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  • Week 38: Statistical analysis, bias-variance tradeoff and resampling methods
  • Exercises week 39
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  • Week 39: Resampling methods and logistic regression
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  • Week 40: Gradient descent methods (continued) and start Neural networks
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  • Week 41 Neural networks and constructing a neural network code
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  • Exercises week 41
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      diff --git a/doc/LectureNotes/_build/html/_sources/exercisesweek41.ipynb b/doc/LectureNotes/_build/html/_sources/exercisesweek41.ipynb index fccc64f45..190c0b96a 100644 --- a/doc/LectureNotes/_build/html/_sources/exercisesweek41.ipynb +++ b/doc/LectureNotes/_build/html/_sources/exercisesweek41.ipynb @@ -327,7 +327,7 @@ "id": "0da7fd52", "metadata": {}, "source": [ - "**d)** Why is a neural network with no activation functions always mathematically equivelent to a neural network with only one layer?\n" + "**d)** Why is a neural network with no activation functions mathematically equivelent to(can be reduced to) a neural network with only one layer?\n" ] }, { @@ -454,7 +454,7 @@ "id": "a6349db6", "metadata": {}, "source": [ - "**b)** Make a matrix of inputs with the shape (number of features, number of inputs), you choose the number of inputs and features per input. Then complete the function `feed_forward_batch` so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)\n" + "**b)** Make a matrix of inputs with the shape (number of inputs, number of features), you choose the number of inputs and features per input. Then complete the function `feed_forward_batch` so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)" ] }, { @@ -480,7 +480,7 @@ "id": "efd07b4e", "metadata": {}, "source": [ - "**c)** Create and evaluate a neural network with 4 inputs and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.\n" + "**c)** Create and evaluate a neural network with 4 input features, and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.\n" ] }, { diff --git a/doc/LectureNotes/_build/html/_static/pygments.css b/doc/LectureNotes/_build/html/_static/pygments.css index 012e6a00a..d7dd57783 100644 --- a/doc/LectureNotes/_build/html/_static/pygments.css +++ b/doc/LectureNotes/_build/html/_static/pygments.css @@ -6,11 +6,11 @@ html[data-theme="light"] .highlight span.linenos.special { color: #000000; backg html[data-theme="light"] .highlight .hll { background-color: #fae4c2 } html[data-theme="light"] .highlight 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*/ +html[data-theme="dark"] .highlight .vi { color: #FFA07A } /* Name.Variable.Instance */ +html[data-theme="dark"] .highlight .vm { color: #FFD900 } /* Name.Variable.Magic */ +html[data-theme="dark"] .highlight .il { color: #FFD900 } /* Literal.Number.Integer.Long */ \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter1.html b/doc/LectureNotes/_build/html/chapter1.html index 41de41ded..767f31eb7 100644 --- a/doc/LectureNotes/_build/html/chapter1.html +++ b/doc/LectureNotes/_build/html/chapter1.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter10.html b/doc/LectureNotes/_build/html/chapter10.html index c69ffbbcf..01bcb40b2 100644 --- a/doc/LectureNotes/_build/html/chapter10.html +++ b/doc/LectureNotes/_build/html/chapter10.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter11.html b/doc/LectureNotes/_build/html/chapter11.html index e7bf014c2..2e993ca42 100644 --- a/doc/LectureNotes/_build/html/chapter11.html +++ b/doc/LectureNotes/_build/html/chapter11.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter12.html b/doc/LectureNotes/_build/html/chapter12.html index d3f429569..20ee15d30 100644 --- a/doc/LectureNotes/_build/html/chapter12.html +++ b/doc/LectureNotes/_build/html/chapter12.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter13.html b/doc/LectureNotes/_build/html/chapter13.html index 6e742d665..ea68cea36 100644 --- a/doc/LectureNotes/_build/html/chapter13.html +++ b/doc/LectureNotes/_build/html/chapter13.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter2.html b/doc/LectureNotes/_build/html/chapter2.html index f63add9f7..4427f4fe8 100644 --- a/doc/LectureNotes/_build/html/chapter2.html +++ b/doc/LectureNotes/_build/html/chapter2.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index 406e63b0c..71f19560d 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter4.html b/doc/LectureNotes/_build/html/chapter4.html index 3a85a17fa..a3526e849 100644 --- a/doc/LectureNotes/_build/html/chapter4.html +++ b/doc/LectureNotes/_build/html/chapter4.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter5.html b/doc/LectureNotes/_build/html/chapter5.html index d5a160188..43426de95 100644 --- a/doc/LectureNotes/_build/html/chapter5.html +++ b/doc/LectureNotes/_build/html/chapter5.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter6.html b/doc/LectureNotes/_build/html/chapter6.html index a07ac5ca3..fd66e1b41 100644 --- a/doc/LectureNotes/_build/html/chapter6.html +++ b/doc/LectureNotes/_build/html/chapter6.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter7.html b/doc/LectureNotes/_build/html/chapter7.html index 5320cec07..1a74833c2 100644 --- a/doc/LectureNotes/_build/html/chapter7.html +++ b/doc/LectureNotes/_build/html/chapter7.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter8.html b/doc/LectureNotes/_build/html/chapter8.html index 7762a1a03..2354e2ec2 100644 --- a/doc/LectureNotes/_build/html/chapter8.html +++ b/doc/LectureNotes/_build/html/chapter8.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapter9.html b/doc/LectureNotes/_build/html/chapter9.html index eaab55eaf..bb2712d21 100644 --- a/doc/LectureNotes/_build/html/chapter9.html +++ b/doc/LectureNotes/_build/html/chapter9.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/chapteroptimization.html b/doc/LectureNotes/_build/html/chapteroptimization.html index d92f1ed5d..2b797ac88 100644 --- a/doc/LectureNotes/_build/html/chapteroptimization.html +++ b/doc/LectureNotes/_build/html/chapteroptimization.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/clustering.html b/doc/LectureNotes/_build/html/clustering.html index da0fe96ee..a8204c1f4 100644 --- a/doc/LectureNotes/_build/html/clustering.html +++ b/doc/LectureNotes/_build/html/clustering.html @@ -28,7 +28,7 @@ - + diff --git a/doc/LectureNotes/_build/html/exercisesweek34.html b/doc/LectureNotes/_build/html/exercisesweek34.html index 33a704f71..33d9ff913 100644 --- a/doc/LectureNotes/_build/html/exercisesweek34.html +++ b/doc/LectureNotes/_build/html/exercisesweek34.html @@ -28,7 +28,7 @@ - + @@ -491,11 +491,11 @@ document.write(`

      b) Compute the mean square error for the line model and for the second degree polynomial model.

      -
      import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.preprocessing import PolynomialFeatures # use the fit_transform method of the created object!
-from sklearn.linear_model import LinearRegression
-from sklearn.metrics import mean_squared_error
+
      import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import PolynomialFeatures # use the fit_transform method of the created object!
+from sklearn.linear_model import LinearRegression
+from sklearn.metrics import mean_squared_error
      @@ -532,7 +532,7 @@ document.write(`

      Hopefully your model fit the data quite well, but to know how well the model actually generalizes to unseen data, which is most often what we care about, we need to split our data into training and testing data.

      -
      from sklearn.model_selection import train_test_split
+
      from sklearn.model_selection import train_test_split
      diff --git a/doc/LectureNotes/_build/html/exercisesweek35.html b/doc/LectureNotes/_build/html/exercisesweek35.html index ded9ae484..279fcd43e 100644 --- a/doc/LectureNotes/_build/html/exercisesweek35.html +++ b/doc/LectureNotes/_build/html/exercisesweek35.html @@ -28,7 +28,7 @@ - + @@ -527,7 +527,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,

      We calculate the optimal intercept by including a feature with the constant value of 1 in our model, which is then multplied by some parameter \(\theta_0\) from the OLS method into the optimal intercept value (which will be \(\theta_0\)). In practice, we include the intercept in our model by adding a column of ones to the start of our feature matrix.

      -
      import numpy as np
+
      import numpy as np
      @@ -556,7 +556,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,

      b) Use the expression from 3d) to find the optimal parameters \(\boldsymbol{\hat{\beta}_{OLS}}\) for predicting spending based on these features. Create a function for this operation, as you are going to need to use it a lot.

      -
      def OLS_parameters(X, y):
+
      def OLS_parameters(X, y):
     return ...
 
 #beta = OLS_parameters(X, y)
@@ -581,7 +581,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,
 
      a) Create a feature matrix \(\boldsymbol{X}\) for the features \(x, x^2, x^3, x^4, x^5\), including an intercept column of ones at the start. Make this into a function, as you will do this a lot over the next weeks.
      
 
      
 
      
-
      def polynomial_features(x, p):
+
      def polynomial_features(x, p):
     n = len(x)
     X = np.zeros((n, p + 1))
     #X[:, 0] = ...
@@ -605,7 +605,7 @@ f_i =\sum_{j=0}^{n-1}a_{ij}x_j,
 
      c) Like in exercise 4 last week, split your feature matrix and target data into a training split and test split.
      
 
      
 
      
-
      from sklearn.model_selection import train_test_split
+
      from sklearn.model_selection import train_test_split
 
 #X_train, X_test, y_train, y_test = ...
 
      
diff --git a/doc/LectureNotes/_build/html/exercisesweek36.html b/doc/LectureNotes/_build/html/exercisesweek36.html
index 414eba618..378d85ecf 100644
--- a/doc/LectureNotes/_build/html/exercisesweek36.html
+++ b/doc/LectureNotes/_build/html/exercisesweek36.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
@@ -479,10 +479,10 @@ defining a new cost function to be optimized, that is
      
 
      Exercise 3 - Scaling data#
      
 
      
 
      
-
      import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn.preprocessing import StandardScaler
+
      import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn.preprocessing import StandardScaler
 
      
 
      
 
      
@@ -499,7 +499,7 @@ defining a new cost function to be optimized, that is
      
 
      a) Adapt your function from last week to only include the intercept column if the boolean argument intercept is set to true.
      
 
      
 
      
-
      def polynomial_features(x, p, intercept=False):
+
      def polynomial_features(x, p, intercept=False):
     n = len(x)
     X = np.zeros((n, p + 1))
     #X[:, 0] = ...
@@ -512,7 +512,7 @@ defining a new cost function to be optimized, that is
      
 
      
 
      
 
      
-
      def polynomial_features(x, p, intercept=False):
+
      def polynomial_features(x, p, intercept=False):
     n = len(x)
     X = np.zeros((n, p))
     X[:, 0] = x[:]
@@ -558,7 +558,7 @@ defining a new cost function to be optimized, that is
      
 
      a) Implement a function for computing the optimal Ridge parameters using the expression from 2a).
      
 
      
 
      
-
      def Ridge_parameters(X, y):
+
      def Ridge_parameters(X, y):
     # Assumes X is scaled and has no intercept column
     return np.linalg.inv(X.T @ X) @ X.T @ y
 
diff --git a/doc/LectureNotes/_build/html/exercisesweek37.html b/doc/LectureNotes/_build/html/exercisesweek37.html
index d3154480e..2103619d4 100644
--- a/doc/LectureNotes/_build/html/exercisesweek37.html
+++ b/doc/LectureNotes/_build/html/exercisesweek37.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
@@ -596,7 +596,7 @@ Then we compute the target values \(y
 
      Below is the code to generate the dataset:
      
 
      
 
      
-
      import numpy as np
+
      import numpy as np
 
 # Set random seed for reproducibility
 np.random.seed(0)
diff --git a/doc/LectureNotes/_build/html/exercisesweek38.html b/doc/LectureNotes/_build/html/exercisesweek38.html
index f506799d9..68f6263b7 100644
--- a/doc/LectureNotes/_build/html/exercisesweek38.html
+++ b/doc/LectureNotes/_build/html/exercisesweek38.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
@@ -537,7 +537,7 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_
 
      a) Using the expression above, compute the mean squared error, bias and variance of the given data. Check that the sum of the bias and variance correctly gives (approximately) the mean squared error.
      
 
      
 
      
-
      import numpy as np
+
      import numpy as np
 
 n = 100
 bootstraps = 1000
@@ -558,15 +558,15 @@ C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_
 
      d) Perform a bias-variance analysis of a polynomial OLS model fit to a one-dimensional function by computing and plotting the bias and variances values as a function of the polynomial degree of your model.
      
