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if someone can please share any report/paper/thesis it will be highly appreciated.
The "technique" leaves the time it finds.
Have you drawn a representative sample from a population?
Did you submit a structured questionnaire to the sample units?
Good: start analyzing the data that describe the phenomenon you are studying and the rest will come by itself.
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I am working on a dataset which contains some of the censored data. As the probabilistic approaches such as the Bayesian estimation technique can be used to consider the censored data, however, I am interested to deploy the Machine Learning using Python. I will appreciate if any literature is shared or provided with any suggestions/guidance. The problem is classification.
Thanks
Take a look at the attached little trick and see if that helps you a bit. David Booth
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If we have multiple experts to get the prior probabilities for the parent nodes how will the experts fill the node probabilities such as low, medium and high and how will we get the consensus of all the expert about the probability distribution of the parent node.
If someone can please share any paper/Questionnaire/expert based Bayesian network where all these queries are explained it will be highly appreciated.
Ette Etuk Thank you so much for the feedback. Actually if you have a lot of stockholders and you want to create a consensus among them then how will we incorporate the probabilities in the parent nodes in the Bayesian network.
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Your guidance and support will be highly appreciated.
Dear Majid Baseer
You may read the articles in the following journals:
Journal of Applied Statistics
The Stata Journal
Review of Economics and Statistics
Annals of Applied Statistics etc etc
I hope that you will find the required stuff.
Good luck
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Could any expert try to examine our novel approach for multi-objective optimization?
The brand new approch was entitled "Probability - based multi - objective optimization for material selection", and published by Springer available at https://link.springer.com/book/9789811933509,
DOI: 10.1007/978-981-19-3351-6.
All experts are welcome to examine the novel approach and book.
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Any technique/method to convert deterministic value into probabilistic values in Bayesian network in order to improve the results.
not clear to me!! why would one need to induce randomness in deterministic values ?
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I'm studying the topic from Probabilistic Robotics by Thrun Burgard and Fox.
In the Extended Kalman Filter algorithm, we linearized the action model in the following way.
𝑔(𝑢(𝑡),𝑥(𝑡-1)) = 𝑔(𝑢(𝑡),𝜇(𝑡−1)) + 𝐺(𝑡)⋅(𝑥(𝑡−1)−𝜇(𝑡−1))
𝑔(𝑢(𝑡),𝑥(𝑡-1)) is the action model and 𝐺(𝑡) is its Jacobian matrix with respect to the state 𝑥(𝑡−1).
I don't see how this guarantees linearity because 𝑔 could be nonlinear in 𝑢(𝑡). The authors don't mention anything about why this is the case.
In other words, I imagined that the multivariate Taylor expansion for this where we get a linear function in both 𝑢(𝑡) and 𝑥(𝑡−1)
Hi,
It should be noted that the Kalman filter is part of the theory of estimation of states of a system represented by a linear model. It is therefore an algorithm that provides estimates of some unknown variables from observed measurements over time. The filter uses as input the command u(t) and the output of the model y(t) and as output it provides an estimate of the output, in other words an estimate of the states of the system. For non-linear systems there is an extension that can deal with these cases: the extended Kalman filter. In my opinion there is no particular requirement on the control when applying this algorithm.
Best regards
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Hi guys,
I am using the negative log likelihood function of a drichilet distribution as my loss function while implementing this paper:
I parameterise the distribution using my network outcome and compute the negative log likelihood of the observed ground truth.
Issue is that I found sometimes the loss is negative, which means that likelihood at the point of observation is greater than 1.
My understanding of this phenomenon comes with two sets:
- This is normal, as likelihood function can be higher 1
- This stands for overfitting, means that the likelihood function probably peaks at the point of the observation so much that other areas of the support would be zero, if we throw an observation there to be test
I would suggest that you are probably using a poor method of fitting. Please look at the entries on the attached Google search. Good luck David Booth PS I note that R has procedures for doing this fit.
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Hello, is there a way for someone who wants to find the parameter which effects the function maximally?
at first, I think, he/she makes a model according to the data in hand then he/she takes the partial derivative with respect to each parameter and according to the result maximum one gives the most effective parameter for that model. Is there any other way especially there is not necessary to make a model before taking a derivative? Such as in conditional probabilistic models such as Bayes networks there is only data and graph network. Thanks.
I think I found an answer to the question about finding the most influential parameter from data without modeling. But it is necessary to get a matrix inverse and there are some shortcomings if parameters have a large gap between their sequential items. In short, now, I think it is better to model the data via known functions or neural networks and find the influential terms via perturbing the parameter. And if necessary make the future space orthogonal to each other otherwise the effects of each other may not be predictable in higher layers.
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Please look at the text of the section on random walk from page 9 to formula 4.7, where you will find mathematical calculations justifying the probabilistic interpretation of the Riemann zeta function.
If the distribution of the zeros of the Riemann zeta function is implicit in your question, then you may find the following paper exciting:
Regards,
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• I have seen the evidence of using network analysis to conduct further cost-effectiveness analysis in two papers. The fractional polynomial models were involved because of nonproportional hazards. I would like to know how to combine the evidence of network meta-analysis with the partitioned survival model in the cost-effectiveness analysis about cancer? What software can be used better? I wonder how the authors connect the two. The abstracts of the two papers are as follows.
• 1. Front Public Health. 2022 Apr 15;10:869960. doi: 10.3389/fpubh.2022.869960.
• eCollection 2022.
• Cost-Effectiveness Analysis of Five Systemic Treatments for Unresectable
• Hepatocellular Carcinoma in China: An Economic Evaluation Based on Network
• Meta-Analysis.
• Zhao M(1)(2), Pan X(1)(2), Yin Y(1)(2), Hu H(1)(2), Wei J(3), Bai Z(3), Tang
• W(1)(2).
• BACKGROUND AND OBJECTIVE: Unresectable hepatocellular carcinoma (uHCC) is the
• main histological subtype of liver cancer and causes a great disease burden in
• China. We aimed to evaluate the cost-effectiveness of five first-line systemic
• treatments newly approved in the Chinese market for the treatment of uHCC,
• namely, sorafenib, lenvatinib, donafenib, sintilimab plus bevacizumab (D + A),
• and atezolizumab plus bevacizumab (T + A) from the perspective of China's
• healthcare system, to provide a basis for decision-making.
