35
36
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- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs027.html b/doc/pub/week40/html/._week40-bs027.html
index 6735ab09f..8c4ec00c6 100644
--- a/doc/pub/week40/html/._week40-bs027.html
+++ b/doc/pub/week40/html/._week40-bs027.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -368,7 +373,7 @@ Note that if n is equal to zero or one, Autograd will give an error message. Thi
36
37
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs028.html b/doc/pub/week40/html/._week40-bs028.html
index f2c9db7c1..97093b17b 100644
--- a/doc/pub/week40/html/._week40-bs028.html
+++ b/doc/pub/week40/html/._week40-bs028.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -356,7 +361,7 @@ Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The
37
38
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs029.html b/doc/pub/week40/html/._week40-bs029.html
index b189849d4..11787e2bc 100644
--- a/doc/pub/week40/html/._week40-bs029.html
+++ b/doc/pub/week40/html/._week40-bs029.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -372,7 +377,7 @@ x = np.a
38
39
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs030.html b/doc/pub/week40/html/._week40-bs030.html
index 59c9a3ace..3d982c2d5 100644
--- a/doc/pub/week40/html/._week40-bs030.html
+++ b/doc/pub/week40/html/._week40-bs030.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -343,7 +348,7 @@ a /=b
39
40
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs031.html b/doc/pub/week40/html/._week40-bs031.html
index 21071f93c..8169e7caf 100644
--- a/doc/pub/week40/html/._week40-bs031.html
+++ b/doc/pub/week40/html/._week40-bs031.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -341,7 +346,7 @@ MathJax.Hub.Config({
40
41
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs032.html b/doc/pub/week40/html/._week40-bs032.html
index 32867cb5f..a84fc9046 100644
--- a/doc/pub/week40/html/._week40-bs032.html
+++ b/doc/pub/week40/html/._week40-bs032.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -344,7 +349,7 @@ a weight variable.
41
42
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs033.html b/doc/pub/week40/html/._week40-bs033.html
index bbe8e6bbb..fddc6c90d 100644
--- a/doc/pub/week40/html/._week40-bs033.html
+++ b/doc/pub/week40/html/._week40-bs033.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -393,7 +398,7 @@ humanities to life science and medicine.
42
43
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- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs034.html b/doc/pub/week40/html/._week40-bs034.html
index 7f7d92c76..e69309a2c 100644
--- a/doc/pub/week40/html/._week40-bs034.html
+++ b/doc/pub/week40/html/._week40-bs034.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -357,7 +362,7 @@ methods we discussed earlier.
43
44
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs035.html b/doc/pub/week40/html/._week40-bs035.html
index a3d5f876e..0d40035d2 100644
--- a/doc/pub/week40/html/._week40-bs035.html
+++ b/doc/pub/week40/html/._week40-bs035.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -348,7 +353,7 @@ to all nodes in the subsequent layer, making this a so-called
44
45
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs036.html b/doc/pub/week40/html/._week40-bs036.html
index 6482ababa..ab801c790 100644
--- a/doc/pub/week40/html/._week40-bs036.html
+++ b/doc/pub/week40/html/._week40-bs036.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -356,7 +361,7 @@ recognition.
45
46
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs037.html b/doc/pub/week40/html/._week40-bs037.html
index 4f0d5ba0d..a6b22fa13 100644
--- a/doc/pub/week40/html/._week40-bs037.html
+++ b/doc/pub/week40/html/._week40-bs037.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -347,7 +352,7 @@ especially well-suited for handwriting and speech recognition.
46
47
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs038.html b/doc/pub/week40/html/._week40-bs038.html
index 8eb1cbed8..94a6a5a07 100644
--- a/doc/pub/week40/html/._week40-bs038.html
+++ b/doc/pub/week40/html/._week40-bs038.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -347,7 +352,7 @@ type of NN due the unusual activation functions.
47
48
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs039.html b/doc/pub/week40/html/._week40-bs039.html
index ded8c2126..46e00c957 100644
--- a/doc/pub/week40/html/._week40-bs039.html
+++ b/doc/pub/week40/html/._week40-bs039.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -344,7 +349,7 @@ Such networks are often called multilayer perceptrons (MLPs).
48
49
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs040.html b/doc/pub/week40/html/._week40-bs040.html
index 514b8e7b1..e190e6231 100644
--- a/doc/pub/week40/html/._week40-bs040.html
+++ b/doc/pub/week40/html/._week40-bs040.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -348,7 +353,7 @@ as to not restrict the range of output values.
49
50
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs041.html b/doc/pub/week40/html/._week40-bs041.html
index 9bb833c66..bd57b40f1 100644
--- a/doc/pub/week40/html/._week40-bs041.html
+++ b/doc/pub/week40/html/._week40-bs041.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -342,7 +347,7 @@ MathJax.Hub.Config({
50
51
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs042.html b/doc/pub/week40/html/._week40-bs042.html
index 9889fb4f2..7d0e250be 100644
--- a/doc/pub/week40/html/._week40-bs042.html
+++ b/doc/pub/week40/html/._week40-bs042.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,20 +312,47 @@ MathJax.Hub.Config({
-Mathematical model
+Examples of XOR, OR and AND gates
-The output \( y \) is produced via the activation function \( f \)
-$$
- y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
-$$
+Let us first try to fit various gates using standard linear regression
-This function receives \( x_i \) as inputs.
-Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \).
-In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
-the neurons in the preceding layer. Furthermore, an MLP is
-fully-connected, which means that each neuron receives a weighted sum
-of the outputs of all neurons in the previous layer.
+
+
+
+
"""
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
+
+import numpy as np
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX matrix:{Xinv}")
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}")
+
+What is happening here?
@@ -348,7 +380,7 @@ of the outputs of all neurons in the previous layer.
