From c8f2aa3dc1eea65765841c838eb6a8298ac619b7 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 25 Nov 2024 08:12:27 +0100 Subject: [PATCH] update --- doc/LectureNotes/DataFiles/cancer.dot | 12 +- doc/LectureNotes/DataFiles/cancer.png | Bin 247219 -> 249475 bytes .../_build/.doctrees/chapter1.doctree | Bin 396486 -> 396486 bytes .../_build/.doctrees/chapter10.doctree | Bin 241083 -> 263045 bytes .../_build/.doctrees/chapter13.doctree | Bin 183970 -> 186502 bytes .../_build/.doctrees/chapter2.doctree | Bin 362377 -> 362363 bytes .../_build/.doctrees/chapter3.doctree | Bin 383363 -> 385343 bytes .../_build/.doctrees/chapter6.doctree | Bin 168815 -> 168807 bytes .../_build/.doctrees/chapter8.doctree | Bin 150133 -> 150127 bytes .../_build/.doctrees/environment.pickle | Bin 829216 -> 871516 bytes .../_build/.doctrees/exercisesweek41.doctree | Bin 95059 -> 95063 bytes .../_build/.doctrees/exercisesweek48.doctree | Bin 0 -> 16624 bytes .../_build/.doctrees/intro.doctree | Bin 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doc/LectureNotes/_build/jupyter_execute/week48.ipynb create mode 100644 doc/LectureNotes/exercisesweek48.ipynb diff --git a/doc/LectureNotes/DataFiles/cancer.dot b/doc/LectureNotes/DataFiles/cancer.dot index 4b5537038..850904527 100644 --- a/doc/LectureNotes/DataFiles/cancer.dot +++ b/doc/LectureNotes/DataFiles/cancer.dot @@ -6,11 +6,11 @@ edge [fontname="helvetica"] ; 0 -> 1 [labeldistance=2.5, labelangle=45, headlabel="True"] ; 2 [label="worst concave points <= 0.135\ngini = 0.031\nsamples = 253\nvalue = [[249, 4]\n[4, 249]]", fillcolor="#e78946"] ; 1 -> 2 ; -3 [label="area error <= 48.975\ngini = 0.008\nsamples = 242\nvalue = [[241, 1]\n[1, 241]]", fillcolor="#e5833c"] ; +3 [label="radius error <= 0.643\ngini = 0.008\nsamples = 242\nvalue = [[241, 1]\n[1, 241]]", fillcolor="#e5833c"] ; 2 -> 3 ; 4 [label="gini = 0.0\nsamples = 239\nvalue = [[239, 0]\n[0, 239]]", fillcolor="#e58139"] ; 3 -> 4 ; -5 [label="radius error <= 0.688\ngini = 0.444\nsamples = 3\nvalue = [[2, 1]\n[1, 2]]", fillcolor="#fdf6f0"] ; +5 [label="worst area <= 566.55\ngini = 0.444\nsamples = 3\nvalue = [[2, 1]\n[1, 2]]", fillcolor="#fdf6f0"] ; 3 -> 5 ; 6 [label="gini = 0.0\nsamples = 1\nvalue = [[0, 1]\n[1, 0]]", fillcolor="#e58139"] ; 5 -> 6 ; @@ -34,7 +34,7 @@ edge [fontname="helvetica"] ; 14 -> 15 ; 16 [label="gini = 0.0\nsamples = 11\nvalue = [[11, 0]\n[0, 11]]", fillcolor="#e58139"] ; 15 -> 16 ; -17 [label="area error <= 23.16\ngini = 0.32\nsamples = 5\nvalue = [[1, 4]\n[4, 1]]", fillcolor="#f6d5bd"] ; +17 [label="radius error <= 0.251\ngini = 0.32\nsamples = 5\nvalue = [[1, 4]\n[4, 1]]", fillcolor="#f6d5bd"] ; 15 -> 17 ; 18 [label="gini = 0.0\nsamples = 1\nvalue = [[1, 0]\n[0, 1]]", fillcolor="#e58139"] ; 17 -> 18 ; @@ -48,10 +48,10 @@ edge [fontname="helvetica"] ; 21 -> 22 ; 23 [label="gini = 0.0\nsamples = 6\nvalue = [[6, 0]\n[0, 6]]", fillcolor="#e58139"] ; 21 -> 23 ; -24 [label="mean smoothness <= 0.079\ngini = 0.015\nsamples = 136\nvalue = [[1, 135]\n[135, 1]]", fillcolor="#e6853f"] 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    Figure 1:

    \n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "71b8eb59", + "metadata": { + "editable": true + }, + "source": [ + "## Topics we have covered this year\n", + "\n", + "The course has two central parts\n", + "\n", + "1. Statistical analysis and optimization of data\n", + "\n", + "2. Machine learning" + ] + }, + { + "cell_type": "markdown", + "id": "473329b7", + "metadata": { + "editable": true + }, + "source": [ + "## Statistical analysis and optimization of data\n", + "\n", + "The following topics have been discussed:\n", + "1. Basic concepts, expectation values, variance, covariance, correlation functions and errors;\n", + "\n", + "2. Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;\n", + "\n", + "3. Central elements from linear algebra, matrix inversion and SVD\n", + "\n", + "4. Gradient methods for data optimization\n", + "\n", + "5. Estimation of errors using cross-validation, bootstrapping and jackknife methods;\n", + "\n", + "6. Practical optimization using Singular-value decomposition and least squares for parameterizing data.\n", + "\n", + "7. Not discussed: Principal Component Analysis to reduce the number of features." + ] + }, + { + "cell_type": "markdown", + "id": "6c94f44a", + "metadata": { + "editable": true + }, + "source": [ + "## Machine learning\n", + "\n", + "* Linear methods for regression and classification:\n", + "\n", + "a. Ordinary Least Squares\n", + "\n", + "b. Ridge regression\n", + "\n", + "c. Lasso regression\n", + "\n", + "d. Logistic regression\n", + "\n", + "* Neural networks and deep learning:\n", + "\n", + "a. Feed Forward Neural Networks\n", + "\n", + "b. Convolutional Neural Networks\n", + "\n", + "c. Recurrent Neural Networks\n", + "\n", + "* Decisions trees and ensemble methods:\n", + "\n", + "a. Decision trees\n", + "\n", + "b. Bagging and voting\n", + "\n", + "c. Random forests\n", + "\n", + "d. Boosting and gradient boosting\n", + "\n", + "* Not discussed this year: Support vector machines\n", + "\n", + "a. Binary classification and multiclass classification\n", + "\n", + "b. Kernel methods\n", + "\n", + "c. Regression" + ] + }, + { + "cell_type": "markdown", + "id": "d9942ded", + "metadata": { + "editable": true + }, + "source": [ + "## Learning outcomes and overarching aims of this course\n", + "\n", + "The course introduces a variety of central algorithms and methods\n", + "essential for studies of data analysis and machine learning. The\n", + "course is project based and through the various projects, normally\n", + "three, you will be exposed to fundamental research problems\n", + "in these fields, with the aim to reproduce state of the art scientific\n", + "results. The students will learn to develop and structure large codes\n", + "for studying these systems, get acquainted with computing facilities\n", + "and learn to handle large scientific projects. A good scientific and\n", + "ethical conduct is emphasized throughout the course. \n", + "\n", + "* Understand linear methods for regression and classification;\n", + "\n", + "* Learn about neural network;\n", + "\n", + "* Learn about bagging, boosting and trees\n", + "\n", + "\n", + "* Learn about basic data analysis;\n", + "\n", + "* Be capable of extending the acquired knowledge to other systems and cases;\n", + "\n", + "* Have an understanding of central algorithms used in data analysis and machine learning;\n", + "\n", + "* Work on numerical projects to illustrate the theory. The projects play a central role." + ] + }, + { + "cell_type": "markdown", + "id": "7818c700", + "metadata": { + "editable": true + }, + "source": [ + "## Perspective on Machine Learning\n", + "\n", + "1. Rapidly emerging application area\n", + "\n", + "2. Experiment AND theory are evolving in many many fields. \n", + "\n", + "3. Requires education/retraining for more widespread adoption\n", + "\n", + "4. A lot of “word-of-mouth” development methods\n", + "\n", + "Huge amounts of data sets require automation, classical analysis tools often inadequate. \n", + "High energy physics hit this wall in the 90’s.\n", + "In 2009 single top quark production was determined via [Boosted decision trees, Bayesian\n", + "Neural Networks, etc.](https://arxiv.org/pdf/0903.0850.pdf)" + ] + }, + { + "cell_type": "markdown", + "id": "cc946602", + "metadata": { + "editable": true + }, + "source": [ + "## Machine Learning Research\n", + "\n", + "Where to find recent results:\n", + "1. Conference proceedings, arXiv and blog posts!\n", + "\n", + "2. **NIPS**: [Neural Information Processing Systems](https://papers.nips.cc)\n", + "\n", + "3. **ICLR**: [International Conference on Learning Representations](https://openreview.net/group?id=ICLR.cc/2018/Conference#accepted-oral-papers)\n", + "\n", + "4. **ICML**: International Conference on Machine Learning\n", + "\n", + "5. [Journal of Machine Learning Research](http://www.jmlr.org/papers/v19/) \n", + "\n", + "6. [Follow ML on ArXiv](https://arxiv.org/list/cs.LG/recent)" + ] + }, + { + "cell_type": "markdown", + "id": "c2b75185", + "metadata": { + "editable": true + }, + "source": [ + "## Starting your Machine Learning Project\n", + "\n", + "1. Identify problem type: classification, regression\n", + "\n", + "2. Consider your data carefully\n", + "\n", + "3. Choose a simple model that fits 1 and 2\n", + "\n", + "4. Consider your data carefully again! Think of data representation more carefully.\n", + "\n", + "5. Based on your results, feedback loop to earliest possible point" + ] + }, + { + "cell_type": "markdown", + "id": "4600986a", + "metadata": { + "editable": true + }, + "source": [ + "## Choose a Model and Algorithm\n", + "\n", + "* Supervised?\n", + "\n", + "* Start with the simplest model that fits your problem\n", + "\n", + "* Start with minimal processing of data" + ] + }, + { + "cell_type": "markdown", + "id": "cfb572d5", + "metadata": { + "editable": true + }, + "source": [ + "## Preparing Your Data\n", + "\n", + "* Shuffle your data\n", + "\n", + "* Mean center your data\n", + "\n", + " * Why?\n", + "\n", + "* Normalize the variance\n", + "\n", + " * Why?\n", + "\n", + "* **Whitening**\n", + "\n", + " * Decorrelates data\n", + "\n", + " * Can be hit or miss\n", + "\n", + " * When to do train/test split?" + ] + }, + { + "cell_type": "markdown", + "id": "3e38f862", + "metadata": { + "editable": true + }, + "source": [ + "## Which activation and weights to choose in neural networks\n", + "\n", + "* RELU? ELU? GELU? etc\n", + "\n", + "* Sigmoid or Tanh?\n", + "\n", + "* Set all weights to 0? Terrible idea\n", + "\n", + "* Set all weights to random values? Small random values" + ] + }, + { + "cell_type": "markdown", + "id": "76abe122", + "metadata": { + "editable": true + }, + "source": [ + "## Optimization Methods and Hyperparameters\n", + "* Stochastic gradient descent\n", + "\n", + " * Stochastic gradient descent + momentum\n", + "\n", + "* State-of-the-art approaches:\n", + "\n", + "a. RMSProp\n", + "\n", + "b. Adam\n", + "\n", + "c. and more\n", + "\n", + "Which regularization and hyperparameters? $L_1$ or $L_2$, soft\n", + "classifiers, depths of trees and many other. Need to explore a large\n", + "set of hyperparameters and regularization methods." + ] + }, + { + "cell_type": "markdown", + "id": "0b246db8", + "metadata": { + "editable": true + }, + "source": [ + "## Resampling\n", + "\n", + "When do we resample?\n", + "\n", + "1. [Bootstrap](https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A)\n", + "\n", + "2. [Cross-validation](https://www.youtube.com/watch?v=fSytzGwwBVw&ab_channel=StatQuestwithJoshStarmer)\n", + "\n", + "3. Jackknife and many other" + ] + }, + { + "cell_type": "markdown", + "id": "9a6555fc", + "metadata": { + "editable": true + }, + "source": [ + "## Other courses on Data science and Machine Learning at UiO\n", + "\n", + "1. [FYS5429 – Advanced machine learning and data analysis for the physical sciences](https://www.uio.no/studier/emner/matnat/fys/FYS5429/index-eng.html)\n", + "\n", + "2. [IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning](https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html). Introductory course in machine learning and AI\n", + "\n", + "3. [STK-INF3000/4000 Selected Topics in Data Science](http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html). The course provides insight into selected contemporary relevant topics within Data Science. \n", + "\n", + "4. [IN4080 Natural Language Processing](https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html). Probabilistic and machine learning techniques applied to natural language processing. \n", + "\n", + "5. [STK-IN4300 – Statistical learning methods in Data Science](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html). An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.\n", + "\n", + "6. [IN-STK5000 Responsible Data Science](https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html). Methods for adaptive collection and processing of data based on machine learning techniques. \n", + "\n", + "7. [IN4310 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN4310/index.html). An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.\n", + "\n", + "8. [IN5310 – Advanced Deep Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5310/index.html)\n", + "\n", + "9. [IN5490 – Advanced Topics in Artificial Intelligence for Intelligent Systems](https://www.uio.no/studier/emner/matnat/ifi/IN5490/index.html)\n", + "\n", + "10. [TEK5040 – Deep learning for autonomous systems](https://www.uio.no/studier/emner/matnat/its/TEK5040/). The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments." + ] + }, + { + "cell_type": "markdown", + "id": "c2561e72", + "metadata": { + "editable": true + }, + "source": [ + "## Additional courses of interest\n", + "\n", + "1. [STK4051 Computational Statistics](https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html)\n", + "\n", + "2. [STK4021 Applied Bayesian Analysis and Numerical Methods](https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html)" + ] + }, + { + "cell_type": "markdown", + "id": "796637c8", + "metadata": { + "editable": true + }, + "source": [ + "## What's the future like?\n", + "\n", + "Based on multi-layer nonlinear neural networks, deep learning can\n", + "learn directly from raw data, automatically extract and abstract\n", + "features from layer to layer, and then achieve the goal of regression,\n", + "classification, or ranking. Deep learning has made breakthroughs in\n", + "computer vision, speech processing and natural language, and reached\n", + "or even surpassed human level. The success of deep learning is mainly\n", + "due to the three factors: big data, big model, and big computing.\n", + "\n", + "In the past few decades, many different architectures of deep neural\n", + "networks have been proposed, such as\n", + "1. Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;\n", + "\n", + "2. Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;\n", + "\n", + "3. Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning." + ] + }, + { + "cell_type": "markdown", + "id": "6afdb59e", + "metadata": { + "editable": true + }, + "source": [ + "## Types of Machine Learning, a repetition\n", + "\n", + "The approaches to machine learning are many, but are often split into two main categories. \n", + "In *supervised learning* we know the answer to a problem,\n", + "and let the computer deduce the logic behind it. On the other hand, *unsupervised learning*\n", + "is a method for finding patterns and relationship in data sets without any prior knowledge of the system.\n", + "Some authours also operate with a third category, namely *reinforcement learning*. This is a paradigm \n", + "of learning inspired by behavioural psychology, where learning is achieved by trial-and-error, \n", + "solely from rewards and punishment.\n", + "\n", + "Another way to categorize machine learning tasks is to consider the desired output of a system.\n", + "Some of the most common tasks are:\n", + "\n", + " * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.\n", + "\n", + " * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", + "\n", + " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", + "\n", + " * Other unsupervised learning algortihms like **Boltzmann machines**" + ] + }, + { + "cell_type": "markdown", + "id": "95097424", + "metadata": { + "editable": true + }, + "source": [ + "## Why Boltzmann machines?\n", + "\n", + "What is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately. \n", + "One of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.\n", + "\n", + "The original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images. \n", + "\n", + "Furthermore, they have been used to solve complicated [quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002)." + ] + }, + { + "cell_type": "markdown", + "id": "31a33cea", + "metadata": { + "editable": true + }, + "source": [ + "## Boltzmann Machines\n", + "\n", + "Why use a generative model rather than the more well known discriminative deep neural networks (DNN)? \n", + "\n", + "* Discriminitave methods have several limitations: They are mainly supervised learning methods, thus requiring labeled data. And there are tasks they cannot accomplish, like drawing new examples from an unknown probability distribution.\n", + "\n", + "* A generative model can learn to represent and sample from a probability distribution. The core idea is to learn a parametric model of the probability distribution from which the training data was drawn. As an example\n", + "\n", + "a. A model for images could learn to draw new examples of cats and dogs, given a training dataset of images of cats and dogs.\n", + "\n", + "b. Generate a sample of an ordered or disordered phase, having been given samples of such phases.\n", + "\n", + "c. Model the trial function for [Monte Carlo calculations](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002)." + ] + }, + { + "cell_type": "markdown", + "id": "d6583875", + "metadata": { + "editable": true + }, + "source": [ + "## Some similarities and differences from DNNs\n", + "\n", + "1. Both use gradient-descent based learning procedures for minimizing cost functions\n", + "\n", + "2. Energy based models don't use backpropagation and automatic differentiation for computing gradients, instead turning to Markov Chain Monte Carlo methods.\n", + "\n", + "3. DNNs often have several hidden layers. A restricted Boltzmann machine has only one hidden layer, however several RBMs can be stacked to make up Deep Belief Networks, of which they constitute the building blocks.\n", + "\n", + "History: The RBM was developed by amongst others [Geoffrey Hinton](https://en.wikipedia.org/wiki/Geoffrey_Hinton), called by some the \"Godfather of Deep Learning\", working with the University of Toronto and Google." + ] + }, + { + "cell_type": "markdown", + "id": "285bbf5d", + "metadata": { + "editable": true + }, + "source": [ + "## Boltzmann machines (BM)\n", + "\n", + "A BM is what we would call an undirected probabilistic graphical model\n", + "with stochastic continuous or discrete units.\n", + "\n", + "It is interpreted as a stochastic recurrent neural network where the\n", + "state of each unit(neurons/nodes) depends on the units it is connected\n", + "to. The weights in the network represent thus the strength of the\n", + "interaction between various units/nodes.\n", + "\n", + "It turns into a Hopfield network if we choose deterministic rather\n", + "than stochastic units. In contrast to a Hopfield network, a BM is a\n", + "so-called generative model. It allows us to generate new samples from\n", + "the learned distribution." + ] + }, + { + "cell_type": "markdown", + "id": "5b51dec9", + "metadata": { + "editable": true + }, + "source": [ + "## A standard BM setup\n", + "\n", + "A standard BM network is divided into a set of observable and visible units $\\hat{x}$ and a set of unknown hidden units/nodes $\\hat{h}$.\n", + "\n", + "Additionally there can be bias nodes for the hidden and visible layers. These biases are normally set to $1$.\n", + "\n", + "BMs are stackable, meaning they cwe can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning\n", + "\n", + "However, they are often hard to train. This leads to the introduction of so-called restricted BMs, or RBMS.\n", + "Here we take away all lateral connections between nodes in the visible layer as well as connections between nodes in the hidden layer. The network is illustrated in the figure below." + ] + }, + { + "cell_type": "markdown", + "id": "4cb37b8f", + "metadata": { + "editable": true + }, + "source": [ + "## The structure of the RBM network\n", + "\n", + "\n", + "\n", + "\n", + "

