This Question is in continuation to a previous one asked Matlab : Plot of entropy vs digitized code length
I want to calculate the entropy of a random variable that is discretized version (0/1) of a continuous random variable x. The random variable denotes the state of a nonlinear dynamical system called as the Tent Map. Iterations of the Tent Map yields a time series of length N.
The code should exit as soon as the entropy of the discretized time series becomes equal to the entropy of the dynamical system. It is known theoretically that the entropy of the system is log_2(2). The code exits but the frst 3 values of the entropy array are erroneous - entropy(1) = 1, entropy(2) = NaN and entropy(3) = NaN. I am scratching my head as to why this is happening and how I can get rid of it. Please help in correcting the code. THank you.
clear all
H = log(2)
threshold = 0.5;
x(1) = rand;
lambda(1) = 1;
entropy(1,1) = 1;
j=2;
tol=0.01;
while(~(abs(lambda-H)<tol))
if x(j - 1) < 0.5
x(j) = 2 * x(j - 1);
else
x(j) = 2 * (1 - x(j - 1));
end
s = (x>=threshold);
p_1 = sum(s==1)/length(s);
p_0 = sum(s==0)/length(s);
entropy(:,j) = -p_1*log2(p_1)-(1-p_1)*log2(1-p_1);
lambda = entropy(:,j);
j = j+1;
end
plot( entropy )
It looks like one of your probabilities is zero. In that case, you'd be trying to calculate 0*log(0) = 0*-Inf = NaN. The entropy should be zero in this case, so you you can just check for this condition explicitly.
Couple side notes: It looks like you're declaring H=log(2), but your post says the entropy is log_2(2). p_0 is always 1 - p_1, so you don't have to count everything up again. Growing the arrays dynamically is inefficient because matlab has to re-copy the entire contents at each step. You can speed things up by pre-allocating them (only worth it if you're going to be running for many timesteps).
Related
The problem:
If a large number of fair N-sided dice are rolled, the average of the simulated rolls is likely to be close to the mean of 1,2,...N i.e. the expected value of one die. For example, the expected value of a 6-sided die is 3.5.
Given N, simulate 1e8 N-sided dice rolls by creating a vector of 1e8 uniformly distributed random integers. Return the difference between the mean of this vector and the mean of integers from 1 to N.
My code:
function dice_diff = loln(N)
% the mean of integer from 1 to N
A = 1:N
meanN = sum(A)/N;
% I do not have any idea what I am doing here!
V = randi(1e8);
meanvector = V/1e8;
dice_diff = meanvector - meanN;
end
First of all, make sure everytime you ask a question that it is as clear as possible, to make it easier for other users to read.
If you check how randi works, you can see this:
R = randi(IMAX,N) returns an N-by-N matrix containing pseudorandom
integer values drawn from the discrete uniform distribution on 1:IMAX.
randi(IMAX,M,N) or randi(IMAX,[M,N]) returns an M-by-N matrix.
randi(IMAX,M,N,P,...) or randi(IMAX,[M,N,P,...]) returns an
M-by-N-by-P-by-... array. randi(IMAX) returns a scalar.
randi(IMAX,SIZE(A)) returns an array the same size as A.
So, if you want to use randi in your problem, you have to use it like this:
V=randi(N, 1e8,1);
and you need some more changes:
function dice_diff = loln(N)
%the mean of integer from 1 to N
A = 1:N;
meanN = mean(A);
V = randi(N, 1e8,1);
meanvector = mean(V);
dice_diff = meanvector - meanN;
end
For future problems, try using the command
help randi
And matlab will explain how the function randi (or other function) works.
Make sure to check if the code above gives the desired result
As pointed out, take a closer look at the use of randi(). From the general case
X = randi([LowerInt,UpperInt],NumRows,NumColumns); % UpperInt > LowerInt
you can adapt to dice rolling by
Rolls = randi([1 NumSides],NumRolls,NumSamplePaths);
as an example. Exchanging NumRolls and NumSamplePaths will yield Rolls.', or transpose(Rolls).
According to the Law of Large Numbers, the updated sample average after each roll should converge to the true mean, ExpVal (short for expected value), as the number of rolls (trials) increases. Notice that as NumRolls gets larger, the sample mean converges to the true mean. The image below shows this for two sample paths.