51
52
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs043.html b/doc/pub/week40/html/._week40-bs043.html
index c67cbfa3e..9a45820fb 100644
--- a/doc/pub/week40/html/._week40-bs043.html
+++ b/doc/pub/week40/html/._week40-bs043.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -310,46 +315,17 @@ MathJax.Hub.Config({
Mathematical model
-First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
-
+The output \( y \) is produced via the activation function \( f \)
$$
-\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
-\tag{7}
-\end{equation}
+ y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z),
$$
-
-Here \( b_i \) is the so-called bias which is normally needed in
-case of zero activation weights or inputs. How to fix the biases and
-the weights will be discussed below. The value of \( z_i^1 \) is the
-argument to the activation function \( f_i \) of each node \( i \), The
-variable \( M \) stands for all possible inputs to a given node \( i \) in the
-first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
-
-$$
-\begin{equation}
- y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
-\tag{8}
-\end{equation}
-$$
-
-
-where we assume that all nodes in the same layer have identical
-activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
-In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
-
-$$
-\begin{equation}
- y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
-\tag{9}
-\end{equation}
-$$
-
-
-where \( N_l \) is the number of nodes in layer \( l \). When the output of
-all the nodes in the first hidden layer are computed, the values of
-the subsequent layer can be calculated and so forth until the output
-is obtained.
+This function receives \( x_i \) as inputs.
+Here the activation \( z=(\sum_{i=1}^n w_ix_i+b_i) \).
+In an FFNN of such neurons, the inputs \( x_i \) are the outputs of
+the neurons in the preceding layer. Furthermore, an MLP is
+fully-connected, which means that each neuron receives a weighted sum
+of the outputs of all neurons in the previous layer.
@@ -377,7 +353,7 @@ is obtained.
52
53
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs044.html b/doc/pub/week40/html/._week40-bs044.html
index 1c1863bae..b2d9ba859 100644
--- a/doc/pub/week40/html/._week40-bs044.html
+++ b/doc/pub/week40/html/._week40-bs044.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -310,29 +315,47 @@ MathJax.Hub.Config({
Mathematical model
-The output of neuron \( i \) in layer 2 is thus,
+First, for each node \( i \) in the first hidden layer, we calculate a weighted sum \( z_i^1 \) of the input coordinates \( x_j \),
$$
-\begin{align}
- y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
-\tag{10}\\
- &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
-\tag{11}
-\end{align}
+\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1
+\tag{7}
+\end{equation}
$$
-where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
+
+Here \( b_i \) is the so-called bias which is normally needed in
+case of zero activation weights or inputs. How to fix the biases and
+the weights will be discussed below. The value of \( z_i^1 \) is the
+argument to the activation function \( f_i \) of each node \( i \), The
+variable \( M \) stands for all possible inputs to a given node \( i \) in the
+first layer. We define the output \( y_i^1 \) of all neurons in layer 1 as
$$
-\begin{align}
- y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
-\tag{12}\\
- &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
- + b_1^3\right]
-\tag{13}
-\end{align}
+\begin{equation}
+ y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right)
+\tag{8}
+\end{equation}
$$
+
+where we assume that all nodes in the same layer have identical
+activation functions, hence the notation \( f \). In general, we could assume in the more general case that different layers have different activation functions.
+In this case we would identify these functions with a superscript \( l \) for the \( l \)-th layer,
+
+$$
+\begin{equation}
+ y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right)
+\tag{9}
+\end{equation}
+$$
+
+
+where \( N_l \) is the number of nodes in layer \( l \). When the output of
+all the nodes in the first hidden layer are computed, the values of
+the subsequent layer can be calculated and so forth until the output
+is obtained.
+
@@ -359,7 +382,7 @@ $$
53
54
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs045.html b/doc/pub/week40/html/._week40-bs045.html
index c7df4af1c..3288b959a 100644
--- a/doc/pub/week40/html/._week40-bs045.html
+++ b/doc/pub/week40/html/._week40-bs045.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -310,19 +315,28 @@ MathJax.Hub.Config({
Mathematical model
-We can generalize this expression to an MLP with \( l \) hidden
-layers. The complete functional form is,
+The output of neuron \( i \) in layer 2 is thus,
$$
\begin{align}
-&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
-\tag{14}
+ y_i^2 &= f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right)
+\tag{10}\\
+ &= f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right]
+\tag{11}
\end{align}
$$
-
-which illustrates a basic property of MLPs: The only independent
-variables are the input values \( x_n \).
+where we have substituted \( y_k^1 \) with the inputs \( x_k \). Finally, the ANN output reads
+
+$$
+\begin{align}
+ y_i^3 &= f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right)
+\tag{12}\\
+ &= f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right)
+ + b_1^3\right]
+\tag{13}
+\end{align}
+$$
@@ -350,7 +364,7 @@ variables are the input values \( x_n \).
54
55
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs046.html b/doc/pub/week40/html/._week40-bs046.html
index fb58e96e4..4e2fc4540 100644
--- a/doc/pub/week40/html/._week40-bs046.html
+++ b/doc/pub/week40/html/._week40-bs046.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -310,28 +315,19 @@ MathJax.Hub.Config({
Mathematical model
-This confirms that an MLP, despite its quite convoluted mathematical
-form, is nothing more than an analytic function, specifically a
-mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
-\hat{y} \in \mathbb{R}^m \).
-
-
-Furthermore, the flexibility and universality of an MLP can be
-illustrated by realizing that the expression is essentially a nested
-sum of scaled activation functions of the form
+We can generalize this expression to an MLP with \( l \) hidden
+layers. The complete functional form is,
$$
-\begin{equation}
- f(x) = c_1 f(c_2 x + c_3) + c_4
-\tag{15}
-\end{equation}
+\begin{align}
+&y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] &&
+\tag{14}
+\end{align}
$$
-where the parameters \( c_i \) are weights and biases. By adjusting these
-parameters, the activation functions can be shifted up and down or
-left and right, change slope or be rescaled which is the key to the
-flexibility of a neural network.
