejecting models not in accord with observed
evidence.
2.6. Numerical methodology
ASA [42] ﬁts shorttime probability distributions to observed data, using a maximum likelihood technique on
the Lagrangian. This algorithm has been developed to ﬁt observed data to a theoretical cost function over a D dimensional parameter space [42], adapting for varying sensitivities of parameters during the ﬁt.
Simulated annealing (SA) was developed in 1983 to deal with highly nonlinear problems [43], as an
extension of a MonteCarlo importancesampling technique developed in 1953 for chemical physics problems. It
helps to visualize the problems presented by such complex systems as a geographical terrain. For example, consider
a mountain range, with two “parameters,” e.g., along the North−South and East−West directions. We wish to ﬁnd
the lowest valley in this terrain. SA approaches this problem similar to using a bouncing ball that can bounce over
mountains from valley to valley. We start at a high “temperature,” where the temperature is an SA parameter that
mimics the effect of a fast moving particle in a hot object like a hot molten metal, thereby permitting the ball to
make very high bounces and being able to bounce over any mountain to access any valley, given enough bounces.
As the temperature is made relatively colder, the ball cannot bounce so high, and it also can settle to become trapped
in relatively smaller ranges of valleys.
We imagine that our mountain range is aptly described by a “cost function.” We deﬁne probability
distributions of the two directional parameters, called generating distributions since they generate possible valleys or
states we are to explore. We deﬁne another distribution, called the acceptance distribution, which depends on the
difference of cost functions of the present generated valley we are to explore and the last saved lowest valley. The
acceptance distribution decides probabilistically whether to stay in a new lower valley or to bounce out of it. All the
generating and acceptance distributions depend on temperatures.
In 1984 [44], it was established that SA possessed a proof that, by carefully controlling the rates of cooling of
temperatures, it could statistically ﬁnd the best minimum, e.g., the lowest valley of our example above. This was
good news for people trying to solve hard problems which could not be solved by other algorithms. The bad news
was that the guarantee was only good if they were willing to run SA forever. In 1987, a method of fast annealing
(FA) was developed [45], which permitted lowering the temperature exponentially faster, thereby statistically
guaranteeing that the minimum could be found in some ﬁnite time. However, that time still could be quite long.
Shortly thereafter, in 1987 the author developed Very Fast Simulated Reannealing (VFSR) [42], now called Adaptive
Simulated Annealing (ASA), which is exponentially faster than FA. It is used worldwide across many
disciplines [8], and the feedback of many users regularly scrutinizing the source code ensures the soundness of the
code as it becomes more ﬂexible and powerful [46].
ASA has been applied to many problems by many people in many disciplines [8,46,47]. The code is
available via anonymous ftp from ftp.ingber.com, which also can be accessed via the worldwide web (WWW) as
http://www.ingber.com/.
3. Fitting SMFM to SP500
3.1. Data processing
For the purposes of this paper, it sufﬁces to consider a twovariable problem, SP500 prices of futures, p1 , and
cash, p2 . (Note that in a previous paper [6], these two variables were inadvertently incorrectly reversed.) Data
included 251 points of 1989 and 252 points of 1990 daily closing data. Time between data was taken as real time t ,
e.g., a weekend added two days to the time between data of a Monday and a previous Friday.
It was decided that relative data should be more important to the dynamics of the SMFM model than absolute
data, and an arbitrary form was developed to preprocess data used in the ﬁts, M i (t ) = pi (t + ∆t )/ pi (t ) , (7) where i = {1, 2} = {futures, cash}, and ∆t was the time between neighboring data points, and t + ∆t is the current
trading time. The ratio served to served to suppress strong drifts in the absolute data.
3.2. ASA ﬁts of SMFM to data
Two source of noise were assumed, so that the equations of this SMFM model are
2
2
dM G
G
= Σ f G ′ M G ′ + Σ gG η i , G = {1, 2} .
ˆi
dt
i =1
G ′=1 (8) G
ˆi
The 8 parameters, { f G ′ , gG } were all taken to be constants.
As discussed previously, the pathintegral representation was used to deﬁne an effective cost function.
Minimization of the cost function was performed using ASA. Some experimentation with the ﬁtting process led to a
scheme whereby after sufﬁcient importancesampling, the optimization was shunted over to a quasilocal code, the Canonical momenta indicators of ﬁnancial markets 5 Lester Ingber BroydenFletcherGoldfarbShanno (BFGS) algorithm [48], to add another decimal of precision. If ASA was
shunted over too quickly to BFGS, then poor ﬁts were obtained, i.e., the ﬁt stopped in a higher local minimum.
