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# Random Walks - Science topic

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I am running a series of state-space density-independent and -dependent population models in PROC MIXED. I have reviewed the literature and found out how to do the gompertz and exponential growth models. I am still struggling to fit a random walk and ricker model. PROC MIXED does not have a restrict statement. How do I restrict parameters in PROC MIXED? Below is the code for the gompertz model:
ODS OUTPUT FITSTATISTICS=FIT_GOMPERTZ2;
ODS OUTPUT CovParms=GOMPERTZ_PARMS2;
PROC MIXED METHOD=REML ALPHA=.05 NOITPRINT NOINFO DATA=FISH_DENSITY;
BY REGION SITE SPECIES;
CLASS TIME;
MODEL LOG_Nt=LOG_Nt_Minus1 / s outp=pred;
RANDOM TIME;
REPEATED / TYPE=AR(1) SUBJECT=INTERCEPT;
ESTIMATE 'INTERCEPT' INTERCEPT 1;
RUN;
good reference to you :
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Photon diffusion happens when photons travelling through a medium undergo repeated scattering events without the photons being absorbed. It can be interpreted as a random walk of photons through the given medium. The question is:
What is then the speed of photon diffusion in a medium with a given density? In other words, how long it takes for the photons to pass through it?
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I need to measure the earnings forecasts accuracy as a dependent variable according to these steps:
1- forecast earning for each year based on the earnings in the prior year
2- calculate the absolute earnings forecasting errors ( the absolute difference between actual and forecasted earnings divided by the actual earnings)
is there any suggestions
The random walk (RW) model is a special case of the autoregressive (AR) model, in which the slope parameter is equal to 1. A very good exposition on the topic of the Random walk model can be found at https://faculty.fuqua.duke.edu/~rnau/Decision411_2007/411rand.htm
A very good exposition on the topic of Time Series Regression and Forecasting can be found at
I hope these 2 articles will be of great help.
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Please suggest for computer based system for processing of raw gyroscope data which method is more suitable for bias drift and noise removal from gyroscope? I need angular velocity in my application.
Thank you Alexandru Isar Sir.
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I'm regressing households' spending (PCE) on disposable income (PDI) between 1980 and 2018 for the U.S economy. I noticed that the coefficient of PDI is higher than 1. I first think that coeff. of PDI may be extreme due to possible autocorrelation or non-stationarity.
While conducting the ADF test, I tried the random walk, drift, and a trend.
However, according to the results of the test, PCE and PDI series are stationary with a drift.
I also applied the Gregory-Hansen cointegration test and find out that series are cointegrated in the long run at the breakpoint.
How can I solve this problem?
@Defne - Do u think that your data has trend or drift? First, ensure it.
If you think that it has drift, only then go ahead with this result otherwise test stationarity without drift. If you find the series I(1) then go ahead with cointegration.
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I am looking for a good couple of reads on game theory, maybe with an historical context. Anything like random walks, or Marvok Process is fine or even, Colonel Blotto, and Prisoners Dilemma. I think my hardship is finding the right literature instead of instructional books. Thank you in advanced.
"Strategies and Games: Theory and Practice" by Prajit Dutta is a good text
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Hi there,
I am relatively new to ANSYS, with a fairly basic module undertaken in university. For my final year project, I am trying to model aerosol droplets (equivalent to water droplets) in an air flow of 3 litres/min entering the nasal cavity, of which I have approximated as a Y shaped tube.
I am using ANSYS FLUENT version 19.0 (student version).
Have been following tutorials from youtube but they are not exactly what I am looking for.
So far I have the following setup:
(Opened via double precision serial)
General: Pressure based - type solver with gravity enabled (9.81/ms^-22) and transiet
Models: Viscous Laminar model with realizable k-epsilon (2 eqns) and enhanced wall treatment (have also tried scalable wall functions and standard)
Discrete Phase is on with interaction with continuous flow enabled and Max number of tracking steps is 50000 (have altered this as I kept getting incomplete particles - this did not make a difference so it must be something else). I have enabled unsteady particle tracking and injecting particles at a particle time step of 0.001s. I have enabled the pressure gradient force, automated tracking scheme, linearized source terms and accuracy control.