+which illustrates a basic property of MLPs: The only independent
+variables are the input values \( x_n \).
@@ -359,7 +355,7 @@ flexibility of a neural network.
55
56
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs047.html b/doc/pub/week40/html/._week40-bs047.html
index ad3135f18..2667b4442 100644
--- a/doc/pub/week40/html/._week40-bs047.html
+++ b/doc/pub/week40/html/._week40-bs047.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,42 +312,32 @@ MathJax.Hub.Config({
-Matrix-vector notation
+Mathematical model
-We can introduce a more convenient notation for the activations in an A NN.
+This confirms that an MLP, despite its quite convoluted mathematical
+form, is nothing more than an analytic function, specifically a
+mapping of real-valued vectors \( \hat{x} \in \mathbb{R}^n \rightarrow
+\hat{y} \in \mathbb{R}^m \).
-Additionally, we can represent the biases and activations
-as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
-is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
+Furthermore, the flexibility and universality of an MLP can be
+illustrated by realizing that the expression is essentially a nested
+sum of scaled activation functions of the form
-
-We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
-With this notation, the sum becomes a matrix-vector multiplication, and we can write
-the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
$$
\begin{equation}
- \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
- f_2\left(\left[\begin{array}{ccc}
- w^2_{11} &w^2_{12} &w^2_{13} \\
- w^2_{21} &w^2_{22} &w^2_{23} \\
- w^2_{31} &w^2_{32} &w^2_{33} \\
- \end{array} \right] \cdot
- \left[\begin{array}{c}
- y^1_1 \\
- y^1_2 \\
- y^1_3 \\
- \end{array}\right] +
- \left[\begin{array}{c}
- b^2_1 \\
- b^2_2 \\
- b^2_3 \\
- \end{array}\right]\right).
-\tag{16}
+ f(x) = c_1 f(c_2 x + c_3) + c_4
+\tag{15}
\end{equation}
$$
+
+where the parameters \( c_i \) are weights and biases. By adjusting these
+parameters, the activation functions can be shifted up and down or
+left and right, change slope or be rescaled which is the key to the
+flexibility of a neural network.
+
@@ -369,7 +364,7 @@ $$
56
57
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs048.html b/doc/pub/week40/html/._week40-bs048.html
index 6ecf9de0b..3d4e9de95 100644
--- a/doc/pub/week40/html/._week40-bs048.html
+++ b/doc/pub/week40/html/._week40-bs048.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,26 +312,42 @@ MathJax.Hub.Config({
-Matrix-vector notation and activation
+Matrix-vector notation
-The activation of node \( i \) in layer 2 is
+We can introduce a more convenient notation for the activations in an A NN.
+
+Additionally, we can represent the biases and activations
+as layer-wise column vectors \( \hat{b}_l \) and \( \hat{y}_l \), so that the \( i \)-th element of each vector
+is the bias \( b_i^l \) and activation \( y_i^l \) of node \( i \) in layer \( l \) respectively.
+
+
+We have that \( \mathrm{W}_l \) is an \( N_{l-1} \times N_l \) matrix, while \( \hat{b}_l \) and \( \hat{y}_l \) are \( N_l \times 1 \) column vectors.
+With this notation, the sum becomes a matrix-vector multiplication, and we can write
+the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as
$$
\begin{equation}
- y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
- f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
-\tag{17}
+ \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) =
+ f_2\left(\left[\begin{array}{ccc}
+ w^2_{11} &w^2_{12} &w^2_{13} \\
+ w^2_{21} &w^2_{22} &w^2_{23} \\
+ w^2_{31} &w^2_{32} &w^2_{33} \\
+ \end{array} \right] \cdot
+ \left[\begin{array}{c}
+ y^1_1 \\
+ y^1_2 \\
+ y^1_3 \\
+ \end{array}\right] +
+ \left[\begin{array}{c}
+ b^2_1 \\
+ b^2_2 \\
+ b^2_3 \\
+ \end{array}\right]\right).
+\tag{16}
\end{equation}
$$
-
-This is not just a convenient and compact notation, but also a useful
-and intuitive way to think about MLPs: The output is calculated by a
-series of matrix-vector multiplications and vector additions that are
-used as input to the activation functions. For each operation
-\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
-
@@ -353,7 +374,7 @@ used as input to the activation functions. For each operation
57
58
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs049.html b/doc/pub/week40/html/._week40-bs049.html
index c6c7306d1..d6c422b75 100644
--- a/doc/pub/week40/html/._week40-bs049.html
+++ b/doc/pub/week40/html/._week40-bs049.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,21 +312,27 @@ MathJax.Hub.Config({
-Activation functions
+Matrix-vector notation and activation
-A property that characterizes a neural network, other than its
-connectivity, is the choice of activation function(s). As described
-in, the following restrictions are imposed on an activation function
-for a FFNN to fulfill the universal approximation theorem
+The activation of node \( i \) in layer 2 is
-
- - Non-constant
- - Bounded
- - Monotonically-increasing
- - Continuous
-
+$$
+\begin{equation}
+ y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) =
+ f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right).
+\tag{17}
+\end{equation}
+$$
+
+This is not just a convenient and compact notation, but also a useful
+and intuitive way to think about MLPs: The output is calculated by a
+series of matrix-vector multiplications and vector additions that are
+used as input to the activation functions. For each operation
+\( \mathrm{W}_l \hat{y}_{l-1} \) we move forward one layer.
