Stochastic - Science topic

The tax affects the budget constraint of the worker by decreasing consumption, unless the salary is increased to a level that restores initial consumption levels despite the tax. On the other hand, since consumption decreases, production also decreases due to a negative demand shock. I don't know if your model considers unemployment, but if agents can seek new positions with a higher salary, then they will, creating a short-term supply shock as workers look for higher wages. At least, that's the logic I consider

Good afternoon Marcus,

Thank you, I will have another look at your paper.

Best wishes Erik

If you cannot get the book, I have a spare copy which I will let you have if you donate a small donation to "Doctors without Borders".

Mario Stipčević you can google them. Although not directly related to the SC, the importance of randomness in mathematics and computer science is being recognized. Since randomness is part of SC, I believe it would gain more media attention in the future as an additional option other than analog or neuromorphic computing, especially in the AI workload.

By the way, regarding how SC could be implemented, there are only 2 options as far as I know, i.e., ASIC or FPGA. Binary computing can simulate the SC operation but it is only meant for case studies.

There is the chillR package, which offers the function temperature_generation(), a wrapper function for the RMAWGEN weather generator mentioned in previous answers. The functions documentation is a bit clearer and the function is also more convenient. The package also contains some useful methods to combine the weather generator with climate change projections.

You can also get some instructions in the following chapter of a masters course accompanying ebook (from the same author as the R package)

And recently I also stumbled upon another R package called wxgenR. But I don't have much experience with that weather generator.

Hey there Gaurav Gangil!

When it comes to tackling multi-stage stochastic problems with a scenario-based approach, scenario reduction techniques are indeed crucial to streamline computations and manage complexity. Now, let me throw some suggestions your way.

1. **Principal Component Analysis (PCA):** Leverage PCA to identify and focus on the most significant scenarios. It helps in capturing the essential variability in your data, allowing you Gaurav Gangil to trim down the scenarios without losing vital information.

2. **Clustering Techniques:** Consider employing clustering algorithms like k-means to group similar scenarios. By consolidating scenarios with similar characteristics, you Gaurav Gangil can represent a cluster with a single scenario, reducing the overall complexity.

3. **Monte Carlo Sampling:** Use Monte Carlo sampling to simulate scenarios and then select a subset that adequately represents the entire distribution. This can be an efficient way to reduce the number of scenarios while preserving the stochastic nature of the problem.

4. **Machine Learning Models:** Train machine learning models to predict the outcomes of potential scenarios. You Gaurav Gangil can then focus on the most influential scenarios identified by the models, discarding less impactful ones.

5. **Adaptive Sampling:** Implement an adaptive sampling strategy where you Gaurav Gangil iteratively refine your scenario set based on the evolving solution. This can be particularly useful in dynamic and uncertain environments.

Remember, these are just a few ideas to get your creative juices flowing. Feel free to adapt or combine these techniques to develop your own scenario reduction method. Good luck with your research!

One of the easiest deterministic models expressing self-organizing responses is

A cellular automaton model for freeway traffic

Kai Nagel & Michael Schreckenberg

https://www.researchgate.net/publication/263504490_A_cellular_automaton_model_for_freeway_traffic/comments

It is astonishing to observe as traffic jams are arising from completely deterministic simulations with increasing density of cars on the road.

I recommend search for experimental data on stadiums done with real cars, it is quite funny to observe that despite the best intentions of all drivers to keep the right distance, traffic jams always arise.

Howdy Marc Fischer,

I applaud your clear statement that "random" and "stochastic" are terms to be applied to human perceptions and efforts, since the underlying dynamics of fluids in turbulence is deterministic. Very rare observation, very welcome! (The fluid knows.)

As far as "quantum mechanics is genuinely indetermistic," some of us think the jury is still out on that because your comments on turbulence apply to human limitations, including ignorance, there also.

Given the experience of "QUANTUM MECHANICAL random fluctuations" by turbulent processes that is possible, I doubt that they could affect the state of the turbulence as having too little energy. "Random" absorption of a photon by a molecule that enhanced the molecule's internal energy would have an effect, of course, by increasing the thermal-motion energy of the fluid, but I doubt it could be noticed in turbulence, even by the flow being triggered to new behavior. This affect is very different from application of human precision in initial conditions that enters mathematical expressions with such strong leverage because of the mathematical non-linearity.