    Figure 1:

    \n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "45fe33d7", + "metadata": { + "editable": true + }, + "source": [ + "## The network\n", + "\n", + "**The network layers**:\n", + "1. A function $\\mathbf{x}$ that represents the visible layer, a vector of $M$ elements (nodes). This layer represents both what the RBM might be given as training input, and what we want it to be able to reconstruct. This might for example be given by the pixels of an image or coefficients representing speech, or the coordinates of a quantum mechanical state function.\n", + "\n", + "2. The function $\\mathbf{h}$ represents the hidden, or latent, layer. A vector of $N$ elements (nodes). Also called \"feature detectors\"." + ] + }, + { + "cell_type": "markdown", + "id": "3bbbaa88", + "metadata": { + "editable": true + }, + "source": [ + "## Goals\n", + "\n", + "The goal of the hidden layer is to increase the model's expressive\n", + "power. We encode complex interactions between visible variables by\n", + "introducing additional, hidden variables that interact with visible\n", + "degrees of freedom in a simple manner, yet still reproduce the complex\n", + "correlations between visible degrees in the data once marginalized\n", + "over (integrated out).\n", + "\n", + "**The network parameters, to be optimized/learned**:\n", + "1. $\\mathbf{a}$ represents the visible bias, a vector of same length as $\\mathbf{x}$.\n", + "\n", + "2. $\\mathbf{b}$ represents the hidden bias, a vector of same lenght as $\\mathbf{h}$.\n", + "\n", + "3. $W$ represents the interaction weights, a matrix of size $M\\times N$." + ] + }, + { + "cell_type": "markdown", + "id": "d6a9f606", + "metadata": { + "editable": true + }, + "source": [ + "## Joint distribution\n", + "\n", + "The restricted Boltzmann machine is described by a Boltzmann distribution" + ] + }, + { + "cell_type": "markdown", + "id": "81aaff6e", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\tP_{rbm}(\\mathbf{x},\\mathbf{h}) = \\frac{1}{Z} e^{-\\frac{1}{T_0}E(\\mathbf{x},\\mathbf{h})},\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9f856eb3", + "metadata": { + "editable": true + }, + "source": [ + "where $Z$ is the normalization constant or partition function, defined as" + ] + }, + { + "cell_type": "markdown", + "id": "f15429d1", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\tZ = \\int \\int e^{-\\frac{1}{T_0}E(\\mathbf{x},\\mathbf{h})} d\\mathbf{x} d\\mathbf{h}.\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e8a8f8e7", + "metadata": { + "editable": true + }, + "source": [ + "It is common to ignore $T_0$ by setting it to one." + ] + }, + { + "cell_type": "markdown", + "id": "4fb17989", + "metadata": { + "editable": true + }, + "source": [ + "## Network Elements, the energy function\n", + "\n", + "The function $E(\\mathbf{x},\\mathbf{h})$ gives the **energy** of a\n", + "configuration (pair of vectors) $(\\mathbf{x}, \\mathbf{h})$. The lower\n", + "the energy of a configuration, the higher the probability of it. This\n", + "function also depends on the parameters $\\mathbf{a}$, $\\mathbf{b}$ and\n", + "$W$. Thus, when we adjust them during the learning procedure, we are\n", + "adjusting the energy function to best fit our problem.\n", + "\n", + "An expression for the energy function is" + ] + }, + { + "cell_type": "markdown", + "id": "0bed5ddd", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "E(\\hat{x},\\hat{h}) = -\\sum_{ia}^{NA}b_i^a \\alpha_i^a(x_i)-\\sum_{jd}^{MD}c_j^d \\beta_j^d(h_j)-\\sum_{ijad}^{NAMD}b_i^a \\alpha_i^a(x_i)c_j^d \\beta_j^d(h_j)w_{ij}^{ad}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f8e64229", + "metadata": { + "editable": true + }, + "source": [ + "Here $\\beta_j^d(h_j)$ and $\\alpha_i^a(x_j)$ are so-called transfer functions that map a given input value to a desired feature value. The labels $a$ and $d$ denote that there can be multiple transfer functions per variable. The first sum depends only on the visible units. The second on the hidden ones. **Note** that there is no connection between nodes in a layer.\n", + "\n", + "The quantities $b$ and $c$ can be interpreted as the visible and hidden biases, respectively.\n", + "\n", + "The connection between the nodes in the two layers is given by the weights $w_{ij}$." + ] + }, + { + "cell_type": "markdown", + "id": "f8f02179", + "metadata": { + "editable": true + }, + "source": [ + "## Defining different types of RBMs\n", + "There are different variants of RBMs, and the differences lie in the types of visible and hidden units we choose as well as in the implementation of the energy function $E(\\mathbf{x},\\mathbf{h})$. \n", + "\n", + "**Binary-Binary RBM:**\n", + "\n", + "RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:" + ] + }, + { + "cell_type": "markdown", + "id": "19008689", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\tE(\\mathbf{x}, \\mathbf{h}) = - \\sum_i^M x_i a_i- \\sum_j^N b_j h_j - \\sum_{i,j}^{M,N} x_i w_{ij} h_j,\n", + "\\label{_auto3} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a3d31b3a", + "metadata": { + "editable": true + }, + "source": [ + "where the binary values taken on by the nodes are most commonly 0 and 1.\n", + "\n", + "**Gaussian-Binary RBM:**\n", + "\n", + "Another varient is the RBM where the visible units are Gaussian while the hidden units remain binary:" + ] + }, + { + "cell_type": "markdown", + "id": "79d29641", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\tE(\\mathbf{x}, \\mathbf{h}) = \\sum_i^M \\frac{(x_i - a_i)^2}{2\\sigma_i^2} - \\sum_j^N b_j h_j - \\sum_{i,j}^{M,N} \\frac{x_i w_{ij} h_j}{\\sigma_i^2}. \n", + "\\label{_auto4} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4d19d779", + "metadata": { + "editable": true + }, + "source": [ + "## More about RBMs\n", + "1. Useful when we model continuous data (i.e., we wish $\\mathbf{x}$ to be continuous)\n", + "\n", + "2. Requires a smaller learning rate, since there's no upper bound to the value a component might take in the reconstruction\n", + "\n", + "Other types of units include:\n", + "1. Softmax and multinomial units\n", + "\n", + "2. Gaussian visible and hidden units\n", + "\n", + "3. Binomial units\n", + "\n", + "4. Rectified linear units\n", + "\n", + "To read more, see [Lectures on Boltzmann machines in Physics](https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook2/ipynb/notebook2.ipynb)." + ] + }, + { + "cell_type": "markdown", + "id": "b8fa06ec", + "metadata": { + "editable": true + }, + "source": [ + "## Autoencoders: Overarching view\n", + "\n", + "Autoencoders are artificial neural networks capable of learning\n", + "efficient representations of the input data (these representations are called codings) without\n", + "any supervision (i.e., the training set is unlabeled). These codings\n", + "typically have a much lower dimensionality than the input data, making\n", + "autoencoders useful for dimensionality reduction. \n", + "\n", + "More importantly, autoencoders act as powerful feature detectors, and\n", + "they can be used for unsupervised pretraining of deep neural networks.\n", + "\n", + "Lastly, they are capable of randomly generating new data that looks\n", + "very similar to the training data; this is called a generative\n", + "model. For example, you could train an autoencoder on pictures of\n", + "faces, and it would then be able to generate new faces. Surprisingly,\n", + "autoencoders work by simply learning to copy their inputs to their\n", + "outputs. This may sound like a trivial task, but we will see that\n", + "constraining the network in various ways can make it rather\n", + "difficult. For example, you can limit the size of the internal\n", + "representation, or you can add noise to the inputs and train the\n", + "network to recover the original inputs. These constraints prevent the\n", + "autoencoder from trivially copying the inputs directly to the outputs,\n", + "which forces it to learn efficient ways of representing the data. In\n", + "short, the codings are byproducts of the autoencoder’s attempt to\n", + "learn the identity function under some constraints.\n", + "\n", + "[Video on autoencoders](https://www.coursera.org/lecture/building-deep-learning-models-with-tensorflow/autoencoders-1U4L3)\n", + "\n", + "See also A. Geron's textbook, chapter 15." + ] + }, + { + "cell_type": "markdown", + "id": "262fc510", + "metadata": { + "editable": true + }, + "source": [ + "## Bayesian Machine Learning\n", + "\n", + "This is an important topic if we aim at extracting a probability\n", + "distribution. This gives us also a confidence interval and error\n", + "estimates.\n", + "\n", + "Bayesian machine learning allows us to encode our prior beliefs about\n", + "what those models should look like, independent of what the data tells\n", + "us. This is especially useful when we don’t have a ton of data to\n", + "confidently learn our model.\n", + "\n", + "[Video on Bayesian deep learning](https://www.youtube.com/watch?v=E1qhGw8QxqY&ab_channel=AndrewGordonWilson)\n", + "\n", + "See also the [slides here](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Articles/lec03.pdf)." + ] + }, + { + "cell_type": "markdown", + "id": "36608f14", + "metadata": { + "editable": true + }, + "source": [ + "## Reinforcement Learning\n", + "\n", + "Reinforcement Learning (RL) is one of the most exciting fields of\n", + "Machine Learning today, and also one of the oldest. It has been around\n", + "since the 1950s, producing many interesting applications over the\n", + "years.\n", + "\n", + "It studies\n", + "how agents take actions based on trial and error, so as to maximize\n", + "some notion of cumulative reward in a dynamic system or\n", + "environment. Due to its generality, the problem has also been studied\n", + "in many other disciplines, such as game theory, control theory,\n", + "operations research, information theory, multi-agent systems, swarm\n", + "intelligence, statistics, and genetic algorithms.\n", + "\n", + "In March 2016, AlphaGo, a computer program that plays the board game\n", + "Go, beat Lee Sedol in a five-game match. This was the first time a\n", + "computer Go program had beaten a 9-dan (highest rank) professional\n", + "without handicaps. AlphaGo is based on deep convolutional neural\n", + "networks and reinforcement learning. AlphaGo’s victory was a major\n", + "milestone in artificial intelligence and it has also made\n", + "reinforcement learning a hot research area in the field of machine\n", + "learning.\n", + "\n", + "[Lecture on Reinforcement Learning](https://www.youtube.com/watch?v=FgzM3zpZ55o&ab_channel=stanfordonline).\n", + "\n", + "See also A. Geron's textbook, chapter 16." + ] + }, + { + "cell_type": "markdown", + "id": "a2b4652f", + "metadata": { + "editable": true + }, + "source": [ + "## Transfer learning\n", + "\n", + "The goal of transfer learning is to transfer the model or knowledge\n", + "obtained from a source task to the target task, in order to resolve\n", + "the issues of insufficient training data in the target task. The\n", + "rationality of doing so lies in that usually the source and target\n", + "tasks have inter-correlations, and therefore either the features,\n", + "samples, or models in the source task might provide useful information\n", + "for us to better solve the target task. Transfer learning is a hot\n", + "research topic in recent years, with many problems still waiting to be studied.\n", + "\n", + "[Lecture on transfer learning](https://www.ias.edu/video/machinelearning/2020/0331-SamoryKpotufe)." + ] + }, + { + "cell_type": "markdown", + "id": "8cef8c03", + "metadata": { + "editable": true + }, + "source": [ + "## Adversarial learning\n", + "\n", + "The conventional deep generative model has a potential problem: the\n", + "model tends to generate extreme instances to maximize the\n", + "probabilistic likelihood, which will hurt its performance. Adversarial\n", + "learning utilizes the adversarial behaviors (e.g., generating\n", + "adversarial instances or training an adversarial model) to enhance the\n", + "robustness of the model and improve the quality of the generated\n", + "data. In recent years, one of the most promising unsupervised learning\n", + "technologies, generative adversarial networks (GAN), has already been\n", + "successfully applied to image, speech, and text.