getting a NaN in correlation coefficient

Hi, i have a simple problem which unfortunately i am unable to understand.

I have matrices and i am trying to calculate correlation coefficient between two variables. A simple example from my code is attched. Why am i getting a NaN here. What does this implies

x=[-7.501899598769999514e-04;-6.501899598769999514e-04;-5.501899598769999514e-04];
y=[-0.414;-0.414;-0.414];
c11=corr2(x,y)

Answer 1

[Adam Danz]

When NaNs appear in the output but are not present in the inputs

Notice that all of the values in y are identical y=[-0.414; -0.414; -0.414];

If you look at the equations for corr2() or Pearson's corr() you'll notice that both have a term in the denominator that subtracts the mean of y from each y-value. When each value of y is identical, the result is a vector of 0s. When you divide by zero, you get NaN.

Another way of putting it, the standard deviation of x or y cannot be 0. When you have a vector of identical values, the std is 0.

The NaN, in this case, is interpretted as no correlation between the two variables. The correlation describes how much one variable changes as the other variable changes. That requires both variables to change.

NaN values in the inputs spreading to the outputs

For r=corr2(x,y):

When there is 1 or more NaN values in the inputs, to corr2(x,y), the output will be NaN. Fill in the missing data before computing the 2D correlation coefficient.

For r=corr(x):

A single NaN value in position (i,j) of input matrix x will result in a full row of NaN values at row i and a full column of NaN values in column j of the output matrix r (see explanation).

x = [
     6     5     1
     3   NaN     9
     5     3     7
     9     5     5 ];
 
 r = corr(x)

            1          NaN     -0.52699
          NaN          NaN          NaN
     -0.52699          NaN            1

For r=corr(x,y):

A single NaN value in position (i,j) of either x or y inputs will results in a column of NaN values in column j of the output matrix r.

x = [
     9     5     1
     1     4     4
     2     6     4
     2     5     9 ];
y = [
     6     5     1
     3   NaN     9   
     5     3     7
     9     5     5 ];
 
 r = corr(x,y)

       0.1623          NaN     -0.92394
       0.3266          NaN     -0.23905
      0.62312          NaN      0.32367

Ignoring NaNs in corr() inputs

The rows option in corr() can be set to complete or pairwise which will ignore NaN values using different methods.

'rows','complete' removes the entire row if the row contains a NaN. In other words, it will remove row 2 from both x and y input matrices. Using the same inputs above,

r = corr(x,y,'rows','complete')

     -0.27735          0.5     -0.94491
     -0.69338           -1      0.75593
      0.81224      0.14286      0.53995
      
      
r2 = corr(x,y) % for comparison

       0.1623          NaN     -0.92394
       0.3266          NaN     -0.23905
      0.62312          NaN      0.32367

Notice that this changes all of the correlation values since the entire row #2 was removed from both inputs x and y. To confirm that, we can remove those rows and recompute the correlation matrix.

% Remove row 2 which contains a NaN in y
r3 = corr(x([1,3,4],:) ,y([1,3,4],:));  

     -0.27735          0.5     -0.94491
     -0.69338           -1      0.75593
      0.81224      0.14286      0.53995

Voila! Outputs r and r3 match.

'rows','pairwise' only removes rows only if a NaN appears in the pairing of two columns. For the same x, y inputs as above, the correlation with columns in x paired with the 2nd column in y will omit the NaN and will be based on the remaining 3 values. All other column-paired correlations will use all 4 rows of values.

r = corr(x,y,'rows','pairwise')

       0.1623          0.5     -0.92394
       0.3266           -1     -0.23905
      0.62312      0.14286      0.32367
      
      
r2 = corr(x,y) % for comparison

       0.1623          NaN     -0.92394
       0.3266          NaN     -0.23905
      0.62312          NaN      0.32367

Notice that values in columns 1 and 3 haven't changed since they do not involve column #2 in y. To confirm the correlation values in column 2 of r,

% Remove row 2 which contains a NaN in y
r3 = corr(x([1,3,4],:) ,y([1,3,4],:)); 
% Replace NaN column in r2 with new r values
r2(:,2) = r3(:,2)

       0.1623          0.5     -0.92394
       0.3266           -1     -0.23905
      0.62312      0.14286      0.32367

Voila! Updated output r2 matches r.

[Sumera Yamin]

I see it now. Many thanks for your explaination. In your last statement you said that for this case, NaN implies no correlation. It means i can take the value 0 for correlation coefficient for furthur processing of the results, as my results are being constant for some of my cases

[Adam Danz]

It depends on what you're doing with those values.

If you're using the correlation values to classify data sets based on whether or not they correlate, sure, you could treat the NaN as 0.

If you're reporting the correlation values or if you're using them on a continuous scale, it would probably be problematic to replace the NaN value with a value of 0.