+
+
diff --git a/doc/pub/week40/html/._week40-bs050.html b/doc/pub/week40/html/._week40-bs050.html
index 0a2d9f4c7..b36ce15e3 100644
--- a/doc/pub/week40/html/._week40-bs050.html
+++ b/doc/pub/week40/html/._week40-bs050.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,29 +312,21 @@ MathJax.Hub.Config({
-Activation functions, Logistic and Hyperbolic ones
+Activation functions
-The second requirement excludes all linear functions. Furthermore, in
-a MLP with only linear activation functions, each layer simply
-performs a linear transformation of its inputs.
+A property that characterizes a neural network, other than its
+connectivity, is the choice of activation function(s). As described
+in, the following restrictions are imposed on an activation function
+for a FFNN to fulfill the universal approximation theorem
-
-Regardless of the number of layers, the output of the NN will be
-nothing but a linear function of the inputs. Thus we need to introduce
-some kind of non-linearity to the NN to be able to fit non-linear
-functions Typical examples are the logistic Sigmoid
+
+ - Non-constant
+ - Bounded
+ - Monotonically-increasing
+ - Continuous
+
-$$
- f(x) = \frac{1}{1 + e^{-x}},
-$$
-
-and the hyperbolic tangent function
-$$
- f(x) = \tanh(x)
-$$
-
-
diff --git a/doc/pub/week40/html/._week40-bs051.html b/doc/pub/week40/html/._week40-bs051.html
index 196f285e3..d9e173b0c 100644
--- a/doc/pub/week40/html/._week40-bs051.html
+++ b/doc/pub/week40/html/._week40-bs051.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,90 +312,28 @@ MathJax.Hub.Config({
-Relevance
+Activation functions, Logistic and Hyperbolic ones
-The sigmoid function are more biologically plausible because the
-output of inactive neurons are zero. Such activation function are
-called one-sided. However, it has been shown that the hyperbolic
-tangent performs better than the sigmoid for training MLPs. has
-become the most popular for deep neural networks
+The second requirement excludes all linear functions. Furthermore, in
+a MLP with only linear activation functions, each layer simply
+performs a linear transformation of its inputs.
+Regardless of the number of layers, the output of the NN will be
+nothing but a linear function of the inputs. Thus we need to introduce
+some kind of non-linearity to the NN to be able to fit non-linear
+functions Typical examples are the logistic Sigmoid
-
-
"""The sigmoid function (or the logistic curve) is a
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
+$$
+ f(x) = \frac{1}{1 + e^{-x}},
+$$
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
+and the hyperbolic tangent function
+$$
+ f(x) = \tanh(x)
+$$
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""Sine Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.sin(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sine function')
-
-plt.show()
-
-"""Plots a graph of the squashing function used by a rectified linear
-unit"""
-z = numpy.arange(-2, 2, .1)
-zero = numpy.zeros(len(z))
-y = numpy.max([zero, z], axis=0)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, y)
-ax.set_ylim([-2.0, 2.0])
-ax.set_xlim([-2.0, 2.0])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('Rectified linear unit')
-
-plt.show()
-
@@ -417,7 +360,7 @@ plt.show()
60
61
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs052.html b/doc/pub/week40/html/._week40-bs052.html
index ba7b61afb..4b47478f3 100644
--- a/doc/pub/week40/html/._week40-bs052.html
+++ b/doc/pub/week40/html/._week40-bs052.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,41 +312,90 @@ MathJax.Hub.Config({
-The multilayer perceptron (MLP)
+Relevance
-The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
-
-
-- A neural network with one or more layers of nodes between the input and the output nodes.
-- The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
-- The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
-
-
-As a convention it is normal to call a network with one layer of input units, one layer of hidden
-units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
+The sigmoid function are more biologically plausible because the
+output of inactive neurons are zero. Such activation function are
+called one-sided. However, it has been shown that the hyperbolic
+tangent performs better than the sigmoid for training MLPs. has
+become the most popular for deep neural networks
-For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
-Hereafter we will call the various entities of a layer for nodes.
-There are also no connections within a single layer.
-
-The number of input nodes does not need to equal the number of output
-nodes. This applies also to the hidden layers. Each layer may have its
-own number of nodes and activation functions.
+
+
"""The sigmoid function (or the logistic curve) is a
+function that takes any real number, z, and outputs a number (0,1).
+It is useful in neural networks for assigning weights on a relative scale.
+The value z is the weighted sum of parameters involved in the learning algorithm."""
-
-The hidden layers have their name from the fact that they are not
-linked to observables and as we will see below when we define the
-so-called activation \( \hat{z} \), we can think of this as a basis
-expansion of the original inputs \( \hat{x} \). The difference however
-between neural networks and say linear regression is that now these
-basis functions (which will correspond to the weights in the network)
-are learned from data. This results in an important difference between
-neural networks and deep learning approaches on one side and methods
-like logistic regression or linear regression and their modifications on the other side.
+import numpy
+import matplotlib.pyplot as plt
+import math as mt
+z = numpy.arange(-5, 5, .1)
+sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
+sigma = sigma_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, sigma)
+ax.set_ylim([-0.1, 1.1])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()
+
+"""Step Function"""
+z = numpy.arange(-5, 5, .02)
+step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
+step = step_fn(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, step)
+ax.set_ylim([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('step function')
+
+plt.show()
+
+"""Sine Function"""
+z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
+t = numpy.sin(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-2*mt.pi,2*mt.pi])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sine function')
+
+plt.show()
+
+"""Plots a graph of the squashing function used by a rectified linear
+unit"""
+z = numpy.arange(-2, 2, .1)
+zero = numpy.zeros(len(z))
+y = numpy.max([zero, z], axis=0)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, y)
+ax.set_ylim([-2.0, 2.0])
+ax.set_xlim([-2.0, 2.0])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('Rectified linear unit')
+
+plt.show()
+
@@ -368,7 +422,7 @@ like logistic regression or linear regression and their modifications on the oth
61
62
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs053.html b/doc/pub/week40/html/._week40-bs053.html
index 18cde57c0..af0b1f751 100644
--- a/doc/pub/week40/html/._week40-bs053.html
+++ b/doc/pub/week40/html/._week40-bs053.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,30 +312,40 @@ MathJax.Hub.Config({
-From one to many layers, the universal approximation theorem
+The multilayer perceptron (MLP)
-A neural network with only one layer, what we called the simple
-perceptron, is best suited if we have a standard binary model with
-clear (linear) boundaries between the outcomes. As such it could
-equally well be replaced by standard linear regression or logistic
-regression. Networks with one or more hidden layers approximate
-systems with more complex boundaries.