The injection is a surface injection at the inlet, I chose a roisin rammler distribution with 5 micron diameter mean sized particles, mass flow rate of 0.05 kg/s and velocity of 0.2 m/s. Have injected using face normal direction. Have also enabled random walk model under turbulent dispersion and random eddy lifetime. I said the stop time was 1000s.
I basically need the aerosol to come in via the two inlets (which have been named accordingly). Both the inlet and outlet have been assigned "escape" under DPM boundary conditions and the walls are assigned with "trap".
What I need from this simulation is to assess how many particles deposit within the airways i.e. number of droplets in versus number out. I also would like to see how they break up via the TAB model - have tried this also to no avail. I cannot seem to get any particle tracklines - however the streamlines come out fine.
I would really really appreciate any input or feedback, as this is my first time doing this complexity of simulation - have very little experience and have spent 9 hours on this to no fruition. If someone could just provide me with some guidance on this configuration or a basis configuration for particle deposition along the walls and particle breakup, I would be really grateful.
So far, I have:
number tracked = 4156, escaped = 4077, aborted = 0, trapped = 0, evaporated = 0, incomplete = 79, incomplete_parallel = 0
I cannot get anything to show up when i try to create a particle tracking plot - nothing will come up at all
Hi;
Plz refer to these two tutorials:
You can also take a look at our previous publications.
BR
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During test of cycle_gate, we ask people to walk, then they pay attention to it and their way of walking is changed, also darkness can change it too.
but that is question, whether mental illness such as depression change the cycle_gate?that could help us to find therapies for these diseases.
Yes,affects,Of course it affect our cycle_gate. Depression (major depressive disorder) is a common and serious medical illness that negatively affects our feels. Depression causes feelings of sadness and/or a loss of interest in activities once enjoyed. It can lead to a variety of emotional and physical problems and can decrease a person’s ability to function at work and at home.
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I am modeling the collective behavior of random walkers (using CTRW) on a 2d lattice and I am having trouble finding a "correct" rule that won't eventually violate the uniformly random motion of my agents. Any ideas?
What about reflecting boundaries? That's what I used in case of the confined CTRW,
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I need random walk code for matrix 20*20 in MATLAB. but without repeatation in each place. I mean we can walk just once in each room.
I do not know why anyone would use MATLAB or python when C compilers are free and you can do anything in C. I have written hundreds of programs and millions of lines of code. I have implemented a random walk in C. If you provide more details, I'd be glad to write the code and send it to you if I don't already have it. My book on Monte Carlo Methods (https://www.amazon.com/dp/B07BHWRDSD) is free on these days: 12/18/19, 12/26/19, 1/3/20, 1/11/20. My book on particle tracking that includes random walk with validation (https://www.amazon.com/dp/B07XRWKS6H) is free on these days: 12/25/19, 1/2/20, 1/10/20, 1/18/20. This simulation of diffusion from the MC text is free online.
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Adaptive Market Hypothesis (AMH) and Random walk model are the underpinning theories of technical analysis of stock market return? Is there any more theories for technical analysis?
You may want to check the following paper out. I think it would be helpful in this regard.
How rewarding is technical analysis? Evidence from Singapore stock market
WK Wong, M Manzur, BK Chew - Applied Financial Economics, 2003 - Taylor & Francis
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I am doing 2D random walk simulation in a confined circle to simulate Brownian motion. I have two questions.
1. When the particle reaches the boundary, which boundary conditions should I use for Brownian motion? Mirror reflection (opposite direction) OR diffuse reflection (using Knudsen law)?
2. If I would like to simulate multiple particles Brownian motion in a confined circle, how should I describe the particle interactions?
Thank you so much for your consideration.
@ Domingo Tavella ,
My case is the real physical particles in a droplet do real Brownian motion.
For the moment, I prefer to use diffuse reflection boundary conditions.
When particles reach the droplet boundary, they must be "bounced" back. But I am not sure which model should be used.