• METHODS: We constructed a network meta-analysis of 4 clinical trials and used
• fractional polynomial models to indirectly compare the effectiveness of
• treatments. The partitioned survival model was used for cost-effectiveness
• analysis. Primary model outcomes included the costs in US dollars and health
• outcomes in quality-adjusted life-years (QALYs) and the incremental
• cost-effectiveness ratio (ICER) under a willingness-to-pay threshold of $33,521 • (3 times the per capita gross domestic product in China) per QALY. We performed • deterministic and probabilistic sensitivity analyses to investigate the • robustness. To test the effect of active treatment duration on the conclusions, • we performed a scenario analysis. • RESULTS: Compared with sorafenib, lenvatinib, donafenib, D + A, and T + A • regimens, it yielded an increase of 0.25, 0.30, 0.95, and 1.46 life-years, • respectively. Correspondingly, these four therapies yielded an additional 0.16, • 0.19, 0.51, and 0.86 QALYs and all four ICERs,$40,667.92/QALY gained,
• $27,630.63/QALY gained,$51,877.36/QALY gained, and $130,508.44/QALY gained, • were higher than$33,521 except for donafenib. T + A was the most effective
• treatment and donafenib was the most economical option. Sensitivity and scenario
• analysis results showed that the base-case analysis was highly reliable.
• CONCLUSION: Although combination therapy could greatly improve patients with
• uHCC survival benefits, under the current WTP, donafenib is still the most
• economical option.
• 2. Value Health. 2022 May;25(5):796-802. doi: 10.1016/j.jval.2021.10.016. Epub 2021
• Dec 1.
• Cost-Effectiveness of Systemic Treatments for Metastatic Castration-Sensitive
• Prostate Cancer: An Economic Evaluation Based on Network Meta-Analysis.
• Wang L(1), Hong H(2), Alexander GC(1), Brawley OW(3), Paller CJ(4), Ballreich
• J(5).
• OBJECTIVES: To assess the cost-effectiveness of systemic treatments for
• metastatic castration-sensitive prostate cancer from the US healthcare sector
• perspective with a lifetime horizon.
• METHODS: We built a partitioned survival model based on a network meta-analysis
• of 7 clinical trials with 7287 patients aged 36 to 94 years between 2004 and
• 2018 to predict patient health trajectories by treatment. We tested parameter
• uncertainties with probabilistic sensitivity analyses. We estimated drug
• acquisition costs using the Federal Supply Schedule and adopted generic drug
• prices when available. We measured cost-effectiveness by an incremental
• cost-effectiveness ratio (ICER).
• RESULTS: The mean costs were approximately $392 000 with androgen deprivation • therapy (ADT) alone and approximately$415 000, $464 000,$597 000, and $959 000 • with docetaxel, abiraterone acetate, enzalutamide, and apalutamide, added to • ADT, respectively. The mean quality-adjusted life-years (QALYs) were 3.38 with • ADT alone and 3.92, 4.76, 3.92, and 5.01 with docetaxel, abiraterone acetate, • enzalutamide, and apalutamide, added to ADT, respectively. As add-on therapy to • ADT, docetaxel had an ICER of$42 069 per QALY over ADT alone; abiraterone
• acetate had an ICER of $58 814 per QALY over docetaxel; apalutamide had an ICER • of$1 979 676 per QALY over abiraterone acetate; enzalutamide was dominated. At
• a willingness to pay below $50 000 per QALY, docetaxel plus ADT is likely the • most cost-effective treatment; at any willingness to pay between$50 000 and
• \$200 000 per QALY, abiraterone acetate plus ADT is likely the most
• cost-effective treatment.
• CONCLUSIONS: These findings underscore the value of abiraterone acetate plus ADT
• given its relative cost-effectiveness to other systemic treatments for
• metastatic castration-sensitive prostate cancer.
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I’m trying to fit some kind of causal model to continuous value data by solving differential equations probabilistically (machine learning).
Currently I’m solving complex-valued vector quadratic differential equation so there are more cross correlations between variables.
dx(t)/dt = diag(Ax(t)x(t)^h) + Bx(t) + c + f(t)
or just
dx(t)/dt = diag(Ax(t)x(t)^h) + Bx(t) + c
diag() takes diagonal of the square matrix.
But my diff. eq. math is rusty because I have studied differential equations 20 years ago. I solved the equation in 1-dimensional case but would need help for vector valued x(t).
Would someone point me to appropriate material?
EDIT: I did edit the question to be a bit more clear to read.
I understand that you want to solve a system of differential equations, but the details you provide is a bit confusing.
A second order differential equation is one that contains the second derivative of the unknown function and maybe, but not necessarily, the first derivative. These equations are linear if the unknown function and its derivatives are raised to the first power only; otherwise they are non-linear. Usually, by quadratic differential equation, we mean one that contains the square of the unknown function. They are different things.
The example you provide seems to be a first order differential equation, but could you write it more clearly?
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Hi for a network traffic analysis task i need a probabilistic model to analysis sequences of network data. Euch observation is here an event consisting of structured Information (e.g. ip adresses, ports, protocol type). Im interested in the dependencies between these observations using a generative model. Any ideas?
Here, you might find a good starting point:
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*what if we only managed to have 1 discharge value (eg. 1.8m3/s) of a river section, is it possible for us to create a predictive hydrograph?
*what are the parameters needed?
*is there any article, journal to support the probabilistic analysis?
The hydrograph is the signature response of stream, or river, to its input as rainfall, snowmelt or springflow in some instances, and is a function of the land use, topography, stream and channel type and morphology, including gradient, sinuosity, storage such as lakes and ponds, etc. If a scientist or hydrologist wants to consider a hydrograph for a stream, it is usually measured, developing a stage discharge relationship at a stable cross section and measuring stage through time Such as with a water level recorder, transducer, etc. The USGS and probably some hydrology books have various papers or books that describe how to accomplish this. For research level work, trying to generate a hydrograph would require applying appropriate standards of measurement and monitoring. Rainfall is also measured at select locations and snow pack if present has water storage that slowly melts with temperature increase, or may quickly melt with a warm rain on snow event. Professor Luna Leopold installed a staff gauge in the nearby stream near his house, and used a telescope to collect stage level during storms through time due to his great interest in hydrology. If your country has nearby gauged streams and rainfall within the same physiographic, climactic and topographic zones, you would be much better off using that data to try to adjust it to the stream of interest. I think I may have uploaded a brief report that developed relationship between short term data in the upper Chattooga River, to the long term stream gauging data in the lower Chattooga River. In most instances, predicting hydrographs also requires predicting rainfall or in some instances snowmelt. Various methods are used for flood prediction, but a common one is taking the annual peak flow for a extended period of time as 25, 50 or more years and ordering the data and plotting on probability paper, extending the curve to some extent for even less frequent events. You might also look up the Unit Hydrograph approach which is based on a hydrograph with 1 inch of water yield. A substantial amount of instrumentation and measurement is the standard approach. Various models may become useful in making estimated to ungauged circumstances when validated to the conditions specific to your area.
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Hello.
I'm a beginner of diffusion tensor imaging.
I want to know pros and cons & similarities and differences between probabilistic tractography of FSL and MRtrix.