+The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of
+
+
+- A neural network with one or more layers of nodes between the input and the output nodes.
+- The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.
+- The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.
+
+
+As a convention it is normal to call a network with one layer of input units, one layer of hidden
+units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.
-As stated earlier,
-an important theorem in studies of neural networks, restated without
-proof here, is the universal approximation
-theorem.
+For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.
+Hereafter we will call the various entities of a layer for nodes.
+There are also no connections within a single layer.
-It states that a feed-forward network with a single hidden layer
-containing a finite number of neurons can approximate continuous
-functions on compact subsets of real functions. The theorem thus
-states that simple neural networks can represent a wide variety of
-interesting functions when given appropriate parameters. It is the
-multilayer feedforward architecture itself which gives neural networks
-the potential of being universal approximators.
+The number of input nodes does not need to equal the number of output
+nodes. This applies also to the hidden layers. Each layer may have its
+own number of nodes and activation functions.
+
+
+The hidden layers have their name from the fact that they are not
+linked to observables and as we will see below when we define the
+so-called activation \( \hat{z} \), we can think of this as a basis
+expansion of the original inputs \( \hat{x} \). The difference however
+between neural networks and say linear regression is that now these
+basis functions (which will correspond to the weights in the network)
+are learned from data. This results in an important difference between
+neural networks and deep learning approaches on one side and methods
+like logistic regression or linear regression and their modifications on the other side.
@@ -357,6 +372,8 @@ the potential of being universal approximators.
61
62
63
+ ...
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs054.html b/doc/pub/week40/html/._week40-bs054.html
index adfe99868..34dac021d 100644
--- a/doc/pub/week40/html/._week40-bs054.html
+++ b/doc/pub/week40/html/._week40-bs054.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,33 +312,30 @@ MathJax.Hub.Config({
-Deriving the back propagation code for a multilayer perceptron model
+From one to many layers, the universal approximation theorem
-As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
-The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
-This leads us to the famous back propagation algorithm.
+A neural network with only one layer, what we called the simple
+perceptron, is best suited if we have a standard binary model with
+clear (linear) boundaries between the outcomes. As such it could
+equally well be replaced by standard linear regression or logistic
+regression. Networks with one or more hidden layers approximate
+systems with more complex boundaries.
-The questions we want to ask are how do changes in the biases and the
-weights in our network change the cost function and how can we use the
-final output to modify the weights?
+As stated earlier,
+an important theorem in studies of neural networks, restated without
+proof here, is the universal approximation
+theorem.
-To derive these equations let us start with a plain regression problem
-and define our cost function as
-
-$$
-{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
-$$
-
-
-where the $t_i$s are our \( n \) targets (the values we want to
-reproduce), while the outputs of the network after having propagated
-all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
-how the basic equations arising from the back propagation algorithm
-can be modified in order to study classification problems with \( K \)
-classes.
+It states that a feed-forward network with a single hidden layer
+containing a finite number of neurons can approximate continuous
+functions on compact subsets of real functions. The theorem thus
+states that simple neural networks can represent a wide variety of
+interesting functions when given appropriate parameters. It is the
+multilayer feedforward architecture itself which gives neural networks
+the potential of being universal approximators.
@@ -359,6 +361,7 @@ classes.
61
62
63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs055.html b/doc/pub/week40/html/._week40-bs055.html
index 422c863d6..79232677b 100644
--- a/doc/pub/week40/html/._week40-bs055.html
+++ b/doc/pub/week40/html/._week40-bs055.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,40 +312,33 @@ MathJax.Hub.Config({
-Definitions
+Deriving the back propagation code for a multilayer perceptron model
-With our definition of the targets \( \hat{t} \), the outputs of the
-network \( \hat{y} \) and the inputs \( \hat{x} \) we
-define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
-\( l \)-th layer as a function of the bias, the weights which add up from
-the previous layer \( l-1 \) and the forward passes/outputs
-\( \hat{a}^{l-1} \) from the previous layer as
+As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.
+The unknowwn quantities are our weights \( w_{ij} \) and we need to find an algorithm for changing them so that our errors are as small as possible.
+This leads us to the famous back propagation algorithm.
+
+
+The questions we want to ask are how do changes in the biases and the
+weights in our network change the cost function and how can we use the
+final output to modify the weights?
+
+
+To derive these equations let us start with a plain regression problem
+and define our cost function as
$$
-z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
+{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2,
$$
-where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
-represents the total number of nodes/neurons/units of layer \( l-1 \). The
-figure here illustrates this equation. We can rewrite this in a more
-compact form as the matrix-vector products we discussed earlier,
-
-$$
-\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
-$$
-
-
-With the activation values \( \hat{z}^l \) we can in turn define the
-output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
-activation function. In the examples here we will use the sigmoid
-function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
-and their nodes. It means we have
-
-$$
-a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
-$$
+where the $t_i$s are our \( n \) targets (the values we want to
+reproduce), while the outputs of the network after having propagated
+all inputs \( \hat{x} \) are given by \( y_i \). Below we will demonstrate
+how the basic equations arising from the back propagation algorithm
+can be modified in order to study classification problems with \( K \)
+classes.
