this equations i wan to transfer in my Mathematica environment and calculate the mean of date the values of this equations
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I have a question regarding transforming one set of random variables to another.
Suppose that we have a sample of data collected from two competing exponential distributions, with rate a and b. We know that the random variables X follow an exponential distribution with rate (a + b).
Now suppose that we know the rate a, and we want to transform the random variables to a new set of random variables, so that the new set of random variables Y so that Y follows the exponential distribution with rate b.
Is there a way to do so? I would appreciate any answers or references.
Best regards.
We have looked into convolution of variables, but it seems not to fit since X cannot be represented as sum of two random variables.
I am no expert in this field but more of a beginner with a bit of experience, so please keep the answer as simple as possible.
I cannot be very specific about this topic but what I am trying to do is predict the velocity of multiple objects(that should have a pattern because they are similar). I am taking the optical flow from every tenth frame and building a histogram of every tenth frame(x and y velocities separate). Then, I convert these histograms to vectors and store them in a CSV file. I am trying to use these vectors in an LSTM for Timeseries forecast. I do not know how to input each x and y velocity vector as a time step to output the next x and y velocities every (let's say) 5 steps.
The tutorials I see are usually about predicting temperature, and the input values(not vectors but single values) of humidity, precipitation, etc. and then output a single value(being the temperature)
please help, I hope I made it relatively clear.
Maybe there is a better approach.
I have to optimize my Parameters in order to get the highest Energy cusumption. I think no Need to explain the physical phenomenon I'm studying but the important informations are: Let's say I have two variables which are the frequency F and the Magnitude A. The Energy consumption Y is not calculated through an equation but with a complex Simulation in Ansys. From Ansys I can have the Energy Y for every Frequency and Magnitude combination I choose. is there a Deep learning technique which allows to use some variables and their Output Energy as Training data to create a Networks which would calculate the Output Energy for every other Parameter combination.
Ideas are welcome …
I'm analysing longitudinal panel data, in which individuals transition between different states in a Markov chain. I'm modelling the transition rates between states using a series of multinomial logistic regressions. This means that I end up with a very large number of regression slopes.
For each regression slope, I obtain a posterior distribution (using WinBUGS). From the posterior distribution, we get the mean, standard deviation, and 95% credible interval associated with the slope in question.
The value I am ultimately interested in is the expected first passage time ('hitting time') through the Markov chain. This is a function of all the different predictor variables, and so is built from the many regression slopes produced by the multinomial logistic regressions.
A simple approach would be to take the mean of each posterior distribution as a point-estimate for each regression slope, and solve for the expected first passage time at a series of different values of the predictor variables. I have now done this, but it is potentially misleading because it doesn't show the uncertainty around the predicted values of expected first passage time.
My question is: how can I calculate a credible interval for the expected first passage time?
My first thought was to approximate the error via simulation, by sampling individual values for the regression slopes from each posterior distribution, obtaining the expected first passage time given those values, and then plotting the standard deviation of all these simulated values. However, I feel like (a) this would make a statistician scream and (b) it doesn't take into account the fact that different posterior distributions will be correlated (it samples from each one independently).
In WinBUGS, you can actually obtain the correlations between the posterior distributions. So if the simulation idea is appropriate, I could in theory simulate the regression slope coefficients incorporating these correlations.
Is there a more direct and less approximate way to find the uncertainty? Could I, for instance, use WinBUGS to find the posterior distribution of the expected first passage time for a given set of values of the predictor variables? Rather like the answer to this question: define a new node and monitor it. I would imagine defining a series of new nodes, where each one is for a different set of actual predictor values, and monitoring each one. Does this make good statistical sense?
Any thoughts about this would be really appreciated!
I have a set of first 25 Zernike polynomials. Below are shown few in Cartesin co-ordinate system.
z2 = 2*x
z3 = 2*y
z4 = sqrt(3)*(2*x^2+2*y^2-1)
:
:
z24 = sqrt(14)*(15*(x^2+y^2)^2-20*(x^2+y^2)+6)*(x^2-y^2)
I am not using 1st since it is piston; so I have these 24 two-dim ANALYTICAL functions expressed in X-Y Cartesian co-ordinate system. All are defined over unit circle, as they are orthogonal over unit circle. The problem which I am describing here is relevant to other 2D surfaces also apart from Zernike Polynomials.
Suppose that origin (0,0) of the XY co-ordinate system and the centre of the unit circle are same.
Next, I take linear combination of these 24 polynomials to build a 2D wavefront shape. I use 24 random input coefficients in this combination.
w(x,y) = sum_over_i a_i*z_i (i=2,3,4,....24)
a_i = random coefficients
z_i = zernike polynomials
Upto this point, everything is analytical part which can be done on paper.
Now comes the discretization!
I know that when you want to re-construct a signal (1Dim/2Dim), your sampling frequency should be at least twice the maximum frequency present in the signal (Nyquist-Shanon principle).
Here signal is w(x,y) as mentioned above which is nothing but a simple 2Dim
function of x & y. I want to represent it on computer now. Obviously I can not take all infinite points from -1 to +1 along x axis and same for y axis.
I have to take finite no. of data points (which are called sample points or just samples) on this analytical 2Dim surface w(x,y)
I am measuring x & y in metres, and -1 <= x <= +1; -1 <= y <= +1.
e.g. If I divide my x-axis from -1 to 1, in 50 sample points then dx = 2/50= 0.04 metre. Same for y axis. Now my sampling frequency is 1/dx i.e. 25 samples per metre. Same for y axis.
But I took 50 samples arbitrarily; I could have taken 10 samples or 1000 samples. That is the crux of the matter here: how many samples points?How will I determine this number?
There is one theorem (Nyquist-Shanon theorem) mentioned above which says that if I want to re-construct w(x,y) faithfully, I must sample it on both axes so that my sampling frequency (i.e. no. of samples per metre) is at least twice the maximum frequency present in the w(x,y). This is nothing but finding power spectrum of w(x,y). Idea is that any function in space domain can be represented in spatial-frequency domain also, which is nothing but taking Fourier transform of the function! This tells us how many (spatial) frequencies are present in your function w(x,y) and what is the maximum frequency out of these many frequencies.
Now my question is first how to find out this maximum sampling frequency in my case. I can not use MATLAB fft2() or any other tool since it means already I have samples taken across the wavefront!! Obviously remaining option is find it analytically ! But that is time consuming and difficult since I have 24 polynomials & I will have to use then continuous Fourier transform i.e. I will have to go for pen and paper.
Any help will be appreciated.
Thanks
Key Assumptions
You want to use the "Nyquist-Shanon" theorem to determine sampling frequency
Obviously remaining option is find it analytically ! But that is time
consuming and difficult since I have 21 polynomials & I have to use
continuous Fourier transform i.e. done by analytically.
Given the assumption I have made (and noting that consideration of other mathematical techniques is out of scope for StackOverflow), you have no option but to calculate the continuous Fourier Transform.
However, I believe you haven't considered all the options for calculating the transform other than a laborious paper exercise e.g.
Numerical approximation of the continuous F.T. using code
Symbolic Integration e.g. Wolfram Alpha
Surely a numerical approximation of the Fourier Transform will be adequate for your solution?
I am assuming this is for coursework or research rather, so all you really care about as a physicist is a solution that is the quickest solution that is accurate within the scope of your problem.
So to conclude, IMHO, don't waste time searching for a more mathematically elegant solution or trick and just solve the problem with one of the above methods