I really appreciate it if someone could help!
Essentially, MRtrix uses spherical harmonics for FOD modelling, whereas FSL uses Bayesian probability theory.
Jerome
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1. My understanding is that when conducting an economic evaluation of clinical trial data, no discounting of costs is applied if the follow-up period of the trial was 12 months or less. Is this still the standard practice and can you please provide a recent reference?
2. How can one adjust for uncertainties/biases when you use historic health outcomes data? If the trial was non-randomised, how can you adjust for that within an economic evaluation other than the usual probabilistic sensitivity analysis?
Thank you so much.
Hi,
Maybe some of these references are of help to you:
Economic Evaluation in Clinical Trials By Henry A. Glick, Jalpa A. Doshi, Seema S. Sonnad, Daniel Polsky 2014 | 272 Pages | ISBN: 0199685029
Economic Evaluation of Cancer Drugs: Using Clinical Trial and Real-World Data by Iftekhar Khan, Ralph Crott, et al. English | 2019 | ISBN: 1498761305 | 442 pages
Design & analysis of clinical trials for economic evaluation & reimbursement: an applied approach using SAS & STATA
Iftekhar Khan
Series: Chapman & Hall/CRC biostatistics series
Publisher: CRC Press, Year: 2015
ISBN: 978-1-4665-0548-3,1466505486
Methods for the Economic Evaluation of Health Care Programmes
Michael F. Drummond, Mark J. Sculpher, Karl Claxton, Greg L. Stoddart, George W. Torrance,ISBN: 0199665877 | 2015 | 461 pages |
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Hi all,
In a experimental investigation, there are two parameters to be measured, say X1 and X2. My goal is to see how X1 varies with X2. Specifically, I am interested in classifying the graph of X1 versus X2 according to a number of characteristic graphs. Each characteristic graph corresponds to a specific state of the system which I need to determine.
The problem is with the graph of X1 vs X2 undergoing significant changes when replicating the test, thus making the classification a perplexing task. A simple approach I could think of is taking the average of these graphs, but I am not sure if this is reasonable; I am looking for a more mathematical framework.
Regards,
Armin
non-reproducible outcomes suggests that there are one or more fundamental flaws in the research. Your sample size might be too small given system variability. You might be missing some key variables. The analysis might not be appropriate. Some combination of all three. Some experiments are difficult, and maybe I can only have three replicates. I run the experiment again and get a different outcome. If you are certain that you have the right experiment, then try running the experiment several times and block each time, but analyze as one experiment.
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Hello everyone,
Could you recommend papers, books or websites about mathematical foundations of artificial intelligence?
Thank you for your attention and valuable support.
Regards,
Cecilia-Irene Loeza-Mejía
Mathematics helps AI scientists to solve challenging deep abstract problems using traditional methods and techniques known for hundreds of years. Math is needed for AI because computers see the world differently from humans. Where humans see an image, a computer will see a 2D- or 3D-matrix. With the help of mathematics, we can input these dimensions into a computer, and linear algebra is about processing new data sets.
Here you can find good sources for this:
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probabilistic metric space which is not a metric space, has been widely developed in theory, but can someone give some example of the applications of such space ?
Please look at this new paper. We investigated a routing protocol for d2d communications based on probabilistic normed spaces:
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I invite you to see the newly-launched website about the Choice Wave and the Theory of Economic Parallel Rationality, tradition and innovation revolusionising economic thought.
Interesting point,,,,
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A hypothetical example (hehe hypothesis), assume we have enough observations, apply both a “frequentist” and “Bayesian'ist'” model (e.g. linear model with Gaussian error distribution and for Bayesian an uninformative prior to keep it rather vague), we look at the intervals, and both models resulted in the same intervals. Then according to [1] we can say it is similar* to suggest the estimate on the population fell between [1] if we know they are similar. Are both than equally “wrong”? And, do they actually quantify uncertainty, as the both “want” to make (or am I wrong, as they really seem to want, although indeed P(data|estimate) and P(estimate|data)) probabilistic statements on the data about the population. Hence, the data is certain, the estimates are based on the data, so it seems is certain the estimate might approximate the population (assuming perfectly sampled population and this description makes sense) might take on a specified value (note Confidence an Credibility intervals have converged). Again, the data is certain, what is uncertain is what is not in the data. I am just curious what more statistical educated people think of this, how they would communicate this, as this seems hardly discussed (or it is my ignorance).
Thank you in advance for you input.
*Not their words. I just remember a part from the text.
I would not count myself to the targeted group of "more statistical educated people" but I would nevertheless participate in this discussion :)
If you use the same probability model for the response and the same structural model (so that the coefficients have the same meaning) in both, frequentists and Bayesian analysis and if you use a flat prior, then the *limits* of the "typical" (1-a) confidence interval are always identical to the *limits* of the "typical" (1-a) credible interval. "Typical" means that the intervals are central, leaving a/2 confidence or credability on either side. So this is not related to having "enough observations". The sample size is relevant only when the prior is "informed". This prior information will be "overridden" or "overruled" by the information from the data in large samples, so that the limits converge with increasing sample size (if the estimators are consistent).
That it "is certain the estimate might approximate the population" is a consequence of the consitency.
So let's take the case that you have a confidence interval and a credible interval with identical limits. Then they still have an entirely different interpretation:
The confidence interval says that all possible estimates outside the interval are deemed statistically incompatible with the (certain) observed data. It stands for itself. It is a random interval (RI) that is derived from the probability distribution of the random variable (RV) that models the response and the sample size (observations are relalizations of the RV, observed confidence intervals are realizations of the RI, which is a function of the RV that returns two limits). A new sample would give a new confidence interval, and this will have different limits. It may not even overlap with the current confidence interval.
The credability interval says that you assign (1-a) probability to the event that the population value is inside this interval. A new sample would add information to your knowledge about the population value (forcing you to *update* your probability distribution assigned to that parameter). There are no two credible intervals - there is only one, and this (always) is based on everything you can resonably know about the population value.
Not sure if I touched your question...
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I have 6 variables with Mean, Stdev, CoV, Min and Max. Find the attached excel file.
You may use the approach for regression analyses. That should work.
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I'm working on a new probabilistic routing protocol based-on k shortest path selection one, and I wish to test it on the simulator NS3 but there's none much literature on the way to process. I'm looking for some tips which can help to do that, I'm opening. Thx!
Nice advice! It is very astucious to exploit the existing algorithm and improve them. I'm going to try it ASAP.
Thx!
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Hi there RG community, I get back with a new couple of questions, I'm trying to implement a probabilistic routing protocol in NS2/NS3, but there is no much literature on it. Is there someone do that before now? If so, How can I proceed? I'm opening to exchange on that. Thx!