Perhaps a better way to interpret a NaN output is "not interpretable" or "not meaningful" whereas a significant correlation of 0 means "no correlation".

[Sumera Yamin]

i am trying to find correlation between a large set of variables and then trying to represent the dependence using correlation coefficient matrix. One strange thing i observed in my study is that if the values of variables doen not change, i gives a NaN, which effectively means no correlation, however, if there is very small differnce, the correlation coefficient is near 1 deoicting strong correlation, although i would assume it to be having weak correlation with values near to zero not to one. I compare the three cases in example below.

x=[-7.501899598769999514e-04;-6.501899598769999514e-04;-5.501899598769999514e-04];
y=[-0.414;-0.414;-0.414];
z=[0.571;0.57;0.571];
t=[4.54;4.59;4.55];
s=[4.39;4.40;4.42];
cy=corr2(x,y)  %no change
cz=corr2(x,z)  % very small but symmetric change w.r.t central value. central value is design value for all cases
ct=corr2(x,t)  % significant change
cs=corr2(x,s)  % small change

%However the correlation coefficients respectively are NaN,0,0.189 and 0.982. Although from data, i would expect them to be 0, near to 0, close to 1 and between 0-5 respectively. Any idea why is there this apparent discrepancy between what i expect by looking at data and what i get from computation?

[Adam Danz]

Why are you using corr2() with column vectors? Use corr() instead.

[rho,pval] = corr(X,Y) returns the p-value for each rho showing the significance of the correlation.

I'm not surprised by these results. Keep in mind you only have 3 data points for each comparison.

x=[-7.501899598769999514e-04;-6.501899598769999514e-04;-5.501899598769999514e-04];
y=[-0.414;-0.414;-0.414];
z=[0.571;0.57;0.571];
t=[4.54;4.59;4.55];
s=[4.39;4.40;4.42];
[cy, py]=corr(x,y)  %no change
[cz, pz]=corr(x,z)  % very small but symmetric change w.r.t central value. central value is design value for all cases
[ct, pt]=corr(x,t)  % significant change
[cs, ps]=corr(x,s)  % small change

tiledlayout(2,2)
nexttile
plot(x,y,'ro')
lsline
title(sprintf('x y r=%.3f p=%.3f', cy, py))

nexttile
plot(x,z,'ro')
lsline
title(sprintf('x z r=%.3f p=%.3f', cz, pz))

nexttile
plot(x,t,'ro')
lsline
title(sprintf('x t r=%.3f p=%.3f', ct, pt))

nexttile
plot(x,s,'ro')
lsline
title(sprintf('x s r=%.3f p=%.3f', cs, ps))

[Sumera Yamin]

Many thanks for your detail answer. When i am running the above example, MATLAB gives me an error. I am using MATLAB R2017a version. I am unable to understand whats the cause of this error

"Undefined function or variable

'tiledlayout'.

Error in ex (line 10)

tiledlayout(2,2)"

[Adam Danz]

If you look at the bottom of the tiledlayout documentation page you'll see that this function wasn't released until r2019b.

Use subplot() instead of tiledlayout & nexttile.

[Sumera Yamin]

i see...Thanks a lot

[Sumera Yamin]

Hi, sorry but the difference between rho and p-values is not very clear to me. If i understand correctly, while using corr p-value is the actual pearson correlation coefficient (value lying between -1,0,1) actually showing the amount of correlation and not the rho values? so what does rho specifies? unfortunately, matlab help does not give much information.

[Adam Danz]

No. The first output is the correlation coefficient and the 2nd output is the p-value indicating the statistical significance of the correlation.

Check out the descriptions of the outputs here

https://www.mathworks.com/help/stats/corr.html#d118e240043

[Sumera Yamin]

many thanks for this excellent explaination. I understand it very well now.

Answer 2

[Jan Pokorny]

Hi, I tried this funkcion as bellow, but it returns all NaN. I don't know why, any idea?

I then lately solved this by making cov() and then corrcov(), but it shouldn't go like that.

Data=[3 0 2 0 3 11 7 6 6 25 31 22 25 48 109 85 67 110 205 124 158 114 126 185 291 259 373 262 160 184 307 281 269 332 282 115 235 195 295];


for Zp=0:21
    X=Data(1,1+Zp:39);
    Y=Data(1,1:39-Zp);
    RMat=corr(X,Y);
    Koef(1+Zp)=RMat(1,2);   
end


plot(0:Zp,Koef,'red*-')

[Adam Danz]

From the documentation,

rho = corr(X,Y) returns a matrix of the pairwise correlation coefficient 
between each pair of columns in the input matrices X and Y.

Note the word columns. You're using rows.

Instead,

RMat=corr(X(:),Y(:))