I also applaud the association you have made as one of the ways to trigger insight, but too often the answer in a specific case is no.

Happy Trails, Len

Yes, STATA provides options to label variables, including company names that are typically represented as string variables. You can use the "label" command to assign labels to variables in STATA. Here's an example of how you can label a variable containing company names:

```

label variable CompanyName "Company Name"

```

In the above example, "CompanyName" is the name of the variable, and "Company Name" is the label assigned to it. You can replace these names with your actual variable name and desired label text.

Once you assign labels to a variable, you can use them for reference and display purposes in STATA. For example, if you want to display the labeled values of the "CompanyName" variable instead of the raw string values, you can use the "tabulate" command with the "nolabel" option:

```

tabulate CompanyName, nolabel

```

This will display the labeled company names instead of the raw string values in the output.

Labeling variables can help enhance the interpretability and readability of your data analysis in STATA, especially when working with string variables such as company names.

CASE A:To begin with, let's admit that nature does not see with our eyes and does not think with our brains.We try to understand how nature performs its own resolutions in space-time x-t like an inseparableunit block.

However, B-Matrix statistical chains (or any other suitable stochastic chains) can answer this question and demonstrate, in a way, how nature works:

i-nature see:the curve as a trapezoidal area.ii-nature see:the square as a cube or a cuboid volume.iii-nature see:the cube as a 4D Hypercube and evaluates its volume as L^4.In all hypotheses i-iii, time t is the additional dimension.However, many people still believe that x-t spacetime is separable and that describing x-t as an inseparable unit block is only essential when the speed of the object concerned approaches that of light.This is the most common error in mathematics as I understand it.The universe has been expanding since the time of the Big Bang at almost the speed of light and this may be the reason why the results of classical mathematics fail and become less accurate than those of the stochastic B-matrix ( or any other suitable matrix) even in the simplest situations like double integration and triple integration.This is the reason why the current definition of double and triple integration is incomplete.Brief,-------Time is part of an inseparable block of space-time and therefore,geometric space is the other part of the inseparable block of space-time.In other words, you can perform integration using the x-t space-time unit with wide applicability and excellent speed and accuracy.On the other hand, the classical mathematical methods of integration using the classical mathematical  technique in the geometric Cartesian space x-alone can still be applied but only in special cases and it is to be expected that their results are only a limit of success.In other words, you can perform integration using the x-t space-time unit with wide applicability and excellent speed and accuracy.1-It is important to understand that mathematics is only a tool for quantitatively describing physical phenomena, it cannot replace physical understanding.2-It is claimed that mathematics is the language of physics, but the reverse is also true, physics can be the language of mathematics as in the case of numerical integration and the derivation of the normal/ Gaussian distribution law via the  statistical  B-matrix chains.However, in a revolutionary technique, chains of B matrices are used to solve numerically PDE, double and triple integrals  as well as the general case of time-dependent partial differential equations with arbitrary Dirichlet BC and initial arbitrary conditions.At first,I=∫∫∫ f(x,y,z) dxdydzwhich has been defined as the limit of the sum of the product f(x,y,z) dxdydz for a small infinitesimal dx,dy,dz is completely ignored in numerical statistical methods as if it never existed.It is obvious that the new B matrix technique ignores the classical 3D integration I=∫∫∫ f(x,y,z) dxdydz.We concentrate below on some results in the field of numerical integration via the theory of the matrix B, which in itself is not complicated but rather long.

------------

7 Free Nodes:

Single finite Integral

I=∫ f(x) dx ... for  a<=x<=b

Briefly, we arrive at,

The statistical integration formula for 7 nodes is given by,

I=6h/77(6.Y1 +11.Y2 + 14.Y3+15.Y4 +14.Y5 + 11.Y6 + 6.Y7)

which is the statistical equivalence of Simpson's rule for 7 nodes.

Now consider the special case,

I=∫ y dx from x=2 to x=8 where y=X^2.

That is,

X = 2 3   4  5   6   7  8

Y = 4 9 16 25 36 49 64

Numerical result via Trapezoidal ruler,

It = Y1/2+Y2+Y3+Y4 +Y5+Y6+Y7/22+9+16 +25+ 36+ 49+32=169 square units.

Analytic integration expression I=X^3/3

Ia=(384-8)/3= 168 square units.

Finally, the statistical integration formula for 7 nodes is given by,

Is=6 h/77 (6*4 +11* 9 + 14* 16+15* 25 +14* 36 + 11*49 + 6*64)

I = 167.455 square units. This means that static integration is quite fast and accurate.

CASE B:

-----------

Double finite Integral  I=∫∫ f(x,y) dx dy... for the domain  a<=x<=b and  c<=y<=d

If we introduce a specific example without loss of generality where ,

the function Z=f(x,y) is defined as,

Z(x,y)= X^2.Y^2 + X^3 . . . . . (1)

defined on the rectangular domain [abcd],

1<=x=>3 and 1<=y=>3 . . . Domain D(1)

The process of double numerical integration (I),

I=∫∫ f(x,y) dxdy

on the D1 domain can be achieved via three different approaches, namely,

1-analytically (a),

Ia=(x^3/3 *y^2 +x^4/4) + (x^2*y^3/3+x^3) . . . (2)

2-Rule of the Double Sympson (ds),

I ds=

h^3.(16f(b+a/2,d+c/2)+4f(b+a/2,d)+4f(b+a/2,c)+4f(b,d+c/ 2) +4f(a,d+c/2)+f(b,d)+f(b,c)+f(a,d)+f(a,c))/36 . . . . (3)

iii- The statistical integration formula via the Cairo technique (ct),

Ict = 9h^3/29.5( 2.75Z(1,1)+3.5Z(1,2)+2.75.Z(1,3)+3.5Z(2,1)+4.5Z(2,2)+ 3 .5Z(2.3)+2.75Z(3.1)+3.5Z(3.2)+2.75Z(3.3)) . . . . (4)

where h is the equidistant interval on the x and y axes.

The numerical results are as follows,

i- Ia =227.25

ii-Ids=226.5

iii-I ct=227.035 ..which is the most accurate.

CASE C:

----------------

Triple finite Integral and Hypercube

I=∫∫∫ f(x,y) dx dy dz... for the domain  a<=x<=b and  c<=y<=d & e<=z<=f.

Again, we present the nature and the matrix of the triple integration on the cube abcdefgh, divided into 27 equidistant nodes,

we present below the supposed nature matrix of the triple integration on the 3D cube abcdefgh (1,1,1 & 2,1,1, .......& 3,3,3)

I=∫∫∫ W(x,y,z) dxdydz

on the cube domain.

I = 27h^4/59( 2.555W(1,1,1)+3.13W(1,2,1)+2.555.W(1,3,1)+3.13W(2,1,1)+3.876 W(2,2,1)+ 3,13W(2,3,1)+2,555W(3,1,1)+2,555W(3,2,1)+3,13Z(3,3,1) . . . etc.)

The question arises, why are statistical forms of integration faster and more accurate than mathematical forms?

We assume that the answer is inherent in the processes of integration, whether they belong to the 3D geometric space or the unitary x-t space.

These lectures: https://irfu.cea.fr/Phocea/file.php?class=page&file=678/QFT-IRFU1.pdf might be a good place to start.

Markov was part of the great tradition of mathematics in Russia and never missed his goal.

He designed and passed on a non-physical stochastic chain and never claimed it could solve physical problems.

We present the following brief comparison between classical stochastic Markov chains and recent B-matrix chains for the statistical transition imposed via the Cairo technique in the year 2020,

Markov defined his original nxn square transition matrix M(i,j) via two assumptions:

(1) the transition probabilities Mi,j are equal to or greater than zero.

(2) the sum of the transition probabilities Sigma Mi,j=1, for all the columns j=1 to n, (called the vertical matrix) OR the sum of the transition probabilities Sigma Mi,j=1, for all the rows i =1 to n , (called horizontal matrix)

Obviously the Markov stochastic chains can be converted into B-matrix chains and vice versa.On the other hand, the well-founded B-matrix and B-transition matrix chains rely on four assumptions. The first two assumptions are those of Markov with a slight modification of the second to allow the insertion of boundary conditions BC.The other two assumptions are physical, namely,(3) the matrix B is symmetric in space like nature itself, i.e.Bij = Bji.condition (3) represents detailed equilibrium and reciprocity.(4) the input elements of the main diagonal (RHS) are all equal to a constant input RO.Where RO is an element of [0,1] and represents the residual or remanence of the energy density at a point in spacetime.It is clear that RO must be zero when solving the Poisson and Laplace PDE and somewhere between zero and one when solving the heat diffusion/conduction PDE.

The question arises as to why Markov stochastic chains cannot solve physical problems (PDE of heat diffusion/conduction, PDE of Poisson and Laplace, derivation of integration and differentiation formulas, etc.) which are simply solved by the Transition-matrix B (B i, j) of the Cairo Technique.

what is missing in the Markov stochastic chains?

The answer is that the physical conditions (3) and (4) are missing.

Decisive difference is that in Markov chains which does not account for space x, the time interval dt is arbitrary prechosen and not necessarily relevant to space while in B-matrix chains time is merged or woven in a unitary 4D x-t  space which means that the time interval dt is imperial.

The main concerns of dynamic performance in stochastic finite element analysis (SFEA) are accurately capturing a system's probabilistic behavior subjected to random input parameters, such as material properties, loading conditions, and geometric uncertainties. Specifically, the focus is on analyzing the system's dynamic response under stochastic loads or uncertainties, including its natural frequencies, mode shapes, and transient behavior. To support informed decision-making, the goal is to provide reliable predictions of the system's dynamic performance, including response variability and probability distribution.

Digital twins can be combined with SFEA to improve the accuracy of dynamic performance predictions. A digital twin is a virtual representation of a physical system that integrates data from various sources, including sensor measurements, simulation models, and historical performance data. By combining SFEA with digital twins, we can obtain real-time feedback on the system's actual behavior and update the model parameters accordingly, leading to improved predictions of the system's dynamic performance. Additionally, digital twins can be used to optimize the system's design, identify potential failure modes, and develop effective maintenance strategies.

Best wishes.

The stochasticity of a system refers to the uncertainty/randomness of its outputs. While the uncertainty of a system could also refer to uncertainties/randomness of the inputs of the systems.

e.g., for a transportation system, the stohcasticity is used to measure its traffic flows/travel times (outputs), the uncertain of the system usually includes the wheather, the travel demand (inputs)

Hi,

I suggest you see the links and attached file on the topic.

Best regards

@ Amer Dababneh

Thank you for sharing the link.

Regards

D. Ghosh

Hello all,

First, kudos to Christian Geiser for furnishing the requested paper.

Second, for requests like this, which involve finding "lost" internet resources, consider using the "Wayback machine" (wayback.archive.org). This site has archived scrapings from the www going back many years.

Sure enough, the requested paper (from the no-longer-active journal site) may be found at: https://web.archive.org/web/20050131073941/http://mpr-online.de/

Good luck with your work.

Dear Ravi,

If your question is specifically with regard to D. Boore's point source stochastic simulation method (1983, 2003) then no you can not simulate two separate horizontal components. Essentially, the method does not distinguish between components. I am pasting a paragraph from 2003 paper p665 in the following. There are studies that allow such distinction as mentioned by Luis Fabian Bonilla . Here is the paragraph from 2003 paper of D. Boore:

"Comparisons of stochastic-method predictions with empirically-determined

ground motions indicates that the stochastic method is useful for simulating mean

ground motions expected for a suite of earthquakes having a specified magnitude and

fault–station distance. Care must be used, however, when the method is used to

simulate site-specific and earthquake-specific ground motions. As described in this

paper, the method does not include any phase effects due to the propagating rupture

and to the wave propagation enroute to the site (including local site response). In

addition, the differences between the various components of motion and different wave

types are ignored. For these reasons, fault-normal effects, phase differences over

horizontal distances, spatial correlations, directivity, etc. are not captured by the

simulated motions. It should be possible to include some of these effects in the method,

and I am aware of some efforts along these lines (e.g., LOH, 1985; LOH and YEH, 1988;

TAMURA et al., 1991; TAMURA and AIZAWA, 1992)."

Hope this helps.

Best regards,

Sanjay

It indicates nothing about the directed version.

Thanks Koen Van de Moortel.

What about it? The opening statement is wrong, that’s all. The fact that the labels of the states are integers doesn’t imply anything about dimensionality. It implies that the states are countable. And that’s all that’s needed.

Hi,

I found this (attached), it is developped to integrate a specific system of SDDE with 2 coupled oscillators, delay is in the coupling. But you can extract the method test it against results obtained by E Buckward, but she uses Ito formulation, here it is more like an Euler formulation; please tell me if it is ok.

can't find the author except:

mw@eml.cc 2013

Regards

Julien

Many thanks, Abebayehu Girma Geffersa . It was two years ago and as you writously said, I followed the procedure that I mentioned here and it is the correct approach. Akbar Muhammad Soetopo please follow a similar procedure that I wrote here as well. For a detailed answer to this, please read this paper. Belotti, F., Daidone, S., Ilardi, G., & Atella, V. (2013). Stochastic frontier analysis using Stata. The Stata Journal, 13(4), 719-758.

Dear Mahgol Motahhari,

I suggest you see the links on the topic.

https://arxiv.org/pdf/2110.03146.pdf

Best regards

AIC and BIC can be two options. Generally, you can develop an arbitrary criterion based on the weight of observations (time points) and the definition of distance (between observation and expectation). For example, you can define distance as the square or absolute value of difference.

Convex optimization theory has an importent aspect: duality gap. This gap is occurring for bad constraints. To search for such problems, we use the analysis of orders of smallness of infinitesimal quantities

bonjour je voudrai connaitre la commande pour décomposer la productivité totale des facteurs en efficacité technique et progrès technologique

Half normal will be a gamma distribution and the hypothesis can be tested if errors positive or magnitude of error is considered .

This article published in the The Stata Journal (2012) talks about Stochastic frontier analysis using Stata

Kindly viait..

Please see the following paper:

Comparison of deterministic, stochastic and fuzzy logic uncertainty modelling for capacity extension projects of DI/WFI pharmaceutical plant utilities with variable/dynamic demand

I have got the full text of your book. It looks great.

Thank you very much for your fruitful comments.

Regards,

Ahmed

Please note that we might not know the probabilities in the environment before we start. We can develop them by averaging while playing.

The learning rate parameter can be adjusted according to our interest.

The question is do we want to explore or exploit the environment??

Long memory volatility data are best modelled as FIGARCH models. There are perhaps no known ways of transformation from long memory to short memory data.

thanks

All this and more is described in van Kampen’s book ``Stochastic processes in physics and chemistry".

Write the equation in a form more useful for calculating anything:

dX/dt=(a(t)+b(X-X0))X+cXη(t)

where η(t) is the noise.

Now write a(t)X+b(X-X0)X=-dU(X)/dX, which shows that this equation describes the motion of a particle in a potential U(X)=-(1/2)(a(t)-bX0)X^2-(b/3)X^3, in the presence of multiplicative noise.

So, for a(t)=0, the term that’s quadratic in X, does describe oscillations with frequency squared bX0, with the cubic term describing escape from the well.

However the multiplicative noise complicates things.

If we divide out by X and set Y(t)=ln X, the noise for Y is additive.

Dear Prof. Dr Imtiaz Husain

Thank you for your answer. I have some papers about the applications of the stochastic arithmetic. But I need to know more and real applications.

Dear Alia Alkameli ,

Stochastic gradient descent is a very popular and common algorithm used in various Machine Learning algorithms, most importantly forms the basis of Neural Networks. In this article, I have tried my best to explain it in detail, yet in simple terms. I highly recommend going through linear regression before proceeding with this article.

Gradient, in plain terms means slope or slant of a surface. So gradient descent literally means descending a slope to reach the lowest point on that surface.

Regards,

Shafagat

Equations (5) and (6) are obtained from eq. (4) by differentiation. The tricky part is that the variables P and Z appear, also, in the limits of integration and not, only, in the integrands. This means the calculation is a bit more complicated. I'll try to verify them.