Good question
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Probability
Causation
In probabilistic approaches to causation, causal relata are represented by events or random variables in a probability space. Since the formalism requires us to make use of negation, conjunction, and disjunction, the relata must be entities (or be accurately represented by entities) to which these operations can be meaningfully applied.
Reference :
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The publicly released THUMS model comes with a standard output of nodal coordinates of certain anatomical locations. Since there has been an already established usergroup, I was wondering if there is a standard output template created with THUMS to output cross sectional forces and moments in different bones (esp the longbones) that could be used to predict the risk of injuries probabilistically ? I could define my own outputs but i was wondering if there is a standard template so the results could be compared across multiple groups.
Thanks!
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I am doing a project named as anomaly detection in videos using matlab. I have to perform data associate with clusters using JPDA, but unfortunately it isn't working fine. I have go through distinct papers of JPDA, but these are all about the tracking of any object.
Kindly guide me how do I proceed, or any research paper in which JPDA is used to perform data association not for tracking purpose.
Regards
Ijaz Durrani
Thanks for your collaboration. This paper is about pedestrian tracking through JPDAF, but I required paper in which JPDA is used only for association not for tracking
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In R-studio, there are many commands of Gumbel package. Arguments are also different.
Im asking about the alpha parameter of the Copula which must be greater than 1. If this is the one used to plot the probability paper, how can I choose the value of alpha?
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Data sets, when structured, can be put in vector form (v(1)...v(n)), adding time dependency it's v(i, t) for i=1...n and t=1...T.
Then we have a matrix of terms v(i, j)...
Matrices are important, they can represent linear operators in finite dimension. Composing such operators f, g, as fog translates into matrix product FxG with obvious notations.
Now a classical matrix M is a table of lines and columns, containing numbers or variables. Precisely at line i and column j of such table, we store term m(i, j), usually belonging to real number set R, or complex number set C, or more generally to a group G.
What about generalising such matrix of numbers into a matrix of set (in any field of science, this could mean storing all data collected for a particular parameter "m(i, j)", which is a set M(i, j) of data?
What can we observe, say, define on such matrices of sets?
If you are as curious as me, in your own field of science or engineering, please follow the link below, and more importantly, feedback here with comments, thoughts, advice on how to take this further.
Ref:
Thank you for sharing this Question
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I am stuck between Quantum mechanics and General relativity. The mind consuming scientific humor ranging from continuous and deterministic to probabilistic seems with no end. I would appreciate anyone for the words which can help me understand at least a bit, with relevance.
Thank you,
Regards,
Ayaz
I guess that the Scattering Theory always will be a trend in QM.
The experimental Neutron Diffraction field for example always is creating new tools where QM is widely used.
Although it is attached to a few experimental facilities around the world, still it is a trend.
We always see new discoveries using neutron diffraction in solid-state.
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To know the details how oi is used in probabilistic metric space . As we know that to generalize trianle inequality, we use Triangular norm but how ? need explanation and also how and where it is used in PM space
It is a good question
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I now have a set of input and output data, and a low-order transfer function model which has several parameters to be identified.
If I use tfest in Matlab, I can identify a set of parameter results, but this is not what I expect. What I expect to get is an interval that can encompass all or most of the observations. Which probability method can solve my problem? Preferably it is a probabilistic method or a prediction interval (PI) method. I would be very grateful if you could give me a paper or website
David Eugene Booth Thank you for your suggestions. Those are useful for me!
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I'm developing a readiness assessment model regarding contractors' preparedness for a specific activity, in order to do so, a survey study was carried out and the data analyzed with PLS-SEM to obtain the CSF contributing to that readiness; nevertheless, due to the subject being too specific, it was impossible to define or quantify a population for it and hence, a probabilistic sample, which can compromise the external validity (generalizability) of my readiness assessment model. Is it feasible trying to reduce that generalizability issue with the minimum sample size requirements (means of power analyses) from Cohen (1992) and the use of PLS predict to determine the prediction power of the model?
I'd be delighted if any colleague could reply to this need
In general, using any rule-of-thumb for sample size planning or assessing statistical power is problematic.
Random sampling provides a model free basis for generalization. Propensity score–based methods for generalization require three assumptions to ensure their validity. First, the stable unit treatment value assumption must hold for all units in the experiment and in the population. Second, generalization using propensity score methods requires strongly ignorable treatment assignment in the experiment. Finally, generalization using propensity score methods requires a strongly ignorable sample selection. The development of rules of thumb that take into account sample size since features of probability samples—the benchmark for generalizability—differ markedly in small samples. This raises the issue of how to judge the adequacy of the match between the experimental sample and the inference population.
Probability sampling is the gold standard for generalizing from samples. The idea is to use the adequacy of matching that would be expected if the experiment had a probability sample to develop benchmarks of adequate matching. There is no reason to expect small experimental samples to match inference populations better than probability samples.
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The birth and death probabilities are p_i and q_i respectively and (1-(p_i+q_i)) is the probability for no change in the process. zero ({0}) is an absorbing state and sate space is {0,1,2, ...}. What are the conditions for {0} to be recurrence (positive or null)? Is the set {1,2,3,...} transient? What we can say about duration of process until absorption and stationary distribution if it exists and etc?
Every comment is appreciated.
There is no logical (reasonable) condition that {0} is not absorbing, so it is always a recurrence state. {1,2,...} is always transient.
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I am looking for some analytical, probabilistic, statistical or any other way to compare the results of a different number of approaches implemented on the same test model. These approaches can be different optimization techniques implemented on the similar problem or different type of sensitivity analysis implemented on a design. I am looking for a metric (generic or application-specific) that can be used to compare the results in a more constructive and structured way.
I would like to hear if there is a technique that you use in your field, as I might able to drive something for my problem.
Thank you very much.
Hi? You might want to have a look at one of my publications - 10.1016/j.envsoft.2020.104800
I recently conducted a similar study where I applied three different sensitivity analysis methods to fire simulations and compared their results!
Cheers!
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Probabilistic sensitivity analysis is criticised for potentially introducing uncertainty itself because of the consideration of the distribution of the parameters. Are there ways of addressing this potential for additional uncertainty?
If you look deeper into the literature, you have some sensitivity analysis methods that are independent of the sampling techniques!
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I have a small dataset and need to learn a mixed probabilistic model (discrete + continuous) and simulate new values taking into account the learned structure.
Mixed graphical models (MGMs) are graphical models learned over a combination of continuous and discrete variables. Mixed variable types are common in biomedical datasets.
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Hello,
I am currently working on Beta Distribution, and I am using the distribution to model a knowledge/opinion into the software. The user adjusts the alpha and beta values of Beta Distribution to come with a graph that resembles his opinion. What I am really interested in is how we can add up two Beta Distribution graphs to come with a more generalized Beta Distribution.