\n", + "\n", + "[Lecture on adversial learning](https://www.youtube.com/watch?v=CIfsB_EYsVI&ab_channel=StanfordUniversitySchoolofEngineering)." + ] + }, + { + "cell_type": "markdown", + "id": "df85a3e6", + "metadata": { + "editable": true + }, + "source": [ + "## Dual learning\n", + "\n", + "Dual learning is a new learning paradigm, the basic idea of which is\n", + "to use the primal-dual structure between machine learning tasks to\n", + "obtain effective feedback/regularization, and guide and strengthen the\n", + "learning process, thus reducing the requirement of large-scale labeled\n", + "data for deep learning. The idea of dual learning has been applied to\n", + "many problems in machine learning, including machine translation,\n", + "image style conversion, question answering and generation, image\n", + "classification and generation, text classification and generation,\n", + "image-to-text, and text-to-image." + ] + }, + { + "cell_type": "markdown", + "id": "f01aaa3c", + "metadata": { + "editable": true + }, + "source": [ + "## Distributed machine learning\n", + "\n", + "Distributed computation will speed up machine learning algorithms,\n", + "significantly improve their efficiency, and thus enlarge their\n", + "application. When distributed meets machine learning, more than just\n", + "implementing the machine learning algorithms in parallel is required." + ] + }, + { + "cell_type": "markdown", + "id": "0005330c", + "metadata": { + "editable": true + }, + "source": [ + "## Meta learning\n", + "\n", + "Meta learning is an emerging research direction in machine\n", + "learning. Roughly speaking, meta learning concerns learning how to\n", + "learn, and focuses on the understanding and adaptation of the learning\n", + "itself, instead of just completing a specific learning task. That is,\n", + "a meta learner needs to be able to evaluate its own learning methods\n", + "and adjust its own learning methods according to specific learning\n", + "tasks." + ] + }, + { + "cell_type": "markdown", + "id": "efcfaccf", + "metadata": { + "editable": true + }, + "source": [ + "## The Challenges Facing Machine Learning\n", + "\n", + "While there has been much progress in machine learning, there are also challenges.\n", + "\n", + "For example, the mainstream machine learning technologies are\n", + "black-box approaches, making us concerned about their potential\n", + "risks. To tackle this challenge, we may want to make machine learning\n", + "more explainable and controllable. As another example, the\n", + "computational complexity of machine learning algorithms is usually\n", + "very high and we may want to invent lightweight algorithms or\n", + "implementations. Furthermore, in many domains such as physics,\n", + "chemistry, biology, and social sciences, people usually seek elegantly\n", + "simple equations (e.g., the Schrödinger equation) to uncover the\n", + "underlying laws behind various phenomena. In the field of machine\n", + "learning, can we reveal simple laws instead of designing more complex\n", + "models for data fitting? Although there are many challenges, we are\n", + "still very optimistic about the future of machine learning. As we look\n", + "forward to the future, here are what we think the research hotspots in\n", + "the next ten years will be.\n", + "\n", + "See the article on [Discovery of Physics From Data: Universal Laws and Discrepancies](https://www.frontiersin.org/articles/10.3389/frai.2020.00025/full)" + ] + }, + { + "cell_type": "markdown", + "id": "f7b109c9", + "metadata": { + "editable": true + }, + "source": [ + "## Explainable machine learning\n", + "\n", + "Machine learning, especially deep learning, evolves rapidly. The\n", + "ability gap between machine and human on many complex cognitive tasks\n", + "becomes narrower and narrower. However, we are still in the very early\n", + "stage in terms of explaining why those effective models work and how\n", + "they work.\n", + "\n", + "**What is missing: the gap between correlation and causation**. Standard Machine Learning is based on what e have called a frequentist approach. \n", + "\n", + "Most\n", + "machine learning techniques, especially the statistical ones, depend\n", + "highly on correlations in data sets to make predictions and analyses. In\n", + "contrast, rational humans tend to reply on clear and trustworthy\n", + "causality relations obtained via logical reasoning on real and clear\n", + "facts. It is one of the core goals of explainable machine learning to\n", + "transition from solving problems by data correlation to solving\n", + "problems by logical reasoning.\n", + "\n", + "**Bayesian Machine Learning is one of the exciting research directions in this field**." + ] + }, + { + "cell_type": "markdown", + "id": "e9142380", + "metadata": { + "editable": true + }, + "source": [ + "## Quantum machine learning\n", + "\n", + "Quantum machine learning is an emerging interdisciplinary research\n", + "area at the intersection of quantum computing and machine learning.\n", + "\n", + "Quantum computers use effects such as quantum coherence and quantum\n", + "entanglement to process information, which is fundamentally different\n", + "from classical computers. Quantum algorithms have surpassed the best\n", + "classical algorithms in several problems (e.g., searching for an\n", + "unsorted database, inverting a sparse matrix), which we call quantum\n", + "acceleration.\n", + "\n", + "When quantum computing meets machine learning, it can be a mutually\n", + "beneficial and reinforcing process, as it allows us to take advantage\n", + "of quantum computing to improve the performance of classical machine\n", + "learning algorithms. In addition, we can also use the machine learning\n", + "algorithms (on classic computers) to analyze and improve quantum\n", + "computing systems.\n", + "\n", + "[Lecture on Quantum ML](https://www.youtube.com/watch?v=Xh9pUu3-WxM&ab_channel=InstituteforPure%26AppliedMathematics%28IPAM%29).\n", + "\n", + "[Read interview with Maria Schuld on her work on Quantum Machine Learning](https://physics.aps.org/articles/v13/179?utm_campaign=weekly&utm_medium=email&utm_source=emailalert). See also [her recent textbook](https://www.springer.com/gp/book/9783319964232)." + ] + }, + { + "cell_type": "markdown", + "id": "cb638c92", + "metadata": { + "editable": true + }, + "source": [ + "## Quantum machine learning algorithms based on linear algebra\n", + "\n", + "Many quantum machine learning algorithms are based on variants of\n", + "quantum algorithms for solving linear equations, which can efficiently\n", + "solve N-variable linear equations with complexity of O(log2 N) under\n", + "certain conditions. The quantum matrix inversion algorithm can\n", + "accelerate many machine learning methods, such as least square linear\n", + "regression, least square version of support vector machine, Gaussian\n", + "process, and more. The training of these algorithms can be simplified\n", + "to solve linear equations. The key bottleneck of this type of quantum\n", + "machine learning algorithms is data input—that is, how to initialize\n", + "the quantum system with the entire data set. Although efficient\n", + "data-input algorithms exist for certain situations, how to efficiently\n", + "input data into a quantum system is as yet unknown for most cases." + ] + }, + { + "cell_type": "markdown", + "id": "676fd56e", + "metadata": { + "editable": true + }, + "source": [ + "## Quantum reinforcement learning\n", + "\n", + "In quantum reinforcement learning, a quantum agent interacts with the\n", + "classical environment to obtain rewards from the environment, so as to\n", + "adjust and improve its behavioral strategies. In some cases, it\n", + "achieves quantum acceleration by the quantum processing capabilities\n", + "of the agent or the possibility of exploring the environment through\n", + "quantum superposition. Such algorithms have been proposed in\n", + "superconducting circuits and systems of trapped ions." + ] + }, + { + "cell_type": "markdown", + "id": "ac1d93d3", + "metadata": { + "editable": true + }, + "source": [ + "## Quantum deep learning\n", + "\n", + "Dedicated quantum information processors, such as quantum annealers\n", + "and programmable photonic circuits, are well suited for building deep\n", + "quantum networks. The simplest deep quantum network is the Boltzmann\n", + "machine. The classical Boltzmann machine consists of bits with tunable\n", + "interactions and is trained by adjusting the interaction of these bits\n", + "so that the distribution of its expression conforms to the statistics\n", + "of the data. To quantize the Boltzmann machine, the neural network can\n", + "simply be represented as a set of interacting quantum spins that\n", + "correspond to an adjustable Ising model. Then, by initializing the\n", + "input neurons in the Boltzmann machine to a fixed state and allowing\n", + "the system to heat up, we can read out the output qubits to get the\n", + "result." + ] + }, + { + "cell_type": "markdown", + "id": "c5047fcb", + "metadata": { + "editable": true + }, + "source": [ + "## Social machine learning\n", + "\n", + "Machine learning aims to imitate how humans\n", + "learn. While we have developed successful machine learning algorithms,\n", + "until now we have ignored one important fact: humans are social. Each\n", + "of us is one part of the total society and it is difficult for us to\n", + "live, learn, and improve ourselves, alone and isolated. Therefore, we\n", + "should design machines with social properties. Can we let machines\n", + "evolve by imitating human society so as to achieve more effective,\n", + "intelligent, interpretable “social machine learning”?\n", + "\n", + "And much more." + ] + }, + { + "cell_type": "markdown", + "id": "bdd66000", + "metadata": { + "editable": true + }, + "source": [ + "## The last words?\n", + "\n", + "Early computer scientist Alan Kay said, **The best way to predict the\n", + "future is to create it**. Therefore, all machine learning\n", + "practitioners, whether scholars or engineers, professors or students,\n", + "need to work together to advance these important research\n", + "topics. Together, we will not just predict the future, but create it." + ] + }, + { + "cell_type": "markdown", + "id": "7fb56aeb", + "metadata": { + "editable": true + }, + "source": [ + "## Best wishes to you all and thanks so much for your heroic efforts this semester\n", + "\n", + "\n", + "\n", + "\n", + "

    Figure 1:

    \n", + "" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/additionweek42.html b/doc/LectureNotes/_build/html/additionweek42.html index e81a40a00..8b6416a7b 100644 --- a/doc/LectureNotes/_build/html/additionweek42.html +++ b/doc/LectureNotes/_build/html/additionweek42.html @@ -257,6 +257,9 @@
  1. Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
  2. Exercise week 47
    3. +
  3. Week 48: Gradient boosting and summary of course
    4. +
  4. Exercises week 48
    5. +

    Projects

    Projects

      @@ -730,7 +733,7 @@ Thereafter we wish to apply it to data which were not included in the training.
      -_images/7208bf883e220945a23bf4a64d7c474c22032a9479d0230f3f7627fde6c870e3.png +_images/01d44d8b3777894a56df96cdebd9411c2a9b9484e40698be0bb84652c7d0f7cf.png

      This example serves several aims. It allows us to demonstrate several @@ -815,7 +818,7 @@ to be dominated by outliers.

      -_images/daf635918e102d6af1572d0f6fcb2bcd9c93425cd41e91dc85022a813e3b1da5.png +_images/326bc2bdd25593d0947bbe11be557f7c976fd425b872153c8c8317ebbb51c5ae.png

      Depending on the parameter in front of the normal distribution, we may @@ -862,16 +865,16 @@ example of the functionality of Scikit-Learn.

