Rearrange distribution function Matlab

Question

I have the following data representing values over a 12 month period:

1. 0
2. 253
3. 168
4. 323
5. 556
6. 470
7. 225
8. 445
9. 98
10. 114
11. 381
12. 187

How can I smooth this line forward?

What I need is that going through the list sequentially any value that is above the mean (268) be evenly distributed amongst the remaining months- but in such a way that it produces as smooth a line as possible. I need to go through from Jan to Dec in order. Looking forward I want to sweep any excess (peaks) into the months still to come such that the distribution is as even as possible (such that the troughs are filled first). So the issue is to, at each point, determine what the "excess" for that month is and secondly how to distribute that amongst the months still to come.

I have used

p = find(Diff>0);
n = find(Diff<=0);
POS = Diff(p,1);
NEG = Diff(n,1)

to see where shortfall/ excesses against the mean exist but unsure how to construct a code that will redistribute forward by allocating to the "troughs" of the distribution first. An analogy is that these numbers represent harvest quantities and I must give out the harvest to the population. How do I redistribute the incoming harvest over the year such that I minimise excess supply/ under supply? I obviously cannot give out anything I haven't received in a particular month unless I have stored some harvest from previous months. e.g. I start in Jan, I see that I cannot give anything to the months Feb to Dec so the value for Jan is 0. In Feb I have 253- do I adjust 253 downwards or give it all out? If so by how much? and where do I redistribute the excess I trim between Mar to Dec? And so on and so forth.. How do I do this to give as smooth (even) a distribution as possible? For any month the new value assigned to that month cannot exceed the current value. The sum for the 12 months must be equal before and after smoothing. As the first position January will always be 0.

Answer 1

Simple version, just loops through and if the next month is lower than the current month, passes value forward to equalise them.

for n = 1:11
    if y(n)>y(n+1);
       y(n:n+1)=(y(n)+y(n+1))/2;
    end
end

Answer 2

It's not very clear to me what you're asking...It sounds a bit like a roundabout way of asking how to fit a straight line to data. If that is the case, see below. Otherwise: please clarify a bit more what you want. Provide a toy example input data, and expected output data.

y = [ 0 253 168 323 556 470 225 445 98 114 381 187 ].';
x = (0:numel(y)-1).';

A = [ones(size(x)) x];

plot(...
    x,       y, 'b.',...
    x, A*(A\y), 'r')
xlabel('Month'), ylabel('Data')
legend('original data', 'fit')

Answer 3

I dont get exactly what you want either, maybe something simple like this?

year= [0 253 168 323 556 470 225 445 98 114 381 187];
m= mean(year);
total_before = sum(year)
linear_year = linspace(0,m*2,12);
toal_after= sum(linear_year)

this gives you a line, the sum stays the same and the line is perfectly smooth ...