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I want to improve the specification performance of my MEMS Gyro, As we know, the measurement errors of a MEMS gyroscope usually contain deterministic errors and stochastic errors. I just focus on stochastic part and so we have:
y(t) = w(t)+b(t)+n(t)
where:
{w(t) is "True Angular Rate"}
{b(t) is "Bias Drift"}
{n(t) is "Measurement Noise"}
The bias drift and other noises are usually modeled in a filtering system to compensate for the outputs of gyroscope to improve accuracy. In order to achieve a considerable noise reduction, there's another solution that the true angular rate and bias drift are both modeled to set as the system state vector to design a KF.
Now if I want model the true angular rate, How could I do this? I just have a real dynamic test of gyro that includes above terms and I don't know how can I determine parameters required by the different models (such as Random Walk, 1st Gauss Markov or AR) for modeling ture angular rate from an unknown true angular rate signal!
You can also model the scaling errors and angular displacement, so the full model would be
y(t) = S R w(t) + b(t) + n(t),
where matrix S is matrix of scaling factors, and R is matrix for angular displacement. However in practice the biggest contributor of error is bias b(t). Errors due to scaling error and angular displacements are nowday usually low, because manufacturing quality of gyro sensors is quite good now.
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While analyzing the trajectories of 40 nm Au nanoparticles diffusing on a surface ( dark-field optical image), I often land up with trajectories which cross. As a result the trajectory of a single particle is identified as several discreet trajectories by the algorithm. I use the standard centre of mass method for tracking. Is there a better technique which can take care of crossed trajectories by taking into consideration velocity/directionality or other parameters of a moving particle? However this is random walk, so this may be difficult. The particles are quite faint, and the background noise is appreciable even after post -processing, so the method also has to be robust in locating two very close spaced particles.
“Faint” particles presumably means low signal-to-noise ratio (SNR) for tracking, which means that there will always be some uncertainty about track assignments for close-passing particles.
Two possible solutions are:
1. to discover some higher quality features in the signals that in effect drives SNR significantly higher; or
2. to discover and include some additional prior information in terms of heuristics that the particles must obey (e.g., particle tracks can cross but they never osculate; particle-track crossings never occur in the close neighborhood of a other particles; the heading of particles do not undergo significant change at the point of crossing the track of another particle; etc.).
Perfect tracking requires perfect information (high SNR and/or faultless heuristics). Your track assignments will always be to uncertain, and therefore subject to errors, while SNR remains low and heuristics are incomplete or imperfect.
Do not fall into the common trap of endlessly revising/tweaking tracking algorithms to deal with yet another error you happened to notice, because then your work will churn on endlessly, with no stopping point, and without confidence in your results.
The way ahead with confidence is to squarely address the question: “How do I use an automatic tracking algorithm that I know is going to make mistakes?”
It has to be addressed statistically and practically:
1. Assess the frequency with which tracking errors occur;
2. Assess the cost and harm of that error frequency to the objectives to your project, or the cost/benefit of using automatic tracking with that frequency of error;
3. Use any tracking algorithm whose error rates and cost of error are acceptable to you. That is your stopping point in tracking algorithm development.
Your goal is presumably not to develop perfect tracking in low SNR conditions. Your goal is rather to address some other more interesting aspects of particle motion or dynamics. The fact that you see errors occurring on rare occasions (assuming they are rare) does not necessarily mean that you have to fix your tracking algorithm. If particle crossings are rare, then your current algorithm may already be at acceptable operation with your current algorithm.
One can imagine situations, for instance, where simply discarding particle tracks that have crossed from further analysis is a serviceable strategy for handling crossings. All you need is a crossing detector. This may or may not apply in your case.
Or you might flag all crossings for visual inspection by you in a manual stage of processing, in which you fix any errors using your expert judgement about each crossing event (assuming that you are a perfectly reliable judge, or at least a better judge that the algorithm, of track continuation).
The points to be made in any case are that, given finite SNR and imperfect heuristics, you generally need to accept that automatic tracking errors will occur, and you need to show that their ultimate negative influence on the objectives of your project is acceptably low and suitably managed.
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For example, such a one-dimensional model, in which the particle is a movable ring on the torus, and the spatial coordinate is the irrational winding of the torus, suitable for the role of the pre-quant model? Even not to mention the random walk of the ring as a quantum behavior of the particle, we can talk about the uniform motion of the ring along the helical lines of the torus, which will immediately give an interpretation of the spin of the particle as the direction of rotation of the ring.