@@ -365,6 +363,7 @@ $$
61
62
63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs056.html b/doc/pub/week40/html/._week40-bs056.html
index 86de71d67..1c500b71d 100644
--- a/doc/pub/week40/html/._week40-bs056.html
+++ b/doc/pub/week40/html/._week40-bs056.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,23 +312,39 @@ MathJax.Hub.Config({
-Derivatives and the chain rule
+Definitions
-From the definition of the activation \( z_j^l \) we have
-$$
-\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
-$$
+With our definition of the targets \( \hat{t} \), the outputs of the
+network \( \hat{y} \) and the inputs \( \hat{x} \) we
+define now the activation \( z_j^l \) of node/neuron/unit \( j \) of the
+\( l \)-th layer as a function of the bias, the weights which add up from
+the previous layer \( l-1 \) and the forward passes/outputs
+\( \hat{a}^{l-1} \) from the previous layer as
-and
$$
-\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
+z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,
$$
-With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
+where \( b_k^l \) are the biases from layer \( l \). Here \( M_{l-1} \)
+represents the total number of nodes/neurons/units of layer \( l-1 \). The
+figure here illustrates this equation. We can rewrite this in a more
+compact form as the matrix-vector products we discussed earlier,
+
$$
-\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
+\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l.
+$$
+
+
+With the activation values \( \hat{z}^l \) we can in turn define the
+output of layer \( l \) as \( \hat{a}^l = f(\hat{z}^l) \) where \( f \) is our
+activation function. In the examples here we will use the sigmoid
+function discussed in our logistic regression lectures. We will also use the same activation function \( f \) for all layers
+and their nodes. It means we have
+
+$$
+a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}.
$$
@@ -348,6 +369,7 @@ $$
61
62
63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs057.html b/doc/pub/week40/html/._week40-bs057.html
index c8e1f2f7f..6f8ea74b8 100644
--- a/doc/pub/week40/html/._week40-bs057.html
+++ b/doc/pub/week40/html/._week40-bs057.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,26 +312,23 @@ MathJax.Hub.Config({
-Derivative of the cost function
+Derivatives and the chain rule
-With these definitions we can now compute the derivative of the cost function in terms of the weights.
+From the definition of the activation \( z_j^l \) we have
+$$
+\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1},
+$$
+
+and
+$$
+\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l.
+$$
-Let us specialize to the output layer \( l=L \). Our cost function is
+With our definition of the activation function we have that (note that this function depends only on \( z_j^l \))
$$
-{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
-$$
-
-The derivative of this function with respect to the weights is
-
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
-$$
-
-The last partial derivative can easily be computed and reads (by applying the chain rule)
-$$
-\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
+\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).
$$
@@ -350,6 +352,7 @@ $$
61
62
63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs058.html b/doc/pub/week40/html/._week40-bs058.html
index 1ef9e7a74..cc89ef9f5 100644
--- a/doc/pub/week40/html/._week40-bs058.html
+++ b/doc/pub/week40/html/._week40-bs058.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,51 +312,26 @@ MathJax.Hub.Config({
-Bringing it together, first back propagation equation
+Derivative of the cost function
-We have thus
-$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
-$$
+With these definitions we can now compute the derivative of the cost function in terms of the weights.
-Defining
+Let us specialize to the output layer \( l=L \). Our cost function is
$$
-\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
+{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2,
$$
-and using the Hadamard product of two vectors we can write this as
-$$
-\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
-$$
-
-
-This is an important expression. The second term on the right handside
-measures how fast the cost function is changing as a function of the $j$th
-output activation. If, for example, the cost function doesn't depend
-much on a particular output node \( j \), then \( \delta_j^L \) will be small,
-which is what we would expect. The first term on the right, measures
-how fast the activation function \( f \) is changing at a given activation
-value \( z_j^L \).
-
-
-Notice that everything in the above equations is easily computed. In
-particular, we compute \( z_j^L \) while computing the behaviour of the
-network, and it is only a small additional overhead to compute
-\( f'(z^L_j) \). The exact form of the derivative with respect to the
-output depends on the form of the cost function.
-However, provided the cost function is known there should be little
-trouble in calculating
+The derivative of this function with respect to the weights is
$$
-\frac{\partial {\cal C}}{\partial (a_j^L)}
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}},
$$
-
-With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+The last partial derivative can easily be computed and reads (by applying the chain rule)
$$
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
+\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},
$$
@@ -374,6 +354,7 @@ $$
61
62
63
+ 64
»
diff --git a/doc/pub/week40/html/._week40-bs059.html b/doc/pub/week40/html/._week40-bs059.html
index 6c11e0f9f..187848e34 100644
--- a/doc/pub/week40/html/._week40-bs059.html
+++ b/doc/pub/week40/html/._week40-bs059.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,21 +312,54 @@ MathJax.Hub.Config({
-Derivatives in terms of \( z_j^L \)
+Bringing it together, first back propagation equation
-It is also easy to see that our previous equation can be written as
-
+We have thus
$$
-\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1},
$$
-which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
+
+Defining
$$
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
+\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
$$
-That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
+and using the Hadamard product of two vectors we can write this as
+$$
+\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}.
+$$
+
+
+This is an important expression. The second term on the right handside
+measures how fast the cost function is changing as a function of the $j$th
+output activation. If, for example, the cost function doesn't depend
+much on a particular output node \( j \), then \( \delta_j^L \) will be small,
+which is what we would expect. The first term on the right, measures
+how fast the activation function \( f \) is changing at a given activation
+value \( z_j^L \).
+
+
+Notice that everything in the above equations is easily computed. In
+particular, we compute \( z_j^L \) while computing the behaviour of the
+network, and it is only a small additional overhead to compute
+\( f'(z^L_j) \). The exact form of the derivative with respect to the
+output depends on the form of the cost function.
