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Monte Carlo Simulation - Science method

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While performing a Monte Carlo simulation I get some "jumps" in the skewness and kurtosis as shown in the picture.
How can you explain this odd behavior and non convergence of the beforementioned moments?
Any advice would do wonders.
Best regards
Edit:
Since the equation I was modelling was hard to analyze for errors or discontinuities, I simulated the random variables of the equation as with random numbers of their corresponding distribution. And, the same thing happened when I simulated a GEV variable and a Lognormal variables (as shown in the attachments)
# Simulations means how many simulations of the same variable are performed using Monte Carlo raw sampling.
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Dear Colleague,
Does your system present a discontinuous phase transition? Jumps in the raw data are compatible with the existence of meta-stable states. I could not see the details, but they do not seem vertical, which implies that the system changes characteristics as localized meta-stable pieces are found out and worked out by the MC simulation.
Does "# of simulations" mean the size of the simulations? Anyway, those jumps in kurtosis, sd, and skewness happen at the same x values suggesting that your simulation is changing specific elements (which seem to be the tail elements) in your configuration distribution, leaving the central ones alone. Are your tail (distribution) elements' behavior important to your system's behavior?
Best regards.
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I wanna simulate data for an observed continuous variable. The values are supposed range between 10 and 20, with mean of 16 and variance of 2.
I would appreciate if anyone could help me with this.
Thanks!
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Thanks Dr. @Christian Geiser. You again came to help! I had gone through your videos and books, before this and they helped a lot. You're a generous person!
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Hello.
I have some questions about CASINO version 3.3.0.4.
Q1. I know that when the electron beam escapes the sample, the scattering stops and the trajectory ends. However, when I ran the simulation, electrons that escaped the specimen were scattering in the vacuum.
(1st. figure: simulatio by me / 2nd figure: reference )
How do I set up the simulation in order for the electrons exiting the specimen to be emitted in a straight line?
Q2. I use a specimen with an area of ​​100 um X 100 um X 50 nm. However, when I run this sample, I get BSE yield and SE yield of 0.
If the area of ​​this sample is set to 100 nm X 100 nm X 50 nm, the yield is not zero.
Why is the yield different depending on the sample dimension, and why is the yield 0?
Q3. What are the criteria for BSE yield and SE yield?
Are you counting the electrons that escaped the sample?
Q4. I use samples with multiple layers. Does the beam interacting with the sample scatter again in the process of emission?
Q5. How does plasmon energy affect the generation of SE?
Q6. Does SE calculated by CASINO include SE2(generated by BSE)?
thanks for all.
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Hi Hyunkyung,
reach me at i.ajayi@lboro.ac.uk and we can have a Microsoft meeting to resolve the problem Hyunkyung Yoon
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I want to determine the temperature dependancy of order to disorder phase transition of cubic perovskite? And I haven’t enough computational resource for that..So, Python or Matlab can be a good option in this case
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All of the software described in the book is free. I attached a zip file containing the examples plus I have more if you don't find what you're looking for.
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Hello everyone, I'm modelling some metallic atoms over a carbon surface. I want to try the UFF published by Rappe et al. However I was checking the VdW potential and they have an uncommon definition of the Lennard Jones potential (paper's formula attached as a picture) Can someone help me to understand why it is so and if these parameters are somehow transferable to the typical definition of the LJ potential.
Many thanks
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Comparing to the more common form 4*epsilon [ (sigma/r)^12 - (sigma/r)^6 ], the factor of 4 has been absorbed into the x_IJ parameter. Substituting x_IJ = sigma * 2^(1/6) (and x=r),
D_IJ [ -2 (sigma*2^(1/6) / r)^6 + (sigma*2^(1/6) / r)^12 ]
= D_IJ [ -2 * 2 (sigma/r)^6 + 2^(12/6) (sigma/r)^12 ]
= 4 D_IJ [ -(sigma/r)^6 + (sigma/r)^12 ]
Thus, the D_IJ parameter is equivalent to the epsilon parameter of the more common form. The x_IJ is 2^(1/6) times the more common sigma parameter.
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Dear all,
I am eager to conduct power analysis by employing the Monte Carlo simulation available in Mplus. I have run a Multivariate Latent Growth Curve model (MLGCM) with my three-wave longitudinal data in my study, and thus, I have followed the guideline presented by Muthen & Muthen (2002; ). However, I guess, I am a systematic mistake (see, the attached output of monte carlo simulation for % sig. coefficient for mean score for SNAT = .134) but, unfortunately, I cannot find out what the problem/mistake is. Is there anyone who might help out to find the mistake(s)?
The out of the Monte Carlo simulation is attached. Besides, you can see the output of my MLGCM below. Thanks in advance for any kind of help!
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
INAT |
ID_T1 1.000 0.000 999.000 999.000
ID_T2 1.000 0.000 999.000 999.000
ID_T3 1.000 0.000 999.000 999.000
SNAT |
ID_T1 0.000 0.000 999.000 999.000
ID_T2 1.000 0.000 999.000 999.000
ID_T3 2.000 0.000 999.000 999.000
IET |
ID_G1 1.000 0.000 999.000 999.000
ID_G2 1.000 0.000 999.000 999.000
ID_G3 1.000 0.000 999.000 999.000
SET |
ID_G1 0.000 0.000 999.000 999.000
ID_G2 1.000 0.000 999.000 999.000
ID_G3 2.000 0.000 999.000 999.000
SNAT WITH
INAT -0.083 0.041 -2.037 0.042
IET WITH
INAT -0.208 0.080 -2.582 0.010
SNAT 0.019 0.034 0.556 0.578
SET WITH
INAT 0.063 0.037 1.732 0.083
SNAT 0.017 0.014 1.178 0.239
IET -0.205 0.087 -2.361 0.018
Means
INAT 3.497 0.061 57.275 0.000
SNAT -0.027 0.026 -1.018 0.309
IET 3.479 0.078 44.763 0.000
SET -0.129 0.035 -3.713 0.000
Intercepts
ID_T1 0.000 0.000 999.000 999.000
ID_T2 0.000 0.000 999.000 999.000
ID_T3 0.000 0.000 999.000 999.000
ID_G1 0.000 0.000 999.000 999.000
ID_G2 0.000 0.000 999.000 999.000
ID_G3 0.000 0.000 999.000 999.000
Variances
INAT 0.706 0.092 7.662 0.000
SNAT 0.079 0.031 2.568 0.010
IET 1.141 0.156 7.316 0.000
SET 0.148 0.069 2.135 0.033
Residual Variances
ID_T1 0.131 0.068 1.929 0.054
ID_T2 0.244 0.040 6.055 0.000
ID_T3 0.057 0.063 0.901 0.368
ID_G1 0.115 0.116 0.998 0.318
ID_G2 0.338 0.059 5.696 0.000
ID_G3 0.139 0.105 1.324 0.186
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Dear Prof. Geiser,
First of all, many thanks for your pretty clear answer.
