=Paper=
{{Paper
|id=Vol-2280/paper-20
|storemode=property
|title=Some Suggestions for Graduate Students and Scholars Undertaking Quantitative Interdisciplinary
Research: Remarks from The Practice
|pdfUrl=https://ceur-ws.org/Vol-2280/paper-20.pdf
|volume=Vol-2280
|authors=Dode Prenga,Safet Sula
|dblpUrl=https://dblp.org/rec/conf/rtacsit/PrengaS18
}}
==Some Suggestions for Graduate Students and Scholars Undertaking Quantitative Interdisciplinary
Research: Remarks from The Practice==
https://ceur-ws.org/Vol-2280/paper-20.pdf
Some suggestions for graduate students and scholars
undertaking quantitative interdisciplinary research:
remarks from the practice
Dode Prenga Safet Sula
Department of Physics, Faculty of natural Sciences, University Department of Physics, Faculty of natural Sciences, University
of Tirana of Tirana
dode.prenga@fshn.edu.al safetsula@yahoo.com.
Abstract literature. However, the proper analysis and
personalized view on concrete problem remain always
Quantitative researches encompass large a in the heart of research work. In this case a good
field of works that embrace mathematics, strategy could be based on avoiding inappropriate
engineering, physics, other natural science , approaches too. It worth to capitulate some aspects of
economy and finance, quantitative sociology this process and below we will discuss it by
and so on. Concerning differences and commenting concrete situation encountered. Notice
benefiting from similarities is a state of art for that in interdisciplinary studies, the quantitative
researchers and it is more highlighted for approach usually starts by assuming a model which
young scholars working in interdisciplinary poses an additional question about it validity. In
applications. Elements of classical and practice many of such aspects would be addressed and
advanced statistics as seen from computing managed by the research team leader and would surely
perspective, simulations, special and general subject of detailed expertise, although it happen that if
techniques and models are the frontward of following a courageously a more independent path of
the start for a successful analysis. In this the research, young scientist might run in inappropriate
aspect there are many challenges for young analysis, and problematic interpretation of the results.
scientist that must be addressed carefully. This In this regard, in the case of interdisciplinary
became more imperative in the framework of researches there are always space for better practice
applied informatics. and strategies. We want to illustrate some such cases in
1. Introduction the following.
Researches in natural science affront students and 2. Standardized models and software
scholars with a permanent challenge, how to shorten Usually in the preparatory work for a concrete
the path from data to the appropriate results. In recent research, scholars try to apply a known model. It is a
years many methods and techniques from natural common advice from mentors and team leaders to use
science have been successfully used on other discipline models that are proven to work on the analysis of the
as econometrics, sociology etc., giving rise to the systems under study. Occasionally the level and routine
interdisciplinary branches as econo-physics, socio of research drop down to the application and therefore
physics and so on. Computing and quantitative analysis the work would be further directed in some verification
have been recommended as initial step of research in of the results; secondary estimation etc. This is
such fields by many guides or books as described in common in spectral or time series analysis,
[Rob16]. Young researchers need to run-through the investigating the reaction coefficients in a model,
simulation techniques that might affront them with measurement or data elaboration and so on. Being
more complicated situations. Some of them mimic those activities so common in data analysis steps, they
physical system as annealing simulation (cooling are considered in dedicated software that attends
Monte Carlo) or use biological behavior to speed up standard models and various thematic issues. Likely, in
procedures of numerical convergences. In this case, physics or chemistry measurement, the instruments are
one needs more solicited knowledge on such natural accompanied with an interface and the software that
science too. However, many calculation problems have perform directly the data analysis. Next, in
been addressed through advanced and specified econometrics, this is realized using more general tools,
techniques that are available as discussed for example the standard statistical software as SPSS, EVIEW,
in [Suz13], [Jan12], [Ott17] etc. So, newly debuting SAS, LISREL, and ONYX. They offer adequate
researcher in interdisciplinary field necessitating modeling and calculation capacities, as described in the
quantitative analysis, computing techniques, algorithms appropriate web pages. Other software wants the user
or simulation procedures would probably find fine to be more active as is the case of R, PYTHON etc.,
solution by carefully reviewing computational
and some others software introduce many randomness is a crucial advice for a good start. In
mathematical and computational tools as MATLAB (or literature there many arguments in testing randomness
its LINUX counterpart, OCTAVE), MATHEMATICA as Dieharder test or other recommended alternatives
etc. Notice that basic languages as C, C++, [Wan03]