It is about such models that I tried to speak in the article about the application of local algebras of vector fields to the modeling of physical phenomena.
experience will tell
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Dear Researchers,
I’m working on a panel N= 13 and T = 54.
I ran fixed and random effects models. In addition, I wanted to run GMM model as a kind of robustness check and to control for potential endogetinty issues, knowing that GMM is preferable with small T and large N.
The problem is : the Arellano – Bond test for autocorrelation AR(1) is non-significant.
I know that the test for AR (1) process in first differences usually rejects the null hypothesis since
Δeit=eit−ei,t−1 and Δei,t−1=ei,t−1−ei,t−2 and both have ei,t−1
What could be the justification behind the failure to reject H0? Does this give any evidence to a Random Walk process? I will appreciate if you can guide me to any interpretation..
Thank you so much.
Follow
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From the point of view of mathematics, there seems to be no obstacle for a noncompact manifold to be embedded in a closed manifold (for example, the winding of a torus everywhere densely filling a torus). However, this is not enough, since such an investment must be "revived", that is, populated with material objects and made to move them. It seems that this area is open for research, but I will be happy if you indicate to me the work corresponding to this topic. In turn, I can offer you a drop, which has the form of a 7-dimensional sphere, which is populated by a vector field of velocities of particles of a continuous medium. In this model, the topological singularities of a vector field should serve as material objects (particles), since linear vector fields form the Dirac matrix algebra. However, interesting consequences of this model from the viewpoint of physics also arise in the two-dimensional case. Thus, in the study of a random walk along broken lines of a winding of a classical sphere, one can obtain a generalized one-dimensional Schrödinger equation.
Mathematical notes on the nature of things (in Russian)
Dear Igor,
I think that you are thinking in the theorem of Banach-Tarski, which goes against our common sense and in fact it corresponds to cut with parts having without measure.
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Hi all,
I have a model that runs a mass balance profile for a glacier (i.e. the net result between melt and snow accumulation). However, I have a set of 7 variables to calibrate the model. I was wondering if there is some kind of small program I can write to run the model several times with all different values for the parameters within a certain range, in order to find the minimum RMSE. For example, parameter 1 from 0.4 to 0.6, parameter 2 from 0.34 to 0.42, etc. I can then check afterwards which combination gives minimized RMSE I was thinking about a random walk method but it is hard to write these kind of things. It would be very much appreciated. Can anybody help me getting started?
Thanks!
Hi Verhaegen,
This really depends on the kind of model you have and the size of parameter space (number of variables and their range).
The previous answers contain some functions which can be really useful, they are a bit more sophisticated methods than the random walk. Also, you might consider evolutionary algorithms like Paricle Swarm Optimization or Ant Colony Optimization.
In addition, I would suggest you to have a look at the Matlab system identification toolbox, there are many algorithms already implemented, and has a GUI as well.
If you would like to learn more, then do the reading: starting from System Identification: Theory for the User, (2nd Edition) by Lennart Ljung.
Also, you will find loads of literature online, look for the following keywords:
"System identification", "Parameter identification", "Dynamic systems"
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I am currently carrying out an assignment in which we need to compare our GARCH( p, q) forecast against a Naive Benchmark model, such as a Random Walk model.
I have considered an IGARCH (1, 1) model to fall under this defintion, due to the coefficents of alpha and beta adding to 1.
Is this therefore a random walk with a drift model, as I am struggling with its interpretation?
I have attached a sample of the Eviews output for reference.
Hi: the random walk equation for sigma_t is
sigma_t = sigma_t-1 + epsilon_t.
so the coefficient of the previous sigma value would be +1.0 if RW for volatility is true.
the "naive model" versus igarch ) is probably that volatility follows a random walk. so, besides what the other people said about forecasting comparisons, another way is to check if volatility follows a random walk by estimating the model above and seeing if the coefficient on sigma_t-1 is +1.
actually, both methods should probably be done. but, the forecasting procedure should be done on out of sample observations that are not part of the respective estimation procedures. so, maybe split the data into two parts and estimate using half and then forecast using the other half. I hope this helps.