Indeed, -0.020 is the slope factor mean in my original model. In fact, theoretically speaking such a stable pattern makes more sense rather than a relatively increased or decreased pattern. As you indicated, %sig coeff might indicate Type-1 error. I will closely monitor the warning messages as well.
Thanks again!
Best,
Savaş
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I would like to run a Monte Carlo Simulation for stocks. Since stock returns are not normally distributed I am wondering, what is the best distribution function?
What about Weilbull, Frechet, Gumbel, Rossi etc.?
My biggest concern is, to incorporate the difference between median and mean. My Data is:
Mean: 10,67% Median: 14,77 % Standarddeviation: 17,41%
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For highest returns, I would recommend the General Extreme Value distribution. Our analysis in the Turkish Stock Exchange by using this distribution showed, however, that highest returns follow the Frechet distribution.
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Can we use Monte-Carlo simulations to understand the practical impact of process variation on Ambit-AND-OR (which requires triple-row activation) in all the components in the subarray (cell capacitance, transistor length/width/resistance, bitline / wordline capacitance and resistance, and voltage levels). ?
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Unfortunatly, i did not use it before for monte carlo simulations. I use cadence
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Hello everyone,
I am trying to run a Monte Carlo simulation to estimate the delivered dose of alpha-particles from Ra-223 decay to cancer cells. My system is simple. Just cells and Ra-223 media. Do you have any recommendations?
Thank you.
Best regards,
Mehran
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Thank you so much dear Ashok. Good to know you.
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Actually, I did not any idea about the Monte Carlo simulation. I would like to add the Monte Carlo simulation to my adsorption study. How can I learn Monte Carlo simulation? Which program should I use? Thanks in advance.
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Good day,
I have a need for carrying out monte-carlo simulation to model the health risk due to heavy metals in groundwater.
I need help figuring it out using Oracle crystal ball.
I also appreciate if you also have an excel template which i can use.
Looking forward to your responses.
Thank you and GOD bless.
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Dear Enoch Ibitogbe,
Look over the article below as well:
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To see complete details, please find the attached file. Thanks.
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Since y|x,z or z|x,y are all easily simulated from. Probably you only need to sample from x|y,z using any appropriate MCMC sampler that you are familiar with. I recommend looking at Slice sampling, Neal 2003.
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There are several methods to check the statistical significance of an absorption feature in the X-ray spectrum of X-ray pulsar. I am looking for the Monte Carlo simulation method. It would be helpful to discuss the detailed method to use this type of simulation. I also looking for more applications of these simulations. Thanks.
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I need some guidance regarding CRLB to compute numerically, and to estimate Doppler frequency, for a synthetic signal, given below.
X =  A*sinc(B*(t-𝜏).*exp(j*2*pi*F*(t-𝜏); whereas   θ = [ F, A, 𝜏  ]
"A" is complex and has amplitude and phase. "F" is doppler and  "𝜏 " is azimuth shift
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Hello to All,
Any suggests in writing LAMMPS input file of pcff forcefield? I couldn't find one in the lammps manual, especially for the pair_style and pair_coeff part.
Thanks in advance.
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The lj/class2 styles compute a 6/9 Lennard-Jones potential.
e,g pair_style lj/class2/coul/cut 10.0 # with cut-off distance=10 angstrom for both interactions
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I have around 1000 images from Micro-CT and from those I want to know the angle made by the non-spherical particles with the column's central axis. I have been able to stack the images in Image J but further, I do not have any clue how to analyze the stack and the angle of the particles.
The other way I have seen to find the orientation is by doing the Markov Monte Carlo simulation on which I am totally clueless about how to proceed.
It will be great if someone can help me using the above either method to calculate the orientation of the particles with the bed's central axis in a packed bed.
I am attaching one such figure from which I want to calculate the orientation and the one stacked volume image from the Micro-CT and also the Complete procedure of Markov Monte Carlo simulation.
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May be you can find your answer using the GrainFind tool:
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How to calculate mean square angular displacement ? Do we need any periodic boundary conditions if at each step angle is updated using theta(t+dt) = theta(t) + eta(t) , where eta(t) is a gaussian noise . Please describe the procedure to calculate this quantity.
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Souvik Sadhukhan Yes: The way to do that is by relating the angular mean square displacement, expressed through <sin2θ>, more precisely, <sinθ(t)sinθ(t')>,
for instance, with the 2-point function of the noise, <η(t)η(t')>. This is the relation that defines the diffusion coefficient (strictly speaking, in the approximation, where the sinθ(t) can be assumed to be drawn from a Gaussian distribution; otherwise it's more complicated) and is obtained by computing the probability distribution of the sinθ(t), from the knowledge of the probability distribution of the η(t) and the relation, dθ(t)/dt = η(t).
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How can I provide Monte-Carlo simulation for power calcilaton of the second order cfa in mplus (40 observed categorial variables, 20 first order latent variables and 5 second order latent variables).
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Thank you much for the answer and your patience. I'm just a beginner in CFA, so some of my questions might be a little bit stupid. I providing the analysis in mplus 7. Here you can look at the output.