 
      
 
      
-
      import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.preprocessing import (
+
      import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import (
     PolynomialFeatures,
 )  # use the fit_transform method of the created object!
-from sklearn.linear_model import LinearRegression
-from sklearn.metrics import mean_squared_error
-from sklearn.model_selection import train_test_split
-from sklearn.utils import resample
+from sklearn.linear_model import LinearRegression
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import train_test_split
+from sklearn.utils import resample
 
      
 
      
 
      
diff --git a/doc/LectureNotes/_build/html/exercisesweek39.html b/doc/LectureNotes/_build/html/exercisesweek39.html
index f06c079e0..a3f52ed0e 100644
--- a/doc/LectureNotes/_build/html/exercisesweek39.html
+++ b/doc/LectureNotes/_build/html/exercisesweek39.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/exercisesweek41.html b/doc/LectureNotes/_build/html/exercisesweek41.html
index 1aabb1657..33749734a 100644
--- a/doc/LectureNotes/_build/html/exercisesweek41.html
+++ b/doc/LectureNotes/_build/html/exercisesweek41.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
@@ -435,29 +435,29 @@ doconce format html exercisesweek41.do.txt  -->
 
      First, here are some functions you are going to need, don’t change this cell. If you are unable to import autograd, just swap in normal numpy until you want to do the final optional exercise.
      
 
      
 
      
-
      import autograd.numpy as np  # We need to use this numpy wrapper to make automatic differentiation work later
-from sklearn import datasets
-import matplotlib.pyplot as plt
-from sklearn.metrics import accuracy_score
+
      import autograd.numpy as np  # We need to use this numpy wrapper to make automatic differentiation work later
+from sklearn import datasets
+import matplotlib.pyplot as plt
+from sklearn.metrics import accuracy_score
 
 
 # Defining some activation functions
-def ReLU(z):
+def ReLU(z):
     return np.where(z > 0, z, 0)
 
 
-def sigmoid(z):
+def sigmoid(z):
     return 1 / (1 + np.exp(-z))
 
 
-def softmax(z):
+def softmax(z):
     """Compute softmax values for each set of scores in the rows of the matrix z.
     Used with batched input data."""
     e_z = np.exp(z - np.max(z, axis=0))
     return e_z / np.sum(e_z, axis=1)[:, np.newaxis]
 
 
-def softmax_vec(z):
+def softmax_vec(z):
     """Compute softmax values for each set of scores in the vector z.
     Use this function when you use the activation function on one vector at a time"""
     e_z = np.exp(z - np.max(z))
@@ -555,7 +555,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      a) Complete the function below so that it returns a list layers of weight and bias tuples (W, b) for each layer, in order, with the correct shapes that we can use later as our network parameters.
      
 
      
 
      
-
      def create_layers(network_input_size, layer_output_sizes):
+
      def create_layers(network_input_size, layer_output_sizes):
     layers = []
 
     i_size = network_input_size
@@ -573,7 +573,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      b) Comple the function below so that it evaluates the intermediary z and activation a for each layer, with ReLU actication, and returns the final activation a. This is the complete feed-forward pass, a full neural network!
      
 
      
 
      
-
      def feed_forward_all_relu(layers, input):
+
      def feed_forward_all_relu(layers, input):
     a = input
     for W, b in layers:
         z = ...
@@ -597,7 +597,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      
 
      
 
      
-
      d) Why is a neural network with no activation functions always mathematically equivelent to a neural network with only one layer?
      
+
      d) Why is a neural network with no activation functions mathematically equivelent to(can be reduced to) a neural network with only one layer?
      
 
 
      
 
      Exercise 4 - Custom activation for each layer#
      
@@ -605,7 +605,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      a) Complete the feed_forward function which accepts a list of activation functions as an argument, and which evaluates these activation functions at each layer.
      
 
      
 
      
-
      def feed_forward(input, layers, activation_funcs):
+
      def feed_forward(input, layers, activation_funcs):
     a = input
     for (W, b), activation_func in zip(layers, activation_funcs):
         z = ...
@@ -639,7 +639,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      a) Complete the function create_layers_batch so that the weight matrix is the transpose of what it was when you only sent in one input at a time.
      
 
      
 
      
-
      def create_layers_batch(network_input_size, layer_output_sizes):
+
      def create_layers_batch(network_input_size, layer_output_sizes):
     layers = []
 
     i_size = network_input_size
@@ -654,13 +654,13 @@ doconce format html exercisesweek41.do.txt  -->
 
      
 
      
 
      
-
      b) Make a matrix of inputs with the shape (number of features, number of inputs), you choose the number of inputs and features per input. Then complete the function feed_forward_batch so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)
      
+
      b) Make a matrix of inputs with the shape (number of inputs, number of features), you choose the number of inputs and features per input. Then complete the function feed_forward_batch so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)
      
 
      
 
      
 
      inputs = np.random.rand(1000, 4)
 
 
-def feed_forward_batch(inputs, layers, activation_funcs):
+def feed_forward_batch(inputs, layers, activation_funcs):
     a = inputs
     for (W, b), activation_func in zip(layers, activation_funcs):
         z = ...
@@ -670,7 +670,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      
 
      
 
      
-
      c) Create and evaluate a neural network with 4 inputs and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.
      
+
      c) Create and evaluate a neural network with 4 input features, and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.
      
 
      
 
      
 
      network_input_size = ...
@@ -715,7 +715,7 @@ doconce format html exercisesweek41.do.txt  -->
     targets[i, t] = 1
 
 
-def accuracy(predictions, targets):
+def accuracy(predictions, targets):
     one_hot_predictions = np.zeros(predictions.shape)
 
     for i, prediction in enumerate(predictions):
@@ -758,11 +758,11 @@ doconce format html exercisesweek41.do.txt  -->
 
      Since we are doing a classification task with multiple output classes, we use the cross-entropy loss function, which can evaluate performance on classification tasks. It sees if your prediction is “most certain” on the correct target.
      
 
      
 
      
-
      def cross_entropy(predict, target):
+
      def cross_entropy(predict, target):
     return np.sum(-target * np.log(predict))
 
 
-def cost(input, layers, activation_funcs, target):
+def cost(input, layers, activation_funcs, target):
     predict = feed_forward_batch(input, layers, activation_funcs)
     return cross_entropy(predict, target)
 
      
@@ -777,7 +777,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      Now we need to compute these gradients. This is pretty hard to do for a neural network, we will use most of next week to do this, but we can also use autograd to just do it for us, which is what we always do in practice. With the code cell below, we create a function which takes all of these gradients for us.
      
 
      
 
      
-
      from autograd import grad
+
      from autograd import grad
 
 
 gradient_func = grad(
@@ -801,7 +801,7 @@ doconce format html exercisesweek41.do.txt  -->
 
      c) Finish the train_network function.
      
 
      
 
      
-
      def train_network(
+
      def train_network(
     inputs, layers, activation_funcs, targets, learning_rate=0.001, epochs=100
 ):
     for i in range(epochs):
diff --git a/doc/LectureNotes/_build/html/genindex.html b/doc/LectureNotes/_build/html/genindex.html
index 79cf126ed..85875624a 100644
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+++ b/doc/LectureNotes/_build/html/genindex.html
@@ -27,7 +27,7 @@
 
 
 
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index 764500be4..3f95f618f 100644
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diff --git a/doc/LectureNotes/_build/html/schedule.html b/doc/LectureNotes/_build/html/schedule.html
index ee62b10b6..ca3c0dfc5 100644
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+++ b/doc/LectureNotes/_build/html/schedule.html
@@ -28,7 +28,7 @@
 
 
 
