Science topic

# Stochastic Models - Science topic

Explore the latest questions and answers in Stochastic Models, and find Stochastic Models experts.
Questions related to Stochastic Models
Question
In robust optimization, random variables are modeled as uncertain parameters belonging to a convex uncertainty set and the decision-maker protects the system against the worst case within that set.
In the context of nonlinear multi-stage max-min robust optimization problems:
What are the best robustness models such as Strict robustness, Cardinality constrained robustness, Adjustable robustness, Light robustness, Regret robustness, and Recoverable robustness?
How to solve max-min robust optimization problems without linearization/approximations efficiently? Algorithms?
How to approach nested robust optimization problems?
For example, the problem can be security-constrained AC optimal power flow.
To tractably reformulate robust nonlinear constraints, you can use the Fenchel duality scheme proposed by Ben Tal, Hertog and Vial in
"Deriving Robust Counterparts of Nonlinear Uncertain Inequalities"
Also, you can use Affine Decision Rules to deal with the multi-stage decision making structure. Check for example: "Optimality of Affine Policies in Multistage Robust Optimization" by Bertsimas, Iancu and Parrilo.
Question
To start a collaborative Project:
The target is to participate to the meeting INNOVATE 2022 Conference at MERLOT University
Perhaps this document which I published one week ago shall contribute to a better understanding of my approach:
Question
I am familiar with the concept of stochastic ordering for two random variables and how we can say if a markov matrix is stochastically monotone. What I'm interested in is if there is a concept for ranking two separate markov matrices.
To illustrate suppose we have two stochastically monotone markov matrices A and B which preserve the ordering of x≿y. Under what circumstances can we say (if any) that matrix A is preferred to matrix B in stochastic order?
Note: The definitions I am using are from this slide deck: http://polaris.imag.fr/jean-marc.vincent/index.html/Slides/asmta09.pdf
Depends on what your mean by two different matrices. If the two different matrices are derived from two different data sets, then no, the comparison is completely meaningless. If your matrices are using the same data but different sets of model parameters then of course you can compare the matrices since this is simple a test of of the fit of one set of parameters vs. another set.
Question
My essay is an attempt to answer the following : « Is the data economy, then, destined to benefit only a few elite firms? » Apparently that would be the issue till now. What are available tools to avoid this false target ? Reference to my essay on Stochastic Models in particular the section « Handling human social technical dimension; in particular man-system interface including positioning technology at man services » you may find guidelines to produce these tools and make BIG DATA exploitable by large majority of users : 1. Engine should trace “player” behaviour, evaluate its capabilities and quickly meet its needs. 2. Immersion generated by simulation enables training and experimentation of behaviour strategies, in particular learning “by doing”. 3. Engine should use following resources : 3.1. Tools to be customized by trainers. 3.2. Applied standards. 3.3. New learning approaches discovery through obtained results, whether these approaches are positive or negative, in the sense of improving technology performance of assembled prototypes. 4. How SPDF (Standard Process Description Format) may produce a universal engine to run the stochastic model ? 4.1. SPDF consists of two parts : 4.1.1. Message structured-data part (including semantics) and, 4.1.2. Process description part (with higher level of semantics). 4.2. Two key outputs of the SPDF research will be a process description specification and framework for the extraction of semantics from legacy systems. 4.2.3. Note that : a)The more we may have semantic rules the more unpredictable events are controlled. b) Acquired knowledge to elaborate semantic rules for unpredictable events requires many occurrences of the stochastic model. c) Convergence shall not be reached until getting more qualitative semantic rules. d) Performing dynamically a given scenario is the goal of the proposed messaging system.
To start our collaborative work, I'll let you propose a case study and we shall try together to apply the knowledge acquired through modelling: what are the challenge facing humanity today: 1) covid 19, 2) Climate change 3) wars in the middle east? but I won't accept to make you select one of these three proposals, since collaborative work requires to share our knowledge equally, should we succeed that should be a great achievement, thank you
Question
I have computed the basic reproduction number for deterministic system by second generation matrix described in the paper . But in this paper with stochastic model , basic reproduction number have been shown for stochastic counterpart. I am totally confused how they computed it. Please put some light on it.
I suggest to read this paper:
Also, for considering stochastic models based on SDEs, our paper can be helpful:
Question
Which one is best in modeling deterministic model, stochastic model or fractional differential equation model?
It would be good to read some intro book on mathematical modeling. The subject is huge. It all started in physics and later shifted within all scientific disciplines.
