diff --git a/chapter03/linPred.m b/chapter03/linPred.m
index c159a54..1ebc439 100644
--- a/chapter03/linPred.m
+++ b/chapter03/linPred.m
@@ -7,18 +7,17 @@
 w = model.w;
 w0 = model.w0;
 y = w'*X+w0;
+
 if nargout > 1
     beta = model.beta;
-    if isfield(model,'U')   % 3.54
-        U = model.U;
-        Xo = bsxfun(@minus,X,model.xbar);
-        XU = U'\Xo;
-        sigma = sqrt(1/beta+dot(XU,XU,1));
-    else
-        sigma = sqrt(1/beta);
-    end
-    if nargin == 3 && nargout == 3
-        p = exp(logGauss(t,y,sigma));
-%         p = exp(-0.5*(((t-y)./sigma).^2+log(2*pi))-log(sigma));
-    end
+    U = model.U;        % 3.54
+    Xo = bsxfun(@minus,X,model.xbar);
+    XU = U'\Xo;
+    sigma = sqrt(1/beta+dot(XU,XU,1));
 end
+
+if nargin == 3 && nargout == 3
+    p = exp(logGauss(t,y,sigma));
+%     p = exp(-0.5*(((t-y)./sigma).^2+log(2*pi))-log(sigma));
+end
+
diff --git a/chapter06/demo.m b/chapter06/demo.m
index a9d0d07..dac4237 100755
--- a/chapter06/demo.m
+++ b/chapter06/demo.m
@@ -4,10 +4,29 @@
 x = linspace(0,2*pi,n);   % test data
 t = sin(x)+rand(1,n)/2;
 
-model = knReg(x,t,1e-4,@knGauss);
-y = knRegPred(model, x);
-figure;
-hold on;
-plot(x,t,'o');
-plot(x,y,'r-');
+% model = knReg(x,t,1e-4,@knGauss);
+% y = knRegPred(model, x);
+% figure;
+% hold on;
+% plot(x,t,'o');
+% plot(x,y,'r-');
 
+
+%% test case for kernel regression
+lambda = 1e-4;
+model_kn = knReg(x,t,lambda,@knLin);
+model_lin = linReg(x,t,lambda);
+
+[y_kn, s_kn] = knRegPred(model_kn, x);
+[y_lin, s_lin] = linPred(model_lin,x);
+
+maxabsdiff(y_kn,y_lin)
+maxabsdiff(s_kn,s_lin)
+%% test case for knCenter
+% kn = @knGauss;
+% X=rand(2,100);
+% X1=rand(2,10);
+% X2=rand(2,5);
+% 
+% isequalf(knCenter(kn,X,X1),diag(knCenter(kn,X,X1,X1)))
+% isequalf(knCenter(kn,X),knCenter(kn,X,X,X));
\ No newline at end of file
diff --git a/chapter06/knPca.m b/chapter06/knPca.m
new file mode 100644
index 0000000..35aa93a
--- /dev/null
+++ b/chapter06/knPca.m
@@ -0,0 +1,6 @@
+function [ output_args ] = knPca( input_args )
+% Kernel PCA
+
+
+end
+
diff --git a/chapter06/knReg.m b/chapter06/knReg.m
index 1f24fe2..d0ae640 100644
--- a/chapter06/knReg.m
+++ b/chapter06/knReg.m
@@ -18,4 +18,5 @@
 model.a = a;
 model.X = X;
 model.tbar = tbar;
-model.beta = beta;
\ No newline at end of file
+model.beta = beta;
+model.U = U;
\ No newline at end of file
diff --git a/chapter06/knRegPred.m b/chapter06/knRegPred.m
index 1d7e058..a2de44c 100755
--- a/chapter06/knRegPred.m
+++ b/chapter06/knRegPred.m
@@ -5,21 +5,15 @@
 a = model.a;
 X = model.X;
 tbar = model.tbar;
-y = a'*knCenterize(kn,X,Xt)+tbar;
+y = a'*knCenter(kn,X,X,Xt)+tbar;
+
 if nargout > 1
     beta = model.beta;
-    if isfield(model,'U')
-        U = model.U;
-        Xo = bsxfun(@minus,X,model.xbar);
-        XU = U'\Xo;
-        sigma = sqrt(1/beta+dot(XU,XU,1));
-        
-        sigma = sqrt(c-k'*C^-1*k);
-    else
-        sigma = sqrt(1/beta);   % 6.67
-    end
-    if nargin == 3 && nargout == 3
-        p = exp(logGauss(t,y,sigma));
-    end
+    U = model.U;
 
+    sigma = sqrt(knCenter(kn,X,Xt)+1/beta-dot(XU,XU,1));
 end
+
+if nargin == 3 && nargout == 3
+    p = exp(logGauss(t,y,sigma));
+end
\ No newline at end of file