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diff --git a/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/sphinxcontrib_bibtex-2.6.5.dist-info/licenses/LICENSE.html b/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/sphinxcontrib_bibtex-2.6.5.dist-info/licenses/LICENSE.html index 130e35f44..3317f8897 100644 --- a/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/sphinxcontrib_bibtex-2.6.5.dist-info/licenses/LICENSE.html +++ b/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/sphinxcontrib_bibtex-2.6.5.dist-info/licenses/LICENSE.html @@ -230,6 +230,7 @@Projects
diff --git a/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/zmq/backend/cffi/README.html b/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/zmq/backend/cffi/README.html index 06c2d4dac..ca2ca70f8 100644 --- a/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/zmq/backend/cffi/README.html +++ b/doc/LectureNotes/_build/html/.venv/lib/python3.13/site-packages/zmq/backend/cffi/README.html @@ -230,6 +230,7 @@Projects
diff --git a/doc/LectureNotes/_build/html/_sources/exercisesweek38.ipynb b/doc/LectureNotes/_build/html/_sources/exercisesweek38.ipynb index 48a19f138..c100028a5 100644 --- a/doc/LectureNotes/_build/html/_sources/exercisesweek38.ipynb +++ b/doc/LectureNotes/_build/html/_sources/exercisesweek38.ipynb @@ -290,7 +290,8 @@ "$$\n", "\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n", "$$\n", - "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$." + "\n", + "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$.\n" ] }, { @@ -321,7 +322,7 @@ }, { "cell_type": "code", - "execution_count": 67, + "execution_count": null, "id": "b5bf581c", "metadata": {}, "outputs": [], @@ -332,7 +333,8 @@ "bootstraps = 1000\n", "\n", "predictions = np.random.rand(bootstraps, n) * 10 + 10\n", - "targets = np.random.rand(bootstraps, n)\n", + "# The definition of targets has been updated, and was wrong earlier in the week.\n", + "targets = np.random.rand(1, n)\n", "\n", "mse = ...\n", "bias = ...\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 { background: #fefefe; color: #080808 } html[data-theme="light"] .highlight .c { color: #515151 } /* Comment */ -html[data-theme="light"] .highlight .err { color: #a12236 } /* Error */ 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+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 352c20ea8..8222a0434 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 b4fae1af9..0780c7635 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 b2830244f..2c99b60da 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 6fc0127d7..90b09e243 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 62d3ba763..32517d292 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 95077a0d7..321eb9de3 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 5129939ab..1264c9a5a 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 f9f8bfc9c..ca68957f2 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 7e5f6dcb0..fd2ee4dc4 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 67f0daf29..b0617491e 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 844acac63..ccfc230fb 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 06cc97005..2b87abeb2 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 ab72857b9..6c792487b 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 4a69ef138..fa046902e 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 823ab375f..3aac114f2 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 04bb626e8..9e18a23c1 100644 --- a/doc/LectureNotes/_build/html/exercisesweek34.html +++ b/doc/LectureNotes/_build/html/exercisesweek34.html @@ -28,7 +28,7 @@ - + @@ -479,11 +479,11 @@ 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
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
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
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)
@@ -569,7 +569,7 @@ 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] = ...
@@ -593,7 +593,7 @@ 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 842d50e9c..9252ef6ea 100644
--- a/doc/LectureNotes/_build/html/exercisesweek36.html
+++ b/doc/LectureNotes/_build/html/exercisesweek36.html
@@ -28,7 +28,7 @@
-
+
@@ -467,10 +467,10 @@ 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
@@ -487,7 +487,7 @@ 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] = ...
@@ -500,7 +500,7 @@ 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[:]
@@ -546,7 +546,7 @@ 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 98bb98d16..52afa1944 100644
--- a/doc/LectureNotes/_build/html/exercisesweek37.html
+++ b/doc/LectureNotes/_build/html/exercisesweek37.html
@@ -28,7 +28,7 @@
-
+
@@ -584,7 +584,7 @@ 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 6286d09a0..760eb54ba 100644
--- a/doc/LectureNotes/_build/html/exercisesweek38.html
+++ b/doc/LectureNotes/_build/html/exercisesweek38.html
@@ -28,7 +28,7 @@
-
+
@@ -525,13 +525,14 @@ 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
predictions = np.random.rand(bootstraps, n) * 10 + 10
-targets = np.random.rand(bootstraps, n)
+# The definition of targets has been updated, and was wrong earlier in the week.
+targets = np.random.rand(1, n)
mse = ...
bias = ...
@@ -545,15 +546,15 @@ 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 150611a5c..9370658b8 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/genindex.html b/doc/LectureNotes/_build/html/genindex.html
index 63018291d..0bb271492 100644
--- a/doc/LectureNotes/_build/html/genindex.html
+++ b/doc/LectureNotes/_build/html/genindex.html
@@ -27,7 +27,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/intro.html b/doc/LectureNotes/_build/html/intro.html
index 6881eaa99..fa66e1185 100644
--- a/doc/LectureNotes/_build/html/intro.html
+++ b/doc/LectureNotes/_build/html/intro.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/linalg.html b/doc/LectureNotes/_build/html/linalg.html
index 6f5fa39f6..7f1073b3e 100644
--- a/doc/LectureNotes/_build/html/linalg.html
+++ b/doc/LectureNotes/_build/html/linalg.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/objects.inv b/doc/LectureNotes/_build/html/objects.inv
index d959024d7..7e793f8f1 100644
Binary files a/doc/LectureNotes/_build/html/objects.inv and b/doc/LectureNotes/_build/html/objects.inv differ
diff --git a/doc/LectureNotes/_build/html/project1.html b/doc/LectureNotes/_build/html/project1.html
index 004c191f4..5dc3d3f97 100644
--- a/doc/LectureNotes/_build/html/project1.html
+++ b/doc/LectureNotes/_build/html/project1.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/schedule.html b/doc/LectureNotes/_build/html/schedule.html
index f186a8461..c5b09073f 100644
--- a/doc/LectureNotes/_build/html/schedule.html
+++ b/doc/LectureNotes/_build/html/schedule.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/search.html b/doc/LectureNotes/_build/html/search.html
index 94e8e71d7..8a8242685 100644
--- a/doc/LectureNotes/_build/html/search.html
+++ b/doc/LectureNotes/_build/html/search.html
@@ -26,7 +26,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 2821bfccf..f244cd5fa 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