G
ˆi
Using 1989 data, the parameters f G ′ were constrained to lie between 1.0 and 1.0. The parameters gG were
1
constrained to lie between 0 and 1.0. The values of the parameters, obtained by this ﬁtting process were: f 1 =
1
1
1
2
2
2
ˆ
ˆ
ˆ
0.0686821, f 2 = −0.068713, g1 = 0.000122309, g2 = 0.000224755, f 1 = 0.645019, f 2 = −0.645172, g1 =
ˆ2
0.00209127, g2 = 0.00122221. 3.3. ASA ﬁts of trading rules
A simple model of trading was developed. Two timeweighted moving averages, of wide and narrow
windows, a w and a n were deﬁned for each of the two momenta variables. During each new epoch of a w , always
using the ﬁts of the SMFM model described in the previous section as a zeroth order estimate, the parameters
G
{ f G ′ , gG } were reﬁt using data within each epoch. Averaged canonical momenta, i.e., using Eq. (5), were
ˆi
calculated for each new set of a w and a n windows. Fluctuation parameters ∆ΠG and ∆ΠG , were deﬁned, such that
w
n
any change in trading position required that there was some reasonable information outside of these ﬂuctuations that
could be used as criteria for trading decisions. No trading was performed for the ﬁrst few days of the year until the
momenta could be calculated. Commissions of $70 were paid every time a new trade of 100 units was taken. Thus,
there were 6 trading parameters used in this example, { a w , a n , ∆ΠG , ∆ΠG }.
w
n
The order of choices made for daily trading are as follows. A 0 represents no positions are open and no
trading is performed until enough data is gathered, e.g., to calculate momenta. A 1 represents entering a long
position, whether from a waiting or a short position, or a current long position was maintained. This was performed
if the both widewindow and narrowwindow averaged momenta of both cash and futures prices were both greater
than their ∆ΠG and ∆ΠG ﬂuctuation parameters. A −1 represents entering a short position, whether from a waiting
w
n
or a long position, or a current short position was maintained. This was performed if the both widewindow and
narrowwindow averaged momenta of both cash and futures prices were both less than their ∆ΠG and ∆ΠG
w
n
ﬂuctuation parameters.
3.4. Insample ASA ﬁts of trading rules
For the data of 1989, recursive optimization was performed. The trading parameters were optimized in an
outer shell, using the negative of the net yearly proﬁt/loss as a cost function. This could have been weighted by
something like the absolute value of maximum loss to help minimize risk, but this was not done here. The inner
shell of optimization ﬁnetuning of the SMFM model was performed daily over the current a w epoch.
At ﬁrst, ASA and shunting over to BFGS was used for each shell, but it was realized that good results could
be obtained using ASA and BFGS on the outer shell, and just BFGS on the inner shell (always using the ASA and
BFGS derived zeroth order SMFM parameters as described above). Thus, recursive optimization was performed to
establish the required goodnessofﬁt, and more efﬁcient local optimization was used only in those instances where
it could replicate the global optimization. This is expected to be quite system dependent.
The tradingrule parameters were constrained to lie within the following ranges: a w integers between 15 and
25, a n integers between 3 and 14, ∆ΠG and ∆ΠG between 0 and 200. The trading parameters ﬁt by this procedure
w
n
were: a w = 18, a n = 11, ∆Π1 = 30.3474, ∆Π2 = 98.0307, ∆Π1 = 11.2855, ∆Π2 = 54.8492.
w
w
n
n
The summary of results was: cumulative proﬁt = $54170, number of proﬁtable long positions = 11, number of
proﬁtable short positions = 8, number of losing long positions = 5, number of losing short positions = 6, maximum
proﬁt of any given trade = $11005, maximum loss of any trade = −$2545, maximum accumulated proﬁt during year
= $54170, maximum loss sustained during year = $0.
3.5. Outofsample SMFM trading
The trading process described above was applied to the 1990 outofsample SP500 data. Note that 1990 was
a “bear” market, while 1989 was a “bull” market. Thus, these two years had quite different overall contexts, and this
was believed to provide a stronger test of this methodology than picking two years with similar contexts.
The inner shell of optimization was performed as described above for 1990 as well. The summary of results
was: cumulative proﬁt = $28300, number of proﬁtable long positions = 10, number of proﬁtable short positions = 6,
number of losing long positions = 6, number of losing short positions = 10, maximum proﬁt of any given trade =
$6780, maximum loss of any trade = −$2450, maximum accumulated proﬁt during year = $29965, maximum loss
sustained during year = −$5945. Tables of results are available as ﬁle markets96_momenta_tbl.txt.Z in
http://www.ingber.com/MISC.DIR/ and ftp.ingber.com/MISC.DIR.
Only one variable, the futures SP500, was actually traded, albeit the code can accommodate trading on
multiple markets. There is more leverage and liquidity in actually trading the futures market. 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 offdiagonal to diagonal terms in Canonical momenta indicators of ﬁnancial markets 6 Lester Ingber 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.
3.6. Reversing data sets
The same procedures described above were repeated, but using the 1990 SP500 data set for training and the
1989 data set for testing.
G
For the training phase, using 1990 data, the parameters f G ′ were constrained to lie between 1.0 and 1.0. The
G
ˆ
parameters gi were constrained to lie between 0 and 1.0. The values of the parameters, obtained by this ﬁtting
1
1
2
2
process were: f 1 = 0.0685466, f 2 = −0.068571, g1 = 7.52368 10−6 , g1 = 0.000274467, f 1 = 0.642585, f 2 =
ˆ1
ˆ2
2
−5
2
−0.642732, g1 = 9.30768 10 , g2 = 0.00265532. Note that these values are quite close to those obtained above
ˆ
ˆ
when ﬁtting the 1989 data.