FORTRAN, BASIC, PASCAL etc., are plenipotentiary
for every computation requirements but they need a. Frequency Test: Monobit
professional skills in programming and computation. b. Frequency Test: Block
Detailed remarks on how to use them are largely c. Runs Test
elaborated and easy reached in open sources. For this d. Test for the Longest Runs of Ones in a Block
reason, in the first tentative, students are advised to e. Binary Matrix Rank Test
apply the preprogramed ones models or to use f. Discrete Fourier Transform (Spectral Test)
functions form the software libraries. No need to spend g. Non-Overlapping Template Matching Test
time for something that others have perfectly done, but h. Overlapping Template Matching Test
one has to know that they exist however. Nevertheless, i. Maurer's Universal Statistical Test
it is crucial that when using dedicated softs and j. Linear Complexity Test
models, each assumption and every condition of the k. Serial Test
models should be completely fulfilled. Sometimes this l. Approximate Entropy Test
is not rigorously possible. So, a good strategy for m. Cumulative Sums Test
valuable quantitative analysis could be the building of n. Random Excursions Test
algorithm from the researcher in a more interactive Each type of tests a-n above and others not included
environment, aside of dedicated software application. herein can be implemented in specific subroutines, but
In following we want to lists some precautions on such comparison between generated random arrays which
cases. have been confirmed by tests, seems to be not an easy
2.1 First stop on random numbers tasks. Moreover it needs detailed knowledge on each.
Random numbers are a key element in calculation To improve the above mentioned calculation, we
algorithms. The process of obtaining unaffected should realize ourselves a better PRN generator. In this
simulated systems outcomes is realized by the help of case we must apply step by step testing to fix the better
random numbers that drive the algorithm to the new generation. To visualize the un-randomness for PRN
unconditioned value. So we pick up a random value for generated in computers let’s start from the evident fact
the variable and calculate the modeled value. In other that in generating random normally distributed
application the probability to select between numbers we would expect that the outcome should be
alternatives is fixed by comparison a given number to a normally distributed. We can test directly for the
random one. It is clear that the quality or randomness Gaussianty as suggested in many textbooks of statistics
for random numbers could be crucial for the unbiased using the kurtosis
(as desired) outcome. Also, it is of great interests in
cryptographic security, where it is necessary to E (x − µ )4
K ( x) = − 3 (1)
examine the real randomness of various “random” σ4
number generators. Next, the randomness of casted Relation (1) is easy to apply but has many
number is decisive for Monte Carlo simulations complementary assumptions that are difficult to be
numerical integration. Some young researchers believe tested. Therefore we have applied another idea by
that the machine random number generators are quite direct measuring the distance of the distribution under
accurate, but in reality this is not the case: it is difficult analysis from normal distribution using q-functions
to get a computer to do something by chance. Remark introduced in [Uma10]
that a computer follows instructions blindly and is
2 1
therefore predictable. In practical simulations, the 1 (1 − q) ⎛ x − µ ⎞ 1−q
random numbers are taken from pseudo-random- p( x) = [(1 + ⎜ ⎟ ] (2)
Z 5 − 3q ⎝ σ ⎠
number generators (PRNG) but in more sensitive
application randomness is based on a very un- Equation (2) reproduce the Gaussian for q=1 so the
predictable physical event as radioactivity or difference q-1 estimate directly the distance from
atmospheric noise. Up here we admit that testing normal distribution. In Fig 1 we show the fit of the
distribution of machine PRN. As routinely practices by 2.1 Avoiding distribution’s assumption
physicist, we use log-log presentation which highlights misuse
the differences in the extremities of the graphs. The Statistical analysis is so common in interdisciplinary
deviance of the generated number’s distribution from modeling and fitting procedures. So, it happens that the
normal ones in the large values limit is easy noticeable assumed theoretical distribution is accepted without
by naked eye. By using (2) for an array of 106 proof as describing the system or process under study.
generated normally distributed random numbers we Or the assumption of the normally distributed
obtained q~1.020 in the generation using for …randn() deviances in the fitting process was not put in doubt
end loop in MATLAB; q~1.017 in the array generation too. However, under some specific circumstances,
using normrnd() command whereas by a simple there are sufficient arguments that the final error
BoxMuller algorithm using rand() as starting points, induced by the violation of normality assumption is not
we had q~1.015. determinant [Gen72]. Other view as in [Hu13] suggest
to the researchers to go deeper in error analysis. In
practice other common assumption are the
homogeneity paradigm; time-invariant processes and
so on. Here one needs a careful evidence for
distributions and other herein mentioned assumption
which in turn result in a quite an easy task, but the
benefit could be remarkable.