+++ b/doc/LectureNotes/_build/html/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"1a)": [[18, "a"]], "3a)": [[18, "id1"]], "3b)": [[18, "b"]], "4a)": [[18, "id2"]], "4b)": [[18, "id3"]], "A Classification Tree": [[9, "a-classification-tree"]], "A Frequentist approach to data analysis": [[0, "a-frequentist-approach-to-data-analysis"], [28, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[8, "a-better-approach"]], "A first summary": [[28, "a-first-summary"]], "A new Cost Function": [[32, "a-new-cost-function"]], "A quick Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "A way to Read the Bias-Variance Tradeoff": [[32, "a-way-to-read-the-bias-variance-tradeoff"]], "ADAM algorithm, taken from Goodfellow et al": [[31, "adam-algorithm-taken-from-goodfellow-et-al"]], "ADAM optimizer": [[13, "adam-optimizer"], [31, "id2"]], "Accuracy": [[31, "accuracy"]], "Activation functions": [[12, "activation-functions"]], "AdaGrad Properties": [[31, "adagrad-properties"]], "AdaGrad Update Rule Derivation": [[31, "adagrad-update-rule-derivation"]], "AdaGrad algorithm, taken from Goodfellow et al": [[31, "adagrad-algorithm-taken-from-goodfellow-et-al"]], "Adam Optimizer": [[31, "adam-optimizer"]], "Adam vs. AdaGrad and RMSProp": [[31, "adam-vs-adagrad-and-rmsprop"]], "Adam: Bias Correction": [[31, "adam-bias-correction"]], "Adam: Exponential Moving Averages (Moments)": [[31, "adam-exponential-moving-averages-moments"]], "Adam: Update Rule Derivation": [[31, "adam-update-rule-derivation"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[10, "adaptive-boosting-adaboost-basic-algorithm"]], "Adaptivity Across Dimensions": [[31, "adaptivity-across-dimensions"]], "Adding error analysis and training set up": [[28, "adding-error-analysis-and-training-set-up"], [29, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms and codes for Adagrad, RMSprop and Adam": [[31, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[28, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[29, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And finally ADAM": [[31, "and-finally-adam"]], "And what about using neural networks?": [[28, "and-what-about-using-neural-networks"]], "Another Example from Scikit-Learn\u2019s Repository": [[32, "another-example-from-scikit-learn-s-repository"]], "Another Example, now with a polynomial fit": [[30, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[21, null]], "Assumptions made": [[32, "assumptions-made"]], "Autocorrelation function": [[25, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[29, "back-to-ridge-and-lasso-regression"], [30, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Background literature": [[23, "background-literature"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[22, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [29, "basic-math-of-the-svd"], [30, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Batches and mini-batches": [[31, "batches-and-mini-batches"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "But none of these can compete with Newton\u2019s method": [[31, "but-none-of-these-can-compete-with-newton-s-method"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Challenge: Choosing a Fixed Learning Rate": [[31, "challenge-choosing-a-fixed-learning-rate"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code Example for Cross-validation and k-fold Cross-validation": [[32, "code-example-for-cross-validation-and-k-fold-cross-validation"]], "Code example for the Bootstrap method": [[32, "code-example-for-the-bootstrap-method"]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Code with a Number of Minibatches which varies": [[31, "code-with-a-number-of-minibatches-which-varies"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [29, "codes-for-the-svd"], [30, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[28, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[30, "comparison-with-ols"]], "Computation of gradients": [[31, "computation-of-gradients"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conditions on convex functions": [[30, "conditions-on-convex-functions"]], "Confidence Intervals": [[32, "confidence-intervals"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convergence rates": [[31, "convergence-rates"]], "Convex function": [[30, "convex-function"]], "Convex functions": [[13, "convex-functions"], [30, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[29, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [29, "correlation-matrix"]], "Correlation Matrix with Pandas": [[29, "correlation-matrix-with-pandas"]], "Course Format": [[28, "course-format"]], "Course setting": [[24, null]], "Covariance Matrix Examples": [[29, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[29, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Cross-validation in brief": [[32, "cross-validation-in-brief"]], "Deadlines for projects (tentative)": [[28, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep Neural Networks": [[31, "deep-neural-networks"]], "Deep learning methods": [[28, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Definitions": [[19, "definitions"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"], [19, "deliverables"], [20, "deliverables"]], "Derivation of the AdaGrad Algorithm": [[31, "derivation-of-the-adagrad-algorithm"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[29, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"], [32, "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": [[29, "deriving-the-lasso-regression-equations"], [30, "deriving-the-lasso-regression-equations"], [30, "id6"]], "Deriving the Ridge Regression Equations": [[29, "deriving-the-ridge-regression-equations"], [30, "deriving-the-ridge-regression-equations"], [30, "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": [[28, "discriminative-modeling"]], "Domains and probabilities": [[25, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[29, "economy-size-svd"], [30, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[25, null]], "Empirical Evidence: Convergence Time and Memory in Practice": [[31, "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": [[28, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[29, "example-2"]], "Example 3": [[29, "example-3"]], "Example 4": [[29, "example-4"]], "Example Matrix": [[29, "example-matrix"], [30, "example-matrix"]], "Example code for Bias-Variance tradeoff": [[32, "example-code-for-bias-variance-tradeoff"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[28, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[28, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[29, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[29, "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": [[28, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "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: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "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: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "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 - 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, 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: 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]], "Expectation value and variance": [[32, "expectation-value-and-variance"]], "Expectation value and variance for \\boldsymbol{\\theta}": [[32, "expectation-value-and-variance-for-boldsymbol-theta"]], "Expectation values": [[25, "expectation-values"]], "Extending to more than one variable": [[30, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[28, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Finding the Limit": [[32, "finding-the-limit"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[29, "fixing-the-singularity"], [30, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[23, "format-for-electronic-delivery-of-report-and-programs"]], "Frequently used scaling functions": [[29, "frequently-used-scaling-functions"], [31, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[30, "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": [[29, "functionality-in-scikit-learn"], [31, "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"], [29, "further-properties-important-for-our-analyses-later"], [30, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[22, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[28, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[28, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [28, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[28, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "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": [[30, "id1"], [31, "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": [[30, "gradient-descent-and-ridge"], [31, "gradient-descent-and-ridge"]], "Gradient