To get the sample mean for each number of dice rolls, I used arrayfun() with
CumulativeAvg1 = arrayfun(#(jj)mean(Rolls(1:jj,1)),[1:NumRolls]);
which is equivalent to using the cumulative sum, cumsum(), to get the same result.
CumulativeAvg1 = (cumsum(Rolls(:,1))./(1:NumRolls).'); % equivalent
% MATLAB R2019a
% Create Dice
NumSides = 6; % positive nonzero integer
NumRolls = 200;
NumSamplePaths = 2;
% Roll Dice
Rolls = randi([1 NumSides],NumRolls,NumSamplePaths);
% Output Statistics
ExpVal = mean(1:NumSides);
CumulativeAvg1 = arrayfun(#(jj)mean(Rolls(1:jj,1)),[1:NumRolls]);
CumulativeAvgError1 = CumulativeAvg1 - ExpVal;
CumulativeAvg2 = arrayfun(#(jj)mean(Rolls(1:jj,2)),[1:NumRolls]);
CumulativeAvgError2 = CumulativeAvg2 - ExpVal;
% Plot
figure
subplot(2,1,1), hold on, box on
plot(1:NumRolls,CumulativeAvg1,'b--','LineWidth',1.5,'DisplayName','Sample Path 1')
plot(1:NumRolls,CumulativeAvg2,'r--','LineWidth',1.5,'DisplayName','Sample Path 2')
yline(ExpVal,'k-')
title('Average')
xlabel('Number of Trials')
ylim([1 NumSides])
subplot(2,1,2), hold on, box on
plot(1:NumRolls,CumulativeAvgError1,'b--','LineWidth',1.5,'DisplayName','Sample Path 1')
plot(1:NumRolls,CumulativeAvgError2,'r--','LineWidth',1.5,'DisplayName','Sample Path 2')
yline(0,'k-')
title('Error')
xlabel('Number of Trials')
I'm trying to verify if my implementation of Logistic Regression in Matlab is good. I'm doing so by comparing the results I get via my implementation with the results given by the built-in function mnrfit.
The dataset D,Y that I have is such that each row of D is an observation in R^2 and the labels in Y are either 0 or 1. Thus, D is a matrix of size (n,2), and Y is a vector of size (n,1)
Here's how I do my implementation:
I first normalize my data and augment it to include the offset :
d = 2; %dimension of data
M = mean(D) ;
centered = D-repmat(M,n,1) ;
devs = sqrt(sum(centered.^2)) ;
normalized = centered./repmat(devs,n,1) ;
X = [normalized,ones(n,1)];
I will be doing my calculations on X.
Second, I define the gradient and hessian of the likelihood of Y|X:
function grad = gradient(w)
grad = zeros(1,d+1) ;
for i=1:n
grad = grad + (Y(i)-sigma(w'*X(i,:)'))*X(i,:) ;
end
end
function hess = hessian(w)
hess = zeros(d+1,d+1) ;
for i=1:n
hess = hess - sigma(w'*X(i,:)')*sigma(-w'*X(i,:)')*X(i,:)'*X(i,:) ;
end
end
with sigma being a Matlab function encoding the sigmoid function z-->1/(1+exp(-z)).
Third, I run the Newton algorithm on gradient to find the roots of the gradient of the likelihood. I implemented it myself. It behaves as expected as the norm of the difference between the iterates goes to 0. I wrote it based on this script.
I verified that the gradient at the wOPT returned by my Newton implementation is null:
gradient(wOP)
ans =
1.0e-15 *
0.0139 -0.0021 0.2290
and that the hessian has strictly negative eigenvalues
eig(hessian(wOPT))
ans =
-7.5459
-0.0027
-0.0194
Here's the wOPT I get with my implementation:
wOPT =
-110.8873
28.9114
1.3706
the offset being the last element. In order to plot the decision line, I should convert the slope wOPT(1:2) using M and devs. So I set :
my_offset = wOPT(end);
my_slope = wOPT(1:d)'.*devs + M ;
and I get:
my_slope =
1.0e+03 *
-7.2109 0.8166
my_offset =
1.3706
Now, when I run B=mnrfit(D,Y+1), I get
B =
-1.3496
1.7052
-1.0238
The offset is stored in B(1).