      The intercept alpha: 
- [1.97386121]
+ [2.00949975]
 Coefficient beta : 
- [[5.12574106]]
-Mean squared error: 0.20
-Variance score: 0.92
-Mean squared log error: 0.01
-Mean absolute error: 0.36
+ [[5.07244691]]
+Mean squared error: 0.30
+Variance score: 0.88
+Mean squared log error: 0.02
+Mean absolute error: 0.42
      -_images/1168b639886622ca16f51897f200d03b2de325bfc3f6688b4b75ff4473160f59.png +_images/b39b8ed8a4aed5f4a94d2137c267e12d64d23f66d1c50c3e3ad1ae8cd8b9ff5e.png

      The function coef gives us the parameter \(\beta\) of our fit while intercept yields @@ -967,8 +970,8 @@ a linear \(x\)-dependence we s

      -_images/2e42f972e6bd064ce2062209501a6f469d0ebd090cf62a0159f888f2deaa3de7.png -
      0.004999999999999993
+_images/99bd4e87bbb54a5fb018ef916c872e3158711414d535c467a50d947ace53c954.png
+
      0.005000000000000009
      diff --git a/doc/LectureNotes/_build/html/chapter10.html b/doc/LectureNotes/_build/html/chapter10.html index d253ae521..76cd1eb3a 100644 --- a/doc/LectureNotes/_build/html/chapter10.html +++ b/doc/LectureNotes/_build/html/chapter10.html @@ -257,6 +257,9 @@
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

      @@ -935,8 +938,9 @@ probability that image 0 is in category 0,1,2,...,9 = 1.10378326e-04 5.08318298e-09 2.03256632e-04 1.92507116e-03 9.84443254e-01 3.11507992e-04] probabilities sum up to: 1.0 - -predictions = (n_inputs) = (1437,) + + +
      predictions = (n_inputs) = (1437,)
 prediction for image 0: 8
 correct label for image 0: 6
      @@ -1114,7 +1118,7 @@ the Hadamard product, meaning element-wise multiplication.

      Old accuracy on training data: 0.1440501043841336
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58742/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1448,7 +1452,7 @@ Lambda =  10.0
 Accuracy score on test set:  0.19166666666666668
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58742/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1457,7 +1461,7 @@ Lambda =  1e-05
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58742/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1466,10 +1470,211 @@ Lambda =  0.0001
 Accuracy score on test set:  0.08611111111111111
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58742/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
+
      Learning rate  =  0.1
+Lambda =  0.001
+Accuracy score on test set:  0.10555555555555556
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+
      
+
      
+
      Learning rate  =  0.1
+Lambda =  0.01
+Accuracy score on test set:  0.08888888888888889
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+
      
+
      
+
      Learning rate  =  0.1
+Lambda =  0.1
+Accuracy score on test set:  0.08611111111111111
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+
      
+
      
+
      Learning rate  =  0.1
+Lambda =  1.0
+Accuracy score on test set:  0.08888888888888889
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+
      
+
      
+
      Learning rate  =  0.1
+Lambda =  10.0
+Accuracy score on test set:  0.09166666666666666
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  1e-05
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  0.0001
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  0.001
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  0.01
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  0.1
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  1.0
+Accuracy score on test set:  0.10555555555555556
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  1.0
+Lambda =  10.0
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  1e-05
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  0.0001
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  0.001
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  0.01
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  0.1
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
 
      ---------------------------------------------------------------------------
 KeyboardInterrupt                         Traceback (most recent call last)
 Cell In[8], line 11
@@ -1480,18 +1685,17 @@ Accuracy score on test set:  0.08611111111111111
      13     DNN_numpy[i][j] = dnn
      15     test_predict = dnn.predict(X_test)
 
-Cell In[6], line 98, in NeuralNetwork.train(self)
-     95 self.X_data = self.X_data_full[chosen_datapoints]
+Cell In[6], line 99, in NeuralNetwork.train(self)
      96 self.Y_data = self.Y_data_full[chosen_datapoints]
----> 98 self.feed_forward()
-     99 self.backpropagation()
+     98 self.feed_forward()
+---> 99 self.backpropagation()
 
-Cell In[6], line 38, in NeuralNetwork.feed_forward(self)
-     36 def feed_forward(self):
-     37     # feed-forward for training
----> 38     self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
-     39     self.a_h = sigmoid(self.z_h)
-     41     self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
+Cell In[6], line 64, in NeuralNetwork.backpropagation(self)
+     61 self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
+     62 self.output_bias_gradient = np.sum(error_output, axis=0)
+---> 64 self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
+     65 self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
+     67 if self.lmbd > 0.0:
 
 KeyboardInterrupt: 
 
      
diff --git a/doc/LectureNotes/_build/html/chapter11.html b/doc/LectureNotes/_build/html/chapter11.html
index f2ad28f93..52d8c5212 100644
--- a/doc/LectureNotes/_build/html/chapter11.html
+++ b/doc/LectureNotes/_build/html/chapter11.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

    Projects

      @@ -558,7 +561,7 @@ systems such as automatic translation and speech-to-text.

      -_images/29361dbec77d5254822f2cf57bdddb701f65d5a7a34bf4d781aba818766c3cbf.png +_images/d213fd49b73afa7db6fe63b54689d9b6c4aff1adac65c06e412ebb9574c86d39.png 
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/src/layers/rnn/rnn.py:204: UserWarning: Do not pass an `input_shape`/`input_dim` argument to a layer. When using Sequential models, prefer using an `Input(shape)` object as the first layer in the model instead.
   super().__init__(**kwargs)
      @@ -584,360 +587,402 @@ systems such as automatic translation and speech-to-text.

      Epoch 1/100
      -
      50/50 - 2s - 41ms/step - loss: 0.5362
+
      50/50 - 2s - 48ms/step - loss: 0.5174
 
      
 
      
 
      Epoch 2/100
 
      
 
      
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+
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      Epoch 3/100
 
      
 
      
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
      
+
      
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+
      
+
      
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      50/50 - 0s - 7ms/step - loss: 0.3825
+
      
+
      
+
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+
      
+
      
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+
      
+
      
+
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+
      
+
      
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+
      
+
      
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+
      
+
      
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+
      
+
      
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+
      
+
      
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+
      
+
      
+
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+
      
+
      
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+
      
+
      
+
      Epoch 67/100
+
      
+
      
 
      ---------------------------------------------------------------------------
 KeyboardInterrupt                         Traceback (most recent call last)
 Cell In[1], line 58
diff --git a/doc/LectureNotes/_build/html/chapter2.html b/doc/LectureNotes/_build/html/chapter2.html
index 1ac08d55e..5264f92ca 100644
--- a/doc/LectureNotes/_build/html/chapter2.html
+++ b/doc/LectureNotes/_build/html/chapter2.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

      @@ -1185,10 +1188,10 @@ covariance matrix through the np.linalg.eig() function.

      -
      -0.07163028969289174
-3.7656278764040367
-[[0.7647107  2.29986727]
- [2.29986727 7.88107866]]
+
      0.00239782448350092
+3.945643027765946
+[[1.04582521 3.00214849]
+ [3.00214849 9.76163974]]
      @@ -1225,10 +1228,10 @@ a more brute force way. Here we scale the mean values for each column of the des
      -
      0.08212703190343323
-2.425866065899094
-[[1.         0.65333306]
- [0.65333306 1.        ]]
+
      0.07878080847813602
+2.1500369042116563
+[[1.         0.63295205]
+ [0.63295205 1.        ]]
      @@ -1258,30 +1261,30 @@ this matrix we easily see that it is a positive definite matrix.

      -
      [[-0.68481734 -2.74241828]
- [-1.03710525 -3.82202137]
- [ 0.71505805  2.54840587]
- [ 0.4645853   0.98764188]
- [-1.95392781 -4.51089358]
- [ 0.79256149  3.37757489]
- [-0.18745641  0.4457749 ]
- [ 2.53950612  7.87543978]
- [ 0.25354074 -0.28008123]
- [-0.90194489 -3.87942287]]
+
      [[-0.05862476  0.45533295]
+ [ 0.84689611  4.01381445]
+ [ 0.21778772  0.62032544]
+ [-1.3117471  -2.69182187]
+ [-0.78512935 -3.44198948]
+ [-0.12030798 -0.93994208]
+ [ 0.60559053  2.2457079 ]
+ [ 1.8170149   4.51652722]
+ [-0.44215901 -3.25646952]
+ [-0.76932106 -1.52148501]]
           0         1
-0 -0.684817 -2.742418
-1 -1.037105 -3.822021
-2  0.715058  2.548406
-3  0.464585  0.987642
-4 -1.953928 -4.510894
-5  0.792561  3.377575
-6 -0.187456  0.445775
-7  2.539506  7.875440
-8  0.253541 -0.280081
-9 -0.901945 -3.879423
-          0         1
-0  1.000000  0.972082
-1  0.972082  1.000000
+0 -0.058625  0.455333
+1  0.846896  4.013814
+2  0.217788  0.620325
+3 -1.311747 -2.691822
+4 -0.785129 -3.441989
+5 -0.120308 -0.939942
+6  0.605591  2.245708
+7  1.817015  4.516527
+8 -0.442159 -3.256470
+9 -0.769321 -1.521485
+         0        1
+0  1.00000  0.92403
+1  0.92403  1.00000
      @@ -1338,37 +1341,37 @@ this matrix we easily see that it is a positive definite matrix.

           0         1         2         3         4         5         6         7   \
 0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.070253  0.070611  0.070163  0.068562  0.067004  0.062136  0.060528   
-2   0.0  0.070611  0.071502  0.070808  0.069529  0.068256  0.062916  0.061526   
-3   0.0  0.070163  0.070808  0.074767  0.073368  0.071996  0.069069  0.067536   
-4   0.0  0.068562  0.069529  0.073368  0.072239  0.071113  0.068033  0.066707   
-5   0.0  0.067004  0.068256  0.071996  0.071113  0.070210  0.067007  0.065872   
-6   0.0  0.062136  0.062916  0.069069  0.068033  0.067007  0.065727  0.064490   
-7   0.0  0.060528  0.061526  0.067536  0.066707  0.065872  0.064490  0.063421   
-8   0.0  0.059023  0.060216  0.066097  0.065457  0.064796  0.063327  0.062411   
-9   0.0  0.057616  0.058984  0.064749  0.064281  0.063778  0.062233  0.061458   
-10  0.0  0.054096  0.054970  0.061945  0.061234  0.060520  0.060261  0.059314   
-11  0.0  0.052750  0.053781  0.060606  0.060054  0.059486  0.059137  0.058322   
-12  0.0  0.051499  0.052672  0.059359  0.058952  0.058518  0.058088  0.057394   
-13  0.0  0.050336  0.051637  0.058198  0.057923  0.057612  0.057109  0.056526   
-14  0.0  0.049255  0.050672  0.057116  0.056962  0.056764  0.056195  0.055714   
+1   0.0  0.081622  0.081406  0.078296  0.079396  0.080527  0.068235  0.069327   
+2   0.0  0.081406  0.081919  0.078612  0.080089  0.081603  0.068822  0.070146   
+3   0.0  0.078296  0.078612  0.080411  0.081717  0.083038  0.073368  0.074578   
+4   0.0  0.079396  0.080089  0.081717  0.083265  0.084831  0.074620  0.075994   
+5   0.0  0.080527  0.081603  0.083038  0.084831  0.086645  0.075870  0.077411   
+6   0.0  0.068235  0.068822  0.073368  0.074620  0.075870  0.069194  0.070300   
+7   0.0  0.069327  0.070146  0.074578  0.075994  0.077411  0.070300  0.071523   
+8   0.0  0.070483  0.071536  0.075839  0.077423  0.079011  0.071441  0.072784   
+9   0.0  0.071702  0.072994  0.077153  0.078908  0.080672  0.072619  0.074087   
+10  0.0  0.059187  0.059821  0.065770  0.066863  0.067945  0.063580  0.064519   
+11  0.0  0.060114  0.060903  0.066767  0.067977  0.069178  0.064472  0.065498   
+12  0.0  0.061096  0.062042  0.067812  0.069142  0.070466  0.065402  0.066517   
+13  0.0  0.062134  0.063240  0.068908  0.070362  0.071812  0.066371  0.067578   
+14  0.0  0.063230  0.064502  0.070057  0.071639  0.073220  0.067381  0.068684   
 
           8         9         10        11        12        13        14  
 0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.059023  0.057616  0.054096  0.052750  0.051499  0.050336  0.049255  
-2   0.060216  0.058984  0.054970  0.053781  0.052672  0.051637  0.050672  
-3   0.066097  0.064749  0.061945  0.060606  0.059359  0.058198  0.057116  
-4   0.065457  0.064281  0.061234  0.060054  0.058952  0.057923  0.056962  
-5   0.064796  0.063778  0.060520  0.059486  0.058518  0.057612  0.056764  
-6   0.063327  0.062233  0.060261  0.059137  0.058088  0.057109  0.056195  
-7   0.062411  0.061458  0.059314  0.058322  0.057394  0.056526  0.055714  
-8   0.061541  0.060718  0.058420  0.057549  0.056732  0.055967  0.055250  
-9   0.060718  0.060014  0.057576  0.056817  0.056103  0.055433  0.054805  
-10  0.058420  0.057576  0.056198  0.055300  0.054459  0.053671  0.052934  
-11  0.057549  0.056817  0.055300  0.054507  0.053763  0.053066  0.052412  
-12  0.056732  0.056103  0.054459  0.053763  0.053109  0.052495  0.051918  
-13  0.055967  0.055433  0.053671  0.053066  0.052495  0.051957  0.051452  
-14  0.055250  0.054805  0.052934  0.052412  0.051918  0.051452  0.051015  
+1   0.070483  0.071702  0.059187  0.060114  0.061096  0.062134  0.063230  
+2   0.071536  0.072994  0.059821  0.060903  0.062042  0.063240  0.064502  
+3   0.075839  0.077153  0.065770  0.066767  0.067812  0.068908  0.070057  
+4   0.077423  0.078908  0.066863  0.067977  0.069142  0.070362  0.071639  
+5   0.079011  0.080672  0.067945  0.069178  0.070466  0.071812  0.073220  
+6   0.071441  0.072619  0.063580  0.064472  0.065402  0.066371  0.067381  
+7   0.072784  0.074087  0.064519  0.065498  0.066517  0.067578  0.068684  
+8   0.074170  0.075601  0.065481  0.066550  0.067660  0.068817  0.070021  
+9   0.075601  0.077163  0.066469  0.067629  0.068835  0.070089  0.071396  
+10  0.065481  0.066469  0.059543  0.060291  0.061066  0.061870  0.062705  
+11  0.066550  0.067629  0.060291  0.061104  0.061946  0.062820  0.063728  
+12  0.067660  0.068835  0.061066  0.061946  0.062859  0.063805  0.064788  
+13  0.068817  0.070089  0.061870  0.062820  0.063805  0.064827  0.065887  
+14  0.070021  0.071396  0.062705  0.063728  0.064788  0.065887  0.067030
      diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html index cd57cea9d..3ba609b3e 100644 --- a/doc/LectureNotes/_build/html/chapter3.html +++ b/doc/LectureNotes/_build/html/chapter3.html @@ -257,6 +257,9 @@
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

      @@ -726,10 +729,10 @@ number \(i\) is left out. Usin
      -
      Runtime: 0.177848 sec
+
      Runtime: 0.154011 sec
 Jackknife Statistics :
 original           bias      std. error
- 99.9735        99.9635        0.149504
+ 100.105        100.095         0.14726
      @@ -948,7 +951,7 @@ theorem.