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What is the difference between Spectral moments of surface and spectral moments of surface profile?
I am not sure if I understand your question correctly, but this is what comes to my mind:
The fractal dimension tells you how the "weight" of your fractal scales with its size. For a surface fractal, the weight will be proportional to the area. So, using the surface profile and its length will give you wrong results.
If by random walk method you mean Monte Carlo integration, that seems like a plausible method for calculating the surface area, given that you use the correct local area element. To get a numerical measure of your fractal dimension, you either would need to use different "box" sizes on the same surface, or use different surfaces produced at different fractal levels.
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At the moment, I'm trying to analyse a difficult set of data that I've been thinking but really don't know how to do it.
So basically, i have two populations of cells and I notice one of them doesn't seem to undergo random migration and are in some kind of patterned movement while the other one looks more like it. There's no external chemotax and the cells are in a homogenous environment, so the non-random migration is potentially intrinsic to one cell type, while the other doesn't have this ability.
So I would like to analyse this but can't think of any way to do so. So I bring the problem on here to ask if anyone knows any kind of way/algorithm/software that will be able to quantify this to confirm my hypothesis?
I can ask to make a simulation, one is random walk, the other is not, but this would take a lot of effort, so I'm looking for a simpler way to do this.
I would really appreciate if anyone can shed some light!
Hi Edward,
The technique might not be of use here as cells are attached to the bottom of a petri dish so not entirely a brownian motion unfortunately. However, the random factor here is that cells move wherever they want, so source of chemoattractant present. So maybe the same algorithm can be applied?
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Is the hypothesis on the representation of the Riemann zeta function true by the product of the sums (or the sum of products) of the probability amplitudes of a random walk along broken lines of the winding of a sphere?
Deleted research item The research item mentioned here has been deleted
Still, I would like to rely not on intuition, faith, but on calculation, proof.
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If we train a data model once on a dataset using a machine learning algorithm, save the model, and then train it again using the same algorithm and the same dataset and data ordering, will the first model be the same as the second?
I would propose a classification of ml algorithms based on their "determinism"
in this respect. On the one extreme we would have:
(i) those which always produce an identical model when trained from the same dataset with the records presented in the same order and on the other end we would have:
(ii) those which produce a different model each time with a very high variability.
Two reasons for why a resulting model varies could be (a) in the machine learning algorithm itself there could be a random walk somewhere, or (b) a sampling of a probability distribution to assign a component of an optimization function. More examples would be welcome !
Also, it would be great to do an inventory of the main ML algorithms based on their "stability" with respect to retraining under the same conditions (i.e. same data in same order). E.g. decision tree induction vs support vector vs neural networks. Any suggestions of an initial list and ranking would be great !
for quite a comprehensive list of methods.
There is an element of chance in the training process. In some software, you can get reproducible answers by using something like set.seed( ) in the R language. Using the seed number again with the same data will then give the same result. Then you can report the software you used with the seed. However in general the different outcomes will be close together, but as with sampling, you will occasionally get outliers (depending on the seed you choose).
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Assuming random walk model with a drift is used.
Depends which method you are using, some methods require stationarity, some not. However, if you are using stock returns in your analysis, they are surely stationary.
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Several theoretical models have been proposed for the study of the lasing behavior in random media such as "Correlated Random Walk", "diffusion with gain", "disorder induced localization coupled with non-linearity". But it seems that none of them have been able to cover the diverse experimental results. I need a model that predicts both localized and extended modes in two and three dimension.
Thanks for the inspiring question and the useful documents.
Iam really excited that I can meet lots of  researchers who deeply devote in the random laser community!
Good luck everyone!
Let's push random lasers to the boundary!
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I want to simulate the flow field in the emitter of drip irrigation and use DPM to get the particle trajectory  in the emitter. when the computation is converged(the mass flow rate of inlet and outlet is tending towards stability), the result of particle trajectory is satisfactory, and there are many eddy currents in the flow field. But if I change convergence condition, the iteration continues iterating several  steps(maybe about 10-50 steps). The new result of particle trajectory becomes different, the residence time of particles also become different. And the flow field doesn't change when the iteration continues iterating.