descent and revisiting Ordinary Least Squares from last week": [[31, "gradient-descent-and-revisiting-ordinary-least-squares-from-last-week"]], "Gradient descent example": [[30, "gradient-descent-example"], [31, "gradient-descent-example"]], "Grading": [[26, "grading"], [26, "id2"], [28, "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": [[32, "identifying-terms"]], "Important Matrix and vector handling packages": [[22, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[29, "important-technicalities-more-on-rescaling-data"]], "Improving gradient descent with momentum": [[31, "improving-gradient-descent-with-momentum"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[26, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"], [31, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Independent and Identically Distributed (iid)": [[32, "independent-and-identically-distributed-iid"]], "Installing R, C++, cython or Julia": [[28, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[28, "installing-r-c-cython-numba-etc"]], "Instructor information": [[26, "instructor-information"]], "Interpretations and optimizing our parameters": [[28, "interpretations-and-optimizing-our-parameters"], [28, "id2"], [28, "id3"], [29, "interpretations-and-optimizing-our-parameters"], [29, "id1"], [29, "id2"]], "Interpreting the Ridge results": [[29, "interpreting-the-ridge-results"], [30, "interpreting-the-ridge-results"], [30, "id4"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [29, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [21, "introduction"], [22, "introduction"]], "Introduction to numerical projects": [[23, "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": [[22, "lu-decomposition-the-inverse-of-a-matrix"]], "Lasso Regression": [[30, "lasso-regression"]], "Lasso case": [[30, "lasso-case"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"], [18, "learning-goals"], [19, "learning-goals"], [20, "learning-goals"]], "Learning outcomes": [[21, "learning-outcomes"], [28, "learning-outcomes"]], "Lectures and ComputerLab": [[28, "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": [[22, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[29, "linear-regression-problems"], [30, "linear-regression-problems"]], "Linear Regression and the SVD": [[30, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "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"], [32, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [29, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[27, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[28, "machine-learning"]], "Machine learning": [[21, "machine-learning"]], "Main textbooks": [[28, "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": [[29, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[29, "material-for-exercises-week-35"]], "Material for lab sessions sessions Tuesday and Wednesday": [[30, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[30, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 8": [[31, "material-for-lecture-monday-september-8"]], "Material for the lab sessions": [[31, "material-for-the-lab-sessions"], [32, "material-for-the-lab-sessions"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [29, "mathematical-interpretation-of-ordinary-least-squares"], [30, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [29, "mathematics-of-the-svd-and-implications"], [30, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[28, "matrices-in-python"]], "Matrix multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation and activation": [[12, "matrix-vector-notation-and-activation"]], "Maximum Likelihood Estimation (MLE)": [[32, "maximum-likelihood-estimation-mle"]], "Meet the covariance!": [[25, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [29, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[29, "meet-the-hessian-matrix"]], "Meet the Pandas": [[28, "meet-the-pandas"]], "Memory Usage and Scalability": [[31, "memory-usage-and-scalability"]], "Memory constraints": [[31, "memory-constraints"]], "Min-Max Scaling": [[29, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"], [31, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More examples on bootstrap and cross-validation and errors": [[32, "more-examples-on-bootstrap-and-cross-validation-and-errors"]], "More interpretations": [[29, "more-interpretations"], [30, "more-interpretations"], [30, "id5"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More on Steepest descent": [[30, "more-on-steepest-descent"]], "More on convex functions": [[30, "more-on-convex-functions"]], "More preprocessing": [[29, "more-preprocessing"], [31, "more-preprocessing"]], "Motivation for Adaptive Step Sizes": [[31, "motivation-for-adaptive-step-sizes"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Non-Convex Problems": [[31, "non-convex-problems"]], "Note about SVD Calculations": [[29, "note-about-svd-calculations"], [30, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[30, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[25, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[22, "numpy-and-arrays"], [28, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[28, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[30, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null]], "Optimizing our parameters": [[28, "optimizing-our-parameters"]], "Optimizing our 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"our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[28, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent (SGD)": [[31, "overview-video-on-stochastic-gradient-descent-sgd"]], "Own code for Ordinary Least Squares": [[28, "own-code-for-ordinary-least-squares"], [29, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[28, "pandas-ai"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[23, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[23, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[23, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and more advanced ways to update the learning the rate": [[23, "part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate"]], "Part e: Writing 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"Resampling methods: More Bootstrap background": [[32, "resampling-methods-more-bootstrap-background"]], "Residual Error": [[29, "residual-error"], [30, "residual-error"]], "Resources on differential equations and deep learning": [[2, "resources-on-differential-equations-and-deep-learning"]], "Revisiting Ordinary Least Squares": [[30, "revisiting-ordinary-least-squares"]], "Revisiting our Linear Regression Solvers": [[13, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[29, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the \\delta-function": [[32, "rewriting-the-delta-function"]], "Rewriting the fitting procedure as a linear algebra problem": [[28, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[28, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[30, 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"and-finally-boldsymbol-x-boldsymbol-x-t"]], "And finally ADAM": [[75, "and-finally-adam"]], "And what about using neural networks?": [[72, "and-what-about-using-neural-networks"]], "Another Example from Scikit-Learn\u2019s Repository": [[76, "another-example-from-scikit-learn-s-repository"]], "Another Example, now with a polynomial