Next, you just learn to define your model inputs. To formalize the observed natural phenomenon using some sort of mathematical formalism.
It is where your question aims. To learn this, I recommend reading many reviews on modeling that are close to your research area. It gives you insights. It gives you intuition. It enables you to write down formalized models.
The next natural step is to write down a computer model or use some library to implement your model into a computer form. A few models are solvable analytically.
Many people think that it is the moment when your research ends. Not so much. You will find out that the output of the model does not agree with the observed phenomenon. That means, you just go back to the beginning to find out what is missing and repeat the whole procedure.
Question
Stochastic Modelling with Optimal Control
I suggest Stochastic Differential Equations - An Introduction with Applications, by Bernt Øksendal, https://www.springer.com/gp/book/9783540047582
Question
Chemical reaction systems can be modeled through a deterministic or stochastic approach. However, it is common to introduce new variables in order to perform dynamic analysis in deterministic systems. Could these reduced forms be directly used to derive stochastic equations or is it necessary to start from the original expressions? It is common for many concentrations to be normalized after these procedures.Does this affect the construction of stochastic models?
I also recommend these two references (below) that explain the relation between microscopic (scale) model, mesoscopic (scale) model, and the macroscopic (scale) model.
1-Pavliotis, Grigoris, and Andrew Stuart. Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008.
2- Lachowicz, Mirosław. "Microscopic, mesoscopic and macroscopic descriptions of complex systems." Probabilistic Engineering Mechanics 26.1 (2011): 54-60.
Question
It is known that the FPE gives the time evolution of the probability density function of the stochastic differential equation.
I could not see any reference that relates the PDF obtain by the FPE with trajectories of the SDE.
for instance, consider the solution of corresponding FPE of an SDE converges to pdf=\delta{x0} asymptotically in time.
does it mean that all the trajectories of the SDE will converge to x0 asymptotically in time?
The Fokker-Plank equation can be treated as a so-called forward Kolmogorov equation for a certain diffusion process.
To derive a stochastic equation for this diffusion process it is very useful if you know a generator of this process. Finally, to find out a form of the generator you have to consider a PDE, dual to the Fokker-Plank equation which is called the backward Kolmogorov equation. The elliptic operator in the backward Kolmogorov equation coincides with the generator of the required disffusion process. Let me give you an example.
Assume that you consider the Cauchy problem for the Fokker-Plank type equation
u_t=Lu, u(0,x)=u_0(x),
where Lu(t,x)=[A^2(x)u(t,x)]_{xx}-[a(x)u(t,x)]_x.
The dual equation is h_t+L^*h=0, where L^*h= A^2(x)h_{xx}+a(x)h_x.
As a result the required diffusion process x(t) satisfies the SDE
dx(t)=a(x(t))dt+A(x(t))dw(t), x(0)= \xi,
where w(t) is a Wiener process and \xi is a random variable independent on w(t) with the distribution density u_0(x).
You may see the book Bogachev V.I., Krylov N.V., Röckner M., Shaposhnikov S.V. "Fokker-Planck-Kolmogorov equations"
Question
Few days earlier on a project presentation on Stochastic Programming Real life applications, i constructed 3 real life scenario based Stochastic Models: A Farmer's Problem, Container Allotment Problem and another on Stochastic Arc Routing. Also solved them for particular scenario.
As stochastic linear programs are lengthy programs with a lot of constraints, it is long-time process to solve a stochastic linear program. And therefore i used LINDO solver to solve the problem. I have a L-shaped algorithm based example too.
But the examiner said me that, why you didn't used the general solving procedure to solve these LP problems? I explained about the long programs and complexity. In reply, I found complement that all credit goes to the LINDO solver, not you.
I am wondering that advances in Science could make our works easier and faster. Shouldn't we take these type of advances in our daily life?
Yes, there are a number of softwares which can solve linear programming problem in a click giving the optimal solution or an indication of in-feasibility or unbounded solution. But, that is second stage. First stage is to learn the basics of linear programming solution procedures i.e. graphical method and Simplex method. The students will have in-depth knowledge of such procedures by manually solving them. Once got acquainted with the procedure, they can use software like Lindo, Matlab etc. to solve the complex LP problems which can not be handled manually.
Question
If existing microgrid energy management system is deterministic, how to design it in real time probabilistic or stochastic model?
How to get expertise in this modeling?