MONTECARLO:
NAMES ARE Item1 Item2 Item3 Item4 Item5 Item6 Item7 Item8 Item9
Item10 Item11 Item12 Item13 Item14 Item15 Item16 Item17 Item18 Item19
Item20 Item21 Item22 Item23 Item24 Item25 Item26 Item27 Item28 Item29
Item30 Item31 Item32 Item33 Item34 Item35 Item36 Item37 Item38 Item39
Item40;
CATEGORICAL = Item1 Item2 Item3 Item4 Item5 Item6 Item7 Item8 Item9
Item10 Item11 Item12 Item13 Item14 Item15 Item16 Item17 Item18 Item19
Item20 Item21 Item22 Item23 Item24 Item25 Item26 Item27 Item28 Item29
Item30 Item31 Item32 Item33 Item34 Item35 Item36 Item37 Item38 Item39
Item40;
GENERATE = Item1-Item40(3 p);
NOBSERVATIONS = 500;
NREPS = 10000;
SEED = 50000;
MODEL POPULATION:
f1 by Item1*.1 Item19*.1;
f2 by Item7*.1 Item25*.1;
f3 by Item13*.1 Item31*.1;
f4 by Item4*.1 Item22*.1;
f5 by Item10*.1 Item28*.1;
f6 by Item16*.1 Item34*.1;
f7 by Item2*.1 Item20*.1;
f8 by Item8*.1 Item26*.1;
f9 by Item14*.1 Item32*.1;
f10 by Item3*.1 Item21*.1;
f11 by Item9*.1 Item27*.1;
f12 by Item15*.1 Item33*.1;
f13 by Item6*.1 Item18*.1;
f14 by Item12*.1 Item24*.1;
f15 by Item30*.1 Item36*.1;
f16 by Item5*.1 Item23*.1;
f17 by Item11*.1 Item29*.1;
f18 by Item17*.1 Item35*.1;
f19 by Item37*.1 Item40*.1;
f20 by Item38*.1 Item39*.1;
F21 by f1*.1 f2*.1 f3*.1 f4*.1;
F22 by f5*.1 f6*.1 f7*.1 f8*.1;
F23 by f9*.1 f10*.1 f11*.1 f12*.1;
F24 by f13*.1 f14*.1 f15*.1 f16*.1;
F25 by f17*.1 f18*.1 f19*.1 f20*.1;
F1-F25@1;
F21 with F22*0.3;
F21 with F23*0.3;
F21 with F24*0.3;
F21 with F25*0.3;
F22 with F23*0.3;
F22 with F24*0.3;
F22 with F25*0.3;
F23 with F24*0.3;
F23 with F25*0.3;
F24 with F25*0.3;
ANALYSIS:
TYPE = general;
ESTIMATOR = WLSMV;
MODEL:
f1 by Item1*.1 Item19*.1;
f2 by Item7*.1 Item25*.1;
f3 by Item13*.1 Item31*.1;
f4 by Item4*.1 Item22*.1;
f5 by Item10*.1 Item28*.1;
f6 by Item16*.1 Item34*.1;
f7 by Item2*.1 Item20*.1;
f8 by Item8*.1 Item26*.1;
f9 by Item14*.1 Item32*.1;
f10 by Item3*.1 Item21*.1;
f11 by Item9*.1 Item27*.1;
f12 by Item15*.1 Item33*.1;
f13 by Item6*.1 Item18*.1;
f14 by Item12*.1 Item24*.1;
f15 by Item30*.1 Item36*.1;
f16 by Item5*.1 Item23*.1;
f17 by Item11*.1 Item29*.1;
f18 by Item17*.1 Item35*.1;
f19 by Item37*.1 Item40*.1;
f20 by Item38*.1 Item39*.1;
F21 by f1*.1 f2*.1 f3*.1 f4*.1;
F22 by f5*.1 f6*.1 f7*.1 f8*.1;
F23 by f9*.1 f10*.1 f11*.1 f12*.1;
F24 by f13*.1 f14*.1 f15*.1 f16*.1;
F25 by f17*.1 f18*.1 f19*.1 f20*.1;
F1-F25@1;
F21 with F22*0.3;
F21 with F23*0.3;
F21 with F24*0.3;
F21 with F25*0.3;
F22 with F23*0.3;
F22 with F24*0.3;
F22 with F25*0.3;
F23 with F24*0.3;
F23 with F25*0.3;
F24 with F25*0.3;
OUTPUT: Tech9;
INPUT READING TERMINATED NORMALLY
*** FATAL ERROR
THE POPULATION COVARIANCE MATRIX THAT YOU GAVE
AS INPUT IS NOT POSITIVE DEFINITE AS IT SHOULD BE.
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MATLAB Monte Carlo simulation is very common and important for UWOC. How did you realize the Matlab Monte Carlo simulation of underwater wireless optical communication? Source code sharing?
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Thanks
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I would like to use a Monte Carlo simulation to calculate base rates of abnormally low scores in cognitive testing. While this has been shown to be possible using the intercorrelations between the tests (Crawford et al., 2007) more recent research has indicated that factors such as IQ and years of education exert a significant influence base rate estimates (e.g. Karr et al., 2017). Is it possible to run Monte Carlo simulations while also adjusting for covariates of these types?
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Many thanks Pablo, I wasn't aware of MCM Alchimia but will definitely check it out.
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I am working on probabilistic human health risks on dumpsite pollution in developing countries. Is there anyone who has an example of an excel file of the Monte Carlo simulation? Any paper that detailed about this issue?
Thank you very much.
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Application of Monte Carlo probabilistic and prognostic methods to estimate the level of pandemic risk, the risk of Coronavirus infection, the risk of developing Cpvid-19 disease, the risk of death or loss of health due to severe Covid-19 disease transition, forecasting the development of the SARS-CoV-2 coronavirus pandemic (Covid -19) can be an interesting and good solution.
Best regards,
Dariusz Prokopowicz
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Dear colleagues
I have tried to conduct a Monte Carlo simulation in Mplus for a latent three-factor model with correlations between the factors. I want to test whether n = 500 would be enough to construct the factors properly and to detect correlations as low as r = .3. (I know it propably is, I just wanted to finally explore Monte Carlo simulations).
However, I keep receiving the following error message, no matter what I do:
*** ERROR in MONTECARLO command
Probabilities of observations for patterns must sum up to 1.
Unfortunately, the Mplus website is no help. I have enclosed my syntax. It is a lot to ask, but those of you who are familiar with Monte Carlo simulations for categorical indicators, would you consider pointing my in the right direction? What do I need to change?
Thank you in advance for your insight.
Best
Marcel
P.S. I have attached a more reader friendly PDF of the syntax as well.