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index 9231737c1..9b1d0dfdb 100644
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Matrix": [[30, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [30, "correlation-matrix"]], "Correlation Matrix with Pandas": [[30, "correlation-matrix-with-pandas"]], "Counting the number of floating point operations": [[36, "counting-the-number-of-floating-point-operations"]], "Course Format": [[29, "course-format"]], "Course setting": [[25, null]], "Covariance Matrix Examples": [[30, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[30, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Cross-validation in brief": [[33, "cross-validation-in-brief"], [34, "cross-validation-in-brief"]], "Deadlines for projects (tentative)": [[29, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep Neural Networks": [[32, "deep-neural-networks"]], "Deep learning methods": [[29, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining intermediate operations": [[36, "defining-intermediate-operations"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Definitions": [[19, "definitions"], [36, "definitions"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"], [19, "deliverables"], [20, "deliverables"], [24, "deliverables"]], "Derivation of the AdaGrad Algorithm": [[32, "derivation-of-the-adagrad-algorithm"]], "Derivative of the cost function": [[36, "derivative-of-the-cost-function"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"], [36, "derivatives-and-the-chain-rule"]], "Derivatives in terms of z_j^L": [[36, "derivatives-in-terms-of-z-j-l"]], "Derivatives of the hidden layer": [[36, "derivatives-of-the-hidden-layer"]], "Derivatives, example 1": [[30, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"], [33, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the  Lasso Regression Equations": [[30, "deriving-the-lasso-regression-equations"], [31, "deriving-the-lasso-regression-equations"], [31, "id6"]], "Deriving the  Ridge Regression Equations": [[30, "deriving-the-ridge-regression-equations"], [31, "deriving-the-ridge-regression-equations"], [31, "id3"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[29, "discriminative-modeling"]], "Discussing the correlation data": [[35, "discussing-the-correlation-data"]], "Does Logistic Regression do a better Job?": [[35, "does-logistic-regression-do-a-better-job"]], "Domains and probabilities": [[26, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[30, "economy-size-svd"], [31, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[26, null]], "Empirical Evidence: Convergence Time and Memory in Practice": [[32, "empirical-evidence-convergence-time-and-memory-in-practice"]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[29, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[30, "example-2"]], "Example 3": [[30, "example-3"]], "Example 4": [[30, "example-4"]], "Example Matrix": [[30, "example-matrix"], [31, "example-matrix"]], "Example code for Bias-Variance tradeoff": [[33, "example-code-for-bias-variance-tradeoff"]], "Example code for Logistic Regression": [[34, "example-code-for-logistic-regression"], [35, "example-code-for-logistic-regression"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[29, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[29, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[30, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[30, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[29, "examples"]], "Examples of XOR, OR and AND gates": [[35, "examples-of-xor-or-and-and-gates"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Examples of likelihood functions used in logistic regression and nueral networks": [[34, "examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks"]], "Exercise 1": [[21, "exercise-1"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1, scale your data": [[18, "exercise-1-scale-your-data"]], "Exercise 1: Creating the report document": [[20, "exercise-1-creating-the-report-document"]], "Exercise 1: Expectation values for ordinary least squares expressions": [[19, "exercise-1-expectation-values-for-ordinary-least-squares-expressions"]], "Exercise 1: Including more data": [[36, "exercise-1-including-more-data"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2": [[21, "exercise-2"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2, calculate the gradients": [[18, "exercise-2-calculate-the-gradients"]], "Exercise 2: Adding good figures": [[20, "exercise-2-adding-good-figures"]], "Exercise 2: Expectation values for Ridge regression": [[19, "exercise-2-expectation-values-for-ridge-regression"]], "Exercise 2: Extended program": [[36, "exercise-2-extended-program"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3": [[21, "exercise-3"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters \\boldsymbol{\\theta}": [[18, "exercise-3-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta"]], "Exercise 3: Deriving the expression for the Bias-Variance Trade-off": [[19, "exercise-3-deriving-the-expression-for-the-bias-variance-trade-off"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 3: Writing an abstract and introduction": [[20, "exercise-3-writing-an-abstract-and-introduction"]], "Exercise 4 - Custom activation for each layer": [[21, "exercise-4-custom-activation-for-each-layer"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4, Implementing the simplest form for gradient descent": [[18, "exercise-4-implementing-the-simplest-form-for-gradient-descent"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 4: Computing the Bias and Variance": [[19, "exercise-4-computing-the-bias-and-variance"]], "Exercise 4: Making the code available and presentable": [[20, "exercise-4-making-the-code-available-and-presentable"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5 - Processing multiple inputs at once": [[21, "exercise-5-processing-multiple-inputs-at-once"]], "Exercise 5, Ridge regression and a new Synthetic Dataset": [[18, "exercise-5-ridge-regression-and-a-new-synthetic-dataset"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise 5: Interpretation of scaling and metrics": [[19, "exercise-5-interpretation-of-scaling-and-metrics"]], "Exercise 5: Referencing": [[20, "exercise-5-referencing"]], "Exercise 6 - Predicting on real data": [[21, "exercise-6-predicting-on-real-data"]], "Exercise 7 - Training on real data (Optional)": [[21, "exercise-7-training-on-real-data-optional"]], "Exercise:  Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function  with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function  with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Exercises week 36": [[17, null]], "Exercises week 37": [[18, null]], "Exercises week 38": [[19, null]], "Exercises week 39": [[20, null]], "Exercises week 41": [[21, null]], "Expectation value and variance": [[33, "expectation-value-and-variance"]], "Expectation value and variance for \\boldsymbol{\\theta}": [[33, "expectation-value-and-variance-for-boldsymbol-theta"]], "Expectation values": [[26, "expectation-values"]], "Explicit derivatives": [[36, "explicit-derivatives"]], "Extending to more predictors": [[34, "extending-to-more-predictors"], [35, "extending-to-more-predictors"]], "Extending to more than one variable": [[31, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[29, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"], [35, "feed-forward-neural-networks"], [36, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"], [36, "final-back-propagating-equation"]], "Final derivatives": [[36, "final-derivatives"]], "Final expression": [[36, "final-expression"]], "Final expressions for the biases of the hidden layer": [[36, "final-expressions-for-the-biases-of-the-hidden-layer"]], "Finding the Limit": [[33, "finding-the-limit"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "First network example, simple percepetron with one input": [[36, "first-network-example-simple-percepetron-with-one-input"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[30, "fixing-the-singularity"], [31, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[24, "format-for-electronic-delivery-of-report-and-programs"]], "Forward and reverse modes": [[36, "forward-and-reverse-modes"]], "Frequently used scaling functions": [[30, "frequently-used-scaling-functions"], [32, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[31, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[30, "functionality-in-scikit-learn"], [32, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [30, "further-properties-important-for-our-analyses-later"], [31, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[23, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[29, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[29, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [29, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[29, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Getting serious, the  back propagation equations for a neural network": [[36, "getting-serious-the-back-propagation-equations-for-a-neural-network"]], "Getting started with project 1": [[20, "getting-started-with-project-1"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient Descent Example": [[31, "id1"], [32, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Gradient descent and Ridge": [[31, "gradient-descent-and-ridge"], [32, "gradient-descent-and-ridge"]], "Gradient descent and revisiting Ordinary Least Squares from last week": [[32, "gradient-descent-and-revisiting-ordinary-least-squares-from-last-week"]], "Gradient descent example": [[31, "gradient-descent-example"], [32, "gradient-descent-example"]], "Gradient expressions": [[36, "gradient-expressions"]], "Grading": [[27, "grading"], [27, "id2"], [29, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Identifying Terms": [[33, "identifying-terms"]], "Illustration of a single perceptron model and a multi-perceptron model": [[35, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"], [36, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"]], "Important Matrix and vector handling packages": [[23, "important-matrix-and-vector-handling-packages"]], "Important observations": [[36, "important-observations"]], "Important technicalities: More on Rescaling data": [[30, "important-technicalities-more-on-rescaling-data"]], "Improving gradient descent with momentum": [[32, "improving-gradient-descent-with-momentum"]], "Improving performance": [[1, "improving-performance"]], "In general not this simple": [[36, "in-general-not-this-simple"]], "In summary": [[27, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"], [32, "including-stochastic-gradient-descent-with-autograd"]], "Including more classes": [[34, "including-more-classes"], [35, "including-more-classes"]], "Incremental PCA": [[11, "incremental-pca"]], "Independent and Identically Distributed (iid)": [[33, "independent-and-identically-distributed-iid"]], "Inputs to the activation function": [[36, "inputs-to-the-activation-function"]], "Installing R, C++, cython or Julia": [[29, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[29, "installing-r-c-cython-numba-etc"]], "Instructor information": [[27, "instructor-information"]], "Interpretations and optimizing our parameters": [[29, "interpretations-and-optimizing-our-parameters"], [29, "id2"], [29, "id3"], [30, "interpretations-and-optimizing-our-parameters"], [30, "id1"], [30, "id2"]], "Interpreting the Ridge results": [[30, "interpreting-the-ridge-results"], [31, "interpreting-the-ridge-results"], [31, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [30, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [22, "introduction"], [23, "introduction"]], "Introduction to Neural networks": [[35, "introduction-to-neural-networks"], [36, "introduction-to-neural-networks"]], "Introduction to numerical projects": [[24, "introduction-to-numerical-projects"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[23, "lu-decomposition-the-inverse-of-a-matrix"]], "Lab sessions Tuesday and Wednesday": [[35, "lab-sessions-tuesday-and-wednesday"]], "Lab sessions on Tuesday and Wednesday": [[36, "lab-sessions-on-tuesday-and-wednesday"]], "Lab sessions week 39": [[34, "lab-sessions-week-39"]], "Lasso Regression": [[31, "lasso-regression"]], "Lasso case": [[31, "lasso-case"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Layout of a neural network with three hidden layers": [[36, "layout-of-a-neural-network-with-three-hidden-layers"]], "Layout of a simple neural network with no hidden layer": [[36, "layout-of-a-simple-neural-network-with-no-hidden-layer"]], "Layout of a simple neural network with one hidden layer": [[36, "layout-of-a-simple-neural-network-with-one-hidden-layer"]], "Layout of a simple neural network with two input nodes, one  hidden layer and one output node": [[36, "layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-and-one-output-node"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"], [18, "learning-goals"], [19, "learning-goals"], [20, "learning-goals"]], "Learning outcomes": [[22, "learning-outcomes"], [29, "learning-outcomes"]], "Lecture Monday  October 6": [[36, "lecture-monday-october-6"]], "Lecture Monday September 29, 2025": [[35, "lecture-monday-september-29-2025"]], "Lecture material": [[34, "lecture-material"]], "Lectures and ComputerLab": [[29, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[23, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[30, "linear-regression-problems"], [31, "linear-regression-problems"]], "Linear Regression and  the SVD": [[31, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linear classifier": [[34, "linear-classifier"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"], [33, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [30, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[28, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"], [34, "logistic-regression"]], "Logistic Regression, from last week": [[35, "logistic-regression-from-last-week"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[29, "machine-learning"]], "Machine learning": [[22, "machine-learning"]], "Main textbooks": [[29, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[30, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[30, "material-for-exercises-week-35"]], "Material for lab sessions  sessions Tuesday and Wednesday": [[31, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[31, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 8": [[32, "material-for-lecture-monday-september-8"]], "Material for the lab sessions": [[32, "material-for-the-lab-sessions"], [33, "material-for-the-lab-sessions"]], "Material for the lecture on Monday October 6, 2025": [[36, "material-for-the-lecture-on-monday-october-6-2025"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [30, "mathematical-interpretation-of-ordinary-least-squares"], [31, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical model": [[35, "mathematical-model"], [35, "id1"], [35, "id2"], [35, "id3"], [35, "id4"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of deep learning": [[36, "mathematics-of-deep-learning"]], "Mathematics of deep learning and neural networks": [[36, "mathematics-of-deep-learning-and-neural-networks"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [30, "mathematics-of-the-svd-and-implications"], [31, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[29, "matrices-in-python"]], "Matrix  multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation": [[35, "matrix-vector-notation"]], "Matrix-vector notation  and activation": [[12, "matrix-vector-notation-and-activation"], [35, "matrix-vector-notation-and-activation"]], "Maximum Likelihood Estimation (MLE)": [[33, "maximum-likelihood-estimation-mle"]], "Maximum likelihood": [[34, "maximum-likelihood"], [35, "maximum-likelihood"]], "Meet the  covariance!": [[26, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [30, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[30, "meet-the-hessian-matrix"]], "Meet the Pandas": [[29, "meet-the-pandas"]], "Memory Usage and Scalability": [[32, "memory-usage-and-scalability"]], "Memory constraints": [[32, "memory-constraints"]], "Min-Max Scaling": [[30, "min-max-scaling"]], "Minimizing the cross entropy": [[34, "minimizing-the-cross-entropy"], [35, "minimizing-the-cross-entropy"]], "Momentum based GD": [[13, "momentum-based-gd"], [32, "momentum-based-gd"]], "More classes": [[34, "more-classes"], [35, "more-classes"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More complicated function": [[36, "more-complicated-function"]], "More considerations": [[36, "more-considerations"]], "More examples on bootstrap and cross-validation and errors": [[33, "more-examples-on-bootstrap-and-cross-validation-and-errors"], [34, "more-examples-on-bootstrap-and-cross-validation-and-errors"]], "More interpretations": [[30, "more-interpretations"], [31, "more-interpretations"], [31, "id5"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More on Steepest descent": [[31, "more-on-steepest-descent"]], "More on convex functions": [[31, "more-on-convex-functions"]], "More on the general approximation theorem": [[36, "more-on-the-general-approximation-theorem"]], "More preprocessing": [[30, "more-preprocessing"], [32, "more-preprocessing"]], "Motivation for Adaptive Step Sizes": [[32, "motivation-for-adaptive-step-sizes"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"], [35, "multilayer-perceptrons"], [36, "multilayer-perceptrons"]], "Multivariable functions": [[36, "multivariable-functions"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural network types": [[35, "neural-network-types"], [36, "neural-network-types"]], "Neural networks": [[12, null]], "New expression for the derivative": [[36, "new-expression-for-the-derivative"]], "Non-Convex Problems": [[32, "non-convex-problems"]], "Note about SVD Calculations": [[30, "note-about-svd-calculations"], [31, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[31, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[26, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[23, "numpy-and-arrays"], [29, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[29, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and Deep learning": [[34, "optimization-and-deep-learning"], [35, "optimization-and-deep-learning"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[31, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null], [34, "optimization-the-central-part-of-any-machine-learning-algortithm"], [35, "optimization-the-central-part-of-any-machine-learning-algortithm"]], "Optimizing our parameters": [[29, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[29, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[1, "optimizing-the-cost-function"]], "Optimizing the parameters": [[36, "optimizing-the-parameters"]], "Organizing our data": [[0, "organizing-our-data"], [29, "organizing-our-data"]], "Other Matrix and Vector Operations": [[23, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[4, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning  at UiO": [[29, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning  at UiO, contn": [[29, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other ingredients of a neural network": [[36, "other-ingredients-of-a-neural-network"]], "Other measures in classification studies": [[35, "other-measures-in-classification-studies"]], "Other parameters": [[36, "other-parameters"]], "Other popular texts": [[29, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"], [35, "other-types-of-networks"], [36, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[29, "our-model-for-the-nuclear-binding-energies"]], "Output layer": [[36, "output-layer"]], "Overarching aims of the exercises this week": [[21, "overarching-aims-of-the-exercises-this-week"]], "Overarching view of a neural network": [[36, "overarching-view-of-a-neural-network"]], "Overview of first week": [[29, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent (SGD)": [[32, "overview-video-on-stochastic-gradient-descent-sgd"]], "Own code for Ordinary Least Squares": [[29, "own-code-for-ordinary-least-squares"], [30, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[29, "pandas-ai"]], "Parameters of neural networks": [[36, "parameters-of-neural-networks"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[24, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[24, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[24, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and more advanced ways to update the learning the rate": [[24, "part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate"]], "Part e: Writing our own code for Lasso regression": [[24, "part-e-writing-our-own-code-for-lasso-regression"]], "Part f: Stochastic gradient descent": [[24, "part-f-stochastic-gradient-descent"]], "Part g: Bias-variance trade-off and resampling techniques": [[24, "part-g-bias-variance-trade-off-and-resampling-techniques"]], "Part h): Cross-validation as resampling techniques, adding more complexity": [[24, "part-h-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Partial Differential Equations": [[2, "partial-differential-equations"]], "Plan for week 39, September 22-26, 2025": [[34, "plan-for-week-39-september-22-26-2025"]], "Plan for week 41, October 6-10": [[36, "plan-for-week-41-october-6-10"]], "Plans for week 35": [[30, "plans-for-week-35"]], "Plans for week 36": [[31, "plans-for-week-36"]], "Plans for week 37, lecture Monday": [[32, "plans-for-week-37-lecture-monday"]], "Plans for week 38, lecture Monday September 15": [[33, "plans-for-week-38-lecture-monday-september-15"]], "Plotting the Histogram": [[33, "plotting-the-histogram"]], "Plotting the mean value for each group": [[34, "plotting-the-mean-value-for-each-group"]], "Practical tips": [[13, "practical-tips"], [32, "practical-tips"]], "Practicalities": [[27, "practicalities"], [27, "id1"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[24, "preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools"]], "Predicting New Points With A Trained Recurrent Neural Network": [[4, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[30, "preprocessing-our-data"]], "Prerequisites": [[29, "prerequisites"]], "Prerequisites and background": [[22, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[3, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[26, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[31, "program-example-for-gradient-descent-with-ridge-regression"], [32, "program-example-for-gradient-descent-with-ridge-regression"]], "Program for stochastic gradient": [[13, "program-for-stochastic-gradient"]], "Project 1 on Machine Learning, deadline October 6 (midnight), 2025": [[24, null]], "Properties of PDFs": [[26, "properties-of-pdfs"]], "Pros and cons": [[32, "pros-and-cons"]], "Pros and cons of trees, pros": [[9, "pros-and-cons-of-trees-pros"]], "Python installers": [[22, "python-installers"], [29, "python-installers"]], "RMS prop": [[13, "rms-prop"]], "RMSProp algorithm, taken from Goodfellow et al": [[32, "rmsprop-algorithm-taken-from-goodfellow-et-al"]], "RMSProp: Adaptive Learning Rates": [[32, "rmsprop-adaptive-learning-rates"]], "RMSprop for adaptive learning rate with Stochastic Gradient Descent": [[32, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "Random Numbers": [[26, "random-numbers"]], "Random forests": [[10, "random-forests"]], "Randomized PCA": [[11, "randomized-pca"]], "Reading material": [[29, "reading-material"]], "Reading recommendations:": [[30, "reading-recommendations"]], "Reading suggestions week 34": [[29, "reading-suggestions-week-34"]], "Readings and Videos": [[33, "readings-and-videos"]], "Readings and Videos, logistic regression": [[34, "readings-and-videos-logistic-regression"]], "Readings and Videos, resampling methods": [[34, "readings-and-videos-resampling-methods"]], "Readings and Videos:": [[32, "readings-and-videos"], [36, "readings-and-videos"]], "Recurrent neural networks": [[12, "recurrent-neural-networks"], [35, "recurrent-neural-networks"], [36, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[4, null]], "Reducing the number of degrees of freedom, overarching view": [[0, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [30, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reducing the number of operations": [[36, "reducing-the-number-of-operations"]], "Reformulating the problem": [[2, "reformulating-the-problem"]], "Regression Case": [[10, "regression-case"]], "Regression analysis and resampling methods": [[24, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[29, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[29, "regression-analysis-overarching-aims-ii"]], "Regularization": [[1, "regularization"]], "Relevance": [[35, "relevance"]], "Reminder from last week": [[30, "reminder-from-last-week"]], "Reminder on Newton-Raphson\u2019s method": [[31, "reminder-on-newton-raphson-s-method"]], "Reminder on Statistics": [[6, "reminder-on-statistics"]], "Reminder on books with hands-on material and codes": [[36, "reminder-on-books-with-hands-on-material-and-codes"]], "Reminder on different scaling methods": [[32, "reminder-on-different-scaling-methods"]], "Reminder on the chain rule and gradients": [[36, "reminder-on-the-chain-rule-and-gradients"]], "Replace or not": [[13, "replace-or-not"], [32, "replace-or-not"]], "Required Technologies": [[22, "required-technologies"]], "Resampling Methods": [[6, null]], "Resampling and the Bias-Variance Trade-off": [[19, "resampling-and-the-bias-variance-trade-off"]], "Resampling approaches can be computationally expensive": [[33, "resampling-approaches-can-be-computationally-expensive"], [34, "resampling-approaches-can-be-computationally-expensive"]], "Resampling methods": [[6, "id1"], [33, "resampling-methods"], [33, "id2"], [34, "resampling-methods"], [34, "id1"]], "Resampling methods: Bootstrap": [[33, "resampling-methods-bootstrap"], [34, "resampling-methods-bootstrap"]], "Resampling methods: Bootstrap approach": [[33, "resampling-methods-bootstrap-approach"]], "Resampling methods: Bootstrap background": [[33, "resampling-methods-bootstrap-background"]], "Resampling methods: Bootstrap steps": [[33, "resampling-methods-bootstrap-steps"]], "Resampling methods: More Bootstrap background": [[33, "resampling-methods-more-bootstrap-background"]], "Residual Error": [[30, "residual-error"], [31, "residual-error"]], "Resources on differential equations and deep learning": [[2, "resources-on-differential-equations-and-deep-learning"]], "Revisiting Ordinary Least Squares": [[31, "revisiting-ordinary-least-squares"]], "Revisiting our Linear Regression Solvers": [[13, "revisiting-our-linear-regression-solvers"]], "Revisiting our Logistic Regression case": [[34, "revisiting-our-logistic-regression-case"], [35, "revisiting-our-logistic-regression-case"]], "Rewriting the Covariance and/or Correlation Matrix": [[30, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the \\delta-function": [[33, "rewriting-the-delta-function"]], "Rewriting the fitting procedure as a linear algebra problem": [[29, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[29, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[31, "ridge-regression"]], "Ridge and LASSO Regression": [[30, "ridge-and-lasso-regression"], [31, "ridge-and-lasso-regression"], [31, "id2"]], "Ridge and Lasso Regression": [[5, null], [5, "id1"]], "SGD example": [[32, "sgd-example"]], "SGD vs Full-Batch GD: Convergence Speed and Memory Comparison": [[32, "sgd-vs-full-batch-gd-convergence-speed-and-memory-comparison"]], "SVD analysis": [[31, "svd-analysis"]], "Same code but now with momentum gradient descent": [[13, "same-code-but-now-with-momentum-gradient-descent"], [32, "same-code-but-now-with-momentum-gradient-descent"], [32, "id3"], [32, "id4"]], "Schedule first week": [[29, "schedule-first-week"]], "Schematic Regression Procedure": [[9, "schematic-regression-procedure"]], "Second moment of the gradient": [[32, "second-moment-of-the-gradient"]], "September 15-19": [[19, "september-15-19"]], "Setting up the Back propagation algorithm": [[12, "setting-up-the-back-propagation-algorithm"]], "Setting up the Back propagation algorithm, part 3": [[36, "setting-up-the-back-propagation-algorithm-part-3"]], "Setting up the Matrix to be inverted": [[30, "setting-up-the-matrix-to-be-inverted"], [31, "setting-up-the-matrix-to-be-inverted"]], "Setting up the back propagation algorithm": [[36, "setting-up-the-back-propagation-algorithm"]], "Setting up the back propagation algorithm, part 2": [[36, "setting-up-the-back-propagation-algorithm-part-2"]], "Setting up the equations for a neural network": [[36, "setting-up-the-equations-for-a-neural-network"]], "Setting up the network using Autograd; The full program": [[2, "setting-up-the-network-using-autograd-the-full-program"]], "Similar (second order function now) problem but now with AdaGrad": [[13, "similar-second-order-function-now-problem-but-now-with-adagrad"], [32, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[9, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[30, "simple-case"], [31, "simple-case"]], "Simple code for solving the above problem": [[31, "simple-code-for-solving-the-above-problem"]], "Simple example": [[34, "simple-example"], [36, "simple-example"]], "Simple example code": [[32, "simple-example-code"]], "Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression": [[31, "simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression"]], "Simple geometric interpretation": [[31, "simple-geometric-interpretation"]], "Simple linear regression model using scikit-learn": [[0, "simple-linear-regression-model-using-scikit-learn"], [29, "simple-linear-regression-model-using-scikit-learn"]], "Simple neural network and the  back propagation equations": [[36, "simple-neural-network-and-the-back-propagation-equations"]], "Simple one-dimensional second-order polynomial": [[18, "simple-one-dimensional-second-order-polynomial"]], "Simple program": [[31, "simple-program"], [32, "simple-program"]], "Simpler examples first, and automatic differentiation": [[36, "simpler-examples-first-and-automatic-differentiation"]], "Slightly different approach": [[32, "slightly-different-approach"]], "Smarter way of evaluating the above function": [[36, "smarter-way-of-evaluating-the-above-function"]], "Sneaking in automatic differentiation using Autograd": [[32, "sneaking-in-automatic-differentiation-using-autograd"]], "Software and needed installations": [[24, "software-and-needed-installations"], [29, "software-and-needed-installations"]], "Solving Differential Equations  with Deep Learning": [[2, null]], "Solving the one dimensional Poisson equation": [[2, "solving-the-one-dimensional-poisson-equation"]], "Solving the wave equation with Neural Networks": [[2, "solving-the-wave-equation-with-neural-networks"]], "Solving using Newton-Raphson\u2019s method": [[34, "solving-using-newton-raphson-s-method"], [35, "solving-using-newton-raphson-s-method"]], "Some famous Matrices": [[23, "some-famous-matrices"]], "Some parallels from real analysis": [[36, "some-parallels-from-real-analysis"]], "Some selected properties": [[34, "some-selected-properties"]], "Some simple problems": [[13, "some-simple-problems"], [31, "some-simple-problems"]], "Some useful matrix and vector expressions": [[30, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[0, "splitting-our-data-in-training-and-test-data"], [30, "splitting-our-data-in-training-and-test-data"]], "Standard Approach based on the Normal Distribution": [[33, "standard-approach-based-on-the-normal-distribution"]], "Standard steepest descent": [[13, "standard-steepest-descent"]], "Statistical analysis": [[33, "statistical-analysis"], [34, "statistical-analysis"]], "Statistical analysis and optimization of data": [[22, "statistical-analysis-and-optimization-of-data"], [29, "statistical-analysis-and-optimization-of-data"]], "Steepest descent": [[13, "steepest-descent"], [31, "steepest-descent"]], "Stochastic Gradient Descent": [[32, "stochastic-gradient-descent"]], "Stochastic Gradient Descent (SGD)": [[13, "stochastic-gradient-descent-sgd"], [32, "stochastic-gradient-descent-sgd"]], "Stochastic variables and the main concepts, the discrete case": [[26, "stochastic-variables-and-the-main-concepts-the-discrete-case"]], "Strongly Convex Case": [[32, "strongly-convex-case"]], "Suggested readings and videos": [[35, "suggested-readings-and-videos"]], "Summing up": [[33, "summing-up"], [34, "summing-up"]], "Support Vector Machines, overarching aims": [[8, null]], "Synthetic data generation": [[34, "synthetic-data-generation"], [35, "synthetic-data-generation"]], "Systematic reduction": [[3, "systematic-reduction"]], "Teachers": [[29, "teachers"]], "Teachers and Grading": [[27, null]], "Teaching Assistants Fall semester 2023": [[27, "teaching-assistants-fall-semester-2023"]], "Tentative deadllines for projects": [[27, "tentative-deadllines-for-projects"]], "Testing the Means Squared Error as function of Complexity": [[0, "testing-the-means-squared-error-as-function-of-complexity"], [30, "testing-the-means-squared-error-as-function-of-complexity"]], "Textbooks": [[28, null]], "The Algorithm before theorem": [[11, "the-algorithm-before-theorem"]], "The Breast Cancer Data, now with Keras": [[1, "the-breast-cancer-data-now-with-keras"]], "The CART algorithm for Classification": [[9, "the-cart-algorithm-for-classification"]], "The CART algorithm for Regression": [[9, "the-cart-algorithm-for-regression"]], "The CIFAR01 data set": [[3, "the-cifar01-data-set"]], "The Central Limit Theorem": [[33, "the-central-limit-theorem"]], "The Hessian matrix": [[31, "the-hessian-matrix"], [32, "the-hessian-matrix"]], "The Hessian matrix for Ridge Regression": [[31, "the-hessian-matrix-for-ridge-regression"], [32, "the-hessian-matrix-for-ridge-regression"]], "The Jacobian": [[30, "the-jacobian"]], "The MNIST dataset again": [[3, "the-mnist-dataset-again"]], "The OLS case": [[31, "the-ols-case"]], "The RELU function family": [[1, "the-relu-function-family"]], "The Ridge case": [[31, "the-ridge-case"]], "The SVD, a Fantastic Algorithm": [[30, "the-svd-a-fantastic-algorithm"], [31, "the-svd-a-fantastic-algorithm"]], "The Softmax function": [[1, "the-softmax-function"]], "The \\chi^2 function": [[0, "the-chi-2-function"], [29, "the-chi-2-function"], [29, "id4"], [29, "id5"], [29, "id6"], [29, "id7"], [29, "id8"]], "The approximation theorem in words": [[36, "the-approximation-theorem-in-words"]], "The bias-variance tradeoff": [[6, "the-bias-variance-tradeoff"], [33, "the-bias-variance-tradeoff"], [34, "the-bias-variance-tradeoff"]], "The code for solving the ODE": [[2, "the-code-for-solving-the-ode"]], "The complete code with a simple data set": [[30, "the-complete-code-with-a-simple-data-set"]], "The cost function rewritten": [[34, "the-cost-function-rewritten"], [35, "the-cost-function-rewritten"]], "The cost/loss function": [[30, "the-cost-loss-function"]], "The course has two central parts": [[22, "the-course-has-two-central-parts"]], "The derivative of the cost/loss function": [[31, "the-derivative-of-the-cost-loss-function"], [32, "the-derivative-of-the-cost-loss-function"]], "The derivatives": [[36, "the-derivatives"]], "The equations": [[31, "the-equations"]], "The equations for ordinary least squares": [[30, "the-equations-for-ordinary-least-squares"]], "The equations to solve": [[34, "the-equations-to-solve"], [35, "the-equations-to-solve"]], "The first Case": [[31, "the-first-case"]], "The gradient step": [[32, "the-gradient-step"]], "The ideal": [[31, "the-ideal"]], "The logistic function": [[7, "the-logistic-function"], [34, "the-logistic-function"]], "The mean squared error and its derivative": [[30, "the-mean-squared-error-and-its-derivative"]], "The moons example": [[8, "the-moons-example"]], "The multilayer  perceptron (MLP)": [[12, "the-multilayer-perceptron-mlp"]], "The network with one input layer, specified number of hidden layers, and one output layer": [[2, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"]], "The optimization problem": [[36, "the-optimization-problem"]], "The ouput layer": [[36, "the-ouput-layer"]], "The plethora  of machine learning algorithms/methods": [[29, "the-plethora-of-machine-learning-algorithms-methods"]], "The same example but now with cross-validation": [[33, "the-same-example-but-now-with-cross-validation"], [34, "the-same-example-but-now-with-cross-validation"]], "The sensitiveness of the gradient descent": [[31, "the-sensitiveness-of-the-gradient-descent"]], "The singular value decomposition": [[5, "the-singular-value-decomposition"], [30, "the-singular-value-decomposition"], [31, "the-singular-value-decomposition"]], "The training": [[36, "the-training"]], "The two-dimensional case": [[8, "the-two-dimensional-case"]], "Theoretical Convergence Speed and convex optimization": [[32, "theoretical-convergence-speed-and-convex-optimization"]], "Time decay rate": [[32, "time-decay-rate"]], "To our real data: nuclear binding energies. 