I get very different values. I would like to know what I am doing wrong. I have some doubt about the normalization and 'un-normalization' process. But I'm not sure, may be I'm doing something else wrong.
Additional Info
When I tape :
B=mnrfit(normalized,Y+1)
I get
-1.3706
110.8873
-28.9114
which is a rearranged version of the opposite of my wOPT. It contains exactly the same elements.
It seems likely that my scaling back of the learnt parameters is wrong. Otherwise, it would have given the same as B=mnrfit(D,Y+1)
I wanna find a root for the following function with an error less than 0.05%
f= 3*x*tan(x)=1
In the MatLab i've wrote that code to do so:
clc,close all
syms x;
x0 = 3.5
f= 3*x*tan(x)-1;
df = diff(f,x);
while (1)
x1 = 1 / 3*tan(x0)
%DIRV.. z= tan(x0)^2/3 + 1/3
er = (abs((x1 - x0)/x1))*100
if ( er <= 0.05)
break;
end
x0 = x1;
pause(1)
end
But It keeps running an infinite loop with error 200.00 I dunno why.
Don't use while true, as that's usually uncalled for and prone to getting stuck in infinite loops, like here. Simply set a limit on the while instead:
while er > 0.05
%//your code
end
Additionally, to prevent getting stuck in an infinite loop you can use an iteration counter and set a maximum number of iterations:
ItCount = 0;
MaxIt = 1e5; %// maximum 10,000 iterations
while er > 0.05 & ItCount<MaxIt
%//your code
ItCount=ItCount+1;
end
I see four points of discussion that I'll address separately:
Why does the error seemingly saturate at 200.0 and the loop continue infinitely?
The fixed-point iterator, as written in your code, is finding the root of f(x) = x - tan(x)/3; in other words, find a value of x at which the graphs of x and tan(x)/3 cross. The only point where this is true is 0. And, if you look at the value of the iterants, the value of x1 is approaching 0. Good.
The bad news is that you are also dividing by that value converging toward 0. While the value of x1 remains finite, in a floating point arithmetic sense, the division works but may become inaccurate, and er actually goes NaN after enough iterations because x1 underflowed below the smallest denormalized number in the IEEE-754 standard.
Why is er 200 before then? It is approximately 200 because the value of x1 is approximately 1/3 of the value of x0 since tan(x)/3 locally behaves as x/3 a la its Taylor Expansion about 0. And abs(1 - 3)*100 == 200.
Divisions-by-zero and relative orders-of-magnitude are why it is sometimes best to look at the absolute and relative error measures for both the values of the independent variable and function value. If need be, even putting an extremely (relatively) small finite, constant value in the denominator of the relative calculation isn't entirely a bad thing in my mind (I remember seeing it in some numerical recipe books), but that's just a band-aid for robustness's sake that typically hides a more serious error.
This convergence is far different compared to the Newton-Raphson iterations because it has absolutely no knowledge of slope and the fixed-point iteration will converge to wherever the fixed-point is (forgive the minor tautology), assuming it does converge. Unfortunately, if I remember correctly, fixed-point convergence is only guaranteed if the function is continuous in some measure, and tan(x) is not; therefore, convergence is not guaranteed since those pesky poles get in the way.
The function it appears you want to find the root of is f(x) = 3*x*tan(x)-1. A fixed-point iterator of that function would be x = 1/(3*tan(x)) or x = 1/3*cot(x), which is looking for the intersection of 3*tan(x) and 1/x. However, due to point number (2), those iterators still behave badly since they are discontinuous.
A slightly different iterator x = atan(1/(3*x)) should behave a lot better since small values of x will produce a finite value because atan(x) is continuous along the whole real line. The only drawback is that the domain of x is limited to the interval (-pi/2,pi/2), but if it converges, I think the restriction is worth it.
Lastly, for any similar future coding endeavors, I do highly recommend #Adriaan's advice. If would like a sort of compromise between the styles, most of my iterative functions are written with a semantic variable notDone like this:
iter = 0;
iterMax = 1E4;
tol = 0.05;
notDone = 0.05 < er & iter < iterMax;
while notDone
%//your code
iter = iter + 1;
notDone = 0.05 < er & iter < iterMax;
end
You can add flags and all that jazz, but that format is what I frequently use.