-------------------------------------------------------------------------------------------
TITLE:
Monte Carlo study
three factors: subjective content knowledge (geography, history, civics)
categorical indicators (4-point Likert scale)
MONTECARLO:
NAMES =
f18_a f18_b f18_c f18_d f18_e f18_f f18_g
f19_a f19_b f19_c f19_d f19_e
f20_a f20_b f20_c f20_d f20_e f20_f f20_h f20_i;
CATEGORICAL =
f18_a f18_b f18_c f18_d f18_e f18_f f18_g
f19_a f19_b f19_c f19_d f19_e
f20_a f20_b f20_c f20_d f20_e f20_f f20_h f20_i;
NOBSERVATIONS = 500;
NREPS = 1000;
!Items measured on a 4-point Likert scale
GENERATE = f18_a-f20_i (3 p);
!5 % missings on each item assumed
PATMISS = f18_a-f20_i (.05);
!50 % of cases assumed to have missings
PATPROBS = .50;
!The next part is probably where I get it wrong
MODEL POPULATION:
SCK_Geo BY f18_a-f18_g*.1
SCK_Geo@1;
SCK_His BY f19_a-f19_e*.1
SCK_His@1;
SCK_Civ BY f20_a-f20_i*.1
SCK_Civ@1;
SCK_Civ WITH SCK_Geo*.3;
SCK_Civ WITH SCK_His* .3;
SCK_His WITH SCK_Geo* .3;
ANALYSIS:
TYPE = general;
ESTIMATOR = WLSMV;
PARAMETERIZATION=THETA;
MODEL:
SCK_Geo BY f18_a-f18_g*.1
SCK_Geo@1;
SCK_His BY f19_a-f19_e*.1
SCK_His@1;
SCK_Civ BY f20_a-f20_i*.1
SCK_Civ@1;
SCK_Civ WITH SCK_Geo*.3;
SCK_Civ WITH SCK_His* .3;
SCK_His WITH SCK_Geo * .3;
OUTPUT: Tech9;
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that makes a lot of sense because defining all those values seemed fairly arbitrary to me. For example, how do I know what value I should assign to the second threshold of the fourth variable? Having some data to fall back on seems advantageous. Still, I would like to be able to set up a population model without any real data, but maybe this is just beyond my abilities. For the moment.
Anyway, I would not even have made it this far without your help. Thanks a million!
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My lab needs to determine a more robust method of determining uncertainty of our gas analysis using a monte carlo simulation which is determined annually . We sample a QC cylinder of known gas concentration before and after every batch of samples. If i assume a normal distribution of the gas analysis, can i simply enter the mean and standard deviation of all the QC results for the respective gas component for the year and then run a simulation off this and calculate the uncertainty? If i assume a rectangular distribution, can i simply use the lowest and highest QC result for the respective component and use these as my highest and lowest bounds, run the simulation and calculate the uncertainty of this? Any guidance or help would be greatly appreciated. Thank you Lachlan
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I want to carry out particle in cell simulation for Electron Cyclotron Resonance (ECR) plasma for the in-house developed reactor at our laboratory. Kindly share any freely available PIC code with instruction guidelines so that I could do the simulation. I am using windows platform. I don't have any information on how to do the simulation. So if there is any tutorial, also share that. Can it be done by MATLAB coding also?
Looking forward for your guidance.
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In the attached file, the Persian Curve is proposed as a certain alternative for the famous Monte Carlo Simulation, MCS. The proposal is a revolution in all areas with application of the MCS.
Do you have any logical reason to oppose the proposal?
If Not, your recommendation, approval and presenting case studies is a great service to human knowledge and academic society.
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23 years in USB?! Great! It is nice to be connected with you.
I will read the draft. In my last paper, I have presented a method with the accuracy of crude MCS and provided a set of examples that mainstream probabilistic approaches fail to solve them. To examine your approach, you may solve them (I can do it if you provide the Matlab code of your approach here):
Evidently if your approach solves all these examples with high accuracy, beside the crude and soft MCS, I will use/suggest your approach in my next papers.
Best regards,
Mohsen
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Dear Colleagues ... Greetings
I would like to compare some robust regression methods based on the bootstrap technique. the comparison will be by using Monte-Carlo simulation (the regression coefficients are known) so, I wonder how can I bootstrap the determination coefficient and MSE.
Thanks in advance
Huda
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Here you can follow:
1- draw the bootstrap sample (size m)
2- Estimate the coefficient by you need from this sample by using the (robust method)
3- find Mse for coefficient
4- Repeat the above bootstarp process B times, here you can estimate Mse for coefficients by taking the average, here you will get the estimation for bootstarp Mse
5- This loop can be repeat N times by Monte-Carlo Simulation
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I am using the HOOMD-blue package, and I was advised to use force-shifted Leanard Jones potential, with a tree neighbor list. I do not know which parameters I should write for those charged molecules (hard spheres) "'N, charge'" The system would also have some counterions, and they have an impact on the system.
classhoomd.md.pair.ForceShiftedLJ(nlist, r_cut=None, r_on=0.0, mode='none')
nlist = hoomd.md.nlist.tree(
r_buff=0.4, check_period=1, d_max=None, dist_check=True, name=None)
Thanks in advance!
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Mohamed-Mourad Lafifi Thanks, those papers are useful, but they do not help me.
I did build the model, and I am struggling to apply electrostatic field to the model.
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Hello All,
Please is it possible to use MATLAB in analysing the sharing and compatibility studies between Satellite Communication systems and 5G systems using the Monte Carlo Simulation approach rather than the Minimum Coupling Loss approach?
OR the Monte Carlo Simulation approach in analysing sharing & compatibility studies is only implemented using Spectrum Engineering software (e.g. SEAMCAT) other than MATLAB?
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I think it is possible to use Monte Carlo simulation, but you need to implement it by yourself in Matlab.
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I am studying free space optical recently, but I don’t know how to generate Málaga fading to Monte-Carlo simulation. Who can help me?
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I am looking for some quality materials for learning the molecular dynamics theory and the use of LAMMPS. Besides the LAMMPS manual from Sandia National Laboratory, which sources can I use for learning LAMMPS?
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Simple geometry was made having concentric cylinders bounded by a square region. The geometry has a reflective boundary. The source was defined at one point in the boundary and when it had been run for criticality calculation, it resulted in an error messaging "the entire source was rejected". I tried to do the simulation by placing the source closer but it still results in an error. Any help will be appreciated.
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You have not properly defined the rectangular box. You have it as the intersection of the "negative" side of planes 1, 2, 3, and 4. The negative is to the "left" of an x plane, and "below" a y plane. So this cell extends to infinity in the third quadrant, effectively. What you want is:
501 5 -0.740582 10 -1 2 -3 4 imp:n=1 tmp= 4.94E-8
As Norbert suggested, you should also add pz planes; their height is irrelevant, but lets say you have the following:
*5 pz 10.0
*6 pz -10.0
Then, you should add "-5 6" to the definition of each cell. You box will become:
501 5 -0.740582 10 -1 2 -3 4 -5 6 imp:n=1 tmp= 4.94E-8
Also, I'm confused about your cell 601. It should probably be:
601 0 1:-2:3:-4:5:-6 imp:n=0
Finally, your ksrc has to contain source locations that are within a fissionable material; otherwise the fission sites will all be rejected when they are sampled (this is your main error, I believe). Your fissionable material is between surfaces 7 and 8, so try:
ksrc 0.3 0.0 0.0
In addition to being inside of a fissionable material, try not to put the ksrc directly on any cell boundaries, as it could get confused about which cell it is in.