Question
(1) What is difference between Fuzzy and Stochastic ?
(2) Fuzzy number converted to Non-Fuzzy. What great advantage is derived in using Fuzzy ?
(3) Can it be Fuzzy and Stochastic model viz. Fuzzy Stochastic TOPSIS ?
Fuzzy can add human knowlege into the modeling process
Question
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.
Question
Please do anybody with an idea on how to estimate linear stochastic models of time series analysis? Be it material or link that might help.
The materials shared are really helpful, i really appreciate you all and thanks a lot.
Question
I want to predict 2013 landuse change based on Markovian stochastic model using CA-MARKOV module.It takes long time, sometimes 24hrs to 3 days. What might be wrong?
I have already run MARKOV module to cross tabulate 2000 and 2006 Landuses. I have suitability maps in Raster group file(each stretched 0-255 integer values), i have also the Markov Transition area file. I am now using 2006 Landuse to predict 2013 Landuse change, where i specified 7 as a Number of Cellular automata iterations, 5 x5 filter.
I have also checked the disk space i have 45GB, RAM is 8GB,
Other computer specifications :Windows7 , Core i3.
Software in Use:IDRISI SELVA v17.00
The process runs well and pass from step 1 to 8, then in step 8 when starts "mola" process, it stays there for so long more 24hrs-5 days, before i terminate it.
Lower the resolution of the input images. I realized that 30*30 resolution generates lots of files while processing that makes your PC temporary memory to burst. The wise and tricky idea try 60*60m 120*120m or 90*90m. If you get further problem please feel free to ask. Lets know about what happened on your side after changing the pixel resulutions
Question
Any stochastic model is very easy derived from a known deterministic model.
The issue is more complicated because we haven't yet developed mine production scheduling models with a direct utilization of ODEs. We use optimization methods of Operations Research with a definition of an economic objective function in most cases. Some researchers, however, use classical formulations of stochastic programming ignoring a lot of important factors of mine production scheduling. This is an engineering rather mathematical problem.
Question
I need help in understanding the role of (random) sampling in implementation of a control system in Simulink. I need a basic, general example to visualize the role of the sampler in a control system, and the way it can be programmed (to be random/event-triggered etc).
Any help in this regard is very much appreciated
Hi Samira,
Referring to the Examples 9.3 and 9.4 in Prof. Lewis' book (Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory, 2e), the attached MATLAB example (m-file) shows how to simulate a stochastic control system.
Hope this helps!
Question
I have 3 objective function in one GAMS code. I have 3 different state stochastic models that each state includes 1000 scenarios which uses in each objective function. (I mean one stochastic data for one objective function). I want to use scented in my code for all objective functions.
It`s possible to do that? How can I do?
Wondering as well
Question
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).
Question
In Time Dependent Model, the definition of foreshock, mainshock and aftershock is not necessary. In this model, every event is potentially triggered by all the previous events and every event can trigger subsequent events according to their relative time-space distance. What do you say about Stochastic models of earthquake clustering?
What are Stochastic models of earthquake clustering?.
Stochastic modeling allows the computation of the expected earthquakes rate density  on a continuous space-time volume, suitable for the validation of a model with respect to others and for real time forecasts.
The significant steps made during the last decades in the physical modeling of earthquake clustering provide a tool for the refinement of these stochastic models.
Jointly with the improvement of the seismological observations, these steps appear as a progress towards the possible practical application for earthquake forecast.
In time dependent model:
The magnitude distribution is the same for all the earthquakes (Gutenberg-Richter law).
The occurrence rate density is the superposition of a time independent (poissonian)  component and that of the triggered seismic activity.
The occurrence rate of triggered events depends exponentially on the magnitude of every preceding event.
The spatial distribution of triggered events is described by an isotropic function around the epicenter of every previous event.
The temporal behavior of triggered events is described by the Omori law starting from the occurrence time of every previous event.
Question
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!
Question
I have hydraulic conductivity data from pumping wells in the area and I try to use this raw data to generate many realizations to see the uncertainty of flow and transport of pollutants. Please suggest the public model to create the stochastic modeling and can be further used as the input data for visual modflow.