      Bootstrap Statistics :
 original           bias      std. error
- 99.9629  14.9658        99.9626        0.148488
+ 100.182  15.0891        100.184         0.15218
      @@ -970,7 +973,7 @@ original bias std. error
      -_images/01ac462cedf743599925f73cc6294e533c9fdc93068eb00daba5834da82ae1f3.png +_images/1f1e999736d47a2811569158c3eb924dab62b0f520b0f854e48f2e2dbb32e047.png
      @@ -1155,9 +1158,7 @@ Error: 0.08426840630693411 Bias^2: 0.0796891867672603 Var: 0.004579219539673834 0.08426840630693411 >= 0.0796891867672603 + 0.004579219539673834 = 0.08426840630693413 - - -
      Polynomial degree: 2
+Polynomial degree: 2
 Error: 0.10398646080125035
 Bias^2: 0.1007711427354898
 Var: 0.0032153180657605116
@@ -1189,9 +1190,7 @@ Error: 0.02760977349102253
 Bias^2: 0.022999498260366312
 Var: 0.004610275230656212
 0.02760977349102253 >= 0.022999498260366312 + 0.004610275230656212 = 0.027609773491022525
-
      -
      -
      Polynomial degree: 8
+Polynomial degree: 8
 Error: 0.017355848195593347
 Bias^2: 0.010331721306655127
 Var: 0.007024126888938232
@@ -1206,14 +1205,14 @@ Error: 0.021592704588025025
 Bias^2: 0.010516485576645508
 Var: 0.011076219011379514
 0.021592704588025025 >= 0.010516485576645508 + 0.011076219011379514 = 0.021592704588025022
-Polynomial degree: 11
+
      +
      +
      Polynomial degree: 11
 Error: 0.07160048164233104
 Bias^2: 0.014436800088904942
 Var: 0.05716368155342608
 0.07160048164233104 >= 0.014436800088904942 + 0.05716368155342608 = 0.07160048164233102
-
      -
      -
      Polynomial degree: 12
+Polynomial degree: 12
 Error: 0.11547777218872497
 Bias^2: 0.01628578269596628
 Var: 0.09919198949275869
@@ -1540,9 +1539,9 @@ Mean squared error on training data: 0.00060704
 Mean squared error on test data: 3262.26814548
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
 
      
 
      
@@ -1776,7 +1775,7 @@ cross-validation (LOOCV).
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
 
      
 
      
@@ -2665,7 +2664,7 @@ linear system as an equation would reduce this down to
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/4162706317.py:7: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/4162706317.py:7: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
      
 
      
@@ -2809,7 +2808,7 @@ with the form utilized in linear regression, viz.
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/3777801602.py:7: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/3777801602.py:7: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
      
 
      
@@ -2849,7 +2848,7 @@ cost function is given by
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/438060758.py:10: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/438060758.py:10: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
      
 
      
@@ -2884,7 +2883,7 @@ cost function is given by
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58812/3544313922.py:9: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23110/3544313922.py:9: UserWarning: set_ticklabels() should only be used with a fixed number of ticks, i.e. after set_ticks() or using a FixedLocator.
   cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)
 
      
 
      
@@ -2937,43 +2936,43 @@ constant as opposed to ridge and OLS. We get a sparse solution with
      -
        0%|                                                                                                 | 0/10 [00:00<?, ?it/s]
+
        0%|                                                                                                                                              | 0/10 [00:00<?, ?it/s]
 
      
 
      
 
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_coordinate_descent.py:628: ConvergenceWarning: Objective did not converge. You might want to increase the number of iterations, check the scale of the features or consider increasing regularisation. Duality gap: 3.924e+00, tolerance: 1.797e+00
   model = cd_fast.enet_coordinate_descent(
 
- 10%|████████▉                                                                                | 1/10 [00:00<00:07,  1.22it/s]
+ 10%|█████████████▍                                                                                                                        | 1/10 [00:01<00:11,  1.33s/it]
 
      
 
      
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       20%|█████████████████▊                                                                       | 2/10 [00:01<00:06,  1.16it/s]
+
       20%|██████████████████████████▊                                                                                                           | 2/10 [00:02<00:09,  1.18s/it]
 
      
 
      
-
       30%|██████████████████████████▋                                                              | 3/10 [00:02<00:05,  1.28it/s]
+
       30%|████████████████████████████████████████▏                                                                                             | 3/10 [00:03<00:07,  1.04s/it]
 
      
 
      
-
       40%|███████████████████████████████████▌                                                     | 4/10 [00:02<00:04,  1.41it/s]
+
       40%|█████████████████████████████████████████████████████▌                                                                                | 4/10 [00:04<00:05,  1.08it/s]
 
      
 
      
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       50%|████████████████████████████████████████████▌                                            | 5/10 [00:03<00:03,  1.46it/s]
+
       50%|███████████████████████████████████████████████████████████████████                                                                   | 5/10 [00:04<00:04,  1.15it/s]
 
      
 
      
-
       60%|█████████████████████████████████████████████████████▍                                   | 6/10 [00:04<00:02,  1.42it/s]
+
       60%|████████████████████████████████████████████████████████████████████████████████▍                                                     | 6/10 [00:05<00:03,  1.18it/s]
 
      
 
      
-
       70%|██████████████████████████████████████████████████████████████▎                          | 7/10 [00:04<00:01,  1.53it/s]
+
       70%|█████████████████████████████████████████████████████████████████████████████████████████████▊                                        | 7/10 [00:06<00:02,  1.20it/s]
 
      
 
      
-
       80%|███████████████████████████████████████████████████████████████████████▏                 | 8/10 [00:05<00:01,  1.64it/s]
+
       80%|███████████████████████████████████████████████████████████████████████████████████████████████████████████▏                          | 8/10 [00:07<00:01,  1.23it/s]
 
      
 
      
-
       90%|████████████████████████████████████████████████████████████████████████████████         | 9/10 [00:06<00:00,  1.52it/s]
+
       90%|████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████▌             | 9/10 [00:08<00:00,  1.19it/s]
 
      
 
      
-
      100%|████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:06<00:00,  1.55it/s]
+
      100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:09<00:00,  1.09it/s]
 
      
 
      
-
      100%|████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:06<00:00,  1.46it/s]
+
      100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 10/10 [00:09<00:00,  1.09it/s]
 
      
 
      
 
      
diff --git a/doc/LectureNotes/_build/html/chapter4.html b/doc/LectureNotes/_build/html/chapter4.html
index 9bbf23ee6..b59ea01be 100644
--- a/doc/LectureNotes/_build/html/chapter4.html
+++ b/doc/LectureNotes/_build/html/chapter4.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

    Projects

      @@ -621,13 +624,13 @@ predicting the target features of query instances is as follows:

      2nd degree coefficients:
-zero power:  -3.6801072677808095
-first power:  0.14054303349959596
-second power:  -0.0002999281168222194
+zero power:  -1.399819665759832
+first power:  -0.04120453230477482
+second power:  0.0007267213049602343
      -_images/928e4d75f6fb1b7a9e9c65e0db52145f075c9aed9b2e97bdef41ed3d957ace12.png -_images/deb9b3ea985ab0b0e0d38d4431927a533e89aac53bf320412eff2b3ca55387f7.png +_images/b27afd9b9c2cb0727463c018ad254fd8be50dfbc3e2597029b6cb3b42667fdb5.png +_images/71a39e681516930bc9bdd50d3be601c8226c85a4b39549563a05614380c05e0c.png
      @@ -1486,10 +1489,10 @@ attributes at each step while growing the tree.

      (426, 30)
 (143, 30)
+Test set accuracy with Logistic Regression: 0.94
      -
      Test set accuracy with Logistic Regression: 0.94
-Test set accuracy with SVM: 0.63
+
      Test set accuracy with SVM: 0.63
 Test set accuracy with Decision Trees: 0.90
 Test set accuracy Logistic Regression with scaled data: 0.96
 Test set accuracy SVM with scaled data: 0.96
diff --git a/doc/LectureNotes/_build/html/chapter7.html b/doc/LectureNotes/_build/html/chapter7.html
index 7c31bda98..a85c4a12c 100644
--- a/doc/LectureNotes/_build/html/chapter7.html
+++ b/doc/LectureNotes/_build/html/chapter7.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

      @@ -617,10 +620,10 @@ covariance matrix through the np.linalg.eig() function.

      -
      -0.08913527419249101
-3.7127415072708665
-[[ 1.13025431  3.39215451]
- [ 3.39215451 11.15061293]]
+
      -0.02745698767039481
+3.9527157086177156
+[[0.80249705 2.35440603]
+ [2.35440603 7.83541057]]
      @@ -660,10 +663,10 @@ a more brute force way. Here we scale the mean values for each column of the des
      -
      0.09335279187105122
-2.2108106787032815
-[[1.         0.66771869]
- [0.66771869 1.        ]]
+
      0.07408022521552643
+1.9519435372439522
+[[1.         0.56323995]
+ [0.56323995 1.        ]]
      @@ -692,30 +695,30 @@ this matrix we easily see that it is a positive definite matrix.

      -
      [[ 0.35582913  0.96196912]
- [ 0.20039135  1.07902642]
- [-0.47708415 -1.39752266]
- [-0.0118395   0.55664957]
- [-0.18990703  1.30646904]
- [ 0.22370063 -0.76173777]
- [-0.25925579 -1.7132548 ]
- [-0.56061802 -1.9921751 ]
- [-1.14025916 -4.44655399]
- [ 1.85904253  6.40713017]]
+
      [[ 0.53340467  1.72530188]
+ [-0.56167889 -3.2544002 ]
+ [ 1.85988539  5.87145508]
+ [-1.23561281 -4.57354138]
+ [ 0.52739566  0.93489572]
+ [ 0.89775088  2.87036267]
+ [-0.71834014 -0.65667853]
+ [ 0.38699758  1.30537358]
+ [-1.48772565 -3.9664681 ]
+ [-0.20207669 -0.25630072]]
           0         1
-0  0.355829  0.961969
-1  0.200391  1.079026
-2 -0.477084 -1.397523
-3 -0.011839  0.556650
-4 -0.189907  1.306469
-5  0.223701 -0.761738
-6 -0.259256 -1.713255
-7 -0.560618 -1.992175
-8 -1.140259 -4.446554
-9  1.859043  6.407130
+0  0.533405  1.725302
+1 -0.561679 -3.254400
+2  1.859885  5.871455
+3 -1.235613 -4.573541
+4  0.527396  0.934896
+5  0.897751  2.870363
+6 -0.718340 -0.656679
+7  0.386998  1.305374
+8 -1.487726 -3.966468
+9 -0.202077 -0.256301
           0         1
-0  1.000000  0.946278
-1  0.946278  1.000000
+0  1.000000  0.966262
+1  0.966262  1.000000
      @@ -772,37 +775,37 @@ this matrix we easily see that it is a positive definite matrix.

           0         1         2         3         4         5         6         7   \
 0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.075894  0.073179  0.073456  0.072746  0.071970  0.064485  0.064123   
-2   0.0  0.073179  0.070833  0.070597  0.070031  0.069409  0.061930  0.061649   
-3   0.0  0.073456  0.070597  0.076555  0.075647  0.074674  0.070295  0.069812   
-4   0.0  0.072746  0.070031  0.075647  0.074821  0.073930  0.069398  0.068968   
-5   0.0  0.071970  0.069409  0.074674  0.073930  0.073126  0.068445  0.068069   
-6   0.0  0.064485  0.061930  0.070295  0.069398  0.068445  0.066551  0.066050   
-7   0.0  0.064123  0.061649  0.069812  0.068968  0.068069  0.066050  0.065589   
-8   0.0  0.063763  0.061372  0.069330  0.068539  0.067694  0.065550  0.065128   
-9   0.0  0.063403  0.061097  0.068844  0.068107  0.067318  0.065047  0.064664   
-10  0.0  0.055942  0.053732  0.062913  0.062085  0.061210  0.060946  0.060463   
-11  0.0  0.055680  0.053525  0.062569  0.061780  0.060945  0.060582  0.060130   
-12  0.0  0.055429  0.053330  0.062237  0.061487  0.060692  0.060229  0.059808   
-13  0.0  0.055190  0.053147  0.061915  0.061205  0.060449  0.059887  0.059497   
-14  0.0  0.054960  0.052974  0.061603  0.060933  0.060217  0.059554  0.059195   
+1   0.0  0.073438  0.077471  0.075458  0.076186  0.076594  0.069080  0.068939   
+2   0.0  0.077471  0.083242  0.080919  0.082408  0.083412  0.074516  0.074729   
+3   0.0  0.075458  0.080919  0.083069  0.084410  0.085279  0.079308  0.079408   
+4   0.0  0.076186  0.082408  0.084410  0.086139  0.087327  0.080792  0.081105   
+5   0.0  0.076594  0.083412  0.085279  0.087327  0.088786  0.081792  0.082290   
+6   0.0  0.069080  0.074516  0.079308  0.080792  0.081792  0.077847  0.078070   
+7   0.0  0.068939  0.074729  0.079408  0.081105  0.082290  0.078070  0.078429   
+8   0.0  0.068716  0.074791  0.079363  0.081242  0.082589  0.078134  0.078618   
+9   0.0  0.068441  0.074753  0.079219  0.081258  0.082750  0.078091  0.078687   
+10  0.0  0.061939  0.066921  0.073159  0.074603  0.075599  0.073240  0.073520   
+11  0.0  0.061580  0.066741  0.072883  0.074456  0.075571  0.073054  0.073431   
+12  0.0  0.061213  0.066524  0.072574  0.074264  0.075487  0.072829  0.073296   
+13  0.0  0.060849  0.066290  0.072252  0.074047  0.075371  0.072584  0.073135   
+14  0.0  0.060497  0.066053  0.071930  0.073822  0.075239  0.072334  0.072965   
 