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Hi, i am conducting a Monte Carlo simulation for my new distribution but the MSE values are increasing while the sample size increase, does any experience this issue before and what should i do to fix it? thank u
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What should i try before concluding that? can i apply an optimization method for the MLE in the MSE simulation? Pablo Constantino
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Dear community,
I want to define a geometry in MCNP with several levels of universes and fills, say:
c cells
10 100 -1 -1 u=1
11 0 1 u=1
c
30 0 -2 fill=1 u=2
31 0 2 u=2
c
40 0 -3 fill=2
41 0 3
Universe 1 consists of cells 10 and 11 and fills cell 30. Cells 30 and 31 make up universe 2 and fill cell 40. (I have not tested the above piece of code for errors but I hope you got the idea.)
Now I want to define an identical geometry where the material in cell 10 is changed from 100 to 200. Is there a way to do this?
"20 like 10 but mat=200 u=3" is possible, of course. However, then the entire remaining code must be copied (with adapted cell and universe numbers) in order to make up new universes. Is there something like "50 like 40 but ***mat of cell 10 within cell 30 within cell 40***=200"?
An alternative solution could be to define the material of 10 as a distribution depending on which higher-level cell cell 10 will be in (like mat=Fcel D1; SI1 L 50 40; SP1 100 200). But I do not see a way to access the cell numbers of higher level either.
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I think the uncertainty function can do so, to give you a wide range of parameter to check and realize which better fit for you
But as Geometry to make a change like a for loop or if function. unfortunately not available
You can read that article for more about uncertainty in MCNP https://mcnp.lanl.gov/pdf_files/la-ur-16-23533.pdf
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I am making a simulation of energy deposited of some radioactive sources in scintillation detector. I have some results, but I don't have the same distribution energy between experimental spectra and simulated spectra. I do not know why.
For instance, the spectra of Cs-137 is shown. However, it does not have the same resolution.
Someone who can help me how to improve the simulation.
The scintillation detector is of NaI:Tl.
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The most significant difference is the width of the photoelectric peak at 662 keV. This width occurs after gamma-ray energy is transferred to the scintillator crystal. Your simulation probably may not include this part.
The number of scintillation photons generated, when 662 keV is totally absorbed by the NaI:Tl crystal, is statistically distributed. It is also a stochastic process whether each scintillation photon is successfully detected by a photodetector (photomultiplier tube or semiconductor device) or lost.
There are two ways to improve the distribution. One simple way is to assume the energy resolution of the detector (crystal + photodetector) and perform a convolution with your results. You can fine the typical energy resolutions in literature: typically ca. 7% [*].
Another detailed method is to add, if necessary, simulation to track the behavior of the scintillation photons.
The other differences seem to be related to Compton scattering; since the NaI:Tl crystal is sealed in a package, some of the energy may be lost by Compton scattering in the package before it enters the crystal. Then, the gamma-ray energy is not 662 keV but less than 475 keV for the detector. Similarly, Compton scattering from surrounding desks, walls, and other objects may enter the crystal and increase the proportion of that component of the experimental data.
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Hello,
My area of research is to valuate Exchange option (Margrabe option). I am looking for Monte Carlo simulation technique that can be applied to some real trading data. I have evaluated them using Liu process and wish to compare my results with simulation. In excel I prepare basic model based on Margrabe formula, but results deviate. Thanks in advance.
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To prepare the Monte Carlo simulation, you need 5,000 results.
Step 1: Dice Rolling Events. First, we develop a range of data with the results of each of the three dice for 50 rolls. ...
Step 2: Range of Outcomes. ...
Step 3: Conclusions. ...
Step 4: Number of Dice Rolls. ...
Step 5: Simulation. ...
Step 6: Probability.
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I am working on molecular simulation of adsorption and I want to design a new molecular simulator software by C++.
The input of this software is some text file. I want to get initial position of each atom in Cartesian coordinate in a text file.
So I want to make this initial position data.
Can I make a structure in Material Studio 2017 and convert/export or extract this data to a text file?
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You can export the Materials Studio generated structure into .car format. It will give you the coordinates of your structure in txt format.
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In general and special for energy materials applications.
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You can read the book for general introduction " SIMULATION AND THE MONTE CARLO METHOD " by Reuven et al
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I want to develop a Markov switching model for "demographic transitions" (discrete time and discrete state) using the estimates (hazard rates or transition rates) from a continuous-time hazard model. The only data I have, is the estimated hazard rates for various age groups e.g. transition rate of marriage for age group 20 to 25 is 0.93, which means the model assumes piecewise linear form of baseline hazard model.
The transition rates have been estimated using following model;
log-hazard rate = log(baselinehazard) + covariate effects
or hazard rate = (baseline hazard)*Exp(covariates)
The first step for developing the Markov switching model is to know the transition probabilities. I have seen in literature that transition probabilities can be calculated using monte carlo simulations. So I will first project the life profiles of a synthetic population and then would randomly let them make transitions. for example, for a single person if there are two possible states in the next period marriage and death with some probability "p", then I generate random numbers and compare them with 'p' and if 'p' is higher than random draw then the individual make the transition otherwise he remains single. (I assume here that 'p' is the transition rate). once simulations are done, i will determine the "transition probabilities", which is the aim of performing this step.
when I run these monte carlo simulations, I also need to keep the track of "durations" like how longer the person remained divorced or married? Now my questions are as follows:
1) Is it appropriate to consider transitions rate as 'p' in the simulations?
2) I am running the simulations on MATLAB, and I do not know how can I keep the track of duration or event history of simulated population.
3) If someone is working in this area, I would be happy to talk in detail about it.
Thank you!
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I'm glad you found some help with the Monte Carlo technique. If you recommend my answers, it will keep me motivated to accelerate your research.
The advantage of MATLAB as a programming language is that matrix calculations are simple and fast. It is better to avoid using any loops (for/while) as much as possible.