For every opinion, there is a set of alpha and beta value. Say two opinions have the following alpha and beta values.
Opinion 1: [α_1 = 2 ; β_1 = 5]
Opinion 2: [α_2 = 4 ; β_2 = 5]
(I have attached the graph of these two sets of alpha and beta for visualization)
It will be much helpful for my undergoing research if I get to know how I can add these two graphs or merge these two graphs into one? Any suggestions would be greatly appreciated. Thank you for your valuable time.
It seems not to be helpful to start by thinking of merging only two curves ... better to think immediately of combining many curves. You also need to decide whether your combined-curve result needs to be within the same family as the "observed" ones
Combining the curves might be done by averaging some characteristic of the curves:
(a) average percentage points, either leaving results raw for many such percentage points, or fitting a distribution to a pair of percentage points
(b) average moments of the "distribution" curves, and fit a distribution to the average moments;
(c) average the parameter values of the "beta distributions" or, for example, logarithms of the parameter values, to get parameters of a combined curve.
In the above you might consider a "robust"version of the average.
It would be good to try something out with some fictitious starting curves and examine the results produced graphically to see what you think most suitable to your problem. Looking at things graphically for many curves may lead you to think about dealing with understanding the differences between opinions, rather than just finding an a average opinion.
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How to calculate the sum and the subtraction of many random variables that follow exponential distributions and have different parameters ?
(The value of Lambda is different for all or some variables).
example :
L(t) = f(t) + g(t) - h(t)
with
f(t) = a.expo(-a.t)
g(t) = b.expo(-b.t)
h(t) = c.expo(-c.t)
st:
a = Lambda_1
b = Lambda_2
c = Lambda_3.
(continued)
In case of more terms (all with different means m_j>0, j=1,2,...,n) the formulas are as follows (ti replaced by -s)
ch.f.(X_1+X_2+...+X_n)(t) = 1/ [ (1+m_1 s) (1 + m_2 s) ... (1 + m_n s)]
= \sum_{j=1}^n A_j / (1+m_j s),
where A_j = \prod_{k\ne j} [ 1 - m_k / m_j]^{-1}
Therefore, in such cases the density of the sum is equal to
\sum_{j=1}^n A_j / m_j \exp( - x/m_j ), for x>0.
If X_j in the sum is preceded by sign -, then the first two formulas remain valid after replacing m_j by - m_j. The last requires replacing the exponential density for positive variable by the opposite one.
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Does anyone know a geotechnical engineering software which can support subset simulation? I need to do some probabilistic analysis of a geotechnical project. However, due to the small probability, I need to use subset simulation instead of the crude Monte Carlo analysis.
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Does the quantum mechanics gives probabilistic results or exact quantified values?
Probabilist results'' are `exact quantified values''-about the pobability distribution in question. Not all probability distributions are delta functions, but this doesn't mean that they are uniform, either.
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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
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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• I wanted to implement Feature selection framework using criterion like probabilistic error or probabilistic distance , i also have doubt like if i have non parametric distribution for my features can i use kernel based estimation techniques to find the class conditional probabilities instead of analytical function to evaluate the criterion like probabilistic error or bayes error rate.
• i was thinking even if we have some non parametric distribution we can use the probabilities value estimated my kernel density estimation and the integration would converge to summation ultimately in the formula i have attached for error rate.
• Is my approach is fine or if anyone have tried this can guide me.
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Do you have any experience with probabilistic software for structural reliability assessment? Any links?
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A question to all you stroke and tDCS / TMS researchers.
I want to visualize the lesion location of my participants in relationship to the stimulation site. In my case I have the lesions as ROIs normalised to the MNI standard space. Now I would like to create a 3 D image with the lesion mask as volume and mark position P4 on top. My objective is to see whether I actually tried to stimulate healthy or affected tissue with my tDCS protocol.
Alternatively marking P4 on the 2D slices would be fine as well. I just don't know how. I found the paper by Okamoto et al, 2004 ( ) which gives coordinates for the MNI templates.
Following
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Dear friends,
Please kindly list the procedure steps shortly on how to make the probabilistic and deterministic earthquake hazard map using GIS or any other related software.
Thanks,
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Suppose we have statistics N(m1, m2), where m1 is the value of the first factor, m2 is the value of the second factor, N(m1, m2) is the number of observations corresponding to the values ​​of factors m1 and m2. In this case, the probability P(m1, m2) = N(m1, m2) /K, where K is the total number of observations. In real situations, detailed statistics N(m1, m2) is often unavailable, and only the normalized marginal values ​​S1(m1) and S2(m2) are known, where S1(m1) is the normalized total number of observations corresponding to the value m1 of the first factor and S2(m2) is the normalized total number of observations corresponding to the value m2 of the second factor. In this case P1(m1) = S1(m1)/K and P2(m2) = S2(m2)/K. It is clear that based on P1(m1) and P2(m2) it is impossible to calculate the exact value of P(m1, m2). But how to do this approximately with the best confidence? Thanks in advance for any advice.
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I am working in small modular nuclear implementation in Europe, initially. We believe that this should be a social enterprise and your work may well enable the 3 legs of the sustainability to be brought to a common currency of economic (already undercut all other sources), social (harder to prove but acid test is municipal and private pension investment on a massive scale), environmental (high temperature gas dry cooled triso fuelled reactors are vastly superior environmentally to PWRs and are also distinctive in being inherently safe provable in real life as opposed to probabilistically safe which was needed only because the consequence of test proving safety is too severe for LWR physics). We envisage 50 MWe units distributed close to domestic and industrial sites with heat, hydrogen and power demands, so your mapping work wold form a strong basis for site selection. I have done similar work for renewables site selection in Scotland.
Robert,
That is good news that people are more comfortable with SMRs. They should be, at least they should be with high temperature 'dry' moderated ones. Thee is a fundamental problem with the physics of wet moderated ones, except in submarines immersed in an infinite heat sink! Unfortunately a decision in USA took the market in the wet direction and only now is the dry route recognised as so much safer and inherently safe at a small scale. Fortunatelyt, UK and Canada knew this and in Uk we only have one wet reactor (Westinghouse Sizewell B) but we have one wet reactor under construction (Hinkley C = EPR) but it is unlikely ever to run.
The cost challenge is not as bleak as you mention. The main reason is that SMRs produce both electricity and low temperature heat in quantities that are saleable. A large reactor cannot do this and it adds 50%+ to the rate of economic return. Furthermore, the electricity is genertaed at the point of use so that the 20% transmission cost is avoided from the delivered price of electricity. Finally, the inherent safety means that most of the ACTIVE safety systems are absent so need neither capital nor maintenance costs.