+However, provided the cost function is known there should be little
+trouble in calculating
+
+$$
+\frac{\partial {\cal C}}{\partial (a_j^L)}
+$$
+
+
+With the definition of \( \delta_j^L \) we have a more compact definition of the derivative of the cost function in terms of the weights, namely
+$$
+\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}.
+$$
+
+
diff --git a/doc/pub/week40/html/._week40-bs060.html b/doc/pub/week40/html/._week40-bs060.html
index a40581438..f067a7937 100644
--- a/doc/pub/week40/html/._week40-bs060.html
+++ b/doc/pub/week40/html/._week40-bs060.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -307,66 +312,21 @@ MathJax.Hub.Config({
-Bringing it together
+Derivatives in terms of \( z_j^L \)
-We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are
-
-
-
-
-
+It is also easy to see that our previous equation can be written as
$$
-\begin{equation}
-\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1},
-\tag{18}
-\end{equation}
+\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L},
$$
-and
+which can also be interpreted as the partial derivative of the cost function with respect to the biases \( b_j^L \), namely
$$
-\begin{equation}
-\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)},
-\tag{19}
-\end{equation}
+\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L},
$$
-and
-
-$$
-\begin{equation}
-\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L},
-\tag{20}
-\end{equation}
-$$
-
-
-An interesting consequence of the above equations is that when the
-activation \( a_k^{L-1} \) is small, the gradient term, that is the
-derivative of the cost function with respect to the weights, will also
-tend to be small. We say then that the weight learns slowly, meaning
-that it changes slowly when we minimize the weights via say gradient
-descent. In this case we say the system learns slowly.
-
-
-Another interesting feature is that is when the activation function,
-represented by the sigmoid function here, is rather flat when we move towards
-its end values \( 0 \) and \( 1 \) (see the above Python codes). In these
-cases, the derivatives of the activation function will also be close
-to zero, meaning again that the gradients will be small and the
-network learns slowly again.
-
-
-We need a fourth equation and we are set. We are going to propagate
-backwards in order to the determine the weights and biases. In order
-to do so we need to represent the error in the layer before the final
-one \( L-1 \) in terms of the errors in the final output layer.
-
-
+That is, the error \( \delta_j^L \) is exactly equal to the rate of change of the cost function as a function of the bias.
diff --git a/doc/pub/week40/html/week40-bs.html b/doc/pub/week40/html/week40-bs.html
index a8251bd75..b5dc13a98 100644
--- a/doc/pub/week40/html/week40-bs.html
+++ b/doc/pub/week40/html/week40-bs.html
@@ -138,6 +138,10 @@ Automatically generated HTML file from DocOnce source
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -271,27 +275,28 @@ MathJax.Hub.Config({
Multilayer perceptrons
Why multilayer perceptrons?
Illustration of a single perceptropn model and a multi-perceptron model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Mathematical model
- Matrix-vector notation
- Matrix-vector notation and activation
- Activation functions
- Activation functions, Logistic and Hyperbolic ones
- Relevance
- The multilayer perceptron (MLP)
- From one to many layers, the universal approximation theorem
- Deriving the back propagation code for a multilayer perceptron model
- Definitions
- Derivatives and the chain rule
- Derivative of the cost function
- Bringing it together, first back propagation equation
- Derivatives in terms of \( z_j^L \)
- Bringing it together
- Final back propagating equation
- Setting up the Back propagation algorithm
+ Examples of XOR, OR and AND gates
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Mathematical model
+ Matrix-vector notation
+ Matrix-vector notation and activation
+ Activation functions
+ Activation functions, Logistic and Hyperbolic ones
+ Relevance
+ The multilayer perceptron (MLP)
+ From one to many layers, the universal approximation theorem
+ Deriving the back propagation code for a multilayer perceptron model
+ Definitions
+ Derivatives and the chain rule
+ Derivative of the cost function
+ Bringing it together, first back propagation equation
+ Derivatives in terms of \( z_j^L \)
+ Bringing it together
+ Final back propagating equation
+ Setting up the Back propagation algorithm
@@ -326,7 +331,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
-
Oct 7, 2021
+Oct 8, 2021
@@ -350,7 +355,7 @@ MathJax.Hub.Config({
9
10
...
- 63
+ 64
»
diff --git a/doc/pub/week40/html/week40-reveal.html b/doc/pub/week40/html/week40-reveal.html
index a4aab3879..c529ab6b9 100644
--- a/doc/pub/week40/html/week40-reveal.html
+++ b/doc/pub/week40/html/week40-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
-
Oct 7, 2021
+Oct 8, 2021
@@ -1438,6 +1438,51 @@ as to not restrict the range of output values.
+
+Examples of XOR, OR and AND gates
+
+
+Let us first try to fit various gates using standard linear regression
+
+
+
+
+
"""
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
+
+import numpy as np
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX matrix:{Xinv}")
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}")
+
+What is happening here?
+
+
+
Mathematical model
diff --git a/doc/pub/week40/html/week40-solarized.html b/doc/pub/week40/html/week40-solarized.html
index c4c59103d..03f304ad5 100644
--- a/doc/pub/week40/html/week40-solarized.html
+++ b/doc/pub/week40/html/week40-solarized.html
@@ -158,6 +158,10 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -254,7 +258,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
-
Oct 7, 2021
+Oct 8, 2021
@@ -1491,6 +1495,51 @@ as to not restrict the range of output values.
+
Examples of XOR, OR and AND gates
+
+
+Let us first try to fit various gates using standard linear regression
+
+
+
+
+
"""
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
+
+import numpy as np
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX matrix:{Xinv}")
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}")
+
+What is happening here?