The tradingrule parameters were constrained to lie within the following ranges: a w integers between 15 and
25, a n integers between 3 and 14, ∆ΠG and ∆ΠG between 0 and 200. The trading parameters ﬁt by this procedure
w
n
were: a w = 11, a n = 8, ∆Π1 = 23.2324, ∆Π2 = 135.212, ∆Π1 = 169.512, ∆Π2 = 9.50857,
w
w
n
n
The summary of results was: cumulative proﬁt = $42405, number of proﬁtable long positions = 11, number of
proﬁtable short positions = 8, number of losing long positions = 7, number of losing short positions = 6, maximum
proﬁt of any given trade = $8280, maximum loss of any trade = −$1895, maximum accumulated proﬁt during year =
$47605, maximum loss sustained during year = −$2915.
For the testing phase, the summary of results was: cumulative proﬁt = $35790, number of proﬁtable long
positions = 10, number of proﬁtable short positions = 6, number of losing long positions = 6, number of losing short
positions = 3, maximum proﬁt of any given trade = $9780, maximum loss of any trade = −$4270, maximum
accumulated proﬁt during year = $35790, maximum loss sustained during year = $0. Tables of results are available
as ﬁle markets96_momenta_tbl.txt.Z in http://www.ingber.com/MISC.DIR/ and ftp.ingber.com/MISC.DIR. 4. Extrapolations to EEG
4.1. Customized Momenta Indicators of EEG
These techniques are quite generic, and can be applied to a model of statistical mechanics of neocortical
interactions (SMNI) which has utilized similar mathematical and numerical algorithms [2023,25,26,29,30,49]. In
this approach, the SMNI model is ﬁt to EEG data, e.g., as previously performed [25]. This develops a zeroth order
guess for SMNI parameters for a given subject’s training data. Next, ASA is used recursively to seek parameterized
predictor rules, e.g., modeled according to guidelines used by clinicians. The parameterized predictor rules form an
outer ASA shell, while regularly ﬁnetuning the SMNI innershell parameters within a moving window (one of the
outershell parameters). The outershell cost function is deﬁned as some measure of successful predictions of
upcoming EEG events.
In the testing phase, the outershell parameters ﬁt in the training phase are used in outofsample data. Again,
the process of regularly ﬁnetuning the innershell of SMNI parameters is used in this phase.
If these SMNI techniques can ﬁnd patterns of such such upcoming activity some time before the trained eye
of the clinician, then the costs of time and pain in preparation for surgery can be reduced. This project will
determine interelectrode and intraelectrode activities prior to spike activity to determine likely electrode circuitries
highly correlated to the onset of seizures. This can only do better than simple averaging or ﬁltering of such activity,
as typically used as input to determine dipole locations of activity prior to the onset of seizures.
If a subset of electrode circuitries are determined to be highly correlated to the onset of seizures, then their
associated regions of activity can be used as a ﬁrst approximate of underlying dipole sources of brain activity
affecting seizures. This ﬁrst approximate may be better than using a spherical head model to deduce such a ﬁrst
guess. Such ﬁrst approximates can then be used for more realistic dipole source modeling, including the actual
shape of the brain surface to determine likely localized areas of diseased tissue.
These momenta indicators should be considered as supplemental to other clinical indicators. This is how they
are being used in ﬁnancial trading systems.
5. Conclusion
A complete sample scenario has been presented: (a) developing a multivariate nonlinear nonequilibrium
model of ﬁnancial markets; (b) ﬁtting the model to data using methods of ASA global optimization; (c) deriving
technical indicators to express dynamics about most likely states; (d) optimizing trading rules using these technical
indicators; (e) trading on outofsample data to determine if steps (a)−(d) are at least sufﬁcient to proﬁt by the
knowledge gained of these ﬁnancial markets, i.e., these markets are not efﬁcient.
Just based the models and representative calculations presented here, no comparisons can yet be made of any
relative superiority of these techniques over other models of markets and other sets of trading rules. Rather, this
exercise should be viewed as an explicit demonstration (1) that ﬁnancial markets can be modeled as nonlinear
nonequilibrium systems, and (2) that ﬁnancial markets are not efﬁcient and that they can be properly ﬁt and Canonical momenta indicators of ﬁnancial markets 7 Lester Ingber proﬁtably traded on real data.
Canonical momenta may offer an intuitive yet detailed coordinate system of some complex systems, which
can be used as reasonable indicators of new and/or strong trends of behavior, upon which reasonable decisions and
actions can be based. A description has been given of a project in progress, using this same methodology to
customize canonical momenta indicators of EEG to human behavioral and physiological states [50].
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This note was uploaded on 10/24/2011 for the course SCIENCE PHY 453 taught by Professor Barnard during the Winter '11 term at BYU.
 Winter '11
 BARNARD
 Physics, mechanics, pH, Statistical Mechanics, The Land

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