First,I use k-e turbulence model, surface injection, interaction with continuous phase, discrete random walk model.
when I discover the problem，I change the surface injection to point injection, don't select the options of interaction with continuous phase and discrete random walk model but the problem still exists.
I've been thinking about it for a long time. I think the particle is sand, and the diameter is 0.01mm-0.1mm, its motion is greatly affected by the flow field, when the iteration continues iterating several steps, the flow field is also changes, although the value changes only little， the influence of particle  trajectory is significant.
So what's the problem?
Please please let me know, if you have any idea about the problem mentioned. Any help is highly appreciated.
Thanks a lot!
Hello Julio Cesar Lelis Alves ,
Thank you so much for your response.
When using steady track, I think each iteration Fluent will calculate the movement
differential equations of particles in the whole calculation domain, and then get
the particles' trajectory. I'm not sure if what I say is correctly.
2、I set wall as reflect condition. It is set up according to the actual situation.
In fact, I want to simulated particle accumulation in the domain. Do you have any suggestions?
Thanks a lot for your response and hope to hear from you ahead.
Thanks again,
Tang.
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Since Random walker is used for both segmentation as well as classification. But it works on supervised learning so is it better to comapre it with other segmentation techniques or classification techniques like SVM.
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I've seen in many places where these terms were addressed separately. But the more i've read about triangulation, it appears the same as mixed methods. Can I get some  help on this?
Dear George Lawrence George,
On the face of it , triangulation and mixed methods seem to be one and the same thing.  Conventionally and by tradition, mixed method research is the one in which quantitative and qualitative methods of research are admixed and hybridized. However, triangulation is a kind of precautionary measure adopted by the researcher to make sure that his/her collection , analysis , and interpretation  of data in QUALITATIVE research is valid and credible. The reason is that in qualitative research, the researcher is very close to the data, and as such, subjectivity is very high. As a consequence, to reduce subjectivity, and to make the qualitative phases of research more objective and credible, the researcher resorts to triangulating methodology, theoretical background underlying the research topic, and the interpretation of data (see Mackey & Gass, 2005, 2016).
Best regards,
R. Biria
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Hello all,
Ackley
Bohachevsky 1
Camel 3 hump
Drop wave
Exponential
Paviani
Are these benchmark functions unimodal or multimodal ?
Thanks to all who contribute to the answer.
great my dear
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A random walker starts at the origin, and experiences unbiased diffusion along a continuous line in 1d. What is the probability for this walker to return to the origin for the first time as a function of time? In other words, I am looking for the probability of first return to the starting point. I would greatly appreciate to learn the functional form of this probability, or to learn of references where it is discussed or stated.
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I am looking for an analytical expression describing the time evolution of the spatially averaged density (in practice, the average concentration in particles/m3) of a system of particles hopping with a given frequency on a d=3 lattice where some traps are randomly placed. The idea is that when a particle meets a trap it disappears from the system, so that the average density decreases to zero with some decay law. I expect to find something like a stretched exponential decay and I would like to relate the parameters of this decay curve with the input data of my system (e.g. intial particle density, traps concentration etc.) The lattice can be ordered (say, cubic) or disordered (like in glassy media or polymers) and the traps can have a finite size. As a further complication, the effect of traps saturation can be also taken into account.
Thanks!
A lot depends on whether your traps are mobile (that is, if they too diffuse) or are fixed, but placed at random. In the former case, decay is eventually exponential in the number of distinct sites visited, S(t). S(t) itself depends on the dimension
S(t)=sqrt(t)     d=1
S(t)=t/ln(t)       d=2
S(t)= const. t for all higher dimensions.
If, on the other hand, the traps are immobile, then you have a stretched exponential
exp[-t^(d/(d+2))]
where d is the dimension. This arises from the fact that you might start in a region where there are exceptionally few traps, and survival is dominated by these events.
However, even in the simplest case of a point particle diffusing on a discrete lattice, there are no analytic expressions to my knowledge.