2.1.1 Some worked example for real systems
Intriguingly the intuitive assumption that distribution
for values of variables arising from a long time natural
process would be a lognormal, has remained in the
basis of many regulation and predictions. So, the
famous Black-Shoe derivation for the distribution of
p − pt −1
the return of prices namely r = t has been
pt −1
found un-applicable even being very attractive in its
Figure 1: Log-log plot of normal random number first appearance. It is suggested that in this case the
distribution could be q-Gaussian of the form (2)
Based on the arguments of [Tsa09] or [Uma10], the [Bor04]. Following this idea, a lognormal analogue of
distribution for numbers generated by the last q-Gaussian (2) has been verified with good statistical
algorithm is more Gaussian. By nature we cannot significance even for exchange rate of ALL as we
measure the randomness directly, but judging from the represented in [Pre14]. In another such analysis
resulting distribution, the random numbers produced in presented in [Sul16] we showed that the probability
the second is expected to be better. To this end, we that an extreme flood in Drini cascade calculated using
suggest to the young researcher to construct themselves lognormal distribution is as 8 times smaller than the
random number generators and hence they would one calculated from the empiric fit distribution
always have a profit from the machine ability to obtained using 20 year daily side floods as registered.
produce in it own the PRN and method perfection to Practically the discharges from the lakes as response of
generate PRN sequences. Next they’d better do near to extreme raining had occurred so frequently last
• test the randomness before application years that coincides to the calculation of expected
• pre-calculate the overall effect of non- occurrence of one time over more than 100 years. In
randomness many other real systems we observed that the best
fitted functions are in the parametric form like (2) or its
lognormal q-counterpart
1 2.1.2 Measurement and data analysis
2
⎛
1 ⎜ ⎛ x1− q − 1 ⎞ ⎞⎟ 1− q assumptions
pq ( x) ~ q ⎜1 − β (1 − q)⎜ ) − µ ⎟ ⎟ (3) Another inadequacy in the data elaboration stage could
x ⎜ ⎜ 1− q ⎟ ⎟
⎝ ⎝ ⎠ ⎠ be the assumption that the distribution is stable. This is
usually fits better than expected functions say worse in the case of real systems with limited number
Gaussian, lognormal, Weibull etc. of points and characteristic heterogeneity. We
specifically mention here
• Data gathered from measurement process in
engineering, natural sciences researches etc.
• Data gathered via inquires in social end economical
sciences
We observed that in some more detailed analysis the
un-verified distributions assumption leads to
speculative conclusions or even in wrong measurement
practice. Scholars report the level of contamination in
an area without offering supporting arguments for
stationary of the state where measurements have been
performed. It seems that mathematically is taken
N
x=
∑ x ↔ E (x ) ≡
i =1 i
xρ (x )dx (4)
Figure 2: Illustration of differences between standard and
alternative distributions approach
N ∫ x − sup port
and thus the mean is the best representative of variable
So, un-proofed assumption that the distribution on the x in its population. Notice that the right side of (4)
data would be Gaussian or lognormal or Weibull etc. exists only if dhe probability density function (the
should be avoided in applications until a test would distribution) of variable x is finite that is the case of
confirmed it. Otherwise it could happen than stationary distribution. If not, value E does not exist at
oversimplification of the systems or tendencies to all, so we cannot perform any statistical report on the
confirm generalized expectation would leads to the measurement. In (4) the variable x could be the direct
following conclusions seen in a paper recently. It is value measured or an output parameter as error in
not surprise if a real erroneous use of normal regression procedures. Hence, in those cases the
distribution paradigm would produce the result of Fig.3 verification stationary for the distribution ρ(x) is
compulsory. Otherwise, the mean could be refereed as
the best value of the sample measured, but not
representative for the population. Mathematically the
stability for distribution would be measured by
parameter α-Levy but in calculation procedures it
requires the fit for a complicated t-Student to the
empirical data. Instead on can suggested an easy way-
out from this situation making use of relations (2)
above and testing parameter q. It is related to the α-
Levy and there is e simple relationship with degree of
3− q
freedom in the T-student by the rule ν = .