When you have some experimental data (Borehole data for hydraulic conductivity) within a stochastic field where the mean and standadrd deviation of permeability is known (obtained from the borehile hydraulic conductivity data) you can use kriging (See e.g., Marsily, G. de, Quantitative Hydrogeology. Groundwater Hydrology for Engineers, Ed. Academic Press, New-York (1986))
Question
I have question regarding simulating under mentioned 1D Stochastic Differential Equation in R using Sim.DiffProc package:
dx1 = (b1*x1 − d1*x1) dt + Sqrt(b1*x1 + d1*x1) dW1(t)
I have taken this equation from book: Modeling with Ito Stochastic Differential Equations by E. Allen. In the deterministic and diffusion part of equation, b1 and d1 are model parameters representing birth and death rates (for single population approximation of two interacting populations compartment model). Relevant lines of my code are as under (note that i,ve used theta's to represent parameters in my code):
Code (1):
> fx <- expression( theta*x1-theta*x1 ) ## drift part
> gx <- expression( (theta*x1+theta*x1)^0.5 ) ## diffusion part
> fitmod <- fitsde(data=mydata,drift=fx,diffusion=gx,start = list(theta1=1,
+ theta2=1,theta3=1,theta4=1),pmle="euler")
Or should I model it like this
Code (2):
>fx <- expression( theta*x1-theta*x1 )
> gx <- expression( (theta*x1+theta*x1)^0.5 )
> fitmod <- fitsde(data=mydata,drift=fx,diffusion=gx,start = list(theta1=1,
+ theta2=1),pmle="euler")
I am not clear whether to use theta, theta, theta, theta as I have used at first place above or should I code it like only using parameters theta and theta (done at second place above) because in original model the parameters b1 and d1(birth and death rates) appearing in the deterministic part are same as appearing in the diffusion part.
I don’t find a single example in Sim.DiffProc package documentation where there is any repetition of parameters just like I have done at second place.
Thanking in anticipation and best regards.
I would use code [2} above with two parameters theta and theta.
Also,  it is very easy to code this directly without using any packages by applying the Euler Maruyama approximation method (which is described in E. Allen's book).
Also, see the book by Linda J.S. Allen which has ALL THE CODES for such examples problems given in the book . So may copy into R directly and implement:
Linda J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Second Edition
Also, the following papers contain more examples of somewhat more complicated stochastic differential equations which have been solve in MATLAB (similar to R) using Euler Maruyama approximation:
A.S. Ackleh and S. Hu, Comparison between Stochastic and Deterministic Selection-Mutation Models. Mathematical Biosciences and Engineering, 4(2007), 133-157.
A.S. Ackleh, K. Deng and Q. Huang, Stochastic Juvenile-Adult Models with Application to a Green Tree Frog Population. Journal of Biological Dynamics, 5(2011), 64-83.
Question
Hello Researchers,
I am trying to model a Noise which is basically difference between Actual and Theoretical Solar radiation. As I am concerned on Long term prediction, models like ARMA is not a suitable option.
Can someone of you suggest models that can be used to model the same. I am looking forward for a stochastic model.
Thanks!
you can use LQG approach. It is good for design any Stochastic model
Question
I am looking for a stochastic model in use clearly for extreme rainfall generation.
the rainfall will be used for hydraulic models and drainage system models.
few concepts are developed in past but it seems there is not any toolkit or software with manual about them, like
Neyman-Scott (NS) or   Bartlett-Lewis or DRIP MODEL
thanks
you can find a Matlab code implementing Neyman-Scott here on RG:
we can support you a bit...as we developed this code :)
Question
Other stochastic modelling processes which can be used to model the data being modelled by dirichlet process
Hi Gaurish, two additional explanations from you can simplify understanding your problem:
- what is the reason your are using the Dirichlet process (simply what are the observed quantities)?
- what is wrong with the Dirichlet process' model, in other words - which parameters cannot be chosen to fit correctly your data?
For an introduction one should take into account that the classical Dirichlet processes appear by normalizing the gamma random measures. Thus the results are random probability measures. As an example, assume that there is reason to choose the a'priori  weights of points  (or - classes of some objects) q1, q2, . . . , qn    as a sequence of independent random variables x1, x2, . . . xn with some (usually - unknown) positive exponents  m1, m2,  . . . , mn, with common value of the scale parmeter.  Then the rv-s     pj := xj/sumx  form random probabilities of the points    qj,  j=1,2,...,n, where  sumx := x1+x2+...+xn . The random variables  pj  possess the extended beta distribution - which is the simplest Dirichlet distribution.  One of the statistical analysis' aim is then to estimate suitably the parameters  m1, m2,. . .,mn. This requires to have a multiple measurement of the random probabilities,  and to have an algorithm leading to estimation of the parameters (I am omitting this step, probably it can be found in some books:).
