Propotional growth in difference equations - difference-equations

Xn denoting the number of affected children after n weeks. Initially, 2 children were affected and 3 more reported affected in the first week of the outbreak.
I know X0 = 2 but I am confused with X1=?
X1 = 5 or X1 = 3 ??

If the original 2 are still affected, then X1=5; if not, X1=3.

Related

matlab - minimum euclidian distances between unique pais of points of two structures

I have this distance matrix between 6 points of one structure and 5 points of the second structure:
a = [2.565 0.394 2.927 2.774 1.600;
0.402 1.950 3.272 2.086 0.985;
2.965 3.250 1.720 0.841 2.305;
2.797 2.050 0.830 0.829 1.585;
3.865 2.662 1.246 2.086 2.634;
1.592 0.977 2.305 1.579 0.274]
I need the minimum distances between points. Sometimes I get one point between two points of the other structure. 0.274 0.394 0.402 0.830 0.829
This means I will get point 4 (from the 6 points structure) to be closest to points 3 and 4 from the other structure. I am not allowed to have one point closest to two others.
How do I get unique pairs of these close points?
I think I should verify if there is a small difference between first 2 minima in a row. The problematic point is always in the middle of other two.
I need to get 0.274 0.394 0.402 0.830 0.841 (see answer 1).
My original code was:
for i = 1 : 6
mins(i) = min(a(i, :));
end
mins = sort(mins);
mins = mins(1 : 5);
Thanks.
So thanks beaker, I do hope the question gets another answer,
[pairs,a1,a2]=matchpairs(a,1,'min')
pairs =
2 1
1 2
4 3
3 4
6 5
a1 =
5
a2 =
0×1 empty double column vector
Also I hope someone explains what the other outputs mean and when can they be useful.

How to get the maximal values and the related coordinates? [duplicate]

suppose that we are determine peaks in vector as follow:
we have real values one dimensional vector with length m,or
x(1),x(2),.....x(m)
if x(1)>x(2) then clearly for first point peak(1)=x(1);else we are then comparing x(3) to x(2),if x(3)
[ indexes,peaks]=function(x,m);
c=[];
b=[];
if x(1)>x(2)
peaks(1)=x(1);
else
for i=2:m-1
if x(i+1)< x(i) & x(i)>x(i-1)
peak(i)=x(i);
end;
end
end
end
peaks are determined also using following picture:
sorry for the second picture,maybe it is not triangle,just A and C are on straight line,but here peak is B,so i can't continue my code for writing algorithm to find peak values in my vector.please help me to continue it
updated.numercial example given
x=[2 1 3 5 4 7 6 8 9]
here because first point is more then second,so it means that peak(1)=2,then we are comparing 1 to 3,because 3 is more then 1,we now want to compare 5 to 3,it is also more,compare 5 to 4,because 5 is more then 4,then it means that peak(2)=5,,so if we continue next peak is 7,and final peak would be 9
in case of first element is less then second,then we are comparing second element to third one,if second is more then third and first elements at the same time,then peak is second,and so on
You could try something like this:
function [peaks,peak_indices] = find_peaks(row_vector)
A = [min(row_vector)-1 row_vector min(row_vector)-1];
j = 1;
for i=1:length(A)-2
temp=A(i:i+2);
if(max(temp)==temp(2))
peaks(j) = row_vector(i);
peak_indices(j) = i;
j = j+1;
end
end
end
Save it as find_peaks.m
Now, you can use it as:
>> A = [2 1 3 5 4 7 6 8 9];
>> [peaks, peak_indices] = find_peaks(A)
peaks =
2 5 7 9
peak_indices =
1 4 6 9
This would however give you "plateaus" as well (adjacent and equal "peaks").
You can use diff to do the comparison and add two points in the beginning and end to cover the border cases:
B=[1 diff(A) -1];
peak_indices = find(B(1:end-1)>=0 & B(2:end)<=0);
peaks = A(peak_indices);
It returns
peak_indices =
1 4 6 9
peaks =
2 5 7 9
for your example.
findpeaks does it if you have a recent matlab version, but it's also a bit slow.
This proposed solution would be quite slow due to the for loop, and you also have a risk of rounding error due to the fact that you compare the maximal value to the central one instead of comparing the position of the maximum, which is better for your purpose.
You can stack the data so as to have three columns : the first one for the preceeding value, the second is the data and the third one is the next value, do a max, and your local maxima are the points for which the position of the max along columns is 2.
I've coded this as a subroutine of my own peak detection function, that adds a further level of iterative peak detection
http://www.mathworks.com/matlabcentral/fileexchange/42927-find-peaks-using-scale-space-approach

k distinct combination of size p without replacement

From the integers 1,...,N I would like to take k random distinct combinations without repetition of size p. For example, if N=10, k=4 and p=3, a possible outcome would be:
1 4 9
9 4 2
3 5 2
1 8 4
But not:
1 4 9
9 4 2
3 5 3
1 9 4
For two reasons:
[1 4 9] and [1 9 4] are the same combination.
[3 5 3] is not without repetition.
Note that getting all possible combinations and the (randomly) picking k of them easily runs into memory problems.
Okay, I have found a solution that works for me. My main concerns were:
I want the k combinations to be random.
processing time.
The below function samples a single random combination of size p, (namely row = randperm(N,p)) each iteration and adds that combination if it isn't already present.
Of the three parameters, mainly k influences the processing time. For not too large k, this codes runs in matters of seconds. The most extreme case I myself will encounter is N = 10^6, k = 2000, p = 10 and it still runs in 1 second.
I hope this also helps other people, as I've come across this question on multiple sites, without a satisfactory answer.
function C = kcombsn(N,k,p)
C = randperm(N,p);
Csort = sort(C,2);
while size(C,1) < k
row = randperm(N,p);
row_sort = sort(row);
if isempty(intersect(row_sort,Csort,'rows'))
C = [C; row];
Csort = [Csort; row_sort];
end
end
end
Edit:
I also posted the code on the MATLAB File Exchange.