           8         9         10        11        12        13        14  
 0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.063763  0.063403  0.055942  0.055680  0.055429  0.055190  0.054960  
-2   0.061372  0.061097  0.053732  0.053525  0.053330  0.053147  0.052974  
-3   0.069330  0.068844  0.062913  0.062569  0.062237  0.061915  0.061603  
-4   0.068539  0.068107  0.062085  0.061780  0.061487  0.061205  0.060933  
-5   0.067694  0.067318  0.061210  0.060945  0.060692  0.060449  0.060217  
-6   0.065550  0.065047  0.060946  0.060582  0.060229  0.059887  0.059554  
-7   0.065128  0.064664  0.060463  0.060130  0.059808  0.059497  0.059195  
-8   0.064706  0.064282  0.059981  0.059679  0.059388  0.059108  0.058837  
-9   0.064282  0.063898  0.059497  0.059226  0.058967  0.058718  0.058478  
-10  0.059981  0.059497  0.056837  0.056475  0.056124  0.055783  0.055452  
-11  0.059679  0.059226  0.056475  0.056139  0.055814  0.055499  0.055194  
-12  0.059388  0.058967  0.056124  0.055814  0.055515  0.055226  0.054947  
-13  0.059108  0.058718  0.055783  0.055499  0.055226  0.054963  0.054709  
-14  0.058837  0.058478  0.055452  0.055194  0.054947  0.054709  0.054481  
+1   0.068716  0.068441  0.061939  0.061580  0.061213  0.060849  0.060497  
+2   0.074791  0.074753  0.066921  0.066741  0.066524  0.066290  0.066053  
+3   0.079363  0.079219  0.073159  0.072883  0.072574  0.072252  0.071930  
+4   0.081242  0.081258  0.074603  0.074456  0.074264  0.074047  0.073822  
+5   0.082589  0.082750  0.075599  0.075571  0.075487  0.075371  0.075239  
+6   0.078134  0.078091  0.073240  0.073054  0.072829  0.072584  0.072334  
+7   0.078618  0.078687  0.073520  0.073431  0.073296  0.073135  0.072965  
+8   0.078919  0.079094  0.073650  0.073652  0.073602  0.073523  0.073430  
+9   0.079094  0.079368  0.073677  0.073765  0.073796  0.073795  0.073777  
+10  0.073650  0.073677  0.069911  0.069801  0.069653  0.069484  0.069308  
+11  0.073652  0.073765  0.069801  0.069768  0.069692  0.069593  0.069484  
+12  0.073602  0.073796  0.069653  0.069692  0.069686  0.069654  0.069611  
+13  0.073523  0.073795  0.069484  0.069593  0.069654  0.069688  0.069709  
+14  0.073430  0.073777  0.069308  0.069484  0.069611  0.069709  0.069791
      @@ -991,10 +994,10 @@ We can write our own code or simply use either the functionaly of numpy< 
                0         1
-0  3.946263  1.971035
-1  1.971035  1.988524
-[[3.94626291 1.97103474]
- [1.97103474 1.98852413]]
+0  4.066103  2.050297
+1  2.050297  2.019626
+[[4.06610301 2.0502966 ]
+ [2.0502966  2.01962561]]
      @@ -1021,11 +1024,11 @@ Our own code here is not very elegant and asks for obvious improvements. It is t 
      Centered covariance using own code
-[[3.94626291 1.97103474]
- [1.97103474 1.98852413]]
+[[4.06610301 2.0502966 ]
+ [2.0502966  2.01962561]]
      -_images/6ca0e6a8c6122c37cbb752b49917fd1e67a1de46298e8c2247a990d083fdb3c7.png +_images/6221ee55bb26dba5cef5540ef9967df02eb119abda11c5c5903f7762c8c9f763.png

      Depending on the number of points \(n\), we will get results that are close to the covariance values defined above. @@ -1082,16 +1085,16 @@ questions.

      Eigenvalues of Covariance matrix
-5.168112312789667
-0.7666747242371026
+5.3343122336302145
+0.7514163800957537
 First eigenvector
-[0.84993979 0.52687982]
+[0.85045481 0.5260481 ]
 Second eigenvector
-[-0.52687982  0.84993979]
+[-0.5260481   0.85045481]
 
      Eigenvector of largest eigenvalue
-[0.84993979 0.52687982]
+[0.85045481 0.5260481 ]
      diff --git a/doc/LectureNotes/_build/html/chapter9.html b/doc/LectureNotes/_build/html/chapter9.html index 2fc3c06e5..c8c75d55e 100644 --- a/doc/LectureNotes/_build/html/chapter9.html +++ b/doc/LectureNotes/_build/html/chapter9.html @@ -257,6 +257,9 @@
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

    Projects

    Projects

    Projects

    Projects

    Projects

    Projects

    Projects

    Projects

    Projects

      @@ -599,18 +602,18 @@ regression.

      Own inversion
-[[4.0010275 ]
- [3.10085547]]
-Eigenvalues of Hessian Matrix:[0.32462943 4.22813513]
+[[4.41622083]
+ [2.74692745]]
+Eigenvalues of Hessian Matrix:[0.26594878 4.45190996]
 theta from own gd
-[[4.0010275 ]
- [3.10085547]]
+[[4.41622083]
+ [2.74692745]]
 theta from own sdg
-[[4.08051993]
- [3.13522576]]
+[[4.38222303]
+ [2.74411787]]
      -_images/28af7ec857f6b0c73278974d8a0f02963d4300b7ede7129fc2e95b8d68a8492b.png +_images/74585d2c5d042ce0364bf21b5a8e9bc5b5504011eaa4a30c445a4415a4c1df98.png

      In the above code, we have use replacement in setting up the @@ -729,17 +732,17 @@ first example shows results with ordinary leats squares.

      Own inversion
-[[3.8679219 ]
- [3.12497024]]
-Eigenvalues of Hessian Matrix:[0.31887646 4.68712364]
+[[4.32147704]
+ [2.78166226]]
+Eigenvalues of Hessian Matrix:[0.32306518 4.38577204]
 
      theta from own gd
-[[3.8679219 ]
- [3.12497024]]
+[[4.32147704]
+ [2.78166226]]
      -_images/634d0baa356c54adef6974350a5d0f31fb981ef2c1199e2bf8abc3dbbd76d38b.png +_images/c61bca31ee6787e0e5d1512e4bf365143d789096f02da0a858ff7e7c3834630c.png
      @@ -807,73 +810,73 @@ Eigenvalues of Hessian Matrix:[0.31887646 4.68712364] 
      Own inversion
 [[4.]
  [3.]]
-Eigenvalues of Hessian Matrix:[0.3011187  4.49781862]
-0 [-10.84111032] [-12.09572749]
-1 [-0.48072858] [0.39646149]
-2 [-0.4485449] [0.3699193]
-3 [-0.41851584] [0.34515405]
-4 [-0.39049716] [0.32204677]
-5 [-0.36435426] [0.30048647]
-6 [-0.33996158] [0.28036959]
-7 [-0.31720192] [0.26159948]
-8 [-0.29596598] [0.24408599]
-9 [-0.27615173] [0.22774499]
-10 [-0.257664] [0.21249799]
-11 [-0.24041398] [0.19827173]
-12 [-0.22431882] [0.18499789]
-13 [-0.20930118] [0.1726127]
-14 [-0.19528895] [0.16105668]
-15 [-0.1822148] [0.1502743]
-16 [-0.17001593] [0.14021378]
-17 [-0.15863375] [0.13082678]
-18 [-0.14801358] [0.12206823]
-19 [-0.13810441] [0.11389604]
-20 [-0.12885864] [0.10627096]
-21 [-0.12023184] [0.09915636]
-22 [-0.1121826] [0.09251807]
-23 [-0.10467223] [0.08632419]
-24 [-0.09766466] [0.08054499]
-25 [-0.09112623] [0.07515268]
-26 [-0.08502554] [0.07012138]
-27 [-0.07933327] [0.06542692]
-28 [-0.07402209] [0.06104673]
-29 [-0.06906648] [0.05695979]
+Eigenvalues of Hessian Matrix:[0.26136301 4.38259367]
+0 [-12.42926102] [-13.7797646]
+1 [-0.35774662] [0.30560116]
+2 [-0.33641182] [0.28737615]
+3 [-0.31634936] [0.27023801]
+4 [-0.29748336] [0.25412193]
+5 [-0.27974246] [0.23896696]
+6 [-0.26305957] [0.22471578]
+7 [-0.24737159] [0.2113145]
+8 [-0.23261919] [0.19871242]
+9 [-0.21874657] [0.18686188]
+10 [-0.20570127] [0.17571808]
+11 [-0.19343394] [0.16523885]
+12 [-0.1818982] [0.15538456]
+13 [-0.17105041] [0.14611795]
+14 [-0.16084954] [0.13740398]
+15 [-0.15125702] [0.12920967]
+16 [-0.14223657] [0.12150404]
+17 [-0.13375406] [0.11425795]
+18 [-0.12577742] [0.107444]
+19 [-0.11827649] [0.1010364]
+20 [-0.11122288] [0.09501094]
+21 [-0.10458992] [0.08934481]
+22 [-0.09835253] [0.08401658]
+23 [-0.09248712] [0.07900612]
+24 [-0.08697151] [0.07429446]
+25 [-0.08178482] [0.06986379]
+26 [-0.07690745] [0.06569735]
+27 [-0.07232095] [0.06177939]
+28 [-0.06800798] [0.05809507]
+29 [-0.06395221] [0.05463048]
 theta from own gd
-[[3.78598925]
- [3.17649673]]
-0 [-0.06444264] [0.05314647]
-1 [-0.06012835] [0.04958843]
-2 [-0.05480861] [0.04520119]
-3 [-0.04954337] [0.0408589]
-4 [-0.04464699] [0.0368208]
-5 [-0.04018906] [0.0331443]
-6 [-0.03616111] [0.02982242]
-7 [-0.03253183] [0.02682931]
-8 [-0.02926511] [0.02413522]
-9 [-0.02632586] [0.02171119]
-10 [-0.02368163] [0.01953047]
-11 [-0.02130293] [0.01756873]
-12 [-0.01916314] [0.01580402]
-13 [-0.01723827] [0.01421657]
-14 [-0.01550675] [0.01278856]
-15 [-0.01394915] [0.011504]
-16 [-0.01254801] [0.01034846]
-17 [-0.0112876] [0.009309]
-18 [-0.0101538] [0.00837394]
-19 [-0.00913389] [0.00753281]
-20 [-0.00821642] [0.00677616]
-21 [-0.00739111] [0.00609552]
-22 [-0.0066487] [0.00548325]
-23 [-0.00598086] [0.00493247]
-24 [-0.0053801] [0.00443702]
-25 [-0.00483969] [0.00399134]
-26 [-0.00435356] [0.00359042]
-27 [-0.00391626] [0.00322978]
-28 [-0.00352289] [0.00290536]
-29 [-0.00316902] [0.00261353]
+[[3.76990501]
+ [3.19655615]]
+0 [-0.06013832] [0.05137251]
+1 [-0.05655187] [0.04830882]
+2 [-0.05210338] [0.04450874]
+3 [-0.04766156] [0.04071437]
+4 [-0.04348664] [0.03714799]
+5 [-0.03964077] [0.0338627]
+6 [-0.03612297] [0.03085765]
+7 [-0.03291338] [0.02811589]
+8 [-0.02998766] [0.02561663]
+9 [-0.02732158] [0.02333916]
+10 [-0.02489239] [0.02126405]
+11 [-0.02267913] [0.0193734]
+12 [-0.02066265] [0.01765085]
+13 [-0.01882546] [0.01608144]
+14 [-0.01715161] [0.01465158]
+15 [-0.01562659] [0.01334885]
+16 [-0.01423717] [0.01216195]
+17 [-0.01297129] [0.01108058]
+18 [-0.01181796] [0.01009536]
+19 [-0.01076718] [0.00919775]
+20 [-0.00980983] [0.00837994]
+21 [-0.0089376] [0.00763484]
+22 [-0.00814292] [0.006956]
+23 [-0.0074189] [0.00633751]
+24 [-0.00675926] [0.00577402]
+25 [-0.00615826] [0.00526063]
+26 [-0.00561071] [0.00479289]
+27 [-0.00511184] [0.00436673]
+28 [-0.00465732] [0.00397847]
+29 [-0.00424322] [0.00362473]
 theta from own gd wth momentum
-[[3.99053294]
- [3.00780757]]
+[[3.98520854]
+ [3.01263545]]
      @@ -926,17 +929,17 @@ theta from own gd wth momentum 
      Own inversion
-[[4.1739262 ]
- [2.85239235]]
-Eigenvalues of Hessian Matrix:[0.27968902 4.49424386]
-0 [-13.16403338] [-14.97007223]
-1 [7.61543606e-16] [2.78435018e-15]
-2 [-4.23272528e-16] [-4.70203152e-16]
-3 [-4.23272528e-16] [-4.70203152e-16]
-4 [-4.23272528e-16] [-4.70203152e-16]
+[[3.96856022]
+ [2.98116672]]
+Eigenvalues of Hessian Matrix:[0.3407072  4.46918198]
+0 [-14.47784776] [-17.95282048]
+1 [2.49106291e-15] [-6.96409595e-15]
+2 [8.32667268e-17] [8.72456212e-17]
+3 [8.32667268e-17] [8.72456212e-17]
+4 [8.32667268e-17] [8.72456212e-17]
 beta from own Newton code
-[[4.1739262 ]
- [2.85239235]]
+[[3.96856022]
+ [2.98116672]]
      @@ -1025,18 +1028,18 @@ beta from own Newton code 
      Own inversion
-[[4.50455587]
- [2.69978846]]
-Eigenvalues of Hessian Matrix:[0.35172802 4.35427615]
+[[4.20603792]
+ [2.82345148]]
+Eigenvalues of Hessian Matrix:[0.26589296 4.18756348]
 theta from own gd
-[[4.50455587]
- [2.69978846]]
+[[4.20603792]
+ [2.82345148]]
      -_images/c8d1a6036144492783b21bafb5df5462fba5688ed1014961e0a3ecd4909a9caf.png +_images/5cfc76ca7e820abfb8e15070adf2fa59d6dcfb6d702ef48441625e67b326e24c.png 
      theta from own sdg
-[[4.51550231]
- [2.6728048 ]]
+[[4.20773884]
+ [2.81706996]]
      @@ -1118,15 +1121,15 @@ theta from own gd 
      Own inversion
-[[3.8301107 ]
- [3.31610195]]
-Eigenvalues of Hessian Matrix:[0.31047897 4.32701817]
+[[3.79928328]
+ [3.09772988]]
+Eigenvalues of Hessian Matrix:[0.33500183 3.88706629]
 theta from own gd
-[[3.82958153]
- [3.31655284]]
+[[3.79937084]
+ [3.09764763]]
 theta from own sdg with momentum
-[[3.82977678]
- [3.25067525]]
+[[3.77947186]
+ [3.08314421]]
      @@ -1201,9 +1204,9 @@ theta from own sdg with momentum 
      theta from own AdaGrad
-[[1.99978312]
- [3.00126504]
- [3.99880408]]
+[[2.00000205]
+ [2.99998905]
+ [4.00001059]]
      @@ -1285,9 +1288,9 @@ theta from own sdg with momentum 
      theta from own RMSprop
-[[1.99827523]
- [2.99976195]
- [3.99585312]]
+[[1.99957289]
+ [3.00371103]
+ [3.99595959]]
      @@ -1373,9 +1376,9 @@ theta from own sdg with momentum 
      theta from own ADAM
-[[2.00000058]
- [2.99998203]
- [4.00001412]]
+[[2.00001509]
+ [2.99992841]
+ [4.00006811]]
      @@ -1448,7 +1451,7 @@ It provides composable transformations of Python+NumPy programs: differentiate, return asarray(x, dtype=self.dtype) -
      [<matplotlib.lines.Line2D at 0x11d60e820>]
+
      [<matplotlib.lines.Line2D at 0x139592670>]
 