q −1
Figure 3: Misuse of standard distribution. Fortunately, from the computing point of view, the
form (2) can be fitted easy with standard nonlinear
To this end, we highlight the logic step of distribution fitting algorithm, whereas T-student is more
analysis to test them starting form un-stationary ones complicated. Next one can perform the evaluation of
which are most likely to be found in real systems. the stability for the distribution under analysis by
simply using the condition of variance finiteness that deviations from the real data were normally
1 distributed which should be analyzed as we discussed
making use of the formula σ = calculated in in the preceding paragraph. We have noticed that in
(5 − 3q )β practice, neglecting (6) unfortunately is not an isolated
5 error and in some cases young researchers have no idea
[Uma10]. Stability requirement is 1 ≤ q ≤ but a
3 about it importance. To complicate things, related to
broader rule say 1 ≤ q ≤ 2 has been suggested therein. relations (6) some programs offered themselves a bin
Moreover, if q>3 there is no distribution at in statistical number (usually 20) or clearly ask to the user to input
sense. In this case the relation (4) became meaningless the bin number. Statistical softs applies directly (6) or
hence the arithmetical average value has to be declared similar formula without signaling us, and so avoiding
as the mean of the data from the measurement and the subjective bin-size. But again, (6) is valid if
never should be confounded with population’s mean deviances are normally distribution that might not be
which does not exist. true. Thereof, a very good suggestion for correctness in
data analysis is the optimization of the bin size.
2.1.3 Bin optimization procedures
Finding the appropriate distribution should not be 2.2 More Flexible when working with
considered as a trivial task. Usually the regressions are
too easy in the first sight. But here is another point to
models and conditions of validity
Using well known models is a good practice but in this
step in. The trick entails the way we approach the
case pre-programmed ones could mislead to wrong
underlying distribution for given data frequencies. So
interpretation. Many dedicated softs as SPSS, SAS,
in practice, the set of data series is ordered in J
EVIEW in statistical analysis or LISREL, ONYX etc.,
categories or classes that (as a rule) are of equal size
in structural equation studies, offers various solutions
x −x for econometric, socio-dynamic problems and related
J = max min
h (5) subjects. Their routine includes many preparatory steps
d ( j ) = n( xi ) ∈ [xmin + ( j − 1)h, xmin + jh,] and assumptions (again some of them need to be tested
separately by the user). In this case a good advice is to
build algorithms ourselves. Here is an example what
The process (5) is called histogram or discretization of can happen. In the calculation of the informal economy
the data distribution. But a (hidden) question remains as a hidden variable, we had in disposal a small portion
mostly unanswered and unreported as well: how is of data, only 18 series (years 1998-2016). The model
chosen the parameter h in (5)? Mathematically known as MIMIC (multi cause, multi indicators)
speaking, the underlying (natural) distribution d should adopted by EVIEW or LISREL have been used by
not be affected from the binning procedure (5) and in other researchers consequently and widely
analytic view one request that moments of variable x recommended in such calculation. But specifically
have not to be affected. So far, this has been considered those programs request a sufficient number of data
straightforwardly and optimization rules have been series for statistical analysis (at least above 50 points in
included in software or programs, but again, there exist our knowledge). Second they apply directly the unit
cases that those steps have not been performed. A roots removing procedures. Next the result obtained as
detailed analysis on methods and techniques for output needs further elaboration. If one tries to program
histogram optimization is provided in [Shi10]. Correct the routine by ourselves, a detailed description is
binning step should use Stokes rules or Friedman- provided in [Jor75]. In our example, we observed that
Diaconincs formula the result obtained using deferent methods does not
match. This was the result of not fulfillment of
1 presumed assumption by our data set. In particular the
−
h ~ (3.49 ÷ 3.73)σN 3 (6) use of differences to remove unit roots as
recommended, from the other side has reduced
significantly the data series from 14 to 12, and for
where σ is standard deviation and N is the total number some variables included in model the stationary has not
of values in the data set. However in (6) it is assumed been verified! However the very small number of
points led to high uncertainty on statistical test. To So we write the algorithm in MATLAB as direct
overcome the problem we preferred the calculation application of the model elaborated in [Jor72], [Gol64]
using our routine that performed those additional steps. including preparatory steps (a-c). The results we
a. analysis of tiny data (monthly records) by which obtained using deferent approach (currency approach
the dynamics of the quantities has been identified and MIMIC model in the concrete work) matched
(in an high level); especially there have been tow much better. Moreover the reproduced variables fit
regimes in the interval considered so we used data very well with original ones confirming the goodness
that belongs to the same regime for the fit of the calculation in this case as seen in the Fig 4. In
b. accounting for those two effects fitting has been another calculation related to the consumer behavior
accepted for lower confidence level we observed inadequate outcomes when using
c. number of factor variables , responses and latent variables directly as from the measured. Calculation
ones have been calculated using factor analysis was performed basing on standard logistic model used
in econometrics and generally in the models involving
categorical variables [Kus18]. Again it is preferred to
construct the program considering specifics of the
system and its variables using the same idea as above.