I believe that the code below achieves what you are after using Newton's method for the convergence. Please leave a comment if I have missed something.
% find x: 3*x*tan(x) = 1
f = #(x) 3*x*tan(x)-1;
dfdx = #(x) 3*tan(x)+3*x*sec(x)^2;
tolerance = 0.05; % your value?
perturbation = 1e-2;
converged = 1;
x = 3.5;
f_x = f(x);
% Use Newton s method to find the root
count = 0;
err = 10*tolerance; % something bigger than tolerance to start
while (err >= tolerance)
count = count + 1;
if (count > 1e3)
converged = 0;
disp('Did not converge.');
break;
end
x0 = x;
dfdx_x = dfdx(x);
if (dfdx_x ~= 0)
% Avoid division by zero
f_x = f(x);
x = x - f_x/dfdx_x;
else
% Perturb x and go back to top of while loop
x = x + perturbation;
continue;
end
err = (abs((x - x0)/x))*100;
end
if (converged)
disp(['Converged to ' num2str(x,'%10.8e') ' in ' num2str(count) ...
' iterations.']);
end
I have the following Markov chain:
This chain shows the states of the Spaceship, which is in the asteroid belt: S1 - is serviceable, S2 - is broken. 0.12 - the probability of destroying the Spaceship by a collision with an asteroid. 0.88 - the probability of that a collision will not be critical. Need to find the probability of a serviceable condition of the ship after the third collision.
Analytical solution showed the response - 0.681. But it is necessary to solve this problem by simulation method using any modeling tool (MATLAB Simulink, AnyLogic, Scilab, etc.).
Do you know what components should be used to simulate this process in Simulink or any other simulation environment? Any examples or links.
First, we know the three step probability transition matrix contains the answer (0.6815).
% MATLAB R2019a
P = [0.88 0.12;
0 1];
P3 = P*P*P
P(1,1) % 0.6815
Approach 1: Requires Econometrics Toolbox
This approach uses the dtmc() and simulate() functions.
First, create the Discrete Time Markov Chain (DTMC) with the probability transition matrix, P, and using dtmc().
mc = dtmc(P); % Create the DTMC
numSteps = 3; % Number of collisions
You can get one sample path easily using simulate(). Pay attention to how you specify the initial conditions.
% One Sample Path
rng(8675309) % for reproducibility
X = simulate(mc,numSteps,'X0',[1 0])
% Multiple Sample Paths
numSamplePaths = 3;
X = simulate(mc,numSteps,'X0',[numSamplePaths 0]) % returns a 4 x 3 matrix
The first row is the X0 row for the starting state (initial condition) of the DTMC. The second row is the state after 1 transition (X1). Thus, the fourth row is the state after 3 transitions (collisions).
% 50000 Sample Paths
rng(8675309) % for reproducibility
k = 50000;
X = simulate(mc,numSteps,'X0',[k 0]); % returns a 4 x 50000 matrix
prob_survive_3collisions = sum(X(end,:)==1)/k % 0.6800
We can bootstrap a 95% Confidence Interval on the mean probability to survive 3 collisions to get 0.6814 ± 0.00069221, or rather, [0.6807 0.6821], which contains the result.
numTrials = 40;
ProbSurvive_3collisions = zeros(numTrials,1);
for trial = 1:numTrials
Xtrial = simulate(mc,numSteps,'X0',[k 0]);
ProbSurvive_3collisions(trial) = sum(Xtrial(end,:)==1)/k;
end
% Mean +/- Halfwidth
alpha = 0.05;
mean_prob_survive_3collisions = mean(ProbSurvive_3collisions)
hw = tinv(1-(0.5*alpha), numTrials-1)*(std(ProbSurvive_3collisions)/sqrt(numTrials))
ci95 = [mean_prob_survive_3collisions-hw mean_prob_survive_3collisions+hw]
maxNumCollisions = 10;
numSamplePaths = 50000;
ProbSurvive = zeros(maxNumCollisions,1);
for numCollisions = 1:maxNumCollisions
Xc = simulate(mc,numCollisions,'X0',[numSamplePaths 0]);
ProbSurvive(numCollisions) = sum(Xc(end,:)==1)/numSamplePaths;
end
For a more complex system you'll want to use Stateflow or SimEvents, but for this simple example all you need is a single Unit Delay block (output = 0 => S1, output = 1 => S2), with a Switch block, a Random block, and some comparison blocks to construct the logic determining the next value of the state.