Random numbers can also be obtained as a matrix. For example
A=rand(4,1)
to obtain multiple random numbers at the same time.
0.9157
0.7922
0.1265
0.6557
If you have p=0.2000, then type
B=A-0.2000
to obtain
0.7157
0.5922
-0.0735
0.4557.
Here, the person with n=3 has a negative value meaning "died" in this cycle. Next, type
C=sign(B)
to obtain
1
1
-1
1.
If positive, it is alive, if negative, it has died. Or.
status=(sign(B)+1)/2
may be easier to use.
1
1
0
1
This results in the new status.
The above is written in summary.
status=(sign(rand(4,1)-0.200)+1)/2
The results are obtained without using any loops (for/while), the new status of n people are updated in a one-line program. That way, your Monte Carlo calculations will be simple and fast. This is my advice for you to get used to matrix calculations.
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Hey!
I am a psychology student working on a small project for one of my seminars. The goal is to present to fellow students how different sample sizes affect different ways of assessing measurement invariance (Chi-square or Alternate Fit Indices).
Additionally, I would like to show them an example of how to do your own Monte Carlo simulation. Someone could theoretically go the extra mile and estimate sample size for the specific experiment they might plan.
However, we all only use R-studio, and we don´t have access to programs like MPlus. Sadly, I am not advanced enough to write some code for this. Therefore, I am looking for researchers who did a Monte Carlo simulation to assess measurement invariance in R-Studio. That way, I could ask for the code they used and hopefully learn something along the way.
I am grateful for any cue and comment. Thank you very much!
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Jonas Brammer You could run regressions datasets to build polynomial models (y = a + bx + cx^2 + dx^3...), and take a look at : https://www.youtube.com/watch?v=xuUMz8exU8Q
Good luck.
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Hello,
I am a newbie in geant4. I'm learning it playing with example. Sometimes, when I go to simulate an example code, it doesn't create an executable file (the green one in my ubuntu terminal). Why does it happen? There is no error while building the file.
Please help.
File lists from terminal, there is no executable file.
anamika@anamika-Inspiron-3421:~/Cyclotron/g4work/B5$ ls
CMakeCache.txt cmake_install.cmake exampleB5.cc exampleB5.out gui.mac icons.mac init.mac Makefile run1.mac run.png vis.mac
CMakeFiles CMakeLists.txt exampleB5.in GNUmakefile History include init_vis.mac README run2.mac src
anamika@anamika-Inspiron-3421:~/Cyclotron/g4work/B5$
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I would recommend to use CRNS's virtual machine. If you run GEANT in Ubuntu, you have to install a lot of dependences for correct running. Also, when you want to run a simulation, you have to set GEANT enviroment.
This virtual machine avoid some thricky problems. On the other hand, for provinding a good solution, it is necessary more information that you have in your CLI interface (terminal) about the error.
Here is the link for virtual machine with GEANT.
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For a Monte Carlo simulation involving photon counting detectors, I am running into the problem of creating random numbers that have a given expectation and a given correlation matrix. Due to the statistics of the detector (double counting), the diagonal elements can be larger than 1 so that the statistics is not exactly Poisson. The off-diagonals of the correlation are positive.
Has anyone come across a similar problem?
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Thank you Nicolai!
This indeed seems to work decently if A and B have the same expectation and if I replace cov(A,B) with 0.5*correlation(A,B).
But I would need a more general algorithm that works for arbitrary A and B, even multidimentionsonal random vectors...
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I created a new test for uniformity, but so far, I've had no luck finding its critical values analytically, I could only obtain them by Monte Carlo simulation. What's worse is that histograms show that the null distribution does not approach normal distribution even at large n, so I cannot approximate it with mean and standard deviation.Is there any sort of "standard procedure" for deriving null distribution of a test statistic? Or at least approximating it with an analytical expression?
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An important question, I am waiting for the answer, too..
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I'm currently working on a project where I deal with a protein of size ~50K atoms. I want to do a computer simulation with the protein using Monte Carlo instead of MD, and looking for any software that can do that.
I have so far tried Cassandra and GOMC but both didn't work out well since my system was too big compared to the system size both software targeted to.
Does anyone have an experience in MC software and know which software is suitable to do the simulation with the above size protein?
Thank you!
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Thank you so much for the suggestions!
I'll look into the links you shared to me and see which one would work the best on my protein simulation.
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Hello All,
I am very new to this. I am tasked with simulating clinical data that is large in volume and spans over a period of time. I will have some data provided to start with but what would be the best approach/method/tool/software to do this. I was told monte carlo simulation is too random for the data. Need a more structured approach.
Any feedback would be greatly appreciated.
Thank you.
-Sachin
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Thanks. Will look into this.
Regards,
-Sachin
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Variance reduction methods are used in Monte Carlo simulation in order to reduce the simulation time. I would appreciate it if anyone suggests the drawbacks of this method.
Regards,
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It depends on the size of your problem. If you are dealing with a deep penetration problem where you have shielding and want to estimate a tally behind a shield or far from the source then you will need variance reduction. Only use it if you know you will need it as in general it will add to your computation time and consume more resources.
One major drawback is, if parameters are not properly chosen, long histories. That happens when histories get split alot so mcnp spends a lot of time tracking particles of low importance. That is common if the lower bound is zero and no proper cut card for particle weights is used.
Another drawback comes also from not choosing he parameters carefully; excessive termination of histories. Particles are split but most of them are below the lower WW bound and the russian roulette game is played. You create a lot of particles but also kill a lot.
There are many methods, mcnp WW generator, MAGIC (both should be OK for less complex problems), CADIS, FW-CADIS (both are advised to be used with complex problems, ITER like models).
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Hi
I am trying to do a multivariate meta-analysis MC simulation in r and I want to control for the sample sizes (studies sample sizes), Within-study correlation, and between-study correlation.
I am using mvmeta and the main problem is where to plug in the between study-correlation!
Any ideas?
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this seems folded matter in R, although there are some packages such as
I have used MATLAB and got the results for small dataset as well have a look here
hope the above points will answer to your question or nearer to it.
Good luck!
Suraj
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I have below model:
X = A*B
Where A = a list of values with Lognormal distribution (size = 13)
and B = another list of values with Lognormal distribution (size = 13)
How can I perform 1st order, 2nd order and total Sobol sensitivity analysis of this model in R programming?
Please help me with the steps.
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Application of Monte-Carlo-Simulation in Structural Reliability, risk analysis or other complicated modeling.
