added section on cumulative densities

statistics/code/cumulative.m

@@ -0,0 +1,18 @@
+x = randn(200, 1);                    % generate some data
+xs = sort(x);                         % sort the data
+cdf = [1:length(x)]/length(x);        % cumulative
+plot(xs, cdf);
+hold on;
+
+dx = 0.01;
+xx = [-4:dx:4];                     % x-values for Gaussian pdf
+gauss = exp(-0.5*xx.^2)/sqrt(2.0*pi);  % Gaussian pdf
+gausscdf = cumsum(gauss)*dx;
+plot(xx, gausscdf);
+hold off;
+
+printf('data : probability of x<-1: %.2f\n', cdf(xs<-1.0)(end))
+printf('gauss: probability of x<-1: %.2f\n', gausscdf(xx<-1.0)(end))
+printf('\n')
+printf('data : 5%% percentile at %.2f\n', xs(cdf<0.05)(end))
+printf('gauss: 5%% percentile at %.2f\n', xx(gausscdf<0.05)(end))

statistics/code/gaussianbins.m

@@ -1,11 +1,11 @@
 x = randn(100, 1);   % generate some data
+db1 = 2;             % large bin width
+db2 = 0.5;           % small bin width
+bins1 = -4:db1:4;    % large bins
+bins2 = -4:db2:4;    % small bins
+[h1,b1] = hist(x, bins1);
+[h2,b2] = hist(x, bins2);
 
-bins1 = -4:2:4;      % large bins
-bins2 = -4:0.5:4;    % small bins
-[h1,b1] = hist(x,bins1);
-[h2,b2] = hist(x,bins2);
-
-subplot( 1, 2, 1 );
 bar(b1, h1)
 hold on 
 bar(b2, h2, 'facecolor', 'r' )

statistics/code/gaussianbinsnorm.m

@@ -1,9 +1,9 @@
 hn1 = h1/sum(h1)/db1;
 hn2 = h2/sum(h2)/db2;
-subplot(  1, 2, 2 )
-bar(b1,hn1)
+
+bar(b1, hn1)
 hold on 
-bar(b2,hn2, 'facecolor', 'r' )
+bar(b2, hn2, 'facecolor', 'r' )
 xlabel('x')
 ylabel('Probability density')
 hold off

statistics/code/gaussiankerneldensity.m

@@ -0,0 +1,42 @@
+data = randn(100, 1);            % generate some data
+sigma = 0.2;                     % standard deviation of Gaussian kernel
+xmin = -4.0;                     % minimum x value for kernel density
+xmax = 4.0;                      % maximum x value for kernel density
+dx = 0.05*sigma;                 % step size for kernel density
+xg = [-4.0*sigma:dx:4.0*sigma];  % x-axis for single Gaussian kernel  
+% single Gaussian kernel:
+kernel = exp(-0.5*(xg/sigma).^2)/sqrt(2.0*pi)/sigma;
+ng = (length(kernel)-1)/2;           % half the length of the Gaussian
+x = [xmin:dx:xmax+0.5*dx];       % x-axis for kernel density
+kd = zeros(1, length(x));        % vector for kernel density
+for i = 1:length(data)           % for every data value ...                
+  xd = data(i);
+  % index of data value in kernel density vector:
+  inx = round((xd-xmin)/dx)+1;
+  % start index for Gaussian in kernel density vector:
+  k0 = inx-ng;
+  % end index for Gaussian in kernel density vector:
+  k1 = inx+ng;
+  g0 = 1;                        % start index in Gaussian
+  g1 = length(kernel);            % end index in Gaussian
+  % check whether left side of Gaussian extends below xmin:
+  if inx < ng+1
+    % adjust start indices accordingly:
+    k0 = 1;
+    g0 = ng-inx+1;
+  end
+  % check whether right side of Gaussian extends above xmax:
+  if inx > length(kd)-ng
+    % adjust end indices accordingly:
+    k1 = length(kd);
+    g1 = length(kernel)-(inx+ng-length(kd));
+  end
+  % add Gaussian on kernel density:
+  kd(k0:k1) = kd(k0:k1) + kernel(g0:g1);
+end
+kd /= length(data);               % normalize by number of data points
+
+% plot kernel density:
+plot(x, kd)
+xlabel('x')
+ylabel('Probability density')