I am optimistic but also realistic and anticipate that the first commercial ones will run in earnest by about 2029 with a huge upsurge in advanced economies by 2035 with developing world from 2035 where the demand growth really occurs.
Regards,
James
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We are working on a large number of building-related time series data sets that display various degrees of 'randomness'. When plotted, some display recognisable diurnal or seasonal patterns that can be correlated to building operational regimes or the weather (i.e. heating energy consumption with distinct signatures as hourly and seasonal intervals). However some appear to be completely random (Lift data that contains a lot of random noise).
Does anyone know if an established method exists that can be deployed on these data sets and provide some 'quantification' or 'ranking' of how much stochasticity exists in each data set?
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.
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Hi, In my city i have seen a discussion among volcanic hazard researchers (Colombian Geologic Service and the local University), the central subject is the quality on the accuracy of volcanic hazard methodologys (i.e. deterministic method vs probabilistic method), i´d like to learn more about works that compare these two methods with observed events. Please, could anybody share me papers or books about the subject? Thanks a lot.
Hello !!
Source Rock of the Volcanic Fragments in Wadi Al-batin, Iraq: Geomorphological, Petrographical and Geochemical Evidences
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In eart slope stability analysis by SLOPE/W Analysis
Dear Dr. Maysam,
I think that the probabilistic and sensitivity analysis may be enough. But, for intensive results and comparisons, you have to consider the statistical analysis. The choice may depend on the nature of your application and research work.
Good Luck!
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Now a days use of AI/ML in disaster modeling is getting quite popular. I would like to know if there exists any such earthquake model which is better than conventional earthquake models for probabilistic seismic hazard analysis (PSHA) and or for deterministic scenario seismic hazard analysis (DSHA).
Machine Learning Approach
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When running a scheduling scheme what probabilistic approach can be use to estimate energy consumption that scheduling algorithm?
According to novel techniques many approaches are used for energy consumption. But i am working on machine learning techniques such as artificial neural networks. In this scheme or algorithm you can easily find the energy consumption or demand and its also can be used for optimization and forecasting of load dealing in smart grid or any electrical system connected with renewable energy resources.
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Let's center the discussion mostly on the fact of characterizing geotechnically those solid wastes.
Are probabilistic analysis the best way for these cases of highly heterogeneous materials?
#Slopes #Geotechnics #GeotechnicalEngineering
Depending upon the site and orientation of rubbish dump, You may assess its physical morphology and depending upon physical/site assessment and heterogeneous material, you may choose preferred locations/section lines, along which you may carry out slope stability assessment with a view to arrive at the status of slope in terms of FoS or Factor of Safety. When < 2, it is normally unstable/critically stable. You need to have the lab-estimated values of select geotechnical parameters used in a particular analysis method. Emperically chosen values from available datasets/ tables etc. may also be used for cross-check, but ideally lab-estimated values would be of more help.
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Many scholars talk about importance of the role of demons in scientific thought- experiments.The most known are Laplace's Demon and Maxwell's Demon.Laplace claimed that world is completely deterministic. Contrary to this Maxwell claimed that the world is probabilistic and indeterministic. What these and other Demons can say to us about our world? Can Demons in science contribute to better understanding of Universe and scientific discoveries?
Hello Dragoljub,
I do not know why technical people are so fond of demons, but let me add a few other demon examples. Schrödinger 's cat, which is used to express the superposition of states, could be considered a demon, though it is not formally given that name. In economics, there is statiscian Maurice Kendall's "Demon of Chance" (circa 1953), which he used to explain the absence of serial correlation as measured by the sliding autocorrelation function of commodity prices such as the time series of weekly wheat or cotton prices. For a discussion of Kendall's Demon, see the following book.
Donald MacKenzie; An Engine, Not a Camera, How Financial Models Shape Markets; MIT Press; 2006; pp. 61-63.
There is also the example of an enormous number of monkeys in a room, each sitting in front of a typewriter and typing furiously. The idea here is that this cohort of simian writers could eventually reproduce the play Hamlet. I believe this demon example, though, again, it is not formally designated as a demon, was supposed to represent Boltzmann's idea, from statistical mechanics, that a possible state of gas molecules in a box would be for them to all colalese into one corner of the box. According to Boltzmann such a trajectory, while unlikely, still has a small but non-zero probability.
My last demon example is Murphy's law, which, and I am paraphrasing here, says that, "Anything the can go wrong will go wrong, and usually at the most inopportune time." As a child, I remember seeing the old Warner Brothers cartoon titled "Falling Hare" (circa 1943) with Bugs Bunny and the Gremlin (a.k.a., demon). This particular demon was the root cause of mechanical and electrical problems at a US Army Air Corps (forerunner of the US Air Force) base during WWII, see https://www.youtube.com/watch?v=ZElJxTCIsJI .
I am sure there are more demon examples out there, somewhere.
Regards,
Tom Cuff
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I need to model a wireless communication system where the absence or presence of the intended receiver should be completely a random process. For instance, when a transmitter sends something, there should be a random probability of whether the receiver would receive the transmitted signals or not. I think I should model the the presence or absence of the receiver as "on/off" modeling. However, I think there might be other useful modeling unknown to me till now. An answer will be highly appreciated.
Dea Abdullah,
I do not know exactly what is your receiver application. But you can find other statistics other than the completely random variable statistics.
I would explain more the physical meaning of my assumed distribution. The t in the equation is the on or the presence time of receiver. tave is the average period of appearance.
This can be implemented by using a switch which is made on for a time issued by random number generator.
For random variables please visit the book by Glover and Grant: Digital communications
Best wishes
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Can any body tell me what is difference between 'probability density function' and 'power spectral density function' for random data like wind speed?
The probability density function p(x) is defined by the following relation:
P(x, x+dx)= p(x) dx,
x is the random variable, P(x,x+dx) is the probability to find x between x and x+dx,
dx is the interval over which one calculates the probability.
So, finally
p(x)=P(x, x+dx)/dx
As for the power spectral density it can be defined by the relation:
P(f, f+df) = p(f) df,
where p(f) is the spectral power density, P(f, f+df) is the power contained in the frequency interval df around the frequency f.
So, p(f)=P(f, f+df)/df,
So the spectral power density is the total power contained in a frequency interval df divide by the frequency interval df.
Best wishes
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We have different metrices to calculate the engieering resilience by comparing disturbed state and mean/equilibrium states. Are then any methods which incorporate the joint probabilistic behaviour of more than one variable to quantify resilience?
I would like to recommend some papers from our group :)
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I am calculating a value that is computed by dividing the derivative of the cumulative distribution function by the value of the distribution function at that point. It is of the form:
𝐽=𝐹′(𝑥) / 𝐹(𝑥)
Where 𝐹(𝑥)is the cumulative distribution function. To get the confidence band on 𝐹(𝑥)I can use the DKF Inequality. How do I get the confidence bands on J?
