K means finding elbow when the elbow plot is a smooth curve

Question

I am trying to plot the elbow of k means using the below code:

load CSDmat %mydata
for k = 2:20
    opts = statset('MaxIter', 500, 'Display', 'off');
    [IDX1,C1,sumd1,D1] = kmeans(CSDmat,k,'Replicates',5,'options',opts,'distance','correlation');% kmeans matlab
    [yy,ii] = min(D1');      %% assign points to nearest center

    distort = 0;
    distort_across = 0;
    clear clusts;
    for nn=1:k
        I = find(ii==nn);       %% indices of points in cluster nn
        J = find(ii~=nn);       %% indices of points not in cluster nn
        clusts{nn} = I;         %% save into clusts cell array
        if (length(I)>0)
            mu(nn,:) = mean(CSDmat(I,:));               %% update mean
            %% Compute within class distortion
            muB = repmat(mu(nn,:),length(I),1);
            distort = distort+sum(sum((CSDmat(I,:)-muB).^2));
            %% Compute across class distortion
            muB = repmat(mu(nn,:),length(J),1);
            distort_across = distort_across + sum(sum((CSDmat(J,:)-muB).^2));
        end
    end
    %% Set distortion as the ratio between the within
    %% class scatter and the across class scatter
    distort = distort/(distort_across+eps);

        bestD(k)=distort;
        bestC=clusts;
end
figure; plot(bestD);

The values of bestD (within cluster variance/between cluster variance) are

[
0.401970132754914
0.193697163350293
0.119427184084282
0.0872681777446508
0.0687948264457301
0.0566215549396577
0.0481117619129058
0.0420491551659459
0.0361696583755145
0.0320384092689509
0.0288948343304147
0.0262373245283877
0.0239462330460614
0.0218350896369853
0.0201506779033703
0.0186757121130685
0.0176258625858971
0.0163239661159014
0.0154933431470081
]

The code is adapted from Lihi Zelnik-Manor, March 2005, Caltech.

The plot ratio of within cluster variance to between cluster variance is a smooth curve with a knee that is smooth like a curve, plot bestD data given above. How do we find the knee for such graphs?

Answer 1

I think that it is better to use only your "within class distortion" as optimization parameter:

%% Compute within class distortion
muB = repmat(mu(nn,:),length(I),1);
distort = distort+sum(sum((CSDmat(I,:)-muB).^2));

Use this without dividing this value by "distort_across". If you calculate the "derivate" of this:

unexplained_error = within_class_distortion;
derivative = diff(unexplained_error);
plot(derivative)

The derivative(k) tells you how much the unexplained error has decreased by adding a new cluster. I suggest that you stop adding clusters when the decrease on this error is less than ten times the first decrease you obtained.

for (i=1:length(derivative))
    if (derivative(i) < derivative(1)/10)
         break
    end
end
k_opt = i+1;

In fact the method to obtain the optimum number of clusters is application dependent, but I think that you can obtain a good value of k using this suggestion.