      
 
      
 _images/02e94753795fba95a52acc4488a9ab82172ba372cbb7c1cc79a3e88c3ba03d69.png
@@ -1483,7 +1486,7 @@ It provides composable transformations of Python+NumPy programs: differentiate,
      -
      <matplotlib.collections.PathCollection at 0x107a83760>
+
      <matplotlib.collections.PathCollection at 0x1394faca0>
 
      
 
      
 _images/60f34a975702165b9275820e0dda6259f453a92c5ad587b37191bb47a9c67ae9.png
diff --git a/doc/LectureNotes/_build/html/exercisesweek42.html b/doc/LectureNotes/_build/html/exercisesweek42.html
index b103c7ae7..6285c55da 100644
--- a/doc/LectureNotes/_build/html/exercisesweek42.html
+++ b/doc/LectureNotes/_build/html/exercisesweek42.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

    Projects

    Projects

    Projects

    Projects

      @@ -573,8 +576,8 @@ matrices and vectors.

      -
      [-0.61405495 -0.32387761  1.12215626 -0.4519346   0.45982886  0.30950817
- -0.99310295  0.38292897  1.44400587 -0.28739213]
+
      [ 1.32917488  0.42655046 -1.04006029 -0.11860214 -0.68913413  0.23546487
+  0.91092242  0.83425723  0.05541192  0.19079267]
      @@ -795,26 +798,26 @@ as (recall that we user lowercase letters for vectors and uppercase letters for
      -
      [[0.56237933 0.1025416  0.22316615 0.85503611 0.99057991 0.94485713
-  0.03845685 0.74210311 0.91828517 0.29234676]
- [0.20570617 0.55663962 0.14395076 0.63318471 0.88506797 0.79793482
-  0.21572768 0.70817694 0.98132273 0.47451039]
- [0.10238488 0.08006058 0.26302678 0.0532921  0.89460147 0.11961481
-  0.90199101 0.17545974 0.89611233 0.76803496]
- [0.02942421 0.99685226 0.12021078 0.72362232 0.02251816 0.67824444
-  0.70638884 0.32589372 0.39505738 0.5378907 ]
- [0.88579143 0.45698195 0.61143771 0.59118363 0.92409815 0.68270325
-  0.58619632 0.77509651 0.93481453 0.87482389]
- [0.40760778 0.62136996 0.7761515  0.76287014 0.81606967 0.17622628
-  0.65211459 0.55721129 0.20955256 0.20757376]
- [0.59910949 0.22876398 0.59992922 0.0586431  0.71238022 0.59293896
-  0.58318087 0.50099904 0.15835637 0.1221625 ]
- [0.3510257  0.5364078  0.15260744 0.21019486 0.53797608 0.92789992
-  0.91110946 0.33914178 0.20822698 0.62281444]
- [0.88111434 0.05117772 0.30607722 0.34939146 0.30612395 0.06384183
-  0.23829156 0.11534663 0.44474604 0.49985968]
- [0.17578147 0.63422131 0.86986521 0.53418362 0.30679815 0.33175971
-  0.83105371 0.97275788 0.37250759 0.05652557]]
+
      [[0.76374594 0.87699656 0.93752045 0.58920669 0.11762693 0.41835272
+  0.14130145 0.22405983 0.63569519 0.57023469]
+ [0.51205632 0.0815808  0.83997706 0.62319185 0.9918084  0.82551589
+  0.23523437 0.5917984  0.26977294 0.22313112]
+ [0.78875244 0.33827328 0.15734346 0.76351169 0.88571475 0.8744876
+  0.8179527  0.96419308 0.28859292 0.21034124]
+ [0.56604693 0.0014122  0.69927222 0.76825307 0.81056827 0.18710226
+  0.96298663 0.51905411 0.75661166 0.19343497]
+ [0.80954821 0.27400301 0.19608145 0.90678518 0.52612126 0.57201534
+  0.9065678  0.72410901 0.77533887 0.73086608]
+ [0.0583118  0.57305201 0.99949947 0.6534883  0.87204797 0.81128625
+  0.39326284 0.67711227 0.61020809 0.31082354]
+ [0.13856233 0.04308921 0.87608062 0.27598375 0.37282031 0.5906656
+  0.05406637 0.09201716 0.39585666 0.16639711]
+ [0.93928366 0.33845772 0.1675106  0.86044604 0.2131759  0.13239107
+  0.06461985 0.25020484 0.93644206 0.93511011]
+ [0.52539786 0.72253483 0.80209029 0.8317496  0.06071723 0.60162183
+  0.41902867 0.01471716 0.75263199 0.47058817]
+ [0.61513099 0.99809525 0.36655653 0.36519275 0.38100512 0.84195998
+  0.06668051 0.41250502 0.54002942 0.66316435]]
      @@ -874,13 +877,13 @@ covariance matrix through the np.linalg.eig() function.

      -
      -0.07252404965482757
-3.80181024148491
--0.6317631821295671
-[[ 0.91103528  2.76447289  2.63577152]
- [ 2.76447289  9.56058946  7.79271587]
- [ 2.63577152  7.79271587 15.61196303]]
-[21.62109277  0.08978688  4.37270812]
+
      0.10396409632756674
+4.152389775605299
+0.1319830915583838
+[[ 1.1503634   3.6073857   3.74515461]
+ [ 3.6073857  12.39943598 12.04015097]
+ [ 3.74515461 12.04015097 18.79739603]]
+[29.01518302  0.08664758  3.24536482]
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    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

