How to get πx (steady-state probabilities of x?in order to plot these equations in MATLAB?

Sir,

I want to plot these attached equations using MATLAB. For plotting these equations, there are three variables. Out of which, I have values of two variables with me. So, I am looking for the value of the third variable 'πx'. How to get the values of πx (also called steady state probability of x). I have attached the screenshot of equations. Actually, if I get the values of πx, I would easily be able to plot the equations in MATLAB. I am unable to get the steady-state probabilities of x.

Please help

I have also attached the values of x along with these equations.

Regards

Surabhi

Whatever, the case is either simulation or plotting. I am confused on how to solve to get steady-state probabilities and hence plot the equations.

[Rik]

This doesn't look like a Matlab question to me. You should always try to understand the question itself before you start writing code.

Have a read here and here. It will greatly improve your chances of getting an answer.

[surabhi sachdeva]

Sir, the authors have written using MATLAB program they have found out the steady-state probability and then plotted the equations in MATLAB.

[KSSV]

In that case...you attach the output images..so that it can be clear to us.

[Jan]

@surabhi sachdeva: What exactly is your question? Which are "these" equations? You did not post the complete definition, but only a screen shot of two equations. But even if you post the full PDF, you cannot expect the readers of a forum to do your work. Please post, what you have tried so far and ask a specific question concerning Matlab.

Please do this by editing the question, not by adding a comment or pseudo-answer. Thanks.

If you consider the two links provided by Rik, your chances to get an answer will grow.

[Walter Roberson]

What would you expect the plot to look like? You have a 729 by 729 numeric array: are you looking to create a 729 x 729 image with the locations color-coded by numeric value? Are you looking to create a 729 x 729 surface with the z being the numeric value?

[surabhi sachdeva]

Sir,

thank you for your suggestion.

I have edited my question.

[surabhi sachdeva]

Sir,

I want to plot these attached equations using MATLAB. For plotting these equations, there are three variables. Out of which, I have values of two variables with me. So, I am looking for the value of the third variable 'πx'. How to get the values of πx (also called steady state probability of x). I have attached the screenshot of equations and the plot.

[Walter Roberson]

I already directed you to the steady-state formulas at https://www.mathworks.com/matlabcentral/answers/359900-how-to-calculate-steady-state-probability#answer_284483 . Use them to calculate your πx values and then you should be able to go on to the calculations to plot.

[surabhi sachdeva]

Sir, I am unable to understand how the transition matrix got defined in he link suggested by you?

how do I have to define the transition matrix?

How have they been marked the values as attached?

Could you suggest something for my 36X36 matrix?

I mean to ask do I have to individually check each of my rows and the column for the values?

How to apply the rules? How to get such values in the transition matrix?

[Walter Roberson]

You would create a matrix of the initial transition probabilities, and you would manipulate it in the way shown in order to arrive at the steady-state probabilities.

The transition matrix would be the usual: entry (J,K) says "Given that you are already in state J, what is the probability of transitioning to state K?". If you have algorithms that allow you to fill out that initial table, then you should use the algorithms.

[surabhi sachdeva]

Sir, I have state transition rules available to me, which I am not able to understand how to apply to the states(J, K)

Could you please give me a hint on how to apply this kind of rules to (J, K)

I have attached rules, kindly suggest something regarding how these rules to be applied.

Here (i,j,k,l,m,n) represents a state (J,K)

Each state is represented by (i,j,k,l,m,n) e.g. 101101, 101110,..... and many more.

As, if we say for all i<Np and there is a transition from (i,j,k,l,m,n) to (i+1, j, k,l,m,n) then it should be displayed 'λp'. How is this possible?

What code is to be written in MATLAB that automatically checks for the same type of transitions?

Kindly suggest

Answer 1

[Walter Roberson]

Constructing the state tables is easy if you use the 12D array that I said you should use. I already showed you how it could be done; it is frustrating that you did not carry through.

