update week41

doc/src/week41/week41.do.txt

@@ -1067,6 +1067,88 @@ plt.show()
 !ec
 
 
+!split
+===== Testing our code for the XOR, OR and AND gates =====
+
+Last week we discussed three different types of gates, the so-called
+XOR, the OR and the AND gates.  Their inputs and outputs can be
+summarized using the following tables, first for the OR gate with
+inputs $x_1$ and $x_2$ and outputs $y$:
+
+|---------------------|
+| $x_1$ | $x_2$ | $y$ |
+|---------------------|
+| 0    |  0 |  0  |
+| 0    | 1  |  1  |
+| 1    | 0  | 1 |
+| 1    | 1  | 1 |
+|---------------------|
+
+!split
+===== The AND and XOR Gates =====
+
+The _AND_ gate is defined as
+
+|---------------------|
+| $x_1$ | $x_2$ | $y$ |
+|---------------------|
+| 0    |  0 |  0  |
+| 0    | 1  |  0  |
+| 1    | 0  | 0  |
+| 1    | 1  | 1 |
+|---------------------|
+
+And finally we have the _XOR_ gate
+
+|---------------------|
+| $x_1$ | $x_2$ | $y$ |
+|---------------------|
+| 0    |  0 |  0  |
+| 0    | 1  |  1  |
+| 1    | 0  | 1 |
+| 1    | 1  | 0 |
+|---------------------|
+
+!split
+===== Representing the Data Sets =====
+
+Our design matrix is defined by the input values $x_1$ and $x_2$. Since we have four possible outputs, our design matrix reads
+
+!bt
+\bm{X}=\begin{bmatrix} 0 & 0 \\
+                       0 & 1 \\
+		       1 & 0 \\
+		       1 & 1 \end{bmatrix},
+!et
+while the vector of outputs is $\bm{y}^T=[0,1,1,0]$ for the XOR gate, $\bm{y}^T=[0,0,0,1]$ for the AND gate and $\bm{y}^T=[0,1,1,1]$ for the OR gate.
+
+!split
+===== Setting up the Neural Network =====
+
+We define first our design matrix and the various input vectors.
+
+!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)
+
+# The XOR gate 
+yXOR = np.array( [ 0, 1 ,1, 0])
+# The OR gate 
+yOR = np.array( [ 0, 1 ,1, 1])
+# The AND gate 
+yAND = np.array( [ 0, 0 ,0, 1])
+
+#print(f"The values of theta for the AND gate:{ThetaAND}")
+#print(f"The linear regression prediction  for the AND gate:{X @ ThetaAND}")
+!ec
+
+
+
 !split
 ===== Building neural networks in Tensorflow and Keras =====
 
@@ -1749,7 +1831,37 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
 
 
 
+!split
+===== Overarching Views, a personal  note =====
 
+The author of these lecture notes has an overarching take on many of
+the machine learning algorithms we discuss here. 
+
+
+If we wish to understand complex systems, we need to find some
+effective degrees of freedom or features that we find essential,
+simply in order to reduce the complexity of the systems we are
+studying.  This leads, in one way or the other to dimensionality
+reductions. Most of the Machine Learning methods we encounter deal
+with this, whether we opt for a principal component analysis, or
+clustering, or convolutional neural networks, or Ridge or Lasso
+regression or random forest, yes, perhaps most machine learning
+methods at large.
+
+For neural networks and our previous discussion, we have seen that we
+in essence end up with matrix-matrix and matrix-vector
+multiplications. In all cases, our matrices are dense ones, and the
+more data we deal with the larger the dimensionalities of the matrices
+and vectors. How can we reduce such dimensionalities?  One possible
+answer is offered by _convolutional neural networks_ (CNN), as
+discussed below. The figure here shows a typical situation of the
+reduction of information in an image and is typical of what CNNs
+actually end up doing.
+
+!split
+===== From a Spherical Cow to a real one =====
+
+FIGURE: [figslides/ImageReduction.png, width=500 frac=0.6]
 
 
 !split
@@ -2148,3 +2260,5 @@ o "Abstract art using convolutional neural networks":"https://deepdreamgenerator
 
 
 
+
+