    Projects

    Projects

    Projects

    Projects

      diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js index 60d7b5686..123007877 100644 --- a/doc/LectureNotes/_build/html/searchindex.js +++ b/doc/LectureNotes/_build/html/searchindex.js @@ -1 +1 @@ -Search.setIndex({"alltitles": {"3D volumes of neurons": [[45, "d-volumes-of-neurons"], [46, "d-volumes-of-neurons"]], "A Classification Tree": [[10, "a-classification-tree"], [47, "a-classification-tree"], [48, "a-classification-tree"]], "A Frequentist approach to data analysis": [[1, "a-frequentist-approach-to-data-analysis"], [35, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[9, "a-better-approach"]], "A deep CNN model (From Raschka et al)": [[45, "a-deep-cnn-model-from-raschka-et-al"], [46, "a-deep-cnn-model-from-raschka-et-al"]], "A first summary": [[35, "a-first-summary"]], "A more advanced example": [[22, "a-more-advanced-example"], [41, "a-more-advanced-example"]], "A more compact expression": [[0, "a-more-compact-expression"], [39, "a-more-compact-expression"]], "A more efficient way of coding the above Convolution": [[45, "a-more-efficient-way-of-coding-the-above-convolution"]], "A new Cost Function": [[37, "a-new-cost-function"], [38, "a-new-cost-function"]], "A possible code using Scikit-Learn": [[47, "a-possible-code-using-scikit-learn"], [48, "a-possible-code-using-scikit-learn"]], "A possible implementation of a neural network": [[44, "a-possible-implementation-of-a-neural-network"]], "A quick Reminder on Lagrangian Multipliers": [[9, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[5, "a-simple-example"], [46, "a-simple-example"]], "A soft classifier": [[9, "a-soft-classifier"]], "A top-down approach, recursive binary splitting": [[47, "a-top-down-approach-recursive-binary-splitting"], [48, "a-top-down-approach-recursive-binary-splitting"]], "A top-down perspective on Neural networks": [[2, "a-top-down-perspective-on-neural-networks"], [42, "a-top-down-perspective-on-neural-networks"], [43, "a-top-down-perspective-on-neural-networks"]], "A typical Decision Tree with its pertinent Jargon, Classification Problem": [[47, "a-typical-decision-tree-with-its-pertinent-jargon-classification-problem"]], "A warm-up example": [[22, "a-warm-up-example"], [41, "a-warm-up-example"]], "A way to Read the Bias-Variance Tradeoff": [[38, "a-way-to-read-the-bias-variance-tradeoff"]], "ADAM algorithm, taken from Goodfellow et al": [[22, "adam-algorithm-taken-from-goodfellow-et-al"], [41, "adam-algorithm-taken-from-goodfellow-et-al"]], "ADAM optimizer": [[14, "adam-optimizer"], [40, "adam-optimizer"], [41, "adam-optimizer"]], "Activation functions": [[13, "activation-functions"], [41, "activation-functions"], [42, "activation-functions"], [43, "activation-functions"], [43, "id3"], [44, "activation-functions"], [44, "id3"], [45, "activation-functions"], [46, "activation-functions"]], "Activation functions, Logistic and Hyperbolic ones": [[41, "activation-functions-logistic-and-hyperbolic-ones"], [42, "activation-functions-logistic-and-hyperbolic-ones"], [43, "activation-functions-logistic-and-hyperbolic-ones"]], "Activation functions, examples": [[44, "activation-functions-examples"]], "AdaBoost Examples": [[47, "adaboost-examples"], [48, "adaboost-examples"]], "AdaGrad algorithm, taken from Goodfellow et al": [[22, "adagrad-algorithm-taken-from-goodfellow-et-al"], [41, "adagrad-algorithm-taken-from-goodfellow-et-al"]], "Adaptive Boosting, AdaBoost": [[47, "adaptive-boosting-adaboost"], [48, "adaptive-boosting-adaboost"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[11, "adaptive-boosting-adaboost-basic-algorithm"], [47, "adaptive-boosting-adaboost-basic-algorithm"], [48, "adaptive-boosting-adaboost-basic-algorithm"]], "Add Dense layers on top": [[45, "add-dense-layers-on-top"], [46, "add-dense-layers-on-top"]], "Adding Neural Networks": [[41, "adding-neural-networks"], [42, "adding-neural-networks"]], "Adding a hidden layer": [[42, "adding-a-hidden-layer"], [43, "adding-a-hidden-layer"]], "Adding error analysis and training set up": [[35, "adding-error-analysis-and-training-set-up"], [36, "adding-error-analysis-and-training-set-up"]], "Additional Remarks": [[45, "additional-remarks"], [46, "additional-remarks"]], "Adjust hyperparameters": [[2, "adjust-hyperparameters"], [43, "adjust-hyperparameters"], [44, "adjust-hyperparameters"]], "Algorithms and codes for Adagrad, RMSprop and Adam": [[22, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"], [40, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"], [41, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"]], "Algorithms for Setting up Decision Trees": [[10, "algorithms-for-setting-up-decision-trees"], [47, "algorithms-for-setting-up-decision-trees"], [48, "algorithms-for-setting-up-decision-trees"]], "Alternative differential equations": [[30, "alternative-differential-equations"]], "An Overview of Ensemble Methods": [[11, "an-overview-of-ensemble-methods"], [47, "an-overview-of-ensemble-methods"], [48, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[5, "an-extrapolation-example"]], "An optimization/minimization problem": [[35, "an-optimization-minimization-problem"]], "Analyzing the last results": [[42, "analyzing-the-last-results"], [43, "analyzing-the-last-results"]], "And with loops": [[40, "and-with-loops"], [41, "and-with-loops"]], "And Logistic Regression": [[40, "and-logistic-regression"], [41, "and-logistic-regression"]], "And a corresponding example using scikit-learn": [[39, "and-a-corresponding-example-using-scikit-learn"], [40, "and-a-corresponding-example-using-scikit-learn"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[36, "and-finally-boldsymbol-x-boldsymbol-x-t"], [37, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And finally ADAM": [[22, "and-finally-adam"], [40, "and-finally-adam"], [41, "and-finally-adam"]], "And what about using neural networks?": [[35, "and-what-about-using-neural-networks"]], "Another Example from Scikit-Learn\u2019s Repository": [[38, "another-example-from-scikit-learn-s-repository"]], "Another Example, now with a polynomial fit": [[37, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[10, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[26, null]], "Artificial neurons": [[41, "artificial-neurons"], [42, "artificial-neurons"]], "Assumptions made": [[37, "assumptions-made"], [38, "assumptions-made"]], "Autocorrelation function": [[32, "autocorrelation-function"]], "Autograd with more complicated functions": [[40, "autograd-with-more-complicated-functions"], [41, "autograd-with-more-complicated-functions"]], "Automatic differentiation": [[14, "automatic-differentiation"], [40, "automatic-differentiation"], [41, "automatic-differentiation"], [42, "automatic-differentiation"]], "Automatic differentiation through examples": [[42, "automatic-differentiation-through-examples"]], "Back propagation": [[44, "back-propagation"]], "Back propagation and automatic differentiation": [[44, "back-propagation-and-automatic-differentiation"]], "Back propagation in time in equations": [[46, "back-propagation-in-time-in-equations"]], "Back propagation in time through figures, part 1": [[46, "back-propagation-in-time-through-figures-part-1"]], "Back propagation in time, part 2": [[46, "back-propagation-in-time-part-2"]], "Back propagation in time, part 3": [[46, "back-propagation-in-time-part-3"]], "Back propagation in time, part 4": [[46, "back-propagation-in-time-part-4"]], "Back to the Cancer Data": [[12, "back-to-the-cancer-data"]], "Background literature": [[28, "background-literature"], [29, "background-literature"]], "Backpropagation in the convolutional layer": [[45, "backpropagation-in-the-convolutional-layer"], [46, "backpropagation-in-the-convolutional-layer"]], "Backpropagation through time": [[46, "backpropagation-through-time"]], "Bagging": [[11, "bagging"], [47, "bagging"], [48, "bagging"]], "Bagging Examples": [[11, "bagging-examples"]], "Basic Matrix Features": [[27, "basic-matrix-features"]], "Basic Steps of AdaBoost": [[47, "basic-steps-of-adaboost"], [48, "basic-steps-of-adaboost"]], "Basic ideas of the Principal Component Analysis (PCA)": [[12, null]], "Basic layout": [[46, "basic-layout"]], "Basic layout, Figures from Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch": [[46, "basic-layout-figures-from-sebastian-rashcka-et-al-machine-learning-with-sickit-learn-and-pytorch"]], "Basic math of the SVD": [[6, "basic-math-of-the-svd"], [36, "basic-math-of-the-svd"]], "Basics": [[8, "basics"], [39, "basics"]], "Basics of a tree": [[10, "basics-of-a-tree"], [47, "basics-of-a-tree"]], "Basics of an NN": [[42, "basics-of-an-nn"]], "Batch Normalization": [[2, "batch-normalization"], [42, "batch-normalization"], [43, "batch-normalization"]], "Batches and mini-batches": [[40, "batches-and-mini-batches"], [41, "batches-and-mini-batches"]], "Bayes\u2019 Theorem": [[37, "bayes-theorem"], [38, "bayes-theorem"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[6, "bayes-theorem-and-ridge-and-lasso-regression"], [37, "bayes-theorem-and-ridge-and-lasso-regression"], [38, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[11, "boosting-a-bird-s-eye-view"], [47, "boosting-a-bird-s-eye-view"], [48, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[7, "bootstrap"]], "Brief reminder on Newton-Raphson\u2019s method": [[39, "brief-reminder-on-newton-raphson-s-method"], [40, "brief-reminder-on-newton-raphson-s-method"]], "Bringing it together": [[42, "bringing-it-together"], [43, "bringing-it-together"]], "Bringing it together, first back propagation equation": [[13, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[2, null]], "Building a neural network code": [[43, "building-a-neural-network-code"], [44, "building-a-neural-network-code"]], "Building a tree, regression": [[10, "building-a-tree-regression"], [47, "building-a-tree-regression"], [48, "building-a-tree-regression"]], "Building convolutional neural networks in Tensorflow and Keras": [[45, "building-convolutional-neural-networks-in-tensorflow-and-keras"], [46, "building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Building neural networks in Tensorflow and Keras": [[2, "building-neural-networks-in-tensorflow-and-keras"], [43, "building-neural-networks-in-tensorflow-and-keras"], [44, "building-neural-networks-in-tensorflow-and-keras"]], "Building our own CNN code": [[45, "building-our-own-cnn-code"], [46, "building-our-own-cnn-code"]], "Building up AdaBoost": [[47, "building-up-adaboost"], [48, "building-up-adaboost"]], "But noen of these can compete with Newton\u2019s method": [[22, "but-noen-of-these-can-compete-with-newton-s-method"]], "But none of these can compete with Newton\u2019s method": [[40, "but-none-of-these-can-compete-with-newton-s-method"], [41, "but-none-of-these-can-compete-with-newton-s-method"]], "CNNs in brief": [[45, "cnns-in-brief"], [46, "cnns-in-brief"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[4, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, simple example": [[45, "cnns-in-more-detail-simple-example"]], "Cancer Data again now with Decision Trees and other Methods": [[10, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Chain rule": [[42, "chain-rule"]], "Chain rule again": [[46, "chain-rule-again"]], "Chain rule, forward and reverse modes": [[42, "chain-rule-forward-and-reverse-modes"]], "Challenge yourself the coming weekend": [[39, "challenge-yourself-the-coming-weekend"]], "Choose cost function and optimizer": [[2, "choose-cost-function-and-optimizer"], [43, "choose-cost-function-and-optimizer"], [44, "choose-cost-function-and-optimizer"]], "Class of functions we can approximate": [[42, "class-of-functions-we-can-approximate"]], "Classical PCA Theorem": [[12, "classical-pca-theorem"]], "Classification and Regression, from linear and logistic regression to neural networks": [[29, "classification-and-regression-from-linear-and-logistic-regression-to-neural-networks"]], "Classification problems": [[39, "classification-problems"]], "Classification tree, how to split nodes": [[47, "classification-tree-how-to-split-nodes"], [48, "classification-tree-how-to-split-nodes"]], "Clustering and Unsupervised Learning": [[15, null]], "Code Example for Cross-validation and k-fold Cross-validation": [[38, "code-example-for-cross-validation-and-k-fold-cross-validation"], [39, "code-example-for-cross-validation-and-k-fold-cross-validation"]], "Code example": [[42, "code-example"], [43, "code-example"]], "Code example for the Bootstrap method": [[38, "code-example-for-the-bootstrap-method"]], "Code examples from week 39 and 40": [[22, "code-examples-from-week-39-and-40"]], "Code for SVD and Inversion of Matrices": [[6, "code-for-svd-and-inversion-of-matrices"], [37, "code-for-svd-and-inversion-of-matrices"]], "Code with a Number of Minibatches which varies": [[40, "code-with-a-number-of-minibatches-which-varies"], [41, "code-with-a-number-of-minibatches-which-varies"]], "Code with a Number of Minibatches which varies, analytical gradient": [[22, "code-with-a-number-of-minibatches-which-varies-analytical-gradient"]], "Codes and Approaches": [[15, "codes-and-approaches"]], "Codes for the SVD": [[6, "codes-for-the-svd"], [36, "codes-for-the-svd"]], "Collect and pre-process data": [[2, "collect-and-pre-process-data"], [43, "collect-and-pre-process-data"], [43, "id2"], [44, "collect-and-pre-process-data"], [44, "id2"]], "Communication channels": [[35, "communication-channels"]], "Commutative process": [[45, "commutative-process"]], "Compact expressions": [[42, "compact-expressions"], [43, "compact-expressions"]], "Compare Bagging on Trees with Random Forests": [[11, "compare-bagging-on-trees-with-random-forests"], [47, "compare-bagging-on-trees-with-random-forests"], [48, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[3, "comparing-with-a-numerical-scheme"], [44, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[37, "comparison-with-ols"]], "Compile and train the model": [[45, "compile-and-train-the-model"], [46, "compile-and-train-the-model"]], "Completing the list": [[42, "completing-the-list"], [43, "completing-the-list"]], "Computation of gradients": [[40, "computation-of-gradients"], [41, "computation-of-gradients"]], "Computing a Tree using the Gini Index": [[47, "computing-a-tree-using-the-gini-index"], [48, "computing-a-tree-using-the-gini-index"]], "Computing the Gini Factor": [[47, "computing-the-gini-factor"], [48, "computing-the-gini-factor"]], "Computing the Gini index": [[10, "computing-the-gini-index"]], "Computing the various Gini Indices": [[47, "computing-the-various-gini-indices"], [48, "computing-the-various-gini-indices"]], "Computing the various Gini Indices, Hours slept": [[47, "computing-the-various-gini-indices-hours-slept"]], "Computing the various Gini Indices, Hours studied": [[47, "computing-the-various-gini-indices-hours-studied"]], "Conditional Probability": [[37, "conditional-probability"], [38, "conditional-probability"]], "Conditions on convex functions": [[39, "conditions-on-convex-functions"], [40, "conditions-on-convex-functions"]], "Confidence Intervals": [[38, "confidence-intervals"]], "Conjugate gradient method": [[14, "conjugate-gradient-method"], [40, "conjugate-gradient-method"], [40, "id2"], [40, "id3"], [40, "id4"], [40, "id5"], [40, "id6"], [40, "id7"]], "Conjugate gradient method and iterations": [[40, "conjugate-gradient-method-and-iterations"]], "Convex function": [[39, "convex-function"], [40, "convex-function"]], "Convex functions": [[14, "convex-functions"], [39, "convex-functions"], [40, "convex-functions"]], "Convolution": [[45, "convolution"], [46, "convolution"]], "Convolution Examples: Polynomial multiplication": [[4, "convolution-examples-polynomial-multiplication"], [45, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[4, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolution in the Fourier domain": [[45, "convolution-in-the-fourier-domain"], [46, "convolution-in-the-fourier-domain"]], "Convolution using separable kernels": [[45, "convolution-using-separable-kernels"], [46, "convolution-using-separable-kernels"]], "Convolution2DLayer: convolution in a hidden layer": [[45, "convolution2dlayer-convolution-in-a-hidden-layer"], [46, "convolution2dlayer-convolution-in-a-hidden-layer"]], "Convolutional Neural Network": [[13, "convolutional-neural-network"], [41, "convolutional-neural-network"], [42, "convolutional-neural-network"]], "Convolutional Neural Networks": [[4, null]], "Convolutional Neural Networks (recognizing images)": [[45, "convolutional-neural-networks-recognizing-images"], [46, "convolutional-neural-networks-recognizing-images"]], "Correlation Function and Design/Feature Matrix": [[36, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[12, "correlation-matrix"], [36, "correlation-matrix"]], "Correlation Matrix with Pandas": [[36, "correlation-matrix-with-pandas"]], "Correlation Matrix with Pandas and the Franke function": [[36, "correlation-matrix-with-pandas-and-the-franke-function"]], "Cost complexity pruning": [[47, "cost-complexity-pruning"], [48, "cost-complexity-pruning"]], "Cost functions": [[43, "cost-functions"], [44, "cost-functions"], [45, "cost-functions"], [46, "cost-functions"]], "Counting the number of floating point operations": [[42, "counting-the-number-of-floating-point-operations"]], "Course Format": [[35, "course-format"]], "Covariance Matrix Examples": [[36, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[36, "covariance-and-correlation-matrix"]], "Cross correlation": [[45, "cross-correlation"]], "Cross-validation": [[7, "cross-validation"]], "Cross-validation in brief": [[38, "cross-validation-in-brief"], [39, "cross-validation-in-brief"]], "Deadlines for projects (tentative)": [[35, "deadlines-for-projects-tentative"]], "Decision trees and Regression": [[47, "decision-trees-and-regression"]], "Decision trees, overarching aims": [[10, null], [47, "decision-trees-overarching-aims"]], "Deep learning methods": [[35, "deep-learning-methods"]], "Define model and architecture": [[2, "define-model-and-architecture"], [43, "define-model-and-architecture"], [44, "define-model-and-architecture"]], "Defining intermediate operations": [[42, "defining-intermediate-operations"]], "Defining the cost function": [[2, "defining-the-cost-function"], [43, "defining-the-cost-function"], [44, "defining-the-cost-function"]], "Defining the data sets to analyze yourself": [[30, "defining-the-data-sets-to-analyze-yourself"]], "Defining the problem": [[44, "defining-the-problem"]], "Definitions": [[42, "definitions"], [43, "definitions"]], "Demonstration": [[45, "demonstration"], [46, "demonstration"]], "Derivative of the cost function": [[42, "derivative-of-the-cost-function"], [43, "derivative-of-the-cost-function"]], "Derivatives and the chain rule": [[13, "derivatives-and-the-chain-rule"], [42, "derivatives-and-the-chain-rule"], [43, "derivatives-and-the-chain-rule"]], "Derivatives in terms of z_j^L": [[42, "derivatives-in-terms-of-z-j-l"], [43, "derivatives-in-terms-of-z-j-l"]], "Derivatives of the hidden layer": [[42, "derivatives-of-the-hidden-layer"], [43, "derivatives-of-the-hidden-layer"]], "Derivatives, example 1": [[36, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[6, "deriving-ols-from-a-probability-distribution"], [37, "deriving-ols-from-a-probability-distribution"], [38, "deriving-ols-from-a-probability-distribution"]], "Deriving the Lasso Regression Equations": [[36, "deriving-the-lasso-regression-equations"], [37, "deriving-the-lasso-regression-equations"]], "Deriving the Ridge Regression Equations": [[36, "deriving-the-ridge-regression-equations"], [37, "deriving-the-ridge-regression-equations"]], "Deriving the back propagation code for a multilayer perceptron model": [[13, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Description of two-dimensional function": [[28, "description-of-two-dimensional-function"]], "Developing a code for doing neural networks with back propagation": [[2, "developing-a-code-for-doing-neural-networks-with-back-propagation"], [43, "developing-a-code-for-doing-neural-networks-with-back-propagation"], [44, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[12, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[9, "different-kernels-and-mercer-s-theorem"]], "Differential equations": [[46, "differential-equations"]], "Disadvantages": [[10, "disadvantages"], [47, "disadvantages"], [48, "disadvantages"]], "Discussing the correlation data": [[0, "discussing-the-correlation-data"], [39, "discussing-the-correlation-data"]], "Does Logistic Regression do a better Job?": [[41, "does-logistic-regression-do-a-better-job"], [42, "does-logistic-regression-do-a-better-job"]], "Doing it correctly": [[37, "doing-it-correctly"], [38, "doing-it-correctly"]], "Domains and probabilