function [K_opt,distance_opt] = kmeans_param_opt(Data,center)
%This function tries to find the sub-optimal number of K for a K-means
%Clustering and the best distance criteria for the data using both
%silhouette method and error 
% Inputs : 
% Data: to be clustered data, rows of the Data are the
% observations and the columns are the variables (features),
% center: 1 for normalizing data (suggested if feature's range are very different)
% Outputs : 
% K: Number of Clusters
% distance : a string to determine the distance method to use 

%% initializing
if center==1
    Data = zscore(Data);
end
K_max = round(size(Data,1)/4); % maximum number of clusters 
Distances_to_consider = {'sqeuclidean','correlation','cityblock','cosine'};
%% Finding Optimum k inside the same distance
% based on error
err_all = zeros([length(Distances_to_consider),K_max]);
err_all_normalized = zeros([length(Distances_to_consider),K_max]);
K_opt_distances = zeros(length(Distances_to_consider),1);
figure();sgtitle('Error vs. Number of Clusters');
for i=1:length(Distances_to_consider)
    distance = Distances_to_consider{i};
    for j=1:K_max
        [~,~,err] = kmeans(Data,j,'Replicates',10,'Start','plus','MaxIter',10000,'Distance',distance,'EmptyAction','drop');
        err_all(i,j) = sum(err);
    end
    err_all_normalized(i,:) = err_all(i,:)./max(err_all(i,:));
    std = stdfilt(err_all_normalized(i,:),ones(1,3));
    K_opt_distances(i) = find(std < 0.05,1,'first');
    subplot(2,ceil(length(Distances_to_consider)/2),i)
    plot(err_all(i,:));title(strcat('Distance: ',Distances_to_consider{i}));xline(K_opt_distances(i),'--r','Stopping Point');
    xlabel('Number of Clusters','FontSize',12);ylabel('Error','FontSize',12);
end

    
%% Comparing across distances
figure()
plot(err_all_normalized','LineWidth',2);xlabel('Number of Clusters','FontSize',12);ylabel('Normalized error','FontSize',12)
legend(Distances_to_consider,'Location','northeastoutside');legend('boxoff')

% based on silhouette
figure()
S = zeros(size(Data,1),length(Distances_to_consider));
for i=1:length(Distances_to_consider)
    distance = Distances_to_consider{i};
    [idx_clus,~,~] = kmeans(Data,K_opt_distances(i),'Replicates',10,'Start','plus','MaxIter',10000,'Distance',distance,'EmptyAction','drop');
    subplot(2,ceil(length(Distances_to_consider)/2),i);
    [S(:,i),h] = silhouette(Data,idx_clus,distance);title(strcat('Distance: ',distance));xline(mean(S(:,i)),'--r','Average');
end

dispersity = var(S);
[~,opt_idx] = min(dispersity);
K_opt = K_opt_distances(opt_idx);
distance_opt = Distances_to_consider{opt_idx};
    
end