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"computation-of-gradients"]], "Computing the Gini index": [[53, "computing-the-gini-index"]], "Conditions on convex functions": [[75, "conditions-on-convex-functions"]], "Confidence Intervals": [[77, "confidence-intervals"]], "Conjugate gradient method": [[57, "conjugate-gradient-method"]], "Content with notebooks": [[24, null]], "Contents": [[30, "contents"]], "Contributors": [[33, "contributors"]], "Convergence rates": [[76, "convergence-rates"]], "Convex function": [[75, "convex-function"]], "Convex functions": [[57, "convex-functions"], [75, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[47, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[47, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[56, "convolutional-neural-network"], [79, "convolutional-neural-network"], [80, "convolutional-neural-network"]], "Convolutional Neural Networks": [[47, null]], "Correlation Function and Design/Feature Matrix": [[74, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[55, "correlation-matrix"], [74, "correlation-matrix"]], "Correlation Matrix with Pandas": [[74, "correlation-matrix-with-pandas"]], "Counting the number of floating point operations": [[80, "counting-the-number-of-floating-point-operations"]], "Course Format": [[73, "course-format"]], "Course setting": [[69, null]], "Covariance Matrix Examples": [[74, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[74, "covariance-and-correlation-matrix"]], "Create a notebook with MyST Markdown": [[23, "create-a-notebook-with-myst-markdown"]], "Creator": [[33, "creator"]], "Cross-validation": [[50, "cross-validation"]], "Cross-validation in brief": [[77, "cross-validation-in-brief"], [78, "cross-validation-in-brief"]], "Deadlines for projects (tentative)": [[73, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[53, null]], "Deep Neural Networks": [[76, "deep-neural-networks"]], "Deep learning methods": [[73, "deep-learning-methods"]], "Define model and architecture": [[45, "define-model-and-architecture"]], "Defining intermediate operations": [[80, "defining-intermediate-operations"]], "Defining the cost function": [[45, "defining-the-cost-function"]], "Definitions": [[63, "definitions"], [80, "definitions"]], "Deliverables": [[59, "deliverables"], [60, "deliverables"], [63, "deliverables"], [64, "deliverables"], [68, "deliverables"]], "Dependencies": [[29, "dependencies"]], "Derivation of the AdaGrad Algorithm": [[76, "derivation-of-the-adagrad-algorithm"]], "Derivative of the cost function": [[80, "derivative-of-the-cost-function"]], "Derivatives and the chain rule": [[56, "derivatives-and-the-chain-rule"], [80, "derivatives-and-the-chain-rule"]], "Derivatives in terms of z_j^L": [[80, "derivatives-in-terms-of-z-j-l"]], "Derivatives of the hidden layer": [[80, "derivatives-of-the-hidden-layer"]], "Derivatives, example 1": [[74, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[49, "deriving-ols-from-a-probability-distribution"], [77, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[60, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[61, "deriving-and-implementing-ridge-regression"]], "Deriving the  Lasso Regression Equations": [[74, "deriving-the-lasso-regression-equations"], [75, "deriving-the-lasso-regression-equations"], [75, "id6"]], "Deriving the  Ridge Regression Equations": [[74, "deriving-the-ridge-regression-equations"], [75, "deriving-the-ridge-regression-equations"], [75, "id3"]], "Deriving the back propagation code for a multilayer perceptron model": [[56, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[45, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[55, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[52, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[53, "disadvantages"]], "Discriminative Modeling": [[73, "discriminative-modeling"]], "Discussing the correlation data": [[79, "discussing-the-correlation-data"]], "Does Logistic Regression do a better Job?": [[79, "does-logistic-regression-do-a-better-job"]], "Domains and probabilities": [[70, "domains-and-probabilities"]], "Dropout": [[45, "dropout"]], "Economy-size SVD": [[74, "economy-size-svd"], [75, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[70, null]], "Empirical Evidence: Convergence Time and Memory in Practice": [[76, "empirical-evidence-convergence-time-and-memory-in-practice"]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[54, null]], "Entropy and the ID3 algorithm": [[53, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[73, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[45, "evaluate-model-performance-on-test-data"]], "Example": [[27, "example"]], "Example 2": [[74, "example-2"]], "Example 3": [[74, "example-3"]], "Example 4": [[74, "example-4"]], "Example Matrix": [[74, "example-matrix"], [75, "example-matrix"]], "Example code for Bias-Variance tradeoff": [[77, "example-code-for-bias-variance-tradeoff"]], "Example code for Logistic Regression": [[78, "example-code-for-logistic-regression"], [79, "example-code-for-logistic-regression"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[73, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[73, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[74, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[74, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[46, "example-exponential-decay"]], "Example: Population growth": [[46, "example-population-growth"]], "Example: The diffusion equation": [[46, "example-the-diffusion-equation"]], "Example: binary classification problem": [[45, "example-binary-classification-problem"]], "Examples": [[73, "examples"]], "Examples of XOR, OR and AND gates": [[79, "examples-of-xor-or-and-and-gates"]], "Examples of likelihood functions used in logistic regression and neural networks": [[51, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Examples of likelihood functions used in logistic regression and nueral networks": [[78, "examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks"]], "Exercise 1": [[65, "exercise-1"]], "Exercise 1 - Choice of model and degrees of freedom": [[61, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[60, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[59, "exercise-1-github-setup"]], "Exercise 1, scale your data": [[62, "exercise-1-scale-your-data"]], "Exercise 1: Creating the report document": [[64, "exercise-1-creating-the-report-document"]], "Exercise 1: Expectation values for ordinary least squares expressions": [[63, "exercise-1-expectation-values-for-ordinary-least-squares-expressions"]], "Exercise 1: Including more data": [[80, "exercise-1-including-more-data"]], "Exercise 1: Setting up various Python environments": [[44, "exercise-1-setting-up-various-python-environments"]], "Exercise 2": [[65, "exercise-2"]], "Exercise 2 - Deriving the expression for OLS": [[60, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[61, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[59, "exercise-2-setting-up-a-github-repository"]], "Exercise 2, calculate the gradients": [[62, "exercise-2-calculate-the-gradients"]], "Exercise 2: Adding good figures": [[64, "exercise-2-adding-good-figures"]], "Exercise 2: Expectation values for Ridge regression": [[63, "exercise-2-expectation-values-for-ridge-regression"]], "Exercise 2: Extended program": [[80, "exercise-2-extended-program"]], "Exercise 2: making your own data and exploring scikit-learn": [[44, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3": [[65, "exercise-3"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[60, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[59, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[61, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[59, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3, using the analytical formulae for OLS and Ridge regression to find the optimal paramters \\boldsymbol{\\theta}": [[62, "exercise-3-using-the-analytical-formulae-for-ols-and-ridge-regression-to-find-the-optimal-paramters-boldsymbol-theta"]], "Exercise 3: Deriving the expression for the Bias-Variance Trade-off": [[63, "exercise-3-deriving-the-expression-for-the-bias-variance-trade-off"]], "Exercise 3: Normalizing our data": [[44, "exercise-3-normalizing-our-data"]], "Exercise 3: Writing an abstract and introduction": [[64, "exercise-3-writing-an-abstract-and-introduction"]], "Exercise 4 - Custom activation for each layer": [[65, "exercise-4-custom-activation-for-each-layer"]], "Exercise 4 - Fitting a polynomial": [[60, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[61, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[61, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[59, "exercise-4-the-train-test-split"]], "Exercise 4, Implementing the simplest form for gradient descent": [[62, "exercise-4-implementing-the-simplest-form-for-gradient-descent"]], "Exercise 4: Adding Ridge Regression": [[44, "exercise-4-adding-ridge-regression"]], "Exercise 4: Computing the Bias and Variance": [[63, "exercise-4-computing-the-bias-and-variance"]], "Exercise 4: Making the code available and presentable": [[64, "exercise-4-making-the-code-available-and-presentable"]], "Exercise 5 - Comparing your code with sklearn": [[60, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5 - Processing multiple inputs at once": [[65, "exercise-5-processing-multiple-inputs-at-once"]], "Exercise 5, Ridge regression and a new Synthetic Dataset": [[62, "exercise-5-ridge-regression-and-a-new-synthetic-dataset"]], "Exercise 5: Analytical exercises": [[44, "exercise-5-analytical-exercises"]], "Exercise 5: Interpretation of scaling and metrics": [[63, "exercise-5-interpretation-of-scaling-and-metrics"]], "Exercise 5: Referencing": [[64, "exercise-5-referencing"]], "Exercise 6 - Predicting on real data": [[65, "exercise-6-predicting-on-real-data"]], "Exercise 7 - Training on real data (Optional)": [[65, "exercise-7-training-on-real-data-optional"]], "Exercise:  Cross-validation as resampling techniques, adding more complexity": [[50, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[50, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[50, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function  with resampling": [[50, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[50, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function  with resampling": [[50, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[44, "exercises"]], "Exercises and Projects": [[50, "exercises-and-projects"]], "Exercises week 34": [[59, null]], "Exercises week 35": [[60, null]], "Exercises week 36": [[61, null]], "Exercises week 37": [[62, null]], "Exercises week 38": [[63, null]], "Exercises week 39": [[64, null]], "Exercises week 41": [[65, null]], "Expectation value and variance": [[77, "expectation-value-and-variance"]], "Expectation value and variance for \\boldsymbol{\\theta}": [[77, "expectation-value-and-variance-for-boldsymbol-theta"]], "Expectation values": [[70, "expectation-values"]], "Explicit derivatives": [[80, "explicit-derivatives"]], "Extending to more predictors": [[78, "extending-to-more-predictors"], [79, "extending-to-more-predictors"]], "Extending to more than one variable": [[75, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[73, "extremely-useful-tools-strongly-recommended"]], "FAQ": [[29, "faq"]], "Features": [[29, "features"]], "Feed-forward neural networks": [[56, "feed-forward-neural-networks"], [79, "feed-forward-neural-networks"], [80, "feed-forward-neural-networks"]], "Feed-forward pass": [[45, "feed-forward-pass"]], "Final back propagating equation": [[56, "final-back-propagating-equation"], [80, "final-back-propagating-equation"]], "Final derivatives": [[80, "final-derivatives"]], "Final expression": [[80, "final-expression"]], "Final expressions for the biases of the hidden layer": [[80, "final-expressions-for-the-biases-of-the-hidden-layer"]], "Finding the Limit": [[77, "finding-the-limit"]], "Fine-tuning neural network hyperparameters": [[45, "fine-tuning-neural-network-hyperparameters"]], "First network example, simple percepetron with one input": [[80, "first-network-example-simple-percepetron-with-one-input"]], "Fitting an Equation of State for Dense Nuclear Matter": [[44, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[74, "fixing-the-singularity"], [75, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[68, "format-for-electronic-delivery-of-report-and-programs"]], "Forward and reverse modes": [[80, "forward-and-reverse-modes"]], "Frequently used scaling functions": [[74, "frequently-used-scaling-functions"], [76, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[75, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[56, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[74, "functionality-in-scikit-learn"], [76, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[47, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[49, "further-properties-important-for-our-analyses-later"], [74, "further-properties-important-for-our-analyses-later"], [75, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[67, "gaussian-elimination"]], "General Features": [[53, "general-features"]], "General linear models and linear algebra": [[73, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[73, 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"History": [[30, "history"]], "How to take derivatives of Matrix-Vector expressions": [[60, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[52, "hyperplanes-and-all-that"]], "Identifying Terms": [[77, "identifying-terms"]], "Illustration of a single perceptron model and a multi-perceptron model": [[79, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"], [80, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"]], "Important Matrix and vector handling packages": [[67, "important-matrix-and-vector-handling-packages"]], "Important observations": [[80, "important-observations"]], "Important technicalities: More on Rescaling data": [[74, "important-technicalities-more-on-rescaling-data"]], "Improving gradient descent with momentum": [[76, "improving-gradient-descent-with-momentum"]], "Improving performance": [[45, "improving-performance"]], "In general not this simple": [[80, "in-general-not-this-simple"]], "In summary": [[71, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[57, "including-stochastic-gradient-descent-with-autograd"], [76, "including-stochastic-gradient-descent-with-autograd"]], "Including more classes": [[78, "including-more-classes"], [79, "including-more-classes"]], "Incremental PCA": [[55, "incremental-pca"]], "Independent and Identically Distributed (iid)": [[77, "independent-and-identically-distributed-iid"]], "Inputs to the activation function": [[80, "inputs-to-the-activation-function"]], "Install": [[28, "install"]], "Installation": [[27, "installation"]], "Installing R, C++, cython or Julia": [[73, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[73, "installing-r-c-cython-numba-etc"]], "Instructor information": [[71, "instructor-information"]], "Interpretations and optimizing our parameters": [[73, "interpretations-and-optimizing-our-parameters"], [73, "id2"], [73, "id3"], [74, "interpretations-and-optimizing-our-parameters"], [74, "id1"], [74, "id2"]], "Interpreting the Ridge results": [[74, "interpreting-the-ridge-results"], [75, "interpreting-the-ridge-results"], [75, "id4"]], "Introducing JAX": [[57, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[55, "introducing-the-covariance-and-correlation-functions"], [74, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[44, "introduction"], [50, "introduction"], [66, "introduction"], [67, "introduction"]], "Introduction to Neural networks": [[79, "introduction-to-neural-networks"], [80, "introduction-to-neural-networks"]], "Introduction to numerical projects": [[68, "introduction-to-numerical-projects"]], "Iterative Fitting, Classification and AdaBoost": [[54, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[54, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[55, "kernel-pca"]], "Kernels and non-linearity": [[52, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[67, "lu-decomposition-the-inverse-of-a-matrix"]], "Lab sessions Tuesday and Wednesday": [[79, "lab-sessions-tuesday-and-wednesday"]], "Lab sessions on Tuesday and Wednesday": [[80, "lab-sessions-on-tuesday-and-wednesday"]], "Lab sessions week 39": [[78, "lab-sessions-week-39"]], "Lasso Regression": [[75, "lasso-regression"]], "Lasso case": [[75, "lasso-case"]], "Layers": [[45, "layers"]], "Layers used to build CNNs": [[47, "layers-used-to-build-cnns"]], "Layout of a neural network with three hidden layers": [[80, "layout-of-a-neural-network-with-three-hidden-layers"]], "Layout of a simple neural network with no hidden layer": [[80, "layout-of-a-simple-neural-network-with-no-hidden-layer"]], "Layout of a simple neural network with one hidden layer": [[80, "layout-of-a-simple-neural-network-with-one-hidden-layer"]], "Layout of a simple neural network with two input nodes, one  hidden layer and one output node": [[80, "layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-and-one-output-node"]], "Learn more": [[22, "learn-more"]], "Learning goals": [[59, "learning-goals"], [60, "learning-goals"], [61, "learning-goals"], [62, "learning-goals"], [63, "learning-goals"], [64, "learning-goals"]], "Learning outcomes": [[66, "learning-outcomes"], [73, "learning-outcomes"]], "Lecture Monday  October 6": [[80, "lecture-monday-october-6"]], "Lecture Monday September 29, 2025": [[79, "lecture-monday-september-29-2025"]], "Lecture material": [[78, "lecture-material"]], "Lectures and ComputerLab": [[73, "lectures-and-computerlab"]], "License": [[27, "license"], [28, "license"], [29, "license"]], "License for Sphinx": [[37, null]], "Licenses for incorporated software": [[37, "licenses-for-incorporated-software"]], "Limitations of supervised learning with deep networks": [[45, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[67, null]], "Linear Regression": [[44, null]], "Linear Regression Problems": [[74, "linear-regression-problems"], [75, "linear-regression-problems"]], "Linear Regression and  the SVD": [[75, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[44, "linear-regression-basic-elements"]], "Linear classifier": [[78, "linear-classifier"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[49, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[49, "linking-the-regression-analysis-with-a-statistical-interpretation"], [77, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[49, "linking-with-the-svd"], [74, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[72, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[51, null], [51, "id1"], [78, "logistic-regression"]], "Logistic Regression, from last week": [[79, "logistic-regression-from-last-week"]], "MNIST and GANs": [[48, "mnist-and-gans"]], "Machine Learning": [[73, "machine-learning"]], "Machine learning": [[66, "machine-learning"]], "Main differences": [[30, "main-differences"]], "Main textbooks": [[73, "main-textbooks"]], "Making a tree": [[53, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[54, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[74, "making-your-own-test-train-splitting"]], "Markdown + notebooks": [[24, "markdown-notebooks"]], "Markdown Files": [[22, null]], "Masked records": [[30, "masked-records"]], "Material for exercises week 35": [[74, "material-for-exercises-week-35"]], "Material for lab sessions  sessions Tuesday and Wednesday": [[75, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[75, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 8": [[76, "material-for-lecture-monday-september-8"]], "Material for the lab sessions": [[76, "material-for-the-lab-sessions"], [77, "material-for-the-lab-sessions"]], "Material for the lecture on Monday October 6, 2025": [[80, "material-for-the-lecture-on-monday-october-6-2025"]], "Mathematical Interpretation of Ordinary Least Squares": [[49, "mathematical-interpretation-of-ordinary-least-squares"], [74, "mathematical-interpretation-of-ordinary-least-squares"], [75, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical model": [[79, "mathematical-model"], [79, "id1"], [79, "id2"], [79, "id3"], [79, "id4"]], "Mathematical optimization of convex functions": [[52, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[47, "mathematics-of-cnns"]], "Mathematics of deep learning": [[80, "mathematics-of-deep-learning"]], "Mathematics of deep learning and neural networks": [[80, "mathematics-of-deep-learning-and-neural-networks"]], "Mathematics of the SVD and implications": [[49, "mathematics-of-the-svd-and-implications"], [74, "mathematics-of-the-svd-and-implications"], [75, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[73, "matrices-in-python"]], "Matrix  multiplication": [[45, "matrix-multiplication"]], "Matrix-vector notation": [[79, "matrix-vector-notation"]], "Matrix-vector notation  and activation": [[56, "matrix-vector-notation-and-activation"], [79, "matrix-vector-notation-and-activation"]], "Maximum Likelihood Estimation (MLE)": [[77, "maximum-likelihood-estimation-mle"]], "Maximum likelihood": [[78, "maximum-likelihood"], [79, "maximum-likelihood"]], "Meet the  covariance!": [[70, "meet-the-covariance"]], "Meet the Covariance Matrix": [[49, "meet-the-covariance-matrix"], [74, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[74, "meet-the-hessian-matrix"]], "Meet the Pandas": [[73, "meet-the-pandas"]], "Memory Usage and Scalability": [[76, "memory-usage-and-scalability"]], "Memory constraints": [[76, "memory-constraints"]], "Min-Max Scaling": [[74, "min-max-scaling"]], "Minimizing the cross entropy": [[78, "minimizing-the-cross-entropy"], [79, "minimizing-the-cross-entropy"]], "Momentum based GD": [[57, "momentum-based-gd"], [76, "momentum-based-gd"]], "More classes": [[78, "more-classes"], [79, "more-classes"]], "More complicated Example: The Ising model": [[50, "more-complicated-example-the-ising-model"]], "More complicated function": [[80, "more-complicated-function"]], "More considerations": [[80, "more-considerations"]], "More examples on bootstrap and cross-validation and errors": [[77, "more-examples-on-bootstrap-and-cross-validation-and-errors"], [78, "more-examples-on-bootstrap-and-cross-validation-and-errors"]], "More interpretations": [[74, "more-interpretations"], [75, "more-interpretations"], [75, "id5"]], "More on Dimensionalities": [[47, "more-on-dimensionalities"]], "More on Rescaling data": [[50, "more-on-rescaling-data"]], "More on Steepest descent": [[75, "more-on-steepest-descent"]], "More on convex functions": [[75, "more-on-convex-functions"]], "More on the general approximation theorem": [[80, "more-on-the-general-approximation-theorem"]], "More preprocessing": [[74, "more-preprocessing"], [76, "more-preprocessing"]], "Motivation for Adaptive Step Sizes": [[76, "motivation-for-adaptive-step-sizes"]], "Multilayer perceptrons": [[56, "multilayer-perceptrons"], [79, "multilayer-perceptrons"], [80, "multilayer-perceptrons"]], "Multivariable functions": [[80, "multivariable-functions"]], "MyST markdown": [[24, "myst-markdown"]], "NCSA Open Source License": [[31, null]], "Network requirements": [[46, "network-requirements"]], "Neural Networks vs CNNs": [[47, "neural-networks-vs-cnns"]], "Neural network types": [[79, "neural-network-types"], [80, "neural-network-types"]], "Neural networks": [[56, null]], "New expression for the derivative": [[80, "new-expression-for-the-derivative"]], "New features": [[30, "new-features"]], "Non-Convex Problems": [[76, "non-convex-problems"]], "Note about SVD Calculations": [[74, "note-about-svd-calculations"], [75, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[75, "note-on-scikit-learn"]], "Notebooks with MyST Markdown": [[23, null]], "Numerical experiments and the covariance, central limit theorem": [[70, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[67, "numpy-and-arrays"], [73, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[73, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and Deep learning": [[78, "optimization-and-deep-learning"], [79, "optimization-and-deep-learning"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[75, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[57, null], [78, "optimization-the-central-part-of-any-machine-learning-algortithm"], [79, "optimization-the-central-part-of-any-machine-learning-algortithm"]], "Optimizing maskedarray": [[30, "optimizing-maskedarray"]], "Optimizing our parameters": [[73, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[73, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[45, "optimizing-the-cost-function"]], "Optimizing the parameters": [[80, "optimizing-the-parameters"]], "Organizing our data": [[44, "organizing-our-data"], [73, "organizing-our-data"]], "Other Matrix and Vector Operations": [[67, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[48, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning  at UiO": [[73, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning  at UiO, contn": [[73, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other ingredients of a neural network": [[80, "other-ingredients-of-a-neural-network"]], "Other measures in classification studies": [[79, "other-measures-in-classification-studies"]], "Other parameters": [[80, "other-parameters"]], "Other popular texts": [[73, "other-popular-texts"]], "Other techniques": [[55, "other-techniques"]], "Other types of networks": [[56, "other-types-of-networks"], [79, "other-types-of-networks"], [80, "other-types-of-networks"]], "Other ways of visualizing the trees": [[53, "other-ways-of-visualizing-the-trees"]], "Our Copyright Policy": [[20, "our-copyright-policy"]], "Our model for the nuclear binding energies": [[73, "our-model-for-the-nuclear-binding-energies"]], "Output layer": [[80, "output-layer"]], "Overarching aims of the exercises this week": [[65, "overarching-aims-of-the-exercises-this-week"]], "Overarching view of a neural network": [[80, "overarching-view-of-a-neural-network"]], "Overview of first week": [[73, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent (SGD)": [[76, "overview-video-on-stochastic-gradient-descent-sgd"]], "Own code for Ordinary Least Squares": [[73, "own-code-for-ordinary-least-squares"], [74, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[55, "pca-and-scikit-learn"]], "Pandas AI": [[73, "pandas-ai"]], "Parameters of neural networks": [[80, "parameters-of-neural-networks"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[68, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[68, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[68, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and more advanced ways to update the learning the rate": [[68, "part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate"]], "Part e: Writing our own code for Lasso regression": [[68, "part-e-writing-our-own-code-for-lasso-regression"]], "Part f: Stochastic gradient descent": [[68, "part-f-stochastic-gradient-descent"]], "Part g: Bias-variance trade-off and resampling techniques": [[68, "part-g-bias-variance-trade-off-and-resampling-techniques"]], "Part h): Cross-validation as resampling techniques, adding more complexity": [[68, "part-h-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Partial Differential Equations": [[46, "partial-differential-equations"]], "Pitaya Smoothie": [[15, null]], "Plan for week 39, September 22-26, 2025": [[78, "plan-for-week-39-september-22-26-2025"]], "Plan for week 41, October 6-10": [[80, "plan-for-week-41-october-6-10"]], "Plans for week 35": [[74, "plans-for-week-35"]], "Plans for week 36": [[75, "plans-for-week-36"]], "Plans for week 37, lecture Monday": [[76, "plans-for-week-37-lecture-monday"]], "Plans for week 38, lecture Monday September 15": [[77, "plans-for-week-38-lecture-monday-september-15"]], "Plotting the Histogram": [[77, "plotting-the-histogram"]], "Plotting the mean value for each group": [[78, "plotting-the-mean-value-for-each-group"]], "Practical tips": [[57, "practical-tips"], [76, "practical-tips"]], "Practicalities": [[71, "practicalities"], [71, "id1"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[68, "preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools"]], "Predicting New Points With A Trained Recurrent Neural Network": [[48, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[74, "preprocessing-our-data"]], "Prerequisites": [[73, "prerequisites"]], "Prerequisites and background": [[66, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[47, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[70, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[75, "program-example-for-gradient-descent-with-ridge-regression"], [76, "program-example-for-gradient-descent-with-ridge-regression"]], "Program for stochastic gradient": [[57, "program-for-stochastic-gradient"]], "Project 1 on Machine Learning, deadline October 6 (midnight), 2025": [[68, null]], "Properties of PDFs": [[70, "properties-of-pdfs"]], "Pros and cons": [[76, "pros-and-cons"]], "Pros and cons of trees, pros": [[53, "pros-and-cons-of-trees-pros"]], "Python installers": [[66, "python-installers"], [73, "python-installers"]], "Quickly add YAML metadata for MyST Notebooks": [[23, "quickly-add-yaml-metadata-for-myst-notebooks"]], "RMS prop": [[57, "rms-prop"]], "RMSProp algorithm, taken from Goodfellow et al": [[76, "rmsprop-algorithm-taken-from-goodfellow-et-al"]], "RMSProp: Adaptive Learning Rates": [[76, "rmsprop-adaptive-learning-rates"]], "RMSprop for adaptive learning rate with Stochastic Gradient Descent": [[76, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "Random Numbers": [[70, "random-numbers"]], "Random forests": [[54, "random-forests"]], "Randomized PCA": [[55, "randomized-pca"]], "Reading material": [[73, "reading-material"]], "Reading recommendations:": [[74, "reading-recommendations"]], "Reading suggestions week 34": [[73, "reading-suggestions-week-34"]], "Readings and Videos": [[77, "readings-and-videos"]], "Readings and Videos, logistic regression": [[78, "readings-and-videos-logistic-regression"]], "Readings and Videos, resampling methods": [[78, "readings-and-videos-resampling-methods"]], "Readings and Videos:": [[76, "readings-and-videos"], [80, "readings-and-videos"]], "Recurrent neural networks": [[56, "recurrent-neural-networks"], [79, "recurrent-neural-networks"], [80, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[48, null]], "Reducing the number of degrees of freedom, overarching view": [[44, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [74, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reducing the number of operations": [[80, "reducing-the-number-of-operations"]], "Reformulating the problem": [[46, "reformulating-the-problem"]], "Regression Case": [[54, "regression-case"]], "Regression analysis and resampling methods": [[68, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[73, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[73, "regression-analysis-overarching-aims-ii"]], "Regularization": [[45, "regularization"]], "Relevance": [[79, "relevance"]], "Reminder from last week": [[74, "reminder-from-last-week"]], "Reminder on Newton-Raphson\u2019s method": [[75, "reminder-on-newton-raphson-s-method"]], "Reminder on Statistics": [[50, "reminder-on-statistics"]], "Reminder on books with hands-on material and codes": [[80, "reminder-on-books-with-hands-on-material-and-codes"]], "Reminder on different scaling methods": [[76, "reminder-on-different-scaling-methods"]], "Reminder on the chain rule and gradients": [[80, "reminder-on-the-chain-rule-and-gradients"]], "Replace or not": [[57, "replace-or-not"], [76, "replace-or-not"]], "Required Technologies": [[66, "required-technologies"]], "Resampling Methods": [[50, null]], "Resampling and the Bias-Variance Trade-off": [[63, "resampling-and-the-bias-variance-trade-off"]], "Resampling approaches can be computationally expensive": [[77, "resampling-approaches-can-be-computationally-expensive"], [78, "resampling-approaches-can-be-computationally-expensive"]], "Resampling methods": [[50, "id1"], [77, "resampling-methods"], [77, "id2"], [78, "resampling-methods"], [78, "id1"]], "Resampling methods: Bootstrap": [[77, "resampling-methods-bootstrap"], [78, "resampling-methods-bootstrap"]], "Resampling methods: Bootstrap approach": [[77, "resampling-methods-bootstrap-approach"]], "Resampling methods: Bootstrap background": [[77, "resampling-methods-bootstrap-background"]], "Resampling methods: Bootstrap steps": [[77, "resampling-methods-bootstrap-steps"]], "Resampling methods: More Bootstrap background": [[77, "resampling-methods-more-bootstrap-background"]], "Residual Error": [[74, "residual-error"], [75, "residual-error"]], "Resources on differential equations and deep learning": [[46, "resources-on-differential-equations-and-deep-learning"]], "Revision notes": [[30, "revision-notes"]], "Revisiting Ordinary Least Squares": [[75, "revisiting-ordinary-least-squares"]], "Revisiting our Linear Regression Solvers": [[57, "revisiting-our-linear-regression-solvers"]], "Revisiting our Logistic Regression case": [[78, "revisiting-our-logistic-regression-case"], [79, "revisiting-our-logistic-regression-case"]], "Rewriting the Covariance and/or Correlation Matrix": [[74, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the \\delta-function": [[77, "rewriting-the-delta-function"]], "Rewriting the fitting procedure as a linear algebra problem": [[73, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[73, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[75, "ridge-regression"]], "Ridge and LASSO Regression": [[74, "ridge-and-lasso-regression"], [75, "ridge-and-lasso-regression"], [75, "id2"]], "Ridge and Lasso Regression": [[49, null], [49, "id1"]], "SGD example": [[76, "sgd-example"]], "SGD vs Full-Batch GD: Convergence Speed and Memory Comparison": [[76, "sgd-vs-full-batch-gd-convergence-speed-and-memory-comparison"]], "SVD analysis": [[75, "svd-analysis"]], "Same code but now with momentum 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"setting-up-the-back-propagation-algorithm-part-2"]], "Setting up the equations for a neural network": [[80, "setting-up-the-equations-for-a-neural-network"]], "Setting up the network using Autograd; The full program": [[46, "setting-up-the-network-using-autograd-the-full-program"]], "Should masked arrays be filled before processing or not?": [[30, "should-masked-arrays-be-filled-before-processing-or-not"]], "Show me": [[29, "show-me"]], "Similar (second order function now) problem but now with AdaGrad": [[57, "similar-second-order-function-now-problem-but-now-with-adagrad"], [76, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[53, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[74, "simple-case"], [75, "simple-case"]], "Simple code for solving the above problem": [[75, "simple-code-for-solving-the-above-problem"]], "Simple example": [[78, "simple-example"], [80, "simple-example"]], "Simple example code": [[76, "simple-example-code"]], "Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression": [[75, "simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression"]], "Simple geometric interpretation": [[75, "simple-geometric-interpretation"]], "Simple linear regression model using scikit-learn": [[44, "simple-linear-regression-model-using-scikit-learn"], [73, "simple-linear-regression-model-using-scikit-learn"]], "Simple neural network and the  back propagation equations": [[80, "simple-neural-network-and-the-back-propagation-equations"]], "Simple one-dimensional second-order polynomial": [[62, "simple-one-dimensional-second-order-polynomial"]], "Simple program": [[75, "simple-program"], [76, "simple-program"]], "Simpler examples first, and automatic differentiation": [[80, "simpler-examples-first-and-automatic-differentiation"]], "Slightly different approach": [[76, "slightly-different-approach"]], "Smarter way of evaluating the above function": [[80, "smarter-way-of-evaluating-the-above-function"]], "Sneaking in automatic differentiation using Autograd": [[76, "sneaking-in-automatic-differentiation-using-autograd"]], "Software and needed installations": [[68, "software-and-needed-installations"], [73, "software-and-needed-installations"]], "Solving Differential Equations  with Deep Learning": [[46, null]], "Solving the one dimensional Poisson equation": [[46, "solving-the-one-dimensional-poisson-equation"]], "Solving the wave equation with Neural Networks": [[46, "solving-the-wave-equation-with-neural-networks"]], "Solving using Newton-Raphson\u2019s method": [[78, "solving-using-newton-raphson-s-method"], [79, "solving-using-newton-raphson-s-method"]], "Some famous Matrices": [[67, "some-famous-matrices"]], "Some parallels from real analysis": [[80, "some-parallels-from-real-analysis"]], "Some selected properties": [[78, "some-selected-properties"]], "Some simple problems": [[57, "some-simple-problems"], [75, "some-simple-problems"]], "Some useful matrix and vector expressions": [[74, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[44, "splitting-our-data-in-training-and-test-data"], [74, "splitting-our-data-in-training-and-test-data"]], "Standard Approach based on the Normal Distribution": [[77, "standard-approach-based-on-the-normal-distribution"]], "Standard steepest descent": [[57, "standard-steepest-descent"]], "Statistical analysis": [[77, "statistical-analysis"], [78, "statistical-analysis"]], "Statistical analysis and optimization of data": [[66, "statistical-analysis-and-optimization-of-data"], [73, "statistical-analysis-and-optimization-of-data"]], "Steepest descent": [[57, "steepest-descent"], [75, "steepest-descent"]], "Stochastic Gradient Descent": [[76, "stochastic-gradient-descent"]], "Stochastic Gradient Descent (SGD)": [[57, "stochastic-gradient-descent-sgd"], [76, "stochastic-gradient-descent-sgd"]], "Stochastic variables and the main concepts, the discrete case": [[70, "stochastic-variables-and-the-main-concepts-the-discrete-case"]], "Strongly Convex Case": [[76, "strongly-convex-case"]], "Structure of translation files": [[41, "structure-of-translation-files"]], "Suggested readings and videos": [[79, "suggested-readings-and-videos"]], "Summing up": [[77, "summing-up"], [78, "summing-up"]], "Support Vector Machines, overarching aims": [[52, null]], "Synthetic data generation": [[78, "synthetic-data-generation"], [79, "synthetic-data-generation"]], "Systematic reduction": [[47, "systematic-reduction"]], "Teachers": [[73, "teachers"]], "Teachers and Grading": [[71, null]], "Teaching Assistants Fall semester 2023": [[71, "teaching-assistants-fall-semester-2023"]], "Tentative deadllines for projects": [[71, "tentative-deadllines-for-projects"]], "Testing the Means Squared Error as function of Complexity": [[44, 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-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html
index dccc5a42f..9253fa3a9 100644
--- a/doc/LectureNotes/_build/html/week35.html
+++ b/doc/LectureNotes/_build/html/week35.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
@@ -907,7 +907,7 @@ We assume our data can represented by a fourth-order polynomial. For the 
 