Remove duplicates appearing next to each other, but keep it if it appears again later

I have a vector that could look like this:
v = [1 1 2 2 2 3 3 3 3 2 2 1 1 1];
that is, the number of equal elements can vary, but they always increase and decrease stepwise by 1.
What I want is an easy way to be left with a new vector looking like this:
v2 = [ 1 2 3 2 1];
holding all the different elements (in the right order as they appear in v), but only one of each. Preferably without looping, since generally my vectors are about 10 000 elements long, and already inside a loop that's taking for ever to run.
Thank you so much for any answers!
You can use diff for this. All you're really asking for is: Delete any element that's equal to the one in front of it.
diff return the difference between all adjacent elements in a vector. If there is no difference, it will return 0. v(ind~=0) will give you all elements that have a value different than zero. The 1 in the beginning is to make sure the first element is counted. As diff returns the difference between elements, numel(diff(v)) = numel(v)-1.
v = [1 1 2 2 2 3 3 3 3 2 2 1 1 1];
ind = [1 diff(v)];
v(ind~=0)
ans =
1 2 3 2 1
This can of course be done in a single line if you want:
v([1, diff(v)]~=0)
You could try using diff which, for a vector X, returns [X(2)-X(1) X(3)-X(2) ... X(n)-X(n-1)] (type help diff for details on this function). Since the elements in your vector always increase or decrease by 1, then
diff(v)
will be a vector (of size one less than v) with zeros and ones where a one indicates a step up or down. We can ignore all the zeros as they imply repeated numbers. We can convert this to a logical array as
logical(diff(v))
so that we can index into v and access its elements as
v(logical(diff(v)))
which returns
1 2 3 2
This is almost what you want, just without the final number which can be added as
[v(logical(diff(v))) v(end)]
Try the above and see what happens!

How to construct a 128x32 scrambled matrix?

How can I construct a scrambled matrix with 128 rows and 32 columns in vb.net or Matlab?
Entries of the matrix are numbers between 1 and 32 with the condition that each row mustn't contain duplicate elements and rows mustn't be duplicates.
This is similar to #thewaywewalk's answer, but makes sure that the matrix has no repeated rows by testing if it does and in that case generating a new matrix:
done = 0;
while ~done
[~, matrix] = sort(rand(128,32),2);
%// generate each row as a random permutation, independently of other rows.
%// This line was inspired by randperm code
done = size(unique(matrix,'rows'),1) == 128;
%// in the event that there are repeated rows: generate matrix again
end
If my computations are correct, the probability that the matrix has repteated rows (and thus has to be generated again) is less than
>> 128*127/factorial(32)
ans =
6.1779e-032
Hey, it's more likely that a cosmic ray will spoil a given run of the program! So I guess you can safely remove the while loop :-)
With randperm you can generate one row:
row = randperm(32)
if this vector wouldn't be that long you could just use perms to find all permutations:
B = perms(randperm(32))
but it's memory-wise too much! ( 32! = 2.6313e+35 rows )
so you can use a little loop:
N = 200;
A = zeros(N,32);
for ii = 1:N
A(ii,:) = randperm(32);
end
B = unique(A, 'rows');
B = B(1:128,:);
For my tests it was sufficient to use N = 128 directly and skip the last two lines, because with 2.6313e+35 possibly permutations the probability that you get a correct matrix with the first try is very high. But to be sure that there are no row-duplicates choose a higher number and select the first 128 rows finally. In case the input vector is relatively short and the number of desired rows close to the total number of possible permutations use the proposed perms(randperm( n )).
small example for intergers from 1 to 4 and a selection of 10 out of 24 possible permutations:
N = 20;
A = zeros(N,4);
for ii = 1:N
A(ii,:) = randperm(4);
end
B = unique(A, 'rows');
B = B(1:10,:);
returns:
B =
1 2 3 4
1 2 4 3
1 3 4 2
2 3 1 4
2 3 4 1
2 4 1 3
2 4 3 1
3 1 2 4
3 1 4 2
3 2 1 4
some additional remarks for the choice of N:
I made some test runs, where I used the loop above to find all permutations like perms does. For vector lengths of n=4 to n=7 and in each case N = factorial(n): 60-80% of the rows are unique.
So for small n I would recommend to choose N as follows to be absolutely on the safe side:
N = min( [Q factorial(n)] )*2;
where Q is the number of permutations you want. For bigger n you either run out of memory while searching for all permutations, or the desired subset is so small compared to the number of all possible permutations that repetition is very unlikely! (Cosmic Ray theory linked by Luis Mendo)
Your requirements are very loose and allow many different possibilities. The most efficient solution I can think off that meets these requirements is as follows:
p = perms(1:6);
[p(1:128,:) repmat(7:32,128,1)]