Drew Lilley We can get the random number of 𝐹′(𝑥) and F(x), right? Then we can get random numbers of J. After that, confidence interval of J is clear then. :)
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I have a small multivariate data ( < 10,000 records) with none of the variable following normal distribution (mix of right-skewed and left-skewed variables). As Gaussian Mixture Model is not suitable for this, what are the other methods that allows computing a probability to belong to each cluster, for every point ?
I think, the batter method is ANN
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Hi,
How to build the Bayesian network and conditional probabilistic tables of the following correlation: x1 → x5 ꓦ x6 ꓦ x7
As in paper "A. Malki, D. Benslimane, S.-M. Benslimane, M. Barhamgi, M. Malki, P. Ghodous, and K. Drira, “Data Services with uncertain and correlated semantics,” World Wide Web, vol. 19, no. 1, pp. 157–175, 2016.".
Cordially.
Hi,
My problem is that I want to build the conditional probability tables from the semantics of the correlation (the propositional formula) x1 → x5 ꓦ x6 ꓦ x7.
The author of the paper gave the semantics of only two operators: the implication and the exclusive or (XOR) (see pages 3 and 4 of the attached document).
Cordially.
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fire index prediction
Dear Leticia Koutchin Reis,
I suggest you to see links and attached files on topic.
Probabilistic Risk Criteria and Safety Goals - Nuclear Energy ...
https://www.oecd-nea.org › nsd › docs › csni-r2009-16
EPRI/NRC-RES Fire PRA Methodology for Nuclear Power ...
https://www.nrc.gov › reading-rm › doc-collections › nuregs › contract
Mapping future fire probability under climate change: Does ...
https://journals.plos.org › plosone › article › journal.po...
Application of fire safety engineering principles to the design ...
Probabilistic Risk Assessment Procedures Guide for NASA ...
www.barringer1.com › mil_files › NASA-PRA-1.1.pdf
Best regards
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Im working with Nataf model trying to fit a joint probabilistic model for circular and Linear variables, but I have some difficulties in calculating the correlation matrix because, I could find an equation for calculating the equivalent correlation between two circular variables or between a circular variable and a linear variable.
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I have applied two different frame-work, i.e. Deterministic (Very fast simulated annealing) and probabilistic (Sampling) algorithms to solve a highly non-linear inverse problem with 6 unknown model parameters to be optimised. As you know, the result of the VFSA is a single solution for the each model parameter and the result of the probabilistic algorithm is a set of solutions sampled from the posterior. I expect that the the solution obtained by the deterministic method should be captured by the samples obtained by the probabilistic approach and the discrepancy should not be significant, but these solutions are far from each other!!!
For both methods, the prior, the forward model and the the noise level are the same. Actually, I defined the prior in probabilistic method uniformly distributed, but in VFSA, the model parameters are updated through sampling of a Cauchy distribution. What causes this discrepancy? Any justification for this matter? is it usual? and what causes this in the problems?
P.N
The attached figure shows the cloud of samples from the probabilistic method, their mean (Black dot) and the optimised value by deterministic algorithm (Red dot)
Hamed Heidari Yes, i think so.
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If existing microgrid energy management system is deterministic, how to design it in real time probabilistic or stochastic model?
How to get expertise in this modeling?
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In this validation process, the team has tried to make sense of the research, devising a working hypothesis, built on scientific bases, such as the reference models that for years have been the pillars of the language sciences, which are: the descriptive method of N. Chomsky ;- The method Lexico-grammatica by M. Gross ; -The Nooj system, according to the Transformational Analysis of Direct Transitive by M. Silberzstein; The probabilistic calculation by Hofmann, according to the Probabilistic latent semantic Analis. The results have given very valid and irrefutable answers such as: - The mathematical laws guide and support the linguistic text, because a language, to be elevated to universal code must be describable, with a rational scientific method. Languages can be converted into a plurality of codes and that formal languages are subjected to techniques of fixity and non-compositionality and therefore guided by mathematical laws pre-established and therefore predictable, was born for market needs and is built in the laboratory Natural languages are subjected to linguistic techniques of causality, and that the first communication and fixed, the second is innate, because ........The homo sapiens transforms the contents of his mental activities into symbols, i.e. letters, numbers, etc. according to anthropology, sociology and natural laws of his culture and therefore semantics belongs only to the man sapiens and to a certain man in the course of that history. The statute of conjecturing that we postulate is that the mathematical laws guide the mind of the homo sapiens in the structuring of the lessies, morphies, dysmorphs in the osmotic voluntary and innate conjecturing of human semantics.
Translated with www.DeepL.com/Translator
Dear Prof. Ritamaria Bucciarelli:
Congratulations and thank you so much for asking this most important question in the context of both Applied and Computational Linguistics! In that regard, as far as the descriptive method is concerned, Chomsky (1956) undertakes highly focused research on several conceptions of linguistic structure to determine whether or not they can provide simple and “revealing” grammars that generate all of the sentences of English and only these. He finds that no finite-state Markov process that produces symbols with transition from state to state can serve as an English grammar. Furthermore, Chomsky (1956) draws attention to the particular subclass of such processes that produce n-order statistical approximations to English do not come closer, with increasing n, to matching the output of an English grammar. Like this manner, he formalizes the notions of phrase structures and shows that this gives us a method for describing language, which is essentially more powerful, though still representable as a rather elementary type of finite-state process. Nevertheless, Chomsky (1956) points out that it is successful only when limited to a small subset of simple sentences. In view of that, he studies the formal properties of a set of grammatical transformations that carry sentences with phrase structure into new sentences with derived phrase structure, showing that transformational grammars are processes of the same elementary type as phrase-structure grammars. Correspondingly, he underlines that the grammar of English is materially simplified if phrase structure description is limited to a kernel of simple sentences from which all other sentences are constructed by repeated transformations, and that this view of linguistic structure gives a certain insight into the use and understanding of language.
Subsequently, Chomsky (1956) states that general linguistic theory can be viewed as a metatheory, which is concerned with the problem of how to choose such a grammar in the case of each particular language on the basis of a finite corpus of sentences. In particular, as indicated by Chomsky (1956), it will consider and attempt to explicate the relation between the set of grammatical sentences and the set of observed sentences. In other words, this author emphasizes that linguistic theory attempts to explain the ability of a speaker to produce and understand- new sentences, and to reject as ungrammatical other new sequences, based on his limited linguistic experience. To this end, Chomsky (1956) gives the following example: suppose that for many languages there are certain clear cases of grammatical sentences and certain clear cases of ungrammatical sequences, e-e., (1) and (2). Respectively, in English.
(1) John ate a sandwich
(2) Sandwich a ate John.