a. Informal Economy by MIMIC 8-1-3 model :
Figure 4: Another example of normalizing models:
Fitting logistic model in consumer behavior
So, the same result has been reported using the
logistics and probit approach, that signify an
improvement of the calculation. In general we can
underline and highlight the importance of carefulness
with models and especially
a. Detailed verification of all dedicated softs
assumption. Avoiding any non-logical operation
on the data series
b. Constructing single-purpose algorithm instead of
b. Reproduction of the indicators: yellow line, unemployment rate, using multi-purpose pre-programed ones
blue line, ln(GDP); red line logarithm of narrow money, c. Going deep in the mathematic of the problem
before applying retunes
Figure 4: Example of easy step by step analysis d. Analyzing the overall state of the system
(case study: Informal Economy estimated by
MIMIC model)
2.3 Trying Calculation challenges the old solution is permanent. In such cases it is very
Some non-linear function or equations cause headache important for the researcher to explore many specific
to the practitioners. Let consider for example the techniques and trying again to challenge the problem
problem of fitting parametrical functions like the ones by madding up routines.
including non-homogenous unit if variable as 3. Non-neglecting Calculation and
y~
(7) Simulation Performance
y0 + (x − xc )m [A + B cos(log ω (x − xc ) + ϕ )...] Advanced studies include simulation and hard
calculation even in the graduate level. Students can try
The form (7) is verified as underlying bobble
directly in open sources as Wolfram Alpha to calculate
dynamics in financial asset or indexes, failures,
difficult integrals or they can use MATHEMATICA,
explosions etc. [Sor01]. Regressions including
MATLAB services etc. In numerical calculus including
nonlinear ones do not work in this case. Taboo search
integration many method exists and with little effort
is not reported as effective too [Sor01]. Moreover, the
nearly all problems for not advanced studies could be
deviation is an Uhlenberg process that cannot be tested
answered using each of them. But choosing the
as we do for chi–deviances; hence the statistics for a fit
appropriate method or algorithm might result in
is not available by standard procedures. To deal with
consuming time and energy for students. Clearly there
numerical analysis of near to characteristic behavior we
exist no general receipt in these cases and it is just the
used recently [Pre16] a more complicated form of (7)
duty of the research leading, but again some advices
by extending relations (5)
P(t ) =
could help. For many purposes the two above
1− q
mentioned software (and surely many others) are really
⎡ (t − t c )1−q − 1 ⎤ (8) mines with opportunities. Just needs to explore them.
a + b(t − t c ) m + c(t − t c ) m cos(ω ⎢ ⎥ + φ)
⎣⎢ 1− q ⎦⎥
But again statistical and mathematical tools are
1− q indispensable. Here are some considerations from a
⎡ (t − t c )1−q − 1 ⎤ recent work.
+ d (t − t c ) m cos(2ω ⎢ ⎥ + φ )...
⎢⎣ 1− q ⎥⎦
3.1.1 More effort in analytic relations
To solve those problems it is suggested a genetic Analytic solutions are always the most desired outcome
algorithm model which is detailed in [Sor01]. It is in the study of systems. Let mention here a simple
based on two step calculation or ‘slaving parameters”. physical system containing two vectors (magnets).