Np = 2;    %last state number for each entry, states are numbered from 0 to Np

%fill in the appropriate values for these probabilities. rand() is used to have _some_ value
lam_p = rand();    %lambda subscript p
gam_p = rand();    %gamma subscript p
gam_f = rand();    %gamma subscript f

%start building the table
Index = @(state_number) 1+state_number;    %helper to allow us to use state numbers instead of indices

Ns = Index(Np);  %number of states

TT = zeros( Ns * ones(1, 12) );   %transition table

case1_src_i = 0:Np-1;
TT(Index(case1_src_i), :, :, :, :, :, Index(case1_src_i+1), :, :, :, :, :) = lam_p;

case2_src_i = 1:Np;
case2_src_j = 0:Np-1;
case2_src_k = 1:Np;
TT(Index(case2_src_i), Index(case2_src_j), Index(case2_src_k), :, :, :, Index(case2_src_i-1), Index(case2_src_j+1), Index(case2_src_k-1), :, :, :) = gam_p;

case3_src_n = 0:Np-1;
case3a_src_j = 1:Np;
case3b_src_k = 1:Np;
case3c_src_l = 1:Np;

TT(:, Index(case3a_src_j), :, :, :, Index(case3_src_n), :, Index(case3a_src_j-1), :, :, :, Index(case3_src_n+1)) = gam_f;
TT(:, :, Index(case3b_src_k), :, :, Index(case3_src_n), :, :, Index(case3b_src_k-1), :, :, Index(case3_src_n+1)) = gam_f;
TT(:, :, :, Index(case3c_src_l), :, Index(case3_src_n), :, :, :, Index(case3c_src_l-1), :, Index(case3_src_n+1)) = gam_f;

T2 = reshape(TT, Ns^6, Ns^6);    %2D transition table

Here, T2 is the 729 x 729 array that you are expecting -- the one that you would proceed to use the numeric steady-state calculations on.

[surabhi sachdeva]

Sorry, sir for the frustration I gave.

And thank you so much for your help.

I will try it right now as said by you

Regards

Surabhi

[surabhi sachdeva]

Sir, Why have you used 12-D array? Can you please specify the reason?

[surabhi sachdeva]

and sir,

I tried all the possible ways but I am not getting the results as expected

not able to get the values of steady-state probabilities.

Please help me out.

Regards

Surabhi

[Walter Roberson]

Your rules are written in terms of (for example)

(i,j,k,l,m,n) to (i+1, j, k,l,m,n)

You have a source which is 6 dimensions long, and a destination that is 6 dimensions long, for a total of 12 dimensions. You can handle all of the (i,j,k,l,m,n) sources at once by using

(1:end-1,:,:,:,:,:)

in your source portion, and

(2:end,:,:,:,:,:)

as your destination

If you do not do this, if you encode the (i,j,k,l,m,n) into base 3 numbers from 0 to 728, then you need to figure out the decimal difference in indices between source and destination 6-tuples. For Case 1 and Case 2 that can be done as constants (same numeric difference for each situation within those two cases), but for the Case 3 situation, the difference in base 3 numbers pretty much needs to be calculated for each individual source because the numeric relationships between source and destination decimal representation are icky.

Using a 12-D representation makes the work easy and clear, and it is simple and fast to switch back to 2-D representation for final presentation.

[surabhi sachdeva]

Thank you sir for making me understand the use of 12-D array

[surabhi sachdeva]

Now, I am trying to code the leftover 3 cases(rules) given

case4_src_n = 1:Np-1;
case4a_src_i = 0;
case4b_src_i = 1:Np;
TT(:, Index(case4a_src_i), :, :, :, Index(case4_src_n), :, Index(case4a_src_k+1), :, :, :, Index(case4a_src_n-1)) = gam_r;
TT(:, :, Index(case4b_src_i), :, :, Index(case4_src_n), :, :, Index(case4b_src_i-1), :, :, Index(case4b_src_j+1)), Index(case4b_src_n-1) = gam_r;

Why is there an error Undefined function or variable 'case4a_src_k'.

I have to code the other 3 rules as well.