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MCS is a general term, if you mean its simplest form, it may not help every time. But, today MCS is enhanced with many machine learning methods and subset simulation techniques that can make it fully applicable in complex problems.
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In our current design of linear binary current steering low-noise CMOS slowDAC, we converged in the use of long channel NMOS transistors. Indeed, a low noise, lower than 10pA/√Hz at 100µA, and good matching of the weights of the binary DAC leads to the use of long channel NMOS transistor in the design of binary coded current mirrors.
It seems better to increase L than W to improve the current mirror matching looking at Monte Carlo Simulations.
NMOS sized with L as long as 20 µm for a W of 8 µm is considered. Assuming that such long channel NMOS transistors are unusual (?) in VLSI design ... have you experienced the use of long-channel transistors in VLSI DAC design? Any papers, or review about the use, and justifications, of long channel MOS transistors for ASIC design?
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There is no unusual dimension in analog design. The question is what is constrained by technology.
In classic CMOS (would say process older than 130nm) channel can be any long. Lower nodes (but not modern ones) like 28-130nm might has limit for channel length and also pocket/halo doping. In such case matching is length dependent and the best approach is to use stack of large number minimum length MOSFETs. Modern technologies (FinFET and FDSOI) has more constraints.
Nevertheless, DAC resolution is determined by matching and for given current designer has to maximize inversion region of current source and the only way to achieve it is by decrease W/L ratio and increase area.
So, back to your consideration 8um/20um dimensions are common rather than unusual.
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I am a beginner to probabilistic forecasting. From my research I have a vague idea that monte carlo simulation can be done for injecting uncertainity in the process. Do i need to get multiple point forecasts doing monte carlo and do postprocessing for obtaining a proabibilistic distribution?.Can any one help with the procedure what steps should i follow to do probabilistic forecasting? It would be helpful if someone can share an example
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The approach presented by Leutbecher and Palmer (2008) aims to assess the sensitivity of the model to initial conditions. The proposed approach can certainly estimate the spread of the trajectories of the model in a phase space and make some rough estimate of the forecast uncertainty, but it should not be confused with probabilistic modelling. The latter can only be performed when the equations used for the forecast are written explicitly for the stochastic variables. The best known model illustrating this principle is that developed for the study of Brownian motions.
The correct mathematical foundation for probabilistic modelling is the Ito calculus. Please kindly consult the following sites:
Ito calculus and Brownian motions:
Itô’s stochastic calculus: Its surprising power for applications:
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In solving an optimisation problem with uncertain input parameters, we are using Monte Carlo simulation (MCS) and scenario reduction to arrive at a number of scenarios with their associated probabilities. Optimisation algorithm outputs decision variables for different scenarios. For example, the power of a generator at different scenarios; Assume that in a generation scheduling problem with Ng generators, at scenario 1 with probability 0.3 power of generator 1 is 12, at scenario 2 with probability 0.2 power of generator 1 is 17 and at scenario 3 with probability 0.5 power of generator 1 is 9. Then in practice, how the decision maker must set the power of this generator?
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This is a problem that needs to be solved with stochastic optimization algorithms. I recommend you take a look at the Benders Decomposition, an algorithm that is well suited for stochastic problems with scenarios. The output of this algorithm is a single value of your variable, given that the algorithm handles the uncertainty in the parameters.
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Hi everyone,
I currently use MCS method to analyze effect of some uncertain parameters on electrical power system and run 10,000 simulations to calculate the output which approximately takes around 1 hour.
I recently read some methods which can reduce the MCS scenarios thus, resulting in low computational time.
So, can our fellow researchers elaborate more on this topic or suggest me any other techniques which has the potential to significantly reduce the computational time of MCS (say around 5 minutes for my work) with reasonable accuracy?
Cheers
Sam
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It depends on what is your application of Monte Carlo simulation. You could have different computational time reduction strategy. But generally speaking, to have a good sampling technique will make your Monte Carlo simulation much easier and more efficient. I would suggest the Latin hypercube sampling (LHS) sampling technique, which I used quite often. It will make the distribution of your samples very close to the expected distribution with small number of sample generation.
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Reliability researchers often consider the result of the crude Monte Carlo simulation as an accurate solution for a reliability problem. However, is this approach really a general accurate reliability method?
Regardless of problems with the small probabilities, can anyone present a reliability problem that crude Monte Carlo simulation is not able to solve it?
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Here now the book Au/Wang is discussed. Unfortunately the method championed there SS [sic!] is some sort of rain dance or voodoo ritual. The authors promise you if you perform the ritual/algorithm you will obtain a meaningful estimate for your failure probability. This is demonstrated by a number of examples. No general proof is given. This works only for limit state functions with a simple geometric structure, for more complicated structures it fails in many cases, as the writer has shown (see the papers on Researchgate).
The maths here is the maths of the ancient Egyptians, who proved the theorem of Pythagoras by examples. But after them came the ancient Greeks and they introduced in maths the concept of a proof. Inspite of these advances here in SS there is a regression to the Egyptians all is based on simple examples and from there it is extrapolated that it works for complicated cases too, no proofs at all. Reminds a little bit of the civil engineer who wanted to prove that all odd numbers are prime.
The writer has tried to explain to the SS people several times that the whole approach is flawed, but they continue to rearrange the deck chairs without taking notice.
The basic misunderstanding of SS and all these fumbling around SS derivates is that they do not understand the problem.
If one wants to calculate failure probabilities, it means to find an integral over a huge domain in the n-dimensional space whose location is not known exactly.
If one wants to avoid to integrate over the whole domain, one mus restrict the integration to a subset (pun intended) of the original domain. And then one invariably ends up in neighborhoods of the points in this domain where the PDF is maximal, i.e. the modes (I avoid the expression design point but they are the same), because somewhere here the main probability mass is located. Now --- to counter objections --- in high dimensional spaces the immediate neighborhood is not quite enough, but one has to go a little bit further.
The problem of SS is that basically it is a MC version of FORM/SORM.
Please have a look at all the SS-examples, the data move always to the modes/design points. Why not to start from these points? We know where they are. Is it necessary to find them by lengthy MC procedures?
Anyway, everybody who understands maths, should see that if one wants to find
P(g(X)<0) it does not make any sense at all to calculate a sequence of probabilities P(g(X)< c_j) with c_j> 0 if the structure of the function g is unknown. This works only if the function g has a simple structure, for example if it is a homogeneous function or a transformation of it, otherwise everything can go south.
