added more spacing entropy estimators

fastlib/u/niche/entropy/jackknifed_m_spacing.m

@@ -0,0 +1,22 @@
+% jackknifed_m_spacing() - get an unbiased estimate of entropy
+% using jackknifed m-spacing
+function H = jackknifed_m_spacing(n);
+
+% the vanilla m-spacing estimator for reference
+%{
+sum_logs = 0;
+for i = 1:n
+  if (i + m) > n
+    sum_logs = sum_logs + log( (n/(2*m)) * (Z(n) - Z(i-m)));
+  elseif (i-m) < 1
+    sum_logs = sum_logs + log( (n/(2*m)) * (Z(i+m) - Z(1)));
+  else    
+    sum_logs = sum_logs + log( (n/(2*m)) * (Z(i+m) - Z(i-m)));
+  end
+end
+%}
+
+H = 1;
+
+% consider leaving out i
+

fastlib/u/niche/entropy/kee/entropy_l2e.tex

@@ -1,4 +1,3 @@
-%\documentclass{amsart}
 \documentclass{article}
 \usepackage{amsmath, amsthm}
 \usepackage[colorlinks=true]{hyperref}
@@ -118,7 +117,7 @@
   \section{Expanded $L_2E$ objective function}
 
   \begin{displaymath}
-    \frac{1}{\vert Q \vert} \sum_{q \in Q} \left( T_{\log} \left( \frac{1}{N} \sum_{i=1}^n K_h(d(q,x_i)) \right) \right) ^2 - 2 \sum_{i=1}^n \log \hat{f}_{h_E,-i}(x_i) \hat{f}_{h,-i}(x_i) \log \hat{f}_{h,-i}(x_i)
+    \arg \min_h \frac{1}{\vert Q \vert} \sum_{q \in Q} \left( T_{\log} \left( \frac{1}{N} \sum_{i=1}^n K_h(d(q,x_i)) \right) \right) ^2 - 2 \sum_{i=1}^n \log \hat{f}_{h_E,-i}(x_i) \hat{f}_{h,-i}(x_i) \log \hat{f}_{h,-i}(x_i)
   \end{displaymath}
 
   where $ T_{\log}(p) = p \log(p) $ and $Q$ is a set of sample points on the range of $X$ (e.g. linear spacing of $M$ points)

fastlib/u/niche/entropy/m_spacing.m

@@ -1,15 +1,45 @@
 % m_spacing() - estimate entropy using m-spacing
-function H = m_spacing(X, m);
+function H = m_spacing(n);
+
+randn('seed', sum(100*clock));
+
+X = normrnd(zeros(n, 1), 1);
+
+m = round(sqrt(n)/2);
 
-X = X / std(X);
 
-N = length(X);
 
 Z = sort(X, 'ascend');
 
-sum = 0;
-for i = 1 : (N - m)
-  sum = sum + log2( (N / m) * (Z(i + m) - Z(i)) );
+sum_logs = 0;
+%for i = (m+1):(n-m)
+for i = 1:n
+  if (i + m) > n
+    sum_logs = sum_logs + log( (n/(2*m)) * (Z(n) - Z(i-m)));
+  elseif (i-m) < 1
+    sum_logs = sum_logs + log( (n/(2*m)) * (Z(i+m) - Z(1)));
+  else    
+    sum_logs = sum_logs + log( (n/(2*m)) * (Z(i+m) - Z(i-m)));
+  end
 end
 
-H = sum / N;
+H = sum_logs / n;
+%H = sum_logs / (n-2*m);
+
+% Vasicek's bias correction
+
+sum_psi = 0;
+for i=1:m
+  sum_psi = sum_psi + psi(i+m-1);
+end
+
+bias = ...
+    + log(n) ...
+    - log(2*m) ...
+    + (1 - 2*m/n) * psi(2*m) ...
+    - psi(n+1) ...
+    + (2/n) * sum_psi;
+
+
+
+H = H - bias;

fastlib/u/niche/entropy/test_m_spacing.m

@@ -0,0 +1,11 @@
+function mean_H_minus_opt = test_m_spacing(num_trials, n);
+
+opt=log(sqrt(2*pi*exp(1)));
+
+h = zeros(num_trials,1);
+
+for i=1:num_trials
+  h(i) = m_spacing(n);
+end
+
+mean_H_minus_opt = mean(h) - opt;