fit": [[74, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[53, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[65, null]], "Assumptions made": [[76, "assumptions-made"]], "Authors": [[33, null]], "Autocorrelation function": [[69, "autocorrelation-function"]], "Automatic differentiation": [[57, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[73, "back-to-ridge-and-lasso-regression"], [74, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[55, "back-to-the-cancer-data"]], "Background literature": [[67, "background-literature"]], "Bagging": [[54, 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"building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[45, "building-neural-networks-in-tensorflow-and-keras"]], "But none of these can compete with Newton\u2019s method": [[75, "but-none-of-these-can-compete-with-newton-s-method"]], "CDN": [[29, "cdn"]], "CHANGELOG": [[29, "changelog"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[47, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[53, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Challenge: Choosing a Fixed Learning Rate": [[75, "challenge-choosing-a-fixed-learning-rate"]], "Choose cost function and optimizer": [[45, "choose-cost-function-and-optimizer"]], "Citations": [[22, "citations"]], "Classical PCA Theorem": [[55, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[58, null]], "Code Example for Cross-validation and k-fold Cross-validation": [[76, "code-example-for-cross-validation-and-k-fold-cross-validation"]], "Code blocks and outputs": [[24, "code-blocks-and-outputs"]], "Code example for the Bootstrap method": [[76, "code-example-for-the-bootstrap-method"]], "Code for SVD and Inversion of Matrices": [[49, "code-for-svd-and-inversion-of-matrices"]], "Code with a Number of Minibatches which varies": [[75, "code-with-a-number-of-minibatches-which-varies"]], "Codes and Approaches": [[58, "codes-and-approaches"]], "Codes for the SVD": [[49, "codes-for-the-svd"], [73, "codes-for-the-svd"], [74, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[59, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[45, "collect-and-pre-process-data"]], "Colors": [[0, "colors"], [1, "colors"], [2, "colors"], [3, "colors"], [4, "colors"], [5, "colors"], [6, "colors"], [7, "colors"], [8, "colors"], [9, "colors"], [10, "colors"], [11, "colors"], [12, "colors"], [13, "colors"], [14, "colors"], [15, "colors"]], "Communication channels": [[72, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[54, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[46, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[74, "comparison-with-ols"]], "Compiled translation files": [[41, "compiled-translation-files"]], "Components": [[31, "components"]], "Computation of gradients": [[75, "computation-of-gradients"]], "Computing the Gini index": [[53, "computing-the-gini-index"]], "Conditions on convex functions": [[74, "conditions-on-convex-functions"]], "Confidence Intervals": [[76, "confidence-intervals"]], "Conjugate gradient method": [[57, "conjugate-gradient-method"]], "Content with notebooks": [[24, null]], "Contents": [[30, "contents"]], "Contributors": [[33, "contributors"]], "Convergence rates": [[75, "convergence-rates"]], "Convex function": [[74, "convex-function"]], "Convex functions": [[57, "convex-functions"], [74, "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"]], "Convolutional Neural Networks": [[47, null]], "Correlation Function and Design/Feature Matrix": [[73, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[55, "correlation-matrix"], [73, "correlation-matrix"]], "Correlation Matrix with Pandas": [[73, "correlation-matrix-with-pandas"]], "Course Format": [[72, "course-format"]], "Course setting": [[68, null]], "Covariance Matrix Examples": [[73, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[73, "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": [[76, "cross-validation-in-brief"]], "Deadlines for projects (tentative)": [[72, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[53, null]], "Deep Neural Networks": [[75, "deep-neural-networks"]], "Deep learning methods": [[72, "deep-learning-methods"]], "Define model and architecture": [[45, "define-model-and-architecture"]], "Defining the cost function": [[45, "defining-the-cost-function"]], "Definitions": [[63, "definitions"]], "Deliverables": [[59, "deliverables"], [60, "deliverables"], [63, "deliverables"], [64, "deliverables"]], "Dependencies": [[29, "dependencies"]], "Derivation of the AdaGrad Algorithm": [[75, "derivation-of-the-adagrad-algorithm"]], "Derivatives and the chain rule": [[56, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[73, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[49, "deriving-ols-from-a-probability-distribution"], [76, "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": [[73, "deriving-the-lasso-regression-equations"], [74, "deriving-the-lasso-regression-equations"], [74, "id6"]], "Deriving the Ridge Regression Equations": [[73, "deriving-the-ridge-regression-equations"], [74, "deriving-the-ridge-regression-equations"], [74, "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": [[72, "discriminative-modeling"]], "Domains and probabilities": [[69, "domains-and-probabilities"]], "Dropout": [[45, "dropout"]], "Economy-size SVD": [[73, "economy-size-svd"], [74, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[69, null]], "Empirical Evidence: Convergence Time and Memory in Practice": [[75, "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": [[72, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[45, "evaluate-model-performance-on-test-data"]], "Example": [[27, "example"]], "Example 2": [[73, "example-2"]], "Example 3": [[73, "example-3"]], "Example 4": [[73, "example-4"]], "Example Matrix": [[73, "example-matrix"], [74, "example-matrix"]], "Example code for Bias-Variance tradeoff": [[76, "example-code-for-bias-variance-tradeoff"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[72, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[72, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[73, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[73, "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": [[72, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[51, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "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: Setting up various Python environments": [[44, "exercise-1-setting-up-various-python-environments"]], "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: making your own data and exploring scikit-learn": [[44, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "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 - 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, 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: 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]], "Expectation value and variance": [[76, "expectation-value-and-variance"]], "Expectation value and variance for \\boldsymbol{\\theta}": [[76, "expectation-value-and-variance-for-boldsymbol-theta"]], "Expectation values": [[69, "expectation-values"]], "Extending to more than one variable": [[74, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[72, "extremely-useful-tools-strongly-recommended"]], "FAQ": [[29, "faq"]], "Features": [[29, "features"]], "Feed-forward neural networks": [[56, "feed-forward-neural-networks"]], "Feed-forward pass": [[45, "feed-forward-pass"]], "Final back propagating equation": [[56, "final-back-propagating-equation"]], "Finding the Limit": [[76, "finding-the-limit"]], "Fine-tuning neural network hyperparameters": [[45, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[44, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[73, "fixing-the-singularity"], [74, "fixing-the-singularity"]], "Format for electronic delivery of report and