The literature is rather huge: the following somewhat random choice may get you started:
Havlin, S., Weiss, G. H., Kiefer, J. E., & Dishon, M. (1984). Exact enumeration of random walks with traps. Journal of Physics A: Mathematical and General, 17(6), L347.
Grassberger, P., & Procaccia, I. (1982). The long time properties of diffusion in a medium with static traps. The Journal of Chemical Physics, 77(12), 6281-6284.
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Also kindly illustrate the key terms in the context of its implementation seeds,masks,labels and beta? what are these terms and what does they represent?
A random walk is a mathematical formalization used for describing a path that consists of a succession of random steps. In the context of image processing, random walker algorithms are considered supervised algorithms because the user has to provide a partial labelling of the image (or seed) and are commonly used for image segmentation. This method (1) treat the image as a graph and (2) minimize certain energy functionals on this graph to produce a segmentation.
I attach you some references for a more detailed specification of the specified terms.
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I'm currently doing some research about oil spill models, focusing on particle tracking and random walk diffusion, but can't find a recent survey discussing the topic. Can anyone suggest one? Thanks in advance.
Thanks you all very much! A lot of helpful information in the answers. I'm actually trying to implement a full system for oil spill modelling, so I guess I'll start reviewing the available documentation for GNOME or OSRA. Thanks again!
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From my understanding Markov clustering (MCL) algorithm is a graph clustering algorithm that takes weighted graph as input and uses Random walk approach for probability assignment of each node on the graph. What is not clear to me is how MCL uses Frobenius norm to perform the final clustering. Also, is there a tool or java codes that I can easily simulate with or adapt? Thanks.
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I have 8 years time sires data on maternal and child mortality rate.
Thank you Anthony G Gordon. Just you have given a clear answer for my question and i will disaggregate the concept respective of the data.
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I am very interested in predictive walks. I am looking for this term where a person would more likely choose to walk in a path with the shortest distance. What do you call that concept?
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I am now working on mobility prediction. I need MATLAB code for simulation of  Random Walk Model.
I have attached two different assignments with their respective solution Matlab codes, hope that it'd be useful.
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What is the difference between the lexicographic max.min optimization and the max.min optimization? I read the difference in two
1-the lexicographic the iteration of max.min .
2-lexicographic is fairly and efficiently.
this is right ?
and what the best algorithm for solving the discrete numbers ?
tab search, random walk?
I am not sure if I get your question correctly, but Lexicographic order is a dictionary order of vectors. Having two vectors X and Y with n components you start comparing both starting from the first component if you can not put greater or less than between that component then you can continue until you get a xi<yi or xi>yi. whenever you arrived there then that inequality will be used for the vectors. if you don't arrive to an inequality sign then they are equal.
Max min is optimizing a certain objective function of several variables (at least two). Maximize with respect to y Minimize with respect to x: means first you minimize the function with respect to x by keeping y as a parameter, and you will have a set of parametric solution then you want to maximize these parametric solutions with respect to x. Ex: maximize satisfaction wile minimizing cost, first you minimize and among those solutions you choose the one which maximize the results with respect to the second variable.
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Start with a reflexion domain  being a polygon A1 A2 ...An, with n summits
A simple computation using formula line 7 page 57 shows that if d(An, An-1)-->0 then P (B5(0)=An)-P(B5(0)=An-1)-->0 as n-->infinity
and P((B5(t+s)€E/B5(0)=An))-P((B5(t+s)€E/B5(0) =An-1))-->0 as n--> infinity
Which allow to derive the distribution of the relected process in a bounded domain with smooth boundaries.
Thanks
Bernard Bellot
l
Why don't you upload the article for easy visibility?
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I want to run Brownian Dynamics simulation and wonder which software package is more popular?
LAMMPS and GROMACS are popular simulation packages for molecular dynamics (MD) simulations. Both can be used to simulate Brownian Dynamics by using Langevin dynamics. See "fix langevin" in LAMMPS or "bd integrator" in GROMACS.
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Hello,
Let us consider the classic random walk:
x0 = 0, xn+1 = xn + rn,
where rn are equally distributed zero-mean Gaussian random variables. It is well-known that the probability distribution of xn is a zero-mean Gaussian with standard deviation proportional to sqrt(n).