+
+
+
+
Mathematical model
diff --git a/doc/pub/week40/html/week40.html b/doc/pub/week40/html/week40.html
index 1a14b3d20..5a17bb5f7 100644
--- a/doc/pub/week40/html/week40.html
+++ b/doc/pub/week40/html/week40.html
@@ -163,6 +163,10 @@ div { text-align: justify; text-justify: inter-word; }
2,
None,
'illustration-of-a-single-perceptropn-model-and-a-multi-perceptron-model'),
+ ('Examples of XOR, OR and AND gates',
+ 2,
+ None,
+ 'examples-of-xor-or-and-and-gates'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
('Mathematical model', 2, None, 'mathematical-model'),
@@ -259,7 +263,7 @@ MathJax.Hub.Config({
[2] Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA
-
Oct 7, 2021
+Oct 8, 2021
@@ -1496,6 +1500,51 @@ as to not restrict the range of output values.
+
Examples of XOR, OR and AND gates
+
+
+Let us first try to fit various gates using standard linear regression
+
+
+
+
+
"""
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
+
+import numpy as np
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX matrix:{Xinv}")
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}")
+
+What is happening here?
+
+
+
+
Mathematical model
diff --git a/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz b/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz
index 78b0feb47..b2dbee390 100644
Binary files a/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz and b/doc/pub/week40/ipynb/ipynb-week40-src.tar.gz differ
diff --git a/doc/pub/week40/ipynb/week40.ipynb b/doc/pub/week40/ipynb/week40.ipynb
index 2c2393e98..b90e931fa 100644
--- a/doc/pub/week40/ipynb/week40.ipynb
+++ b/doc/pub/week40/ipynb/week40.ipynb
@@ -10,7 +10,7 @@
" \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Ion Beams, Michigan State University, USA\n",
"\n",
- "Date: **Oct 7, 2021**\n",
+ "Date: **Oct 8, 2021**\n",
"\n",
"Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -1540,6 +1540,59 @@
"\n",
"\n",
"\n",
+ "## Examples of XOR, OR and AND gates\n",
+ "\n",
+ "Let us first try to fit various gates using standard linear regression"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\"\"\"\n",
+ "Simple code that tests XOR, OR and AND gates with linear regression\n",
+ "\"\"\"\n",
+ "\n",
+ "import numpy as np\n",
+ "# Design matrix\n",
+ "X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)\n",
+ "print(f\"The X.TX matrix:{X.T @ X}\")\n",
+ "Xinv = np.linalg.pinv(X.T @ X)\n",
+ "print(f\"The invers of X.TX matrix:{Xinv}\")\n",
+ "\n",
+ "# The XOR gate \n",
+ "yXOR = np.array( [ 0, 1 ,1, 0])\n",
+ "ThetaXOR = Xinv @ X.T @ yXOR\n",
+ "print(f\"The values of theta for the XOR gate:{ThetaXOR}\")\n",
+ "print(f\"The linear regression prediction for the XOR gate:{X @ ThetaXOR}\")\n",
+ "\n",
+ "\n",
+ "# The OR gate \n",
+ "yOR = np.array( [ 0, 1 ,1, 1])\n",
+ "ThetaOR = Xinv @ X.T @ yOR\n",
+ "print(f\"The values of theta for the OR gate:{ThetaOR}\")\n",
+ "print(f\"The linear regression prediction for the OR gate:{X @ ThetaOR}\")\n",
+ "\n",
+ "\n",
+ "# The OR gate \n",
+ "yAND = np.array( [ 0, 0 ,0, 1])\n",
+ "ThetaAND = Xinv @ X.T @ yAND\n",
+ "print(f\"The values of theta for the AND gate:{ThetaAND}\")\n",
+ "print(f\"The linear regression prediction for the AND gate:{X @ ThetaAND}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "What is happening here?\n",
+ "\n",
+ "\n",
"## Mathematical model\n",
"\n",
"The output $y$ is produced via the activation function $f$"
diff --git a/doc/src/week40/week40.do.txt b/doc/src/week40/week40.do.txt
index d1176fcc7..30a054afe 100644
--- a/doc/src/week40/week40.do.txt
+++ b/doc/src/week40/week40.do.txt
@@ -1103,6 +1103,47 @@ as to not restrict the range of output values.
FIGURE: [figures/nns.png, width=600 frac=0.8] In a) we show a single perceptron model while in b) we dispay a network with two hidden layers, an input layer and an output layer.
+!split
+===== Examples of XOR, OR and AND gates =====
+
+Let us first try to fit various gates using standard linear regression
+
+!bc pycod
+"""
+Simple code that tests XOR, OR and AND gates with linear regression
+"""
+
+import numpy as np
+# Design matrix
+X = np.array([ [1, 0, 0], [1, 0, 1], [1, 1, 0],[1, 1, 1]],dtype=np.float64)
+print(f"The X.TX matrix:{X.T @ X}")
+Xinv = np.linalg.pinv(X.T @ X)
+print(f"The invers of X.TX matrix:{Xinv}")
+
+# The XOR gate
+yXOR = np.array( [ 0, 1 ,1, 0])
+ThetaXOR = Xinv @ X.T @ yXOR
+print(f"The values of theta for the XOR gate:{ThetaXOR}")
+print(f"The linear regression prediction for the XOR gate:{X @ ThetaXOR}")
+
+
+# The OR gate
+yOR = np.array( [ 0, 1 ,1, 1])
+ThetaOR = Xinv @ X.T @ yOR
+print(f"The values of theta for the OR gate:{ThetaOR}")
+print(f"The linear regression prediction for the OR gate:{X @ ThetaOR}")
+
+
+# The OR gate
+yAND = np.array( [ 0, 0 ,0, 1])
+ThetaAND = Xinv @ X.T @ yAND
+print(f"The values of theta for the AND gate:{ThetaAND}")
+print(f"The linear regression prediction for the AND gate:{X @ ThetaAND}")
+!ec
+
+What is happening here?
+
+
!split
===== Mathematical model =====