programs": [[67, "format-for-electronic-delivery-of-report-and-programs"]], "Frequently used scaling functions": [[73, "frequently-used-scaling-functions"], [75, "frequently-used-scaling-functions"]], "From OLS to Ridge and Lasso": [[74, "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": [[73, "functionality-in-scikit-learn"], [75, "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"], [73, "further-properties-important-for-our-analyses-later"], [74, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[66, "gaussian-elimination"]], "General Features": [[53, "general-features"]], "General linear models and linear algebra": [[72, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[72, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [72, "id1"]], "Generative Adversarial Networks": [[48, "generative-adversarial-networks"]], "Generative Models": [[48, "generative-models"]], "Generative Versus Discriminative Modeling": [[72, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[55, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Getting started with project 1": [[64, "getting-started-with-project-1"]], "Github Dark": [[6, null]], "Github Dark Colorblind": [[7, null]], "Github Dark High Contrast": [[8, null]], "Github Light": [[9, null]], "Github Light Colorblind": [[10, null]], "Github Light High Contrast": [[11, null]], "Gotthard Dark": [[12, null]], "Gotthard Light": [[13, null]], "Gradient Boosting, Classification Example": [[54, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[54, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[45, "gradient-clipping"]], "Gradient Descent Example": [[74, "id1"], [75, "id1"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[54, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[46, "gradient-descent"]], "Gradient descent and Ridge": [[74, "gradient-descent-and-ridge"], [75, "gradient-descent-and-ridge"]], "Gradient descent and revisiting Ordinary Least Squares from last week": [[75, "gradient-descent-and-revisiting-ordinary-least-squares-from-last-week"]], "Gradient descent example": [[74, "gradient-descent-example"], [75, "gradient-descent-example"]], "Grading": [[70, "grading"], [70, "id2"], [72, "grading"]], "Greative": [[14, null]], "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": [[76, "identifying-terms"]], "Important Matrix and vector handling packages": [[66, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[73, "important-technicalities-more-on-rescaling-data"]], "Improving gradient descent with momentum": [[75, "improving-gradient-descent-with-momentum"]], "Improving performance": [[45, "improving-performance"]], "In summary": [[70, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[57, "including-stochastic-gradient-descent-with-autograd"], [75, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[55, "incremental-pca"]], "Independent and Identically Distributed (iid)": [[76, "independent-and-identically-distributed-iid"]], "Install": [[28, "install"]], "Installation": [[27, "installation"]], "Installing R, C++, cython or Julia": [[72, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[72, "installing-r-c-cython-numba-etc"]], "Instructor information": [[70, "instructor-information"]], "Interpretations and optimizing our parameters": [[72, "interpretations-and-optimizing-our-parameters"], [72, "id2"], [72, "id3"], [73, "interpretations-and-optimizing-our-parameters"], [73, "id1"], [73, "id2"]], "Interpreting the Ridge results": [[73, "interpreting-the-ridge-results"], [74, "interpreting-the-ridge-results"], [74, "id4"]], "Introducing JAX": [[57, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[55, "introducing-the-covariance-and-correlation-functions"], [73, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[44, "introduction"], [50, "introduction"], [65, "introduction"], [66, "introduction"]], "Introduction to numerical projects": [[67, "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": [[66, "lu-decomposition-the-inverse-of-a-matrix"]], "Lasso Regression": [[74, "lasso-regression"]], "Lasso case": [[74, "lasso-case"]], "Layers": [[45, "layers"]], "Layers used to build CNNs": [[47, "layers-used-to-build-cnns"]], "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": [[65, "learning-outcomes"], [72, "learning-outcomes"]], "Lectures and ComputerLab": [[72, "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": [[66, null]], "Linear Regression": [[44, null]], "Linear Regression Problems": [[73, "linear-regression-problems"], [74, "linear-regression-problems"]], "Linear Regression and the SVD": [[74, "linear-regression-and-the-svd"]], "Linear Regression, basic elements": [[44, "linear-regression-basic-elements"]], "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"], [76, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[49, "linking-with-the-svd"], [73, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[71, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[51, null], [51, "id1"]], "MNIST and GANs": [[48, "mnist-and-gans"]], "Machine Learning": [[72, "machine-learning"]], "Machine learning": [[65, "machine-learning"]], "Main differences": [[30, "main-differences"]], "Main textbooks": [[72, "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": [[73, "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": [[73, "material-for-exercises-week-35"]], "Material for lab sessions sessions Tuesday and Wednesday": [[74, "material-for-lab-sessions-sessions-tuesday-and-wednesday"]], "Material for lecture Monday September 2": [[74, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 8": [[75, "material-for-lecture-monday-september-8"]], "Material for the lab sessions": [[75, "material-for-the-lab-sessions"], [76, "material-for-the-lab-sessions"]], "Mathematical Interpretation of Ordinary Least Squares": [[49, "mathematical-interpretation-of-ordinary-least-squares"], [73, "mathematical-interpretation-of-ordinary-least-squares"], [74, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[52, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[47, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[49, "mathematics-of-the-svd-and-implications"], [73, "mathematics-of-the-svd-and-implications"], [74, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[72, "matrices-in-python"]], "Matrix multiplication": [[45, "matrix-multiplication"]], "Matrix-vector notation and activation": [[56, "matrix-vector-notation-and-activation"]], "Maximum Likelihood Estimation (MLE)": [[76, "maximum-likelihood-estimation-mle"]], "Meet the covariance!": [[69, "meet-the-covariance"]], "Meet the Covariance Matrix": [[49, "meet-the-covariance-matrix"], [73, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[73, "meet-the-hessian-matrix"]], "Meet the Pandas": [[72, "meet-the-pandas"]], "Memory Usage and Scalability": [[75, "memory-usage-and-scalability"]], "Memory constraints": [[75, "memory-constraints"]], "Min-Max Scaling": [[73, "min-max-scaling"]], "Momentum based GD": [[57, "momentum-based-gd"], [75, "momentum-based-gd"]], "More complicated Example: The Ising model": [[50, "more-complicated-example-the-ising-model"]], "More examples on bootstrap and cross-validation and errors": [[76, "more-examples-on-bootstrap-and-cross-validation-and-errors"]], "More interpretations": [[73, "more-interpretations"], [74, "more-interpretations"], [74, "id5"]], "More on Dimensionalities": [[47, "more-on-dimensionalities"]], "More on Rescaling data": [[50, "more-on-rescaling-data"]], "More on Steepest descent": [[74, "more-on-steepest-descent"]], "More on convex functions": [[74, "more-on-convex-functions"]], "More preprocessing": [[73, "more-preprocessing"], [75, "more-preprocessing"]], "Motivation for Adaptive Step Sizes": [[75, "motivation-for-adaptive-step-sizes"]], "Multilayer