[Walter Roberson]

You wrote,

TT(:, Index(case4a_src_i), :, :, :, Index(case4_src_n), :, Index(case4a_src_k+1), :, :, :, Index(case4a_src_n-1)) = gam_r;

Notice this includes Index(case4a_src_k+1) but you did not define case4a_src_k .

I recommend that you re-examine the line, marking each field

     i            j          k  l  m           n          I
  TT(:, Index(case4a_src_i), :, :, :, Index(case4_src_n), :, 

               J         K  L  M            N
  Index(case4a_src_k+1), :, :, :, Index(case4a_src_n-1))

Your variable names disagree with the slot they correspond to.

[surabhi sachdeva]

Thanks, sir,

I will get back to you after coding all the rules in the way you taught me to do.

Thanks again.

Regards

Surabhi

[surabhi sachdeva]

Sir

Please see once

I have got this table.

I wanted to know how can I change the values of the input rows and columns from 1,2,3,4,........... to 100001,110011,...........?

Are these values in the file states is the initial input of the matrix?

I want the entities of the file states as an input for the matrix, How can it be done?

Kindly suggest

Regards Surabhi

[surabhi sachdeva]

Sir, I Request you to please help

[Walter Roberson]

dec2base(decimal_index - 1, 3, 6)

[Walter Roberson]

I do not know where that "states" matrix came from, or why it has duplicates. If for some reason you need to convert those to 2D index numbers, then

base2dec(states, 3) + 1

[surabhi sachdeva]

Sir, this doesn't work out. I am not getting the steady-state probabilities as expected. What can I do? I am feeling helpless, not able to draw the graphs.

Please help.

[Walter Roberson]

Could you remind me of the full set of rules? I think I've only seen 6 of them including the three case 3s.

Also please post your current code.

[surabhi sachdeva]

Sir, I want to tell you the complete scenario starting from the first.

Actually, I have been given a state-transition matrix Q having only 7 states as input. And it was mentioned it's just a part of state space and transition rates. Also, there are rules defined about these transitions.

I thought to generate the complete state space first. For which I used the MATLAB code and got 729 states with me.

Now, in order to generate the complete state transition matrix, I have to apply the transition rules to these 729 states I got. As from this state transition matrix only I will get the steady-state probabilities so as to plot the equations.

Firstly, sir I am confused do I have to generate the complete transition matrix for 729 elements or I have to use the Q (having 7 states) given to me? Can you help me regarding this?

Secondly, * If I have to generate the complete state transition matrix, please help me generating it for the entire 729 states.*

I have attached the sets of rules and the state space I got. I have also attached the state transition matrix which is given to me and I am expecting the same kind of transition rate matrix for the 729 elements.

Regards

Surabhi

Request you to also please suggest whether my idea of generating state space is right or do I have to use only the state space of 7 elements given to me?

[Walter Roberson]

You need to use the full 729 states.

You did not post your current code.

[surabhi sachdeva]

Currently, by using the code suggested by you, I made the code for 5 out of 6 cases given which I have attached.

I don't know whether I am going the right way or not? Please help.

I am expecting a 729 X729 matrix representation to be of the same kind as that of state space file I m uploading.

Please help me generating a 729X729 state transition matrix sir.

I would be very thankful.

Regards

Surabhi

[surabhi sachdeva]

Sir, Kindly reply.

Please help out.

Regards

Surabhi

[Walter Roberson]

I haven't even had breakfast yet.

[surabhi sachdeva]

I am so sorry, sir

Please continue

Whenever you get free kindly respond

Regards

Surabhi

[surabhi sachdeva]

Sir,

Kindly help me in the coding. I have attached the code file as well.

I want to use all the 729 states generated in the form of state transition matrix file I uploaded before as well.

 Please help in writing the code for the same.

Regards

Surabhi

[surabhi sachdeva]

Sir,

I need your help. Request you to kindly help me out solving the problem.

Regards

Surabhi

[surabhi sachdeva]

@Walter Roberson sir,

You have not responded. Request you to please help me.

Regards

Surabhi