Thus: what can be changed with respect to the above idea? Frankly speaking - everything!
A. However a kind of a simple generalization is to choose another  positive process of independent increments, e.g.
A1. some stable processes proposed in the above answer by George, if the heavy tails are preferred;
A2. a short sum (say of  two) gamma processes with different  scales (then additionaly the ratio of the scale parameters are to be estimated, too!).
B. Another possibility (which is much more complicated if compared to A) is to assume a correlation between the independent terms  x1, x2, . . .  , xn.
Regards
Question
Hello Researchers,
I have a developed a "Stochastic solar model" for the purpose of long term distribution system planning. I am aware about the indices researchers commonly use to validate Solar models like; RMSE, MBE etc.
But I face the challenge to find similar literature's (or Similar Solar prediction models) and the same Solar data set that they had used for validating the models. I'm also confused whether it is logical to use aforementioned indices for validating a "Stochastic model", because indices values are not constant.
Kindly let me know your suggestions in this regard. Thanks in advance!
Check This Attached article if it can help , Best Wishes
Question
I have the data and want to find some rule of thumbs to conclude that the data point distribution is either highly, moderately or approximately skewed. I have used Skewness and Kurtosis. I have calculated the mean, median already.
Any idea ... ?
Thanks
.
rules of thumb are by definition somewhat aritrary and therefore do not really need strong grounding in references to prior work !
(moreover, books stating such "rules of thumb" always give the same caveat : "it is somewhat arbitrary, use at your own risks" !)
.
anyway, if you want to hide behind a reference, here is one :
An Introduction to Statistical Concepts: Third Edition
Richard G. Lomax,Debbie L. Hahs-Vaughn
Routledge / Taylor and Francis 2012
pp 89-91
you'll note that their "rules of thumb" are more liberal than Brown U's one ...
.
screen copies from a google book query :
Question
when posteriori probabilities are 0 or 1, Stochastic EM is usable?
I attached this complete question.
You seem to be working with probability .. likelihood .. instead of log-likelihood.  This will cause underflow errors.  Viewed as log-likelihood then each Gaussian is a paraboloid, the GMM "E" log-function is slightly greater than Max of individual paraboloid log functions log(p+q)>max(log(p)+log(q)), the "M" step of a single GMM does not require you to compute "E" for all points it requires only to find the "most likely" classification for each sample point given a set of parameters, this is just Max(p).  With "double GMM" block method this should work too if you are updating only one whilst holding the other constant, but I am sorry I am not familiar with this.
Question
I am using a birth and death process.
I prefer Papoulis, "Probability, Random Variables, and Stochastic Processes" 2nd ed., McGraw-Hill 1984
Question
I am considering the scenario as a birth and death process
@ Alexander, thanks for the suggestions. Would do that.
Question
Suppose we have different candidate models proposed for a time series based on ACF and PACF. Now the basic equation has the white noise term. In MATLAB, u have "randn" command to generate normal random numbers. The parameters can be estimated from "armax" command.
After parameter estimation (calibration), the validation involves comparing the observed data with the predicted values. Now the problem which I m facing is that figuring out whether the white noise should be generated of what length (data length or a bigger population). Secondly, the white noise sequence should be preserved for all candidate model validation or subject to change? If they are changed, then the performance indicators such as RMSE, ML, AIC, BIC will also change.
So what should i do?
I'd say that if your sample size / number of repetitions is small enough that you're worried about dramatic shifts in your summary statistics, then you should be looking at doing ensemble-type runs to ensure that you are in the realm of suitably-sized numbers.  I would assume that you would want your results to be robust enough that they are reproducible without specifying your RNG scheme or seed.
Question
Developing models in past to what extent have supported real life situation...
Vinoth,
When building a model, you need to consider the operational assumptions of the model (e.g., what time period the model will cover, how long will the model run, how many iterations will the model run, what pseudorandom seed will be used if the model is stochastic in nature, etc.), its initial conditions (time), its boundary conditions (space), and the behavior of the system you are attempting to model (e,g., equations of motion, equations of state, etc.).
You can develop a model that uses an extensive number of parameters and attempts to model your system in an exacting manner, but that will lead to a model which has a long run time and requires a number of assumptions that may or may not be realistic. A person can also develop a model that has only a small number of the required parameters, that does not run very long, but may oversimplify some of the assumptions made to determine the key parameter values you are looking for. In most situations, we approach modeling somewhere in between these two extremes.
When modeling TB and HIV, the models are usually an increasing exponential to track the growth curves of these viruses and negative exponentials to track the destruction of immune system antibodies. These models can usually be linearized with some simplifying assumptions (for a small range). Cancer modeling is usually stochastic, where there is an amount of genetic damage to a cell and at some point, a threshold is crossed where one additional 'hit' provides enough cellular damage to transform a normal cell into a cancer cell. I have included some references for you that will help.
Eric
Question
How can one simulate a stationary Gaussian process through its spectral density?
Thank you for answers. Let me explain more precisely my question: How can one simulate the following Ornstein Uhlenbeck stationary Gaussian process,
dX_t=X_0-\alpha X_tdt +dB_t, where X_0=\int_0^\infty e^{\alpha s} dB_s and B is a fractional brownian motion.
Question
Hello, I am currently working on models where energy can be produced using either a clean or dirty technology and investment (in knowledge) reduces the average cost of the clean technology or backstop. A steady state involves using both the dirty and clean technologies when their marginal costs are equal.
I am thinking of including a stochastic process for change in energy prices such that investment in the backstop is feasible only when energy prices are above a certain level (that is to say, investment in knowledge now reduces the future average cost of the backstop but there is also a huge fixed cost in actually using the backstop). Theoretically, I believe that this would involve switching back and forth between clean and dirty technologies. I am looking for any ideas in how to model this. I am attaching my recent publication (basically including stochasticity as I said in my current model).
I am interested in collaborating! any ideas?
Supratim
Hi Joaquim, thanks for your answer! I am not aware of the Baum-Welch algorithm but would definitely look into it.
I would definitely be in touch if I need more help regarding the stochastic derivation of my model.
Supratim
Question
Should we use just one value for each realization? Or different quartiles? What if one of the quartiles is eliminated in the reduced model?
You may use the two stage stochastic modeling technique. Some decisions are made before realizations of the uncertain events and some of them are after them.
Question
I am working on a forced stochastic model and needs to simulate a two-parameter plot indicating where synchrony happens. Been searching literatures about simulating Arnold's tongues but not satisfied with my search. Looking forward to your help. :)
This is a good question with more than answer.
Arnold's tongue diagrams are investigated in
S.-B. Shim, M. Imboden, P. Mohanty, Synchronized oscillation in coupled nanomechanical oscillations, Science 316, 2007:
Synchronized states are often represented by Arnold's tongues (regions of frequency locking in the parameter space) [see page 96].   In particular, see Fig. 3, page 97.
More to the point, a thorough investigation of Arnold's tongues is given in
P. Kaira, Resonance Forcing in Catalytic Surface Reactions, Ph.D.thesis, Technische Universitaet Berlin, 2009:
See the introduction to single oscillators, page 15.    In an extended system, it is observed that close to the boundaries of Arnold's tongues stable frequency locked patterns may exist (page 16).   A summary of the findings is given on page 111.
Question
I wish to know whether I could the adopt time-varying stochastic frontier model when the values of each cross section are not available for the complete time period selected for the analysis. Could someone help me in this regard?
Thanking you.
Question
I would like to apply the Integer Value Autoregressive model for predicting pest count.
Can anybody help me in estimating the INAR model using R software, or any other software?
See eg Count time Series Models by K Fokianos (fokianios@ucy.ac.cy) Handbook of TS analysis.
Question
For instance, are there some references about the calculation of the invasion speed of a disease using an IBM?
Dear Nadja: for analysis did you check the papers of Méléard and collaborators about trait invasion, e.g. "Invasion and adaptive evolution for individual-based spatially structured populations" by Nicolas Champagnat, Sylvie Méléard. On the numerical point of view : if you have a few individual A among a lot B individual, to simulate the invasion of the A's among the B's you can hybridize : a PDE or integro-differential equations for the B's coupled with an IBM for the A's.
Question
Can we code Lagrangian stochastic dispersion models in Matlab with providing particle emission data?
Why would you want to do this in MatLab? It would take forever! You should use a computer code designed to do this. Also, I wouldn't suggest using Lagrangian based particle tracking, because it's slower than a herd of snails stampeding up the side of a salt dome. In the Lagrangian method, the time step is controlled by the fastest movement in the smallest element, which is why it's so slow. The Hamiltonian method is orders of magnitude faster. Where time is the independent variable in Lagrangian particle tracking, velocity is the independent variable in Hamiltonian particle tracking. PTRAX is the fastest and most versatile program available and it has been extensively validated against analytical solutions and large-scale field data. It is used by several big DOE contractors for contaminant transport. Here's a link to the report http://dudleybenton.altervista.org/publications/Development%20of%20the%20Fast%203D%20Particle%20Tracker%20PTRAX.pdf If you're interested I could pull together some examples and put a ZIP file out where you can download it.  It handles 2D and 3D domains and 5 shapes of elements (triangles, qudralaterals, tetrahedra, prisms, and bricks) as well as finite element or finite difference. It will create graphs of each particle track, snapshots of the concentrations, and even animations. I can put one of the animations somewhere so you can see what they look like. They're animated GIFs that will play in any web browser. You can also insert capture wells (circles in 2D or cylinders in 3D) and walls that will report every particle that passes through them.
Question
I am using an individual based stochastic model to explore the impact of environmental changes on speciation. In this model individuals forage in two different habitats with a probability determined by the profitability of the habitat. Apart from the ecological and mating loci, I have a certain number of non-coding neutral loci that are used to follow genetic signatures of evolutionary divergence. The neutral loci act like microsatellites, with high maturation rates. My question is how to calculate Fst in Matlab? Are there any softwares that can be used in Matlab not in R?
Hi Lai
I will suggest for the case of your study PGEToolbox, which is Matlab-based open-sourced software package for data analysis in population genetics (see pdf attached for more informations)!!
PGEToolbox is available free of charge at http://bioinformatics.org/pgetoolbox.
Question
I am facing a problem regarding the stationary distribution of mobility models in revising my paper.
In a general sense, my question is like this.
• Given a stochastic process {Xt: t>=0}  such that the initial state Xis uniformly distributed in the state space and for all t > 0, Xt = X0,namely, the process does not evolve with time. For this specific process, can we say that {Xt} has a stationary distribution that is uniformly distributed in the state space?
The answer is quite simple: Yes.
The process "inherits" the initial distribution for all time instants through the perfect correlation, so the initial distribution equals the stationary distribution. And your process is perfectly well defined.
Question
Could anyone update me about the differences between a chaotic and stochastic system? Suppose for a certain mathematical model, we have the chaotic behaviour. How could do the similar kind of analysis through stochastic modelling? Do I need to find the SDE and just solve it through EM Method? Please do suggest me relevant papers, books and software to perform all these kinds of analysis.
Stochastic motion is random at all times and distances. Chaotic motion is predictable in the very short term, but appears random for longer periods. A good example is the sedimentation of three spheres in a viscous fluid. The governing equations are very simple. Three spheres arranged in a horizontal equilateral triangle maintain their relative positions indefinitely. Three in a horizontal isosceles triangle settle with a periodic motion. Three in a horizontal line settle with a chaotic motion .
The chaotic motion of a sphere in a suspension of identical spheres can be approximated by finding the sequence of position-velocity values of a joint Markov process and interpolating a smooth curve .
I.M. Janosi, T. Tel, D.E. Wolf, J.A.C. Gallas, Chaotic particle dynamics in viscous flows: the three-particle Stokeslet problem, Phys. Rev. E 56 (1997) 2858-2868. doi.org/10.1103/PhysRevE.56.2858
M. Bargiel and E.M. Tory, A five-parameter Markov model for simulating the paths of sedimenting particles, Appl. Math. Modelling 31 (2007) 2080-2094.
doi:10.1016/j.apm.2006.08.023
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Suppose we have a metric space (X,d) such that the points in this space are distributed by a probability distribution function (for example Poisson d.f). How we can calculate the probability of d(p,q)<t; such that p,q in X and t in R.
This is one of the classic problems in geometric probability. It (and many variations) are "covered" (pardon the pun) in books like Hall (1988) "Introduction to the theory of coverage processes" and Solomon (1978) "Geometric probability". However, I think you should first check an old paper by Cyril Domb (1947) "The problem of random intervals on a line" Proc. Cambr. Phil. Soc., vol 43, 329-341. If I remember correctly, this paper deals with the problem you describe.
Question
In statistical inference our aim is to study the population characteristic that is the parameter. But if it is said that 'T' is an estimator of the function 'Ѱ(ϴ)' (which is of interest) then what is the meaning of it.
You have ϴ, a parameter of the population, and Ѱ(.), that is a function of that parameter, then T is an estimator derived over a random sample of Ѱ(ϴ). If Ѱ(.) is the identity function, then Ѱ(ϴ)=ϴ and T is an estimator of ϴ.
For example:
Let Y1, Y2, ....Yn be n independent Bernoulli(P) random variables which value 0 or 1.
And let X=Sum(Y1, Y2, ....Yn) the total of the random sample, then it may be shown that X has a Binomial distribution with ϴ=(n, P) where n is the number of independent bernoulli trials and P is the probability of success ( then Pr(Yi=1)=P ) of each Bernoulli.
In this example ϴ is a vector of parameters. Let µ be the mean of X, because E(X)=µ, but µ is a parametric function of ϴ=(n, P) because in the Binomial Distribution E(X)=µ= n P
If we want estimate µ we can obtain a MLE estimator using T= sample mean (of Y1, Y2, ....Yn), then we have an estimator of Ѱ(ϴ)=µ = n P.
Question
A metric can be used to evaluate performance
The residual sum of squares (rss) can be calculated using ctree's "predict" function, so if "model_ctree" is your ctree model and "y" is your dependent variable:
tss <- sum((y-mean(y))^2)
Note that this R2 is a prediction on the learning data. If you want to predict on new data, you will have to specify the test set conaining the new data using the "newdata" parameter in the "predict" function. If your test set is "testset", use:
predict(model_ctree, newdata = testset)
If you do not have fresh data for the test set, you can hold out an amout of cases from your entire sample by randomly splitting it up into a learning sample (that you use to fit the ctree model) and a test sample (that you use for prediction).
Question
Could we determine with certainty a species to have reached a point of no recovery?
It depends strongly of the population size. A "small" population of insects has little to do with a population of bears. To detect precisely the Allee effect, you have to show that the population growth rate is inverse density dependent: the smaller the population, the lower the growth rate. You need to have at hand the data of several generations. Compute the lambda (or r equally) for each couple of successive generations , and plot it against the population density of the first generation (or year) of the couple. A linear relation indicates no density dependence, a tendency toward a sill for high density indicate positive density dependence, an exageration of the slope at low densities indicates an Allee effect. I agree with Jabi: stochasticity prevents from predicting the future of small populations, and if few data are available, viability analysis will not give much insight...
Question
Normally we have to assume that the service time of the queueing process has an exponential distribution. How do we assume the service time follows heavy tail and what is the situation for adopting heavy tail distribution? Moreover, what is the reason beyond that?
We don't *have* to assume the service time is exponential. People often *do* assume the service time is exponential simply because this leads to a more easy analysis.
You should study the service time distribution empirically. Find out if it really is exponentially distributed (memory-less) or not. Heavy tails means that rarely there are extremely large service times, much larger than would be the case with the exponential (memoryless) distribution. Memoryless: however long you have waited, it is still equally likely that the service is finished in one hour. Heavy tails: the longer you have waited, the more likely it is that you will have to wait longer still.
Theory which is based on exponential distributions can give severely wrong answers when the actual distribution is heavy-tailed.
Question
I think the photons generated in SPDC have a stochastic phase to the pump light because they are generated from the vacuum state. Is it true? I have looked up some references but haven't found anything.
I can not give you an earlier reference, since this question was probably discussed for the first time long ago. Partially it is admitted e. g. , in Optics Express, Vol. 18, Issue 26, pp. 27130-27135 (2010) http://dx.doi.org/10.1364/OE.18.027130. From general point of view, any spontaneously emitted photon have random phase with respect to photon emitted in separate spontaneous emission act, despite photons generated in the same SPDS event are correlated with each other
Question
More precisely, I want to bound
|P(\sup_{t\in S_1}X(t)\sup_{t\in S_2}X(t)-P((\sup_{t\in S_1}X(t))P((\sup_{t\in S_1}X(t))|
where S_1,S_2 are two compact set and (d(S_1,S_2)=k with k\to\infty
X is a stationary weakly dependent gaussian process
I don't get the question. You have probabilities - of what? There is no event.
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I am doing a research on a Markov chain approach on the behavior of rainfall and am considering first and second order chains. Using the Bayesian Information Criterion I want to determine the optimal model and use the optimal model to determine the expected length of dry (rainy) spell. So for second order models I am not sure of how I calculate the expected length of dry (rainy) spells?
The expect length of higher Markov model can be found from my paper 2007, AISM.
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