perceptrons": [[56, "multilayer-perceptrons"]], "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 networks": [[56, null]], "New features": [[30, "new-features"]], "Non-Convex Problems": [[75, "non-convex-problems"]], "Note about SVD Calculations": [[73, "note-about-svd-calculations"], [74, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[74, "note-on-scikit-learn"]], "Notebooks with MyST Markdown": [[23, null]], "Numerical experiments and the covariance, central limit theorem": [[69, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[66, "numpy-and-arrays"], [72, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[72, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and gradient descent, the central part of any Machine Learning algortithm": [[74, "optimization-and-gradient-descent-the-central-part-of-any-machine-learning-algortithm"]], "Optimization, the central part of any Machine Learning algortithm": [[57, null]], "Optimizing maskedarray": [[30, "optimizing-maskedarray"]], "Optimizing our parameters": [[72, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[72, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[45, "optimizing-the-cost-function"]], "Organizing our data": [[44, "organizing-our-data"], [72, "organizing-our-data"]], "Other Matrix and Vector Operations": [[66, "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": [[72, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[72, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[72, "other-popular-texts"]], "Other techniques": [[55, "other-techniques"]], "Other types of networks": [[56, "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": [[72, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[72, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent (SGD)": [[75, "overview-video-on-stochastic-gradient-descent-sgd"]], "Own code for Ordinary Least Squares": [[72, "own-code-for-ordinary-least-squares"], [73, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[55, "pca-and-scikit-learn"]], "Pandas AI": [[72, "pandas-ai"]], "Part a : Ordinary Least Square (OLS) for the Runge function": [[67, "part-a-ordinary-least-square-ols-for-the-runge-function"]], "Part b: Adding Ridge regression for the Runge function": [[67, "part-b-adding-ridge-regression-for-the-runge-function"]], "Part c: Writing your own gradient descent code": [[67, "part-c-writing-your-own-gradient-descent-code"]], "Part d: Including momentum and more advanced ways to update the learning the rate": [[67, "part-d-including-momentum-and-more-advanced-ways-to-update-the-learning-the-rate"]], "Part e: Writing our own code for Lasso regression": [[67, "part-e-writing-our-own-code-for-lasso-regression"]], "Part f: Stochastic gradient descent": [[67, "part-f-stochastic-gradient-descent"]], "Part g: Bias-variance trade-off and resampling techniques": [[67, "part-g-bias-variance-trade-off-and-resampling-techniques"]], "Part h): Cross-validation as resampling techniques, adding more complexity": [[67, "part-h-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Partial Differential Equations": [[46, "partial-differential-equations"]], "Pitaya Smoothie": [[15, null]], "Plans for week 35": [[73, "plans-for-week-35"]], "Plans for week 36": [[74, "plans-for-week-36"]], "Plans for week 37, lecture Monday": [[75, "plans-for-week-37-lecture-monday"]], "Plans for week 38, lecture Monday September 15": [[76, "plans-for-week-38-lecture-monday-september-15"]], "Plotting the Histogram": [[76, "plotting-the-histogram"]], "Practical tips": [[57, "practical-tips"], [75, "practical-tips"]], "Practicalities": [[70, "practicalities"], [70, "id1"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[67, "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": [[73, "preprocessing-our-data"]], "Prerequisites": [[72, "prerequisites"]], "Prerequisites and background": [[65, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[47, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[69, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[74, "program-example-for-gradient-descent-with-ridge-regression"], [75, "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": [[67, null]], "Properties of PDFs": [[69, "properties-of-pdfs"]], "Pros and cons": [[75, "pros-and-cons"]], "Pros and cons of trees, pros": [[53, "pros-and-cons-of-trees-pros"]], "Python installers": [[65, "python-installers"], [72, "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": [[75, "rmsprop-algorithm-taken-from-goodfellow-et-al"]], "RMSProp: Adaptive Learning Rates": [[75, "rmsprop-adaptive-learning-rates"]], "RMSprop for adaptive learning rate with Stochastic Gradient Descent": [[75, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "Random Numbers": [[69, "random-numbers"]], "Random forests": [[54, "random-forests"]], "Randomized PCA": [[55, "randomized-pca"]], "Reading material": [[72, "reading-material"]], "Reading recommendations:": [[73, "reading-recommendations"]], "Reading suggestions week 34": [[72, "reading-suggestions-week-34"]], "Readings and Videos": [[76, "readings-and-videos"]], "Readings and Videos:": [[75, "readings-and-videos"]], "Recurrent neural networks": [[56, "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"], [73, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[46, "reformulating-the-problem"]], "Regression Case": [[54, "regression-case"]], "Regression analysis and resampling methods": [[67, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[72, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[72, "regression-analysis-overarching-aims-ii"]], "Regularization": [[45, "regularization"]], "Reminder from last week": [[73, "reminder-from-last-week"]], "Reminder on Newton-Raphson\u2019s method": [[74, "reminder-on-newton-raphson-s-method"]], "Reminder on Statistics": [[50, "reminder-on-statistics"]], "Reminder on different scaling methods": [[75, "reminder-on-different-scaling-methods"]], "Replace or not": [[57, "replace-or-not"], [75, "replace-or-not"]], "Required Technologies": [[65, "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": [[76, "resampling-approaches-can-be-computationally-expensive"]], "Resampling methods": [[50, "id1"], [76, "resampling-methods"], [76, "id2"]], "Resampling methods: Bootstrap": [[76, "resampling-methods-bootstrap"]], "Resampling methods: Bootstrap approach": [[76, "resampling-methods-bootstrap-approach"]], "Resampling methods: Bootstrap background": [[76, "resampling-methods-bootstrap-background"]], "Resampling methods: Bootstrap steps": [[76, "resampling-methods-bootstrap-steps"]], "Resampling methods: More Bootstrap background": [[76, "resampling-methods-more-bootstrap-background"]], "Residual Error": [[73, "residual-error"], [74, "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": [[74, "revisiting-ordinary-least-squares"]], "Revisiting our Linear Regression Solvers": [[57, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[73, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the \\delta-function": [[76, "rewriting-the-delta-function"]], "Rewriting the fitting procedure as a linear algebra problem": [[72, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[72, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[74, "ridge-regression"]], "Ridge and LASSO Regression": [[73, "ridge-and-lasso-regression"], [74, "ridge-and-lasso-regression"], [74, "id2"]], "Ridge and Lasso Regression": [[49, null], [49, "id1"]], "SGD example": [[75, "sgd-example"]], "SGD vs Full-Batch GD: Convergence Speed and Memory Comparison": [[75, "sgd-vs-full-batch-gd-convergence-speed-and-memory-comparison"]], "SVD analysis": [[74, "svd-analysis"]], "Same code but now with momentum gradient descent": [[57, "same-code-but-now-with-momentum-gradient-descent"], [75, "same-code-but-now-with-momentum-gradient-descent"], [75, "id3"], [75, "id4"]], "Sample