We write an ad hoc such a routine and the fit has been Later on it is proposed to model opinion formation in a
found very accurately in the case of the dynamics of pair of individuals. Statistical mechanics calculation
exchange rates [Pre16], [Pre14] or anxious-like start with partition function that in this case reads
behavior in the water level during intensive floods in
Komani Lake [Sul16]. Genetic algorithm is found ⎛ H ⎞
Z = exp⎜ − ⎟dΓ (9)
successful for many such fitting difficulties. For ∫ ⎝ kT ⎠
interest of the readers we motioned that genetic Γ
algorithm mimics the Darwinian evolution. So in the Where
core of the program, one impose by a given probability
a mutation in the solution vectors v = m, xc , ω , ϕ , [ ] H =− ∑ Jm m + ∑ µBm (10)
i, j
i j
j
i
and if the result is not good, one changes the
distribution of the random numbers used to impose the the Hamiltonian, m is magnet vectors and B is the
mutation. We realized that by using beta distribution to magnetic inductions. Here m2=1. Physical quantities in
produce random numbers, the convergence of our ad- principle will be calculated using appropriate formula
hoc algorithm has been realized even for more of physics, once the partition function Z is evaluated in
complicated forms of (7) resulting in (8) which we analytic form. Calculation of (9) having H given by
called near to characteristic behavior in [Pre16]. (10) a genuine trick proposed in [Cif99] just to replace
Similarly, the taboo searches can work for other m1m2 = (m1 + m2 )2 − 2 for 2-continus spin magnet
situation especially where the possibility of returning in system rends (10) the form
J
H =− (
2
)
M 2 − 2 − BM cos( B, M ) (11)
that turns calculation (9) to be in analytic form! In
statistical physics analytic forms of Z are the most
“wanted” cases! Here M=m1+m2 is the sum of tow
vectors. All calculation has been performed in [Cif16].
So, in a case-application in socio-dynamics, and
practically for calculating of the opinion using an ad-
hoc model, we used a more complicated inter-coupled
Hamiltonian in the form proposed in [Pre18]
O2 J
U=J−
2
⎡
⎣ 2
( ⎤
⎦
)
− FO cos(F , O )⎢1 − α O 2 − 2 ⎥ (12)
where O=O1+O2 is the resulting vector of opinion and
U is the utility function using terms proposed in Figure 5: Illustration of the calculation of opinion (13)
performed by directly using Matlab routines.
[Sta09]. Here making use of properties of Bessel
function, some adding integrals realized in [Cif16], one
realized to find analytic form of the Z integral and 3.1.2 Actual softs offer near everything: symbolic
following statistical mechanics formula finally we
operations might help significantly
obtained the average opinion per individuals as
A full analysis of the system having utility (12)
following
2
Similar calculation could strain the researcher because
⎛ β JO 2 ⎞ I1 (βFA(O) ) the calculation of Hessian needs differencing (20) and
exp⎜⎜
∫ ⎟
⎟ A(O)dO analyzing the behavior of parametric equation,
1 0 ⎝ 2 ⎠ 4 − O2 studying the logic solution, imposing constraints etc.
Ox =
2 2 ⎛ β JO 2 ⎞ I 0 (β FA(O) ) Fortunately this is not a case: by using symbolic
exp⎜⎜
∫ ⎟ dO equation and differencing in MATLAB
⎝ 2 ⎟⎠ 4 − O 2
0 (13) (MATHEMATICA etc.) we easily identified fixed
points, null clines and everything from nonlinear
⎛ αJ 2
where A(O) = O⎜1 −
⎝ 2
[⎞
O −2 ⎟.