      # matrix inversion to find theta
 # First we set up the data
-import numpy as np
+import numpy as np
 x = np.random.rand(100)
 y = 2.0+5*x*x+0.1*np.random.randn(100)
 # and then the design matrix X including the intercept
@@ -941,7 +941,7 @@ We assume our data can represented by a fourth-order polynomial. For the Scikit-Learn here we can define our own \(R2\) function as
      
 
      
 
      
-
      def R2(y_data, y_model):
+
      def R2(y_data, y_model):
     return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
 
      
 
      
@@ -958,7 +958,7 @@ Since we are not using Scikit-Learn here we can define our own
 
      We can easily add our MSE score as
      
 
      
 
      
-
      def MSE(y_data,y_model):
+
      def MSE(y_data,y_model):
     n = np.size(y_model)
     return np.sum((y_data-y_model)**2)/n
 
@@ -970,7 +970,7 @@ Since we are not using Scikit-Learn here we can define our own
 
      and finally the relative error as
      
 
      
 
      
-
      def RelativeError(y_data,y_model):
+
      def RelativeError(y_data,y_model):
     return abs((y_data-y_model)/y_data)
 print(RelativeError(y, ytilde))
 
      
@@ -997,16 +997,16 @@ but now splitting the data into a training set and a test set.
      
 
      
 
      %matplotlib inline
 
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
+import os
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
 
 
-def R2(y_data, y_model):
+def R2(y_data, y_model):
     return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
+def MSE(y_data,y_model):
     n = np.size(y_model)
     return np.sum((y_data-y_model)**2)/n
 
@@ -1047,7 +1047,7 @@ but now splitting the data into a training set and a test set.
      
 
      
 
      
 
      # equivalently in numpy
-def train_test_split_numpy(inputs, labels, train_size, test_size):
+def train_test_split_numpy(inputs, labels, train_size, test_size):
     n_inputs = len(inputs)
     inputs_shuffled = inputs.copy()
     labels_shuffled = labels.copy()
@@ -1152,13 +1152,13 @@ simple test design matrix with random numbers. Each column could then
 represent a specific feature whose mean value is subracted.
      
 
      
 
      
-
      import sklearn.linear_model as skl
-from sklearn.metrics import mean_squared_error
-from sklearn.model_selection import  train_test_split
-from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer
-import numpy as np
-import pandas as pd
-from IPython.display import display
+
      import sklearn.linear_model as skl
+from sklearn.metrics import mean_squared_error
+from sklearn.model_selection import  train_test_split
+from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer
+import numpy as np
+import pandas as pd
+from IPython.display import display
 np.random.seed(100)
 # setting up a 10 x 5 matrix
 rows = 10
@@ -1214,12 +1214,12 @@ the aims is to reproduce Figure 2.11 of \(X\) defined by a fourth-order polynomial.  Scale your data and split it in training and test data.
      