YES, I think, you can use an alternative approach, such as saddle point approximation. See e.g.,

See also

1. Introduction to Stochastic Calculus for Finance: A New Didactic Approach (Lecture Notes in Economics and Mathematical Systems)

2.From Measures to Itô Integrals (AIMS Library of Mathematical Sciences)

3. Stochastic Integration by Parts and Functional Itô Calculus

Dear Harish Mangilal Choudhary ,

There are some really good papers on this topic aside from the Adrian paper you referring to :

The first paper by José Hugo Elsa and L. Moriconihas the details you are looking for.

There is another paper by Jonathan H. Tu that you can see

You can also read the dissertation by Yangzi Huang of Syracuse University to read more about vortex detection:

Discrete optimization would be something like the classic Traveling Salesman problem - you are finding a sequence of discrete points that satisfies some optimization criteria. Continuous optimization involves finding a set of extreme points on a continuous hypersurface that is defined by a continuous cost function. Hamiltonian and Lagrangian approaches from physics are very old forms of this type of optimization.

Online search engines provide a treasure-trove of hits on stochastic and deterministic approaches. Give it a whirl!

Dear Mathusuthanan Mari

There are several stochastic-based crack distribution models being researched upon in recent years. However, there are some fundamental papers in this area that still influences research in this area as their models are simple, elegant, and effective. Some of them are

I hope these articles will be of some help to your research.

Based on the substitution of personal goals for market goals - these are more theories from the field of behavioral finance. As we say: a tit in the hand is better than a crane in the sky!

Thanks @Ette Etuk

One needs to be more attentive and not just believe on politicians.

For example, according to France and Brazil, the airplane was invented by Santos Dumont, who flew around the Eiffel Tower, and though he won the International prize for it, this is not accepted in the US, it is not even mentioned.

I see that other countries do not believe in "America First." The leadership stands divided.

Some are just grateful to be born and live here, hopeful to make contributions. China is passing the US on 5G. In short, the more one advances, the more opportunities for others to contribute and have success.

Here, elections must be based on trust, not just voting. As Stalin said, "who counts the votes control the voting." Many countries don't even consider the US a democracy, because there is no direct vote here yet -- no people voting. The president is voted indirectly. Maybe the problem yesterday will do away with the slave-time Electoral College. Will see.

Second, from size, India has a bigger claim to "largest democracy."

As you mentioned, parameter tuning studies for a metaheuristic is quite important. Researchers should determine proper control parameters for their optimization problem to increase the success of the algorithm. However, many researchers uses algorithm parameters suggested by their developers as this is can be a time consuming task via a trial and error approach. Also, I agree that self-adaptive versions of these algorithms can increase both effectiveness and performance compared to their original versions. However, they can require definition of extra parameters as well in the algorithm. In my cases, I prefer to use original versions of the algorithms via a parameter tuning study. Besides, I use two termination criteria including a predefined maksimun generation number and a tolerans value. If the algorithm provides a misfit value less than the tolerans, it stops before the reaching maksimum number of generation. Sometimes I take into account a number of successive generations. For instance, if the solution do not improve during the last 30 generations, I stop the algorithm. This provides relatively decrease the high computation cost due to much execution of the forward equation. This is the biggest drawback of the global optimization compared to derivative-based approaches considering high-dimensional optimization problems.

In python you can use sckitlearn library or You can use other software, like Maple or GAMS. It depends on your need.

1- Stochastic Model Predictive Control: An Overview and Perspectives for Future Research

DOI: 10.1109/MCS.2016.2602087

2- Stochastic model predictive control — how does it work?

By: panelTor Aksel N.HeirungJoel A.Paulson

3- Stochastic Nonlinear Model Predictive Control with Probabilistic Constraints Ali Mesbah1 , Stefan Streif , Rolf Findeisen , and Richard D. Braatz

For software refer following article...

Dear;

You can solve the integrals manually or by a software.

Regards

see

Random Ordinary Differential Equations and Their Numerical Solution

Springer SingaporeXiaoying Han, Peter E. Kloeden (auth.)Year:2017

Random Differential Equations in Science and Engineering

Academic PressT.T. Soong (Eds.)Year:1973

Random Differential Equations in Science and Engineering

Academic PressT.T. Soong (Eds.)Year:1973

I hope so too. Thanks Dr. @Hom Nath Chalise.

Dear Sandeep,

These are the dual objectives to be achieved in particular, a major difficulty consists in resolving the Exploration / Exploitation (E / E) compromise.