Roles and Directives": [[22, "sample-roles-and-directives"]], "Schedule first week": [[72, "schedule-first-week"]], "Schematic Regression Procedure": [[53, "schematic-regression-procedure"]], "Second moment of the gradient": [[75, "second-moment-of-the-gradient"]], "September 15-19": [[63, "september-15-19"]], "Setting up the Back propagation algorithm": [[56, "setting-up-the-back-propagation-algorithm"]], "Setting up the Matrix to be inverted": [[73, "setting-up-the-matrix-to-be-inverted"], [74, "setting-up-the-matrix-to-be-inverted"]], "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"], [75, "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": [[73, "simple-case"], [74, "simple-case"]], "Simple code for solving the above problem": [[74, "simple-code-for-solving-the-above-problem"]], "Simple example code": [[75, "simple-example-code"]], "Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression": [[74, "simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression"]], "Simple geometric interpretation": [[74, "simple-geometric-interpretation"]], "Simple linear regression model using scikit-learn": [[44, "simple-linear-regression-model-using-scikit-learn"], [72, "simple-linear-regression-model-using-scikit-learn"]], "Simple one-dimensional second-order polynomial": [[62, "simple-one-dimensional-second-order-polynomial"]], "Simple program": [[74, "simple-program"], [75, "simple-program"]], "Slightly different approach": [[75, "slightly-different-approach"]], "Sneaking in automatic differentiation using Autograd": [[75, "sneaking-in-automatic-differentiation-using-autograd"]], "Software and needed installations": [[67, "software-and-needed-installations"], [72, "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"]], "Some famous Matrices": [[66, "some-famous-matrices"]], "Some simple problems": [[57, "some-simple-problems"], [74, "some-simple-problems"]], "Some useful matrix and vector expressions": [[73, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[44, "splitting-our-data-in-training-and-test-data"], [73, "splitting-our-data-in-training-and-test-data"]], "Standard Approach based on the Normal Distribution": [[76, "standard-approach-based-on-the-normal-distribution"]], "Standard steepest descent": [[57, "standard-steepest-descent"]], "Statistical analysis": [[76, "statistical-analysis"]], "Statistical analysis and optimization of data": [[65, "statistical-analysis-and-optimization-of-data"], [72, "statistical-analysis-and-optimization-of-data"]], "Steepest descent": [[57, "steepest-descent"], [74, "steepest-descent"]], "Stochastic Gradient Descent": [[75, "stochastic-gradient-descent"]], "Stochastic Gradient Descent (SGD)": [[57, "stochastic-gradient-descent-sgd"], [75, "stochastic-gradient-descent-sgd"]], "Stochastic variables and the main concepts, the discrete case": [[69, "stochastic-variables-and-the-main-concepts-the-discrete-case"]], "Strongly Convex Case": [[75, "strongly-convex-case"]], "Structure of translation files": [[41, "structure-of-translation-files"]], "Summing up": [[76, "summing-up"]], "Support Vector Machines, overarching aims": [[52, null]], "Systematic reduction": [[47, "systematic-reduction"]], "Teachers": [[72, "teachers"]], "Teachers and Grading": [[70, null]], "Teaching Assistants Fall semester 2023": [[70, "teaching-assistants-fall-semester-2023"]], "Tentative deadllines for projects": [[70, "tentative-deadllines-for-projects"]], "Testing the Means Squared Error as function of Complexity": [[44, "testing-the-means-squared-error-as-function-of-complexity"], [73, "testing-the-means-squared-error-as-function-of-complexity"]], "Textbooks": [[71, null]], "Thanks": [[30, "thanks"]], "The Algorithm before theorem": [[55, "the-algorithm-before-theorem"]], "The Breast Cancer Data, now with Keras": [[45, "the-breast-cancer-data-now-with-keras"]], "The CART algorithm for Classification": [[53, "the-cart-algorithm-for-classification"]], "The CART algorithm for Regression": [[53, "the-cart-algorithm-for-regression"]], "The CIFAR01 data set": [[47, "the-cifar01-data-set"]], "The Central Limit Theorem": [[76, "the-central-limit-theorem"]], "The Hessian matrix": [[74, "the-hessian-matrix"], [75, "the-hessian-matrix"]], "The Hessian matrix for Ridge Regression": [[74, "the-hessian-matrix-for-ridge-regression"], [75, "the-hessian-matrix-for-ridge-regression"]], "The IPython licensing terms": [[20, null]], "The Jacobian": [[73, "the-jacobian"]], "The MIT License (MIT)": [[19, null]], "The MNIST dataset again": [[47, "the-mnist-dataset-again"]], "The OLS case": [[74, "the-ols-case"]], "The RELU function family": [[45, "the-relu-function-family"]], "The Ridge case": [[74, "the-ridge-case"]], "The SVD, a Fantastic Algorithm": [[73, "the-svd-a-fantastic-algorithm"], [74, "the-svd-a-fantastic-algorithm"]], "The Softmax function": [[45, "the-softmax-function"]], "The \\chi^2 function": [[44, "the-chi-2-function"], [72, "the-chi-2-function"], [72, "id4"], [72, "id5"], [72, "id6"], [72, "id7"], [72, "id8"]], "The bias-variance tradeoff": [[50, "the-bias-variance-tradeoff"], [76, "the-bias-variance-tradeoff"]], "The code for solving the ODE": [[46, "the-code-for-solving-the-ode"]], "The complete code with a simple data set": [[73, "the-complete-code-with-a-simple-data-set"]], "The cost/loss function": [[73, "the-cost-loss-function"]], "The course has two central parts": [[65, "the-course-has-two-central-parts"]], "The derivative of the cost/loss function": [[74, "the-derivative-of-the-cost-loss-function"], [75, "the-derivative-of-the-cost-loss-function"]], "The equations": [[74, "the-equations"]], "The equations for ordinary least squares": [[73, "the-equations-for-ordinary-least-squares"]], "The first Case": [[74, "the-first-case"]], "The gradient step": [[75, "the-gradient-step"]], "The ideal": [[74, "the-ideal"]], "The logistic function": [[51, "the-logistic-function"]], "The mean squared error and its derivative": [[73, "the-mean-squared-error-and-its-derivative"]], "The moons example": [[52, "the-moons-example"]], "The multilayer perceptron (MLP)": [[56, "the-multilayer-perceptron-mlp"]], "The network with one input layer, specified number of hidden layers, and one output layer": [[46, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"]], "The plethora of machine learning algorithms/methods": [[72, "the-plethora-of-machine-learning-algorithms-methods"]], "The same example but now with cross-validation": [[76, "the-same-example-but-now-with-cross-validation"]], "The sensitiveness of the gradient descent": [[74, "the-sensitiveness-of-the-gradient-descent"]], "The singular value decomposition": [[49, "the-singular-value-decomposition"], [73, "the-singular-value-decomposition"], [74, "the-singular-value-decomposition"]], "The two-dimensional case": [[52, "the-two-dimensional-case"]], "Theoretical Convergence Speed and convex optimization": [[75, "theoretical-convergence-speed-and-convex-optimization"]], "Time decay rate": [[75, "time-decay-rate"]], "To our real data: nuclear binding energies. Brief reminder on masses and binding energies": [[72, "to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies"]], "To update a translation": [[41, "to-update-a-translation"]], "ToDo": [[29, "todo"]], "Topics covered in this course: Statistical analysis and optimization of data": [[72, "topics-covered-in-this-course-statistical-analysis-and-optimization-of-data"]], "Towards the PCA theorem": [[55, "towards-the-pca-theorem"]], "Train and test datasets": [[45, "train-and-test-datasets"]], "Translation source files": [[41, "translation-source-files"]], "Translation workflow": [[41, null]], "Two-dimensional Objects": [[47, "two-dimensional-objects"]], "Type of problem": [[46, "type-of-problem"]], "Types of Machine