 
      
 
      
-
      import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.linear_model import LinearRegression
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
+
      import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.linear_model import LinearRegression
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.model_selection import train_test_split
+from sklearn.pipeline import make_pipeline
 
 
 np.random.seed(2018)
@@ -1499,13 +1499,13 @@ discussion of Ridge regression.
      
 
      The code here is a simple demonstration of how to implement Ridge regression with our own code and compare this with scikit-learn.
      
 
      
 
      
-
      import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
+
      import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
 
-def MSE(y_data,y_model):
+def MSE(y_data,y_model):
     n = np.size(y_model)
     return np.sum((y_data-y_model)**2)/n
 
@@ -1663,9 +1663,9 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
 
      Codes for the SVD#
      
 
      
 
      
-
      import numpy as np
+
      import numpy as np
 # SVD inversion
-def SVD(A):
+def SVD(A):
     ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
     SVD is numerically more stable than the inversion algorithms provided by
     numpy and scipy.linalg at the cost of being slower.
@@ -2038,7 +2038,7 @@ covariance matrix through the np.linalg.eig() function.
      
 
      
 
      
 
      # Importing various packages
-import numpy as np
+import numpy as np
 n = 100
 x = np.random.normal(size=n)
 print(np.mean(x))
@@ -2061,7 +2061,7 @@ code which sets up the correlations matrix for the previous example in
 a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \(2\times 2\) correlation matrix (since we have only two vectors).
      
 
      
 
      
-
      import numpy as np
+
      import numpy as np
 n = 100
 # define two vectors                                                                                           
 x = np.random.random(size=n)
@@ -2096,8 +2096,8 @@ this matrix we easily see that it is a positive definite matrix.
      
 
      We whow here how we can set up the correlation matrix using pandas, as done in this simple code
      
 
      
 
      
-
      import numpy as np
-import pandas as pd
+
      import numpy as np
+import pandas as pd
 n = 10
 x = np.random.normal(size=n)
 x = x - np.mean(x)
@@ -2535,20 +2535,20 @@ and \(\tilde{X}_{ij} = X_{ij} - \frac
 Note also that we do not split the data into training and test.
      
 
      
 
      
-
      import numpy as np
-import matplotlib.pyplot as plt
+
      import numpy as np
+import matplotlib.pyplot as plt
 
-from sklearn.linear_model import LinearRegression
+from sklearn.linear_model import LinearRegression
 
 
 np.random.seed(2021)
 
-def MSE(y_data,y_model):
+def MSE(y_data,y_model):
     n = np.size(y_model)
     return np.sum((y_data-y_model)**2)/n
 
 
-def fit_beta(X, y):
+def fit_beta(X, y):
     return np.linalg.pinv(X.T @ X) @ X.T @ y
 
 
@@ -2678,13 +2678,13 @@ intercept.
      
 
      Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known  vanilla data set.
      
 
      
 
      
-
      import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
+
      import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
 
-def MSE(y_data,y_model):
+def MSE(y_data,y_model):
     n = np.size(y_model)
     return np.sum((y_data-y_model)**2)/n
 
@@ -2753,14 +2753,14 @@ What happens if we do not include the intercept in our fit?
 Let us see how we can change this code by zero centering.
      
 
      
 
      
-
      import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-from sklearn import linear_model
-from sklearn.preprocessing import StandardScaler
+
      import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+from sklearn.preprocessing import StandardScaler
 
-def MSE(y_data,y_model):
+def MSE(y_data,y_model):
     n = np.size(y_model)
     return np.sum((y_data-y_model)**2)/n
 # A seed just to ensure that the random numbers are the same for every run.
diff --git a/doc/LectureNotes/_build/html/week36.html b/doc/LectureNotes/_build/html/week36.html
index e2927b8d5..f0252609b 100644
--- a/doc/LectureNotes/_build/html/week36.html
+++ b/doc/LectureNotes/_build/html/week36.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/week37.html b/doc/LectureNotes/_build/html/week37.html
index 785bc0432..20b503629 100644
--- a/doc/LectureNotes/_build/html/week37.html
+++ b/doc/LectureNotes/_build/html/week37.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/week38.html b/doc/LectureNotes/_build/html/week38.html
index c7f8d63fc..c257c34e5 100644
--- a/doc/LectureNotes/_build/html/week38.html
+++ b/doc/LectureNotes/_build/html/week38.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/week39.html b/doc/LectureNotes/_build/html/week39.html
index e96aad05d..262bb6357 100644
--- a/doc/LectureNotes/_build/html/week39.html
+++ b/doc/LectureNotes/_build/html/week39.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/week40.html b/doc/LectureNotes/_build/html/week40.html
index 814622be4..3ec4779e4 100644
--- a/doc/LectureNotes/_build/html/week40.html
+++ b/doc/LectureNotes/_build/html/week40.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/html/week41.html b/doc/LectureNotes/_build/html/week41.html
index bb300e769..7cf6055a7 100644
--- a/doc/LectureNotes/_build/html/week41.html
+++ b/doc/LectureNotes/_build/html/week41.html
@@ -28,7 +28,7 @@
 
 
 
-    
+    
     
     
     
diff --git a/doc/LectureNotes/_build/jupyter_execute/.venv/lib/python3.13/site-packages/jupyter_book/book_template/markdown-notebooks.ipynb b/doc/LectureNotes/_build/jupyter_execute/.venv/lib/python3.13/site-packages/jupyter_book/book_template/markdown-notebooks.ipynb
index 5e6c28bbf..ff55b6fa1 100644
--- a/doc/LectureNotes/_build/jupyter_execute/.venv/lib/python3.13/site-packages/jupyter_book/book_template/markdown-notebooks.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/.venv/lib/python3.13/site-packages/jupyter_book/book_template/markdown-notebooks.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "da3b753e",
+   "id": "1232311e",
    "metadata": {},
    "source": [
     "# Notebooks with MyST Markdown\n",
@@ -19,7 +19,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "2bea4705",
+   "id": "f961e284",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -28,7 +28,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1d36b822",
+   "id": "3b5f5a93",
    "metadata": {},
    "source": [
     "When your book is built, the contents of any `{code-cell}` blocks will be\n",
diff --git a/doc/LectureNotes/_build/jupyter_execute/exercisesweek41.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek41.ipynb
index 57b063993..d5378765f 100644
--- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek41.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek41.ipynb
@@ -327,7 +327,7 @@
    "id": "0da7fd52",
    "metadata": {},
    "source": [
-    "**d)** Why is a neural network with no activation functions always mathematically equivelent to a neural network with only one layer?\n"
+    "**d)** Why is a neural network with no activation functions mathematically equivelent to(can be reduced to) a neural network with only one layer?\n"
    ]
   },
   {
@@ -454,7 +454,7 @@
    "id": "a6349db6",
    "metadata": {},
    "source": [
-    "**b)** Make a matrix of inputs with the shape (number of features, number of inputs), you choose the number of inputs and features per input. Then complete the function `feed_forward_batch` so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)\n"
+    "**b)** Make a matrix of inputs with the shape (number of inputs, number of features), you choose the number of inputs and features per input. Then complete the function `feed_forward_batch` so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)"
    ]
   },
   {
@@ -480,7 +480,7 @@
    "id": "efd07b4e",
    "metadata": {},
    "source": [
-    "**c)** Create and evaluate a neural network with 4 inputs and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.\n"
+    "**c)** Create and evaluate a neural network with 4 input features, and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.\n"
    ]
   },
   {
diff --git a/doc/LectureNotes/exercisesweek41.ipynb b/doc/LectureNotes/exercisesweek41.ipynb
index fccc64f45..190c0b96a 100644
--- a/doc/LectureNotes/exercisesweek41.ipynb
+++ b/doc/LectureNotes/exercisesweek41.ipynb
@@ -327,7 +327,7 @@
    "id": "0da7fd52",
    "metadata": {},
    "source": [
-    "**d)** Why is a neural network with no activation functions always mathematically equivelent to a neural network with only one layer?\n"
+    "**d)** Why is a neural network with no activation functions mathematically equivelent to(can be reduced to) a neural network with only one layer?\n"
    ]
   },
   {
@@ -454,7 +454,7 @@
    "id": "a6349db6",
    "metadata": {},
    "source": [
-    "**b)** Make a matrix of inputs with the shape (number of features, number of inputs), you choose the number of inputs and features per input. Then complete the function `feed_forward_batch` so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)\n"
+    "**b)** Make a matrix of inputs with the shape (number of inputs, number of features), you choose the number of inputs and features per input. Then complete the function `feed_forward_batch` so that you can process this matrix of inputs with only one matrix multiplication and one broadcasted vector addition per layer. (Hint: You will only need to swap two variable around from your previous implementation, but remember to test that you get the same results for equivelent inputs!)"
    ]
   },
   {
@@ -480,7 +480,7 @@
    "id": "efd07b4e",
    "metadata": {},
    "source": [
-    "**c)** Create and evaluate a neural network with 4 inputs and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.\n"
+    "**c)** Create and evaluate a neural network with 4 input features, and layers with output sizes 12, 10, 3 and activations ReLU, ReLU, softmax.\n"
    ]
   },
   {
diff --git a/doc/LectureNotes/requirements.txt b/doc/LectureNotes/requirements.txt
new file mode 100644
index 000000000..54b882503
--- /dev/null
+++ b/doc/LectureNotes/requirements.txt
@@ -0,0 +1,93 @@
+accessible-pygments==0.0.5
+alabaster==0.7.16
+appnope==0.1.4
+asttokens==3.0.0
+attrs==25.3.0
+babel==2.17.0
+beautifulsoup4==4.13.5
+certifi==2025.8.3
+charset-normalizer==3.4.3
+click==8.2.1
+comm==0.2.3
+debugpy==1.8.16
+decorator==5.2.1
+docutils==0.21.2
+executing==2.2.1
+fastjsonschema==2.21.2
+idna==3.10
+imagesize==1.4.1
+importlib_metadata==8.7.0
+ipykernel==6.30.1
+ipython==9.5.0
+ipython_pygments_lexers==1.1.1
+jedi==0.19.2
+Jinja2==3.1.6
+jsonschema==4.25.1
+jsonschema-specifications==2025.9.1
+jupyter-book==1.0.4.post1
+jupyter-cache==1.0.1
+jupyter_client==8.6.3
+jupyter_core==5.8.1
+latexcodec==3.0.1
+linkify-it-py==2.0.3
+markdown-it-py==3.0.0
+MarkupSafe==3.0.2
+matplotlib-inline==0.1.7
+mdit-py-plugins==0.5.0
+mdurl==0.1.2
+myst-nb==1.3.0
+myst-parser==3.0.1
+nbclient==0.10.2
+nbformat==5.10.4
+nest-asyncio==1.6.0
+numpy==2.3.3
+packaging==25.0
+parso==0.8.5
+pexpect==4.9.0
+platformdirs==4.4.0
+prompt_toolkit==3.0.52
+psutil==7.0.0
+ptyprocess==0.7.0
+pure_eval==0.2.3
+pybtex==0.25.1
+pybtex-docutils==1.0.3
+pydata-sphinx-theme==0.15.4
+Pygments==2.19.2
+python-dateutil==2.9.0.post0
+PyYAML==6.0.2
+pyzmq==27.0.2
+referencing==0.36.2
+requests==2.32.5
+rpds-py==0.27.1
+setuptools==80.9.0
+six==1.17.0
+snowballstemmer==3.0.1
+soupsieve==2.8
+Sphinx==7.4.7
+sphinx-book-theme==1.1.4
+sphinx-comments==0.0.3
+sphinx-copybutton==0.5.2
+sphinx-jupyterbook-latex==1.0.0
+sphinx-multitoc-numbering==0.1.3
+sphinx-thebe==0.3.1
+sphinx-togglebutton==0.3.2
+sphinx_design==0.6.1
+sphinx_external_toc==1.0.1
+sphinxcontrib-applehelp==2.0.0
+sphinxcontrib-bibtex==2.6.5
+sphinxcontrib-devhelp==2.0.0
+sphinxcontrib-htmlhelp==2.1.0
+sphinxcontrib-jsmath==1.0.1
+sphinxcontrib-qthelp==2.0.0
+sphinxcontrib-serializinghtml==2.0.0
+SQLAlchemy==2.0.43
+stack-data==0.6.3
+tabulate==0.9.0
+tornado==6.5.2
+traitlets==5.14.3
+typing_extensions==4.15.0
+uc-micro-py==1.0.3
+urllib3==2.5.0
+wcwidth==0.2.13
+wheel==0.45.1
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