Best regards

You might try looking at the Continuous Ranked Probability Score (CRPS) . You would need the version of this that compares two probability distributions, and would also need to extend the definition from 1 to multiple dimensions. Essentially it is just the integral of the squared difference between the two cumulative distributions functions. I don't know if there are formulae for univariate or multivariate normal distributions. In the univariate case, it is an extension of the mean absolute error.

Otherwise, you might look through the list of distance measures at https://en.wikipedia.org/wiki/Statistical_distance .

You can find some dataset in the following link:

https://www.kaggle.com/search?q=covid

All the best.

Nice Dear Anup Kumar Yadava

Find it here; Good luck!

Hi, my suggestions are:

The Euler scheme with Brownian increments and step T/n strongly converges to the underlying diffusion rate n^{-1/2} for the sup norm over [0,T] in every L^p such tat X_0\in L^p.

The Milstein scheme with Brownian increments and step T/n strongly converges to the underlying diffusion rate n^{-1} for the sup norm over [0,T] in every L^p such tat X_0\in L^p.

Hence it t converges faster BUT, first it applies to SDEs with smoother drifts and diffusions coefficients, it is not simulable when the dimension of the Brownian motion is greater than 1 because in higher dimension it involves the simulation of Levy areas which is not possible at reasonable computational cost.

Finally both schemes have the same weak rate of convergence (O(1/n)) ie rate of rate of convergence

\E f(X_T) -\Ef(\bar X_T)

under various regularity or uniform ellipticity assumptions assumptions on teh coefficients of the diffusions and /or the function f. If I may, all this is detailed and proved in my book

Numerical Probability: an introduction with appliaction to Finance

Universitext, Springer.

No, there is nothing precisely like that.

"Random" is what we can not explain or predict (for whatever reason; it does not matter if there is no such possible explanation or if we are just not aware of one).

The model uses some predictors (known to us; like the time of the day, the wether conditions including the day in the year, etc.) and makes a prediction of the response (the energy consumption) - the response value we should expect, given the corresponding values of the predictors. You can see the model as a mathematical formula of the predictor values. The formula contains parameters that make the model flexible and adjustable to observed data (think of an intercept and a slope of a regression line, or the frequency and amplitude of a sinusoidal wave).

The deviation of observations from these expected values are called residuals. They are not explained by the model and are thus considered "random". This randomness is mathematically handled by a probability distribution: we don't say that a particular resudual will be this or that large; instead we give a probability distribution (more correctly, we give the probability distribution of the response, conditional on the predictors). Using this probability model allows us to find the probability of the observed data (what is called the likelihood) given any combination of chosen values of the model parameters. Usually, we "fit" these parameters to maximize this likelihood (-> maximum likelihood estimates).

Thus, given a fitted model (on a given set of observations), we have a (maximized) likelihood (which depends on the data and on the functional model and on the probability model).

This can be used to compare different models. One might just see which of the models has the largest (maximized) likelihood. There are a few practical problems, because models with more parameters can get higher likelihood s just because they are more flexible - not more "correct". This is tried to be accounted for in by giving penalties for the model flexibility. This leads to the formulation of different information criteria (AIC, BIC, DIC and alikes, that all differ in the way the penalties are counted).

So, after that long post, you may look for such ICs to compare different models. The limitation remains that the models are all compared only on the data that was used to fit them, without guarantee that they will behave similar for new data. So if you have enough data it might be wise to fit the models using only a subset of the available data and then check how well these models predict the rest of the data. It does not really matter how you quantify this; I would plot the differences of the models side-by-side in a boxplot or a scatterplot.

You can use MESO-NH, with SURFEX and model pollutant dispersion. The advantage of MESO-NH is that it simulates well the turbulence.

I have been using LINGO for a while, and yes stochastic mixed-integer non-linear model is difficult to be solved, are you sure that writing the constraints with uniform distribution in the sub-model will solve the problem?

There is quite useful lingo library which you can access from the lingo website for free. There are also some examples of writing submodels if you look at SCM planning problem. You might find your exact answer there.

That is interesting regarding the poor Treg function in RAG1ko‘s. Sorry that the paper was of no help to you, as I said it was a quick google search but is obviously 2013! Would you be able to let me know what you find out?