Presumably you must execute the simulation a (very) large number of times and average the results to get a statistically significant output.
You'll need to change the "seed" of the random generator each time you run the simulation.
This can be done by setting the seed to be "now" (or something similar to that).
Alternatively you could quite easily vectorize the model so that you only need to execute it once.
If you want to simulate this, it is fairly easy in matlab:
servicable = 1;
t = 0;
while servicable =1
t = t+1;
servicable = rand()<=0.88
end
Now t represents the amount of steps before the ship is broken.
Wrap this in a for loop and you can do as many simulations as you like.
Note that this can actually give you the distribution, if you want to know it after 3 times, simply add && t<3 to the while condition.
I have two signals, let's call them 'a' and 'b'. They are both nearly identical signals (recorded from the same input and contain the same information) however, because I recorded them at two different 'b' is time shifted by an unknown amount. Obviously, there is random noise in each.
Currently, I am using cross correlation to compute the time shift, however, I am still getting improper results.
Here is the code I am using to calculate the time shift:
function [ diff ] = FindDiff( signal1, signal2 )
%FINDDIFF Finds the difference between two signals of equal frequency
%after an appropritate time shift is applied
% Calculates the time shift between two signals of equal frequency
% using cross correlation, shifts the second signal and subtracts the
% shifted signal from the first signal. This difference is returned.
length = size(signal1);
if (length ~= size(signal2))
error('Vectors must be equal size');
end
t = 1:length;
tx = (-length+1):length;
x = xcorr(signal1,signal2);
[mx,ix] = max(x);
lag = abs(tx(ix));
shifted_signal2 = timeshift(signal2,lag);
diff = signal1 - shifted_signal2;
end
function [ shifted ] = timeshift( input_signal, shift_amount )
input_size = size(input_signal);
shifted = (1:input_size)';
for i = 1:input_size
if i <= shift_amount
shifted(i) = 0;
else
shifted(i) = input_signal(i-shift_amount);
end
end
end
plot(FindDiff(a,b));
However the result from the function is a period wave, rather than random noise, so the lag must still be off. I would post an image of the plot, but imgur is currently not cooperating.
Is there a more accurate way to calculate lag other than cross correlation, or is there a way to improve the results from cross correlation?
Cross-correlation is usually the simplest way to determine the time lag between two signals. The position of peak value indicates the time offset at which the two signals are the most similar.
%// Normalize signals to zero mean and unit variance
s1 = (signal1 - mean(signal1)) / std(signal1);
s2 = (signal2 - mean(signal2)) / std(signal2);
%// Compute time lag between signals
c = xcorr(s1, s2); %// Cross correlation
lag = mod(find(c == max(c)), length(s2)) %// Find the position of the peak
Note that the two signals have to be normalized first to the same energy level, so that the results are not biased.
By the way, don't use diff as a name for a variable. There's already a built-in function in MATLAB with the same name.
Now there are two functions in Matlab:
one called finddelay
and another called alignsignals that can do what you want, I believe.
corr finds a dot product between vectors (v1, v2). If it works bad with your signal, I'd try to minimize a sum of squares of differences (i.e. abs(v1 - v2)).
signal = sin(1:100);
signal1 = [zeros(1, 10) signal];
signal2 = [signal zeros(1, 10)];
for i = 1:length(signal1)
signal1shifted = [signal1 zeros(1, i)];
signal2shifted = [zeros(1, i) signal2];
d2(i) = sum((signal1shifted - signal2shifted).^2);
end
[fval lag2] = min(d2);
lag2
It is computationally worse than cross-calculation which can be speeded up by using FFT. As far as I know you can't do this with euclidean distance.
UPD. Deleted wrong idea about cross-correlation with periodic signals
You can try matched filtering in frequency domain
function [corr_output] = pc_corr_processor (target_signal, ref_signal)
L = length(ref_signal);
N = length(target_signal);
matched_filter = flipud(ref_signal')';
matched_filter_Res = fft(matched_filter,N);
corr_fft = matched_filter_Res.*fft(target_signal);
corr_out = abs(ifft(corr_fft));
The peak of the matched filter maximum-index of corr_out above should give you the lag amount.