In this case, we can test the adequacy of a proposed linguistic theory by determining, for each language, whether or not the clear cases are handled properly by the grammars constructed in accordance with this theory. For example, if a large corpus of English does not happen to contain either (1) or (2), we ask whether the grammar that is determined for this corpus will project the corpus to include (1) and exclude (2). In this sense, Chomsky (1956) points out that even though such clear cases may provide only a weak test of adequacy for the grammar of a given language taken in isolation. They provide a very strong test for any general linguistic theory and for the set of grammars to which it leads, since, as Chomsky (1956) claims, in the case of each language the clear cases be handled properly in a fixed and predetermined manner. Like so, Chomsky (1956) does foreground that we can take certain steps towards the construction of an operational characterization of “grammatical sentence” that will provide us with the clear cases required to set the task of linguistics significantly.
More to the point, Odlin (1994: 45) asserts that, nevertheless, at the practical level, the Universal Grammar Model by Chomsky suggests that more attention must be paid by teachers to the teaching of specifically syntactic aspects of vocabulary acquisition.
Bibliographical references
• Chomsky, N. (1956). Three models for the description of language. IRE Transactions on information theory, 2(3), 113-124. Retrieved from: (https://www.princeton.edu/~wbialek/rome/refs/chomsky_3models.pdf). [Accessed August 06, 2019].
• Odlin, T. (Ed.). (1994). Perspectives on pedagogical grammar. Cambridge University Press.
Best wishes,
Javier.
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Let X = [x1,...,xn] and Y = [y1,...,yn] be two random points in the n-dimensional hypercube with the constraint that x1 + ... + xn = 1 and y1 + ... + yn = 1
Let also the coordinates of the points be uniformly distributed in the [0,1] interval, i.e.: xi,yi ~ U[0,1].
My problem is that of determining the probability distribution (pdf) of the Euclideand distance between X and Y.
Any suggestion will be greatly appreciated.
Best,
f.
Notice that for n = 2, under the proposed constraints x1 + x2 = 1 and y1 + y2 = 1, the condition that the Euclidean distance betweeen points X(x1, y1) and Y(x2, y2) is less than d (0 ≤ d ≤ √2) reduces to the inequality |x1 - y1| < d/√2. Hence, the detrrmination of CDF F2(x) of the desired random variable reduced to the known problem in Geometric Probability “Probability that two persons will meet” (presented in my previous answer, with x/√2 instead of x and √2 instead of 1).
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I have a dataset and trying to apply soft clustering, preferably Multivariate Gaussian Mixture model, but I have following doubts :
1. Does Multivariate GMM assume the underlying data to follow multivariate normality ? I guess, even if the individual components are Gaussian, their mixture can still be non-gaussian and thus violating this condition. Is this so ?
2. If multivariate normality is indeed required, what are the other ways to attain probabilistic clustering. Will be really helpful, if someone could refer a python or r based implementation.
No problem. Well, let's see what this really is about.
Probably the simplest example is fitting a simple univariate Gaussian distribution to some data. The free parameters are the mean m and the standard deviation sigma. One just writes the likelihood function of the samples, takes the logarithm of it and maximizes by equating the derivative to zero. This leads to the well known formulae as the estimated mean is the average of all samples, and the estimated sigma parameter is the empirical standard deviation of the samples. So far so good.
If we have a multivariate Gaussian, one can do the same, although it will be a bit more complicated as matrix inversion needs to be solved. Anyways, the steps are the same.
When it comes the a Guassian mixture, it has as many free parameters as all of its components plus the weight of each component. For this problem, there is no closed form solution, however, one can apply the expectation maximization (EM) technique to iteratively approach and converge to the maximum a-posteriori estimation of parameters. This is what is usually implemented under the hood in Gaussian mixture models.
Now, the thing that all this works for the Gaussian is by chance, or nature, or God. For a more general distribution, for example, like the Gamma or Beta distributions, one can hardly derive any formula to fit its parameters to the data, not even in the univariate case. This is just an example, I didn't check, there might be some closed formula for the Gamma distribution, but what I want to illustrate is that it is not natural that for a complex, analytically described distribution one easily estimate the parameters. In many cases it leads to unsolvable equations. And it becomes even more tricky with multivariate distributions, and even more tricky with mixtures of multivariate distributions.
What I want to say, is that what works with the Guassian is due to its special properties, generally it does not work with other distributions, not even in the univariate case.
However, there is a solution, which is called probabilistic programming, closely related to Bayesian statistics. This is not an analytic but a purely probabilistic approach to fit the parameters of any distributions, and if you have enough data compared to the number of free parameters to be fitted, it works fine, and theoretically, it converges.
This probabilistic programming needs a different mindset. Unlike in frequentist statistics (what I described above), in this case you won't get a single point estimate that "hey, this is the mean of the distribution". Rather, you get a distribution for each free parameter, like the mean of a Gaussian. So what you get from a probabilistic programming framework is that the mean of the Gaussian generating your samples is 1 with 10% probability, 2 with 45% probability, 3 with 35% probability and 4 with 10% probability. You get a distribution like this for each parameter -- to be exact, you get a joint distribution. And you can take the most likely parameter combination (for example, 2 as it has 45% probability), this is related to the frequentist choice - the maximum a-posteriori, but it is even better to take the weighted mean and compute the expected value of this parameter, which will be something like 2.4.
This is only an illustration, but this approach works with any distribution with any complexity, and is based on Monte Carlo simulation. So this can be used to fit the parameters of any distribution of any complexity to any dataset. The thing that Gaussian Mixture Model works is very unique. In Python, the most widely used package is pymc3. You can find further description here https://docs.pymc.io/notebooks/getting_started.html
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OPF; Plugin electric vehicle
Thanks for the suggestion.
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I want to model the spatial variability of soil in numerical reliability slope stability analysis.
Dear Amir
Slide2 2018 is the only available commercial software to carry out slope stability analysis considering spatial variability of soil properties. It is a LEM based software. You can export the random field from Slide to RS2 software as well to do FEM analysis. Please let me know if you have any question. Thanks
Thanks Ashkan Khosravi Hajivand Khosravi Hajivand
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I am carrying out an investigation with a methodological triangulation:
1st step: quantitative. surveys of students from schools in Buenos Aires. sampling by conglomerates (neighborhoods) with similar socio-educational index.
2nd step: qualitative. In-depth interviews with students from Buenos Aires schools. they will be chosen by the different neighborhoods / conglomerates.
The quantitative sampling is not entirely probabilistic (I believe): it does not take all the conglomerates, and I group them under my criteria (which is theoretical, but I still have to develop).
Finally, as the sampling of the quantitative part has certain problems, I do not know if I should do the qualitative part with the same students that were surveyed, and deepen with the interview. but can I take students from other schools and other conglomerates with similar indexes?