⎠
] dynamics analysis of the system. So if we try to obtain
the solution of
Next we proceeded with numeric integration of (13) (Oc , φc )i =
concerning in the zeros and infinite values. For the
⎧∂ ⎡ J 2 J 2 ⎤⎤
interest of the reader, we mention that the MATLAB ( ) ⎡
(
⎪ ⎢− O − 2 − FO cos φ ⎢1 − α O − 2 ⎥ ⎥ = 0;
∂ϕ 2 2
)
offer adding facilities when dealing with integrands so ⎪ ⎣ ⎣ ⎦⎦
(13) have been calculated numerically and the result is ⎪⎪ ∂ ⎡ J J 2 ⎤⎤
represented in the Fig.7. This problem solved by using Arg ⎨
∂O
( 2
) ⎡
( )
⎢− O − 2 − FO cos φ ⎢1 − α O − 2 ⎥ ⎥ = 0;
2 2
⎪ ⎣ ⎣ ⎦⎦
Matlab and some adding knowledge about functions ⎪0 ≤ ϕ ≤ π
c
involved in there, is a good argument for suggesting ⎪
crossing of the methods and techniques. We observe ⎪⎩0 ≤ Oc ≤ 2
that without mathematical the trick offered in [Cif99]
analytic forms weren’t impossible and so the following
which give null clines and in the analysis of second
calculation in (14). However, exploring about solutions
order derivatives involved in the Hessian, we observe
of specific problems even very difficult could result
that traditional effort are very likely to fail. Moreover,
successful because there is always someone that can
symbolic operation in this case facilitate remarkably
solve easy our problem.
the analysis by giving the opportunity of solving
complicated systems including inequalities, plotting
complicated graphs. In the Fig.8 is shows such a step 4. Calculate the energy in new configuration, if
on searching for stationary state for the system (12) at it is smaller, the move is accepted, else it is
zero degree temperature. accepted with Metropolis probability.
5. Stop if no more improvement could be done
Basically this algorithm is fruitful for complexes
calculation and it worked for some simplified case of
equation (12). Other alternatives are available too. But
if one use the simplified Metropolis-Hasting method
we observe a non-sufficient convergence for simples
XY2D model. Notice that new researchers want to
follow the simplified MH procedure (11) instead of
taking care of full detailed balance assumption. In this
case a good advice is to measure directly the
acceptance ratio. In many Monte Carlo algorithm
would have acceptance ratio around 0.5 or lower, but it
is not a receipt however. The suggestion on those cases
is to explore patiently on possible specific algorithms
rather using general algorithm. In our example in first
Figure 6: Qualitative Analysis of stationary state
tentative we had an acceptation ratio as high as 0.8, and
This example suggest that a better knowledge about by using right formula on probability detailed balance
particular programs would be a very helpful when dealing this ratio was decreased to 0.5. Later on a modified
with complicated algebra in calculation.
version called MALA (Metropolis-Adjusted Langeven
3.1.3 Exploration on simulation platforms Algorithm) as detailed in [Jan12], [Suz13] etc. However
In many applications, the first idea coming in mind the solution of the problem was not finally concluded
could be speeding up the study, so practically one start until we used the Wolf algorithm. Surely this could be
with general algorithm and easiest ones. Not a common circumstance for many students or new
surprisingly this can lead the research on some valley researcher therefore we insist in the suggestion of
of the solution, making every effort to amend properly being real careful in implementation of every specific
the algorithm, useless. As routinely used in numerical of quantitative methods. This is very useful for such
simulation, Monte Carlo technique is the broadest researcher dealing in the interdisciplinary studies and
method used. In those similar cases it very important to especially for them that have in their basic background
explore as many as possible algorithms and methods. do not have a solid mathematical programming
Typically algorithm might slow down or might never formation.
converge due to the number of states around particular
point in the solution space. We will explain in short
4. Conclusions
this idea by just evoking the calculation of the average Successful quantitative studies needs for important and
opinion of system (12). According to the literature individual efforts in informatics, programing, applied
suggestions, we used the WOLF algorithm. The core mathematics and computation techniques. In the data
algorithm has the following steps: analysis researches, graduate students and new scientist
must pay much effort for a prior deep knowledge of the
1. one start from a random configuration of system, its characteristics and the nature of the state
magnets assimilated in the angles between a where the measurements have been made. In general,
vector and exterior field (φ ) using preprogramed algorithm or programs would not
2. pick a magnet (i) and calculate the energy of be the first choice and the benefit of building algorithm
the cell involving all surrounding magnets themselves could be apparent for new researchers.
Investing in a deeper mathematical model analysis
3. randomly select a direction θ, and turn all
would be a very good start in the case of young
spins upward to this direction
researchers with solid natural science background. New
scientist dealing with interdisciplinary studies would Comput Neurosci (2010) 29:171–182. DOI
have better results if exploring patiently on the 10.1007/s10827-009-0180-4
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