Now, I will say that the instant k is a zero-crossing for a particular realization of the random walk iff  xk has opposite sign than xk+1, and I will consider the random variable T defined as the temporal distance between two consecutive zero-crossing.
What can we say about the probability distribution of T?
My intuition would suggest such probability distribution is time dependent and its expected value would also grow as sqrt(n). Am I correct or wrong?
I thank you all, Giuseppe Papari
.
there is no reason for the distribution of the duration beween successive zero-crossings of a simple random walk to be time-dependent : every time the random walker crosses the zero line, you can imagine it is "reset" : as there is no memory, crossing the zero line at time 100 or crossing the zero line at time 100000 leaves the random walker in the same - physical time-independent - situation (see the notion of "random walk local time" for technical details)
in addition to Markus references above, you might be interested in
and references therein
.
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Well, I'm a novice for both matlab and research on machine learning. Can anyone tell me how to represent affinity matrix in random walk graph matching case (i.e. what will be its structure?). e.g. Wia,jb  will it be two dimensional or four dimensional array (matrix)? As you see from the example the affinity matrix W has four indexes (ia, jb). Does it mean the matrix should be four dimensional? Thanks in advance for any kind of explanation.
you could use the cell matrix
search in matlab help about cell();
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I am confused with Random walk and Red noise. Some people said that random walk is red noise. Both of them can be generated by integrating white noise. However, Random walk is a nonstationary process, which doesn't have PSD. But red noise has its own PSD, which is 1/f^2. How should I understand the difference between them? Is there anybody has some comments on this?
Random walks and noises are very different stochastic processes.
White (or red, or pink or whatever colour) noise have values that are independent: the value of the noise at time t is a random variable that is independent of the value at time s, provided t and s are not equal. In fact this is an approximation because the notion of value here is not well defined, but this is the idea. Your PSD is the distribution function, that is the same at any time (stationary process). Again this is not a proper distribution function but this is not the point.
A random walk is a Markov process, and the most classical is the Brownian motion. This means that the values at times t and s are not independent. But the increases are independent. To make it simple, a Brownian motion is the integral of white noise, and more generally random walks are the integral of some "noise". Since this integral defines the random walk, there is a relationship (integration/derivation) between "noises" and random walks, but they really differ. E.g. a random walk is continuous while a noise is discontinuous.
I hope it helps.
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I am aware of one method - for assessing whether a time-series variable follows a random walk (using differences and comparing the standard deviations). However, I am interested in possible explanations for other more rigorous methods.
First, let us differentiate between a random walk process and a random set of observations. A random walk process is modeled by y(t)=y(t-1) +\eta, where $\eta$ is i.i.d (white noise) series. Taking the first difference of a random walk model results in a random series.
Now the traditional ARFIMA modeling methods assume the data generating processes are linear and stochastic. However, nonlinear tests show many financial time series are nonlinear. Accordingly, tests such as unit root cannot be relied upon for determining whether a series is random (contain a unit root) or not if the series is nonlinear. Methods from nonlinear dynamical systems must be used in testing for non-linearity and determinism or stochastic nature  of the time series under investigation.
Along with Leonardo's recommended reading list let me inform the distinguished reader, and at the same time promote ( even though the royalty on sales of the book is contributed to libraries in Africa by the publisher),  our edited volume Soofi, A. and L. Cao Modelling and Forecasting Financial Data: Techniques of Deterministic Nonlinear Dynamics, Kluwer Publisher 2002.
Also please see  Soofi, A.,et al. " Applications of Methods and Algorithms of Nonlinear Dynamics in Economics and Finance, in Faggini, M and A. Parziale (eds.) Complexity in Economics, Springer, 2014.
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Pagerank works on a graph which only has direct edges between vertices and there is no weight on edges of the graph but random walk's graph edges are not directed and they have weight. Then, how we should change pagerank which could be applied to these kinds of graphs.
Use two things:
1) undirected edges are equivalent to edges in both directions
2) scale the probability to surf from a node to another according to edge weight
Here is a paper that uses pagerank on undirected, weighted graphs:
cf: Section on "continuous LexRank", and LexRank is a PageRank variant for undirected graphs.