Learning": [[72, "types-of-machine-learning"]], "Understanding what happens": [[76, "understanding-what-happens"]], "Use": [[28, "use"]], "Use in Browser": [[29, "use-in-browser"]], "Use the books!": [[63, "use-the-books"]], "Use with node.js": [[29, "use-with-node-js"]], "Useful Python libraries": [[65, "useful-python-libraries"], [72, "useful-python-libraries"]], "Using Autograd": [[57, "using-autograd"]], "Using forward Euler to solve the ODE": [[46, "using-forward-euler-to-solve-the-ode"]], "Using gradient descent methods, limitations": [[57, "using-gradient-descent-methods-limitations"], [74, "using-gradient-descent-methods-limitations"], [75, "using-gradient-descent-methods-limitations"]], "Using maskedarray with matplotlib": [[30, "using-maskedarray-with-matplotlib"]], "Using the new package with numpy.core.ma": [[30, "using-the-new-package-with-numpy-core-ma"]], "Various steps in cross-validation": [[76, "various-steps-in-cross-validation"]], "Visualization": [[45, "visualization"], [45, "id1"]], "Visualizing the Tree, Classification": [[53, "visualizing-the-tree-classification"]], "Week 34: Introduction to the course, Logistics and Practicalities": [[72, null]], "Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression": [[73, null]], "Week 36: Linear Regression and Gradient descent": [[74, null]], "Week 37: Gradient descent methods": [[75, null]], "Week 38: Statistical analysis, bias-variance tradeoff and resampling methods": [[76, null]], "Welcome to your Jupyter Book": [[21, null]], "What Is Generative Modeling?": [[72, "what-is-generative-modeling"]], "What does it mean?": [[73, "what-does-it-mean"], [74, "what-does-it-mean"]], "What is Machine Learning?": [[44, "what-is-machine-learning"]], "What is MyST?": [[22, "what-is-myst"]], "What is a good model?": [[44, "what-is-a-good-model"], [72, "what-is-a-good-model"]], "What is a good model? 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"up": [44, 46, 53, 56, 59, 72, 73, 74, 76], "updat": [41, 67, 75], "us": [28, 29, 30, 44, 45, 46, 47, 51, 57, 60, 62, 63, 65, 67, 72, 73, 74, 75], "usag": 75, "v": [47, 75], "valid": [50, 67, 76], "valu": [49, 55, 63, 69, 73, 74, 76], "vari": 75, "variabl": [69, 74], "varianc": [50, 63, 67, 76], "variou": [44, 76], "vector": [52, 56, 60, 66, 72, 73], "versu": 72, "video": [75, 76], "view": [44, 48, 54, 73], "virtual": 59, "visual": [45, 53], "wai": [53, 67, 76], "wave": 46, "we": [72, 75], "wednesdai": 74, "week": [59, 60, 61, 62, 63, 64, 72, 73, 74, 75, 76], "weekli": [], "welcom": 21, "what": [22, 44, 72, 73, 74, 76], "when": 75, "which": [45, 75], "why": [72, 75, 76], "wisconsin": 51, "workflow": 41, "wrap": 76, "write": [48, 55, 64, 67, 74], "x": 73, "xgboost": 54, "yaml": 23, "yet": 74, "your": [21, 44, 54, 60, 62, 67, 73]}})
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/statistics.html b/doc/LectureNotes/_build/html/statistics.html
index 88b5655fc..26c249fb9 100644
--- a/doc/LectureNotes/_build/html/statistics.html
+++ b/doc/LectureNotes/_build/html/statistics.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/teachers.html b/doc/LectureNotes/_build/html/teachers.html
index f07ee847a..2f927a9d7 100644
--- a/doc/LectureNotes/_build/html/teachers.html
+++ b/doc/LectureNotes/_build/html/teachers.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/textbooks.html b/doc/LectureNotes/_build/html/textbooks.html
index 60ac2e10c..395e86015 100644
--- a/doc/LectureNotes/_build/html/textbooks.html
+++ b/doc/LectureNotes/_build/html/textbooks.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/week34.html b/doc/LectureNotes/_build/html/week34.html
index 564c11e4b..4c1440e30 100644
--- a/doc/LectureNotes/_build/html/week34.html
+++ b/doc/LectureNotes/_build/html/week34.html
@@ -28,7 +28,7 @@
-
+
diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html
index ff2da35cc..6fe5c65b9 100644
--- a/doc/LectureNotes/_build/html/week35.html
+++ b/doc/LectureNotes/_build/html/week35.html
@@ -28,7 +28,7 @@
-
+
@@ -895,7 +895,7 @@ 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
@@ -929,7 +929,7 @@ 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)
@@ -946,7 +946,7 @@ 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
@@ -958,7 +958,7 @@ 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))
@@ -985,16 +985,16 @@ 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
@@ -1035,7 +1035,7 @@ 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()
@@ -1140,13 +1140,13 @@ 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
@@ -1202,12 +1202,12 @@ 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)
@@ -1487,13 +1487,13 @@ 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
@@ -1651,9 +1651,9 @@ 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.
@@ -2026,7 +2026,7 @@ 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))
@@ -2049,7 +2049,7 @@ 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)
@@ -2084,8 +2084,8 @@ 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)
@@ -2523,20 +2523,20 @@ 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
@@ -2666,13 +2666,13 @@ 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
@@ -2741,14 +2741,14 @@ 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 f5de8a3c0..c1d4a86e5 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 5e2eeeb7b..3cc219e03 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 6f81f5ea5..0bfa4cf19 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/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 fe0b695d3..bfc403c58 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": "089950cb",
+ "id": "23486404",
"metadata": {},
"source": [
"# Notebooks with MyST Markdown\n",
@@ -19,7 +19,7 @@
{
"cell_type": "code",
"execution_count": null,
- "id": "87c2f6e9",
+ "id": "cd6246ff",
"metadata": {},
"outputs": [],
"source": [
@@ -28,7 +28,7 @@
},
{
"cell_type": "markdown",
- "id": "04f8b494",
+ "id": "da76d871",
"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/exercisesweek38.ipynb b/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb
index c8894b983..93d8969c2 100644
--- a/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/exercisesweek38.ipynb
@@ -290,7 +290,8 @@
"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$\n",
- "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$."
+ "\n",
+ "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$.\n"
]
},
{
@@ -321,7 +322,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": null,
"id": "b5bf581c",
"metadata": {},
"outputs": [],
@@ -332,7 +333,8 @@
"bootstraps = 1000\n",
"\n",
"predictions = np.random.rand(bootstraps, n) * 10 + 10\n",
- "targets = np.random.rand(bootstraps, n)\n",
+ "# The definition of targets has been updated, and was wrong earlier in the week.\n",
+ "targets = np.random.rand(1, n)\n",
"\n",
"mse = ...\n",
"bias = ...\n",
diff --git a/doc/LectureNotes/exercisesweek38.ipynb b/doc/LectureNotes/exercisesweek38.ipynb
index 48a19f138..c100028a5 100644
--- a/doc/LectureNotes/exercisesweek38.ipynb
+++ b/doc/LectureNotes/exercisesweek38.ipynb
@@ -290,7 +290,8 @@
"$$\n",
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
"$$\n",
- "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$."
+ "\n",
+ "In order to arrive at the equation for the bias, we have to approximate the unknown function $f$ with the output/target values $y$.\n"
]
},
{
@@ -321,7 +322,7 @@
},
{
"cell_type": "code",
- "execution_count": 67,
+ "execution_count": null,
"id": "b5bf581c",
"metadata": {},
"outputs": [],
@@ -332,7 +333,8 @@
"bootstraps = 1000\n",
"\n",
"predictions = np.random.rand(bootstraps, n) * 10 + 10\n",
- "targets = np.random.rand(bootstraps, n)\n",
+ "# The definition of targets has been updated, and was wrong earlier in the week.\n",
+ "targets = np.random.rand(1, n)\n",
"\n",
"mse = ...\n",
"bias = ...\n",