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Dear All,
I want to calculate the proton primary dose and yield in water for 250 Mev proton beam using Fluka Mc Code but I couldn't. I think it's because detector data entered incorrectly.
So what should I do in this situation?
best regards.
NB1: I'm a beginner user
NB2: attached you will find the corresponding input and output files
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thank you Docs, M.Radovan can you show me how i can install it ?
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Dear researchers,
According to the central limit theorem for a large number of random samples as X1, X2, ..., Xn, with expected value mu and standard deviation sigma, the following confidence interval is defined:
[mu - Z*sigma/sqrt(n), mu + Z*sigma/sqrt(n)]
As it can be seen in the interval above, as the number of samples n, increases the interval becomes narrower, and as a result so many samples stay out of the confidence interval, which is not good for our prediction of the problem. I have a question that put it in the following forms:
What is the main definition of a confidence interval? Is it a confidence interval for the average of Xi or Xi itself? In other words, should all Xi(s) lie within the interval with the probability of say 95%? or this is not necessary?
I have already attached a MATLAB code for randn function, one can see that as n increases the CI becomes narrower and so many Xi(s) stay out of the CI, however, for lower value of n, the results seem logical.
thanks for your responses.
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The confidence interval refers to (the unknown) mu.Having larger sample sizes means having more information about the unknown what translates into narrower confidence intervals (on average).
What you want is probably a prediction interval (https://en.wikipedia.org/wiki/Prediction_interval). This is about where one may expext realizations of the random variable (X). These will be somewhere around mu (which is estimated just as for the confidence interval), but acknowledges that the variance (or sigma) will always add to the uncertainty, no matter how well mu is estimated.
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How can I validate computational findings using Monte Carlos simulation?
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It is good question for research!
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Dear scientific researchers,
I am working with SOLTRACE software to simulate the optical performance of a solar parabolic trough collector and I would like to know if there are possibilities to include a DNI file on the simulation of the studied collector instead of a fixed DNI value. Moreover, I calculate the ratio between the rays hitting the tube receiver and the ones reflected by the mirrors. I would like to know if this ratio is corresponding to the optical efficiency or to the intercept factor.
Thank you in advance for your help and collaboration.
Best regards.
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Thank you Dr
Ali Alwaeli
for your replay. based on your answer and the published paper, the calculated ratio therefore corresponds to the total optical efficiency. In addition, I kidnly ask you if I can use a DNI file instead of a single value, if you have already worked with the soltrace software.
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Hello I am new to molecular dynamics simulations. I am trying to calculate the chemical potential for iron and phosphorus to do some grand canonical monte carlo simulations. Please help me. Thank you very much in advance.
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I agree with the previous two answers. We know the chemical potentials of Fe and P (see the CRC Chem Physics Handbook or Lange's Handbook of Chemistry). If you are attempting to replicate and/or utilize these through molecular modeling, then you want to "tune" the software to match the known behavior, which depends on the software. It can be quite a trick to get software to reproduce desired behavior. A good reference is "Perturbation Theories for the Thermodynamic Properties of Fluids and Solids," by J. R. Solana, from CRC Press.
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Fortran code of kinetic monte carlo simulation for study hetero epitaxiale deposition
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thank you very much for reply
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I want to use a rssa file in a MCNPX code several time , my rssa file scored 1e6 history and i want to use it 8 times(8×1e6) to reach valid variance .
When i ran that , just 1 time it used.
How can i use that 8 times ?
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The MCNP code must be compiled with I8 (Integer 8) switch for NPS greater than 1e6.
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I have the topology file for a polymer, i.e. I know all the bonded and non-bonded interaction parameters, but I am unaware of the well-accepted methodology to generate the initial structure. I can think of three methods, to perform this but I am not sure which is the best, or if there are better methods out there.
Method 1: I generate a random initial structure and I do an energy minimization in vacuum (the polymer will shrink in size) and then energy minimization in water (the polymer will grow in size). (I may get stuck in local minima)
Method 2: Use Method 1, then perform Monte-Carlo simulations to find global minima.
Method 3: Use Method 1, then increase the temperature and the reduce to temperature so that I can sample the global minima.
What algorithms does a software like Schrodinger use to find the 3-d structure of the molecule from the chemical formula?
My intention is to develop a python script to do the same work that Schrodinger does, but I can't seem to find a book/paper to do which discusses the available algorithms.
PS: I use GROMACS simulation package.
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Method #1 is just fine. There is no rationale in using global minimum as a starting point for a simulation, especially that, for a large polymer, finding the global minimum may take hundreds of years.
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When dealing with an uncertainty characterization problem, we do not know what the true value of an uncertain variable is. So, how can we evaluate which method (e.g. Monte Carlo simulation, Chance-constrained programming, etc.) could better estimate the true value of an uncertain variable?
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At first I would say that the true value cannot be known. What you can get is a mean value of an interval that will represent the interval of values which can be be attributed (with a probability) to the measurand (or result or measurement).
There are several reference document to know how to estimate and express the uncertainty and most of them have forms of calculation and variants that end up being equivalent. There are TWO main approach for uncertainty estimation. GUM (from JCGM100:2008) and Monte Carlo MCM (JCGM 101:2008). The fact is in most of cases both yield equivalent results, above all when math model have several quantities with similar uncertainty contribution. However there are cases when the measuran probability distribution when the distribution obtained by monte carlo is not gaussian or assimetric, GUM and MCM could be quite different. JCGM101:2008 states that in these cases GUM is not a good estimator of measurand, and so you should lean towards MCM approach. Having said that, and answering your question, I would say that the approach you should take, deppends on which is your business:
a) If you are a researcher that need to know truly your process, I think MCM in all cases is the best approach since is not based on numerical approximations.
b) If you are giving a calibration or test service, or even participating in proficiency test or comparisons I think you should estimate by GUM. Don't forget that GUM is more than a calculation approach, is an international agreement. So, if you don't clarify, everybody will supose you used that approach and everybody will know what are you talking about. For example, In a proficiency test if you use MCM and other laboratories use GUM and your results are different, I guess it is unimportant who calculates it better. Even if you're the best you will be out with no complaint.
c) For a routinary work with low precision, you can use eurachem 2012 simplified method (it's GUM), which is easy an fast, and you don't use partial derivatives nor hard equations.
d) Just in case I recommend you check out the freeware mcmalchimia (www.mcmalchimia.com), that makes all calculations for you in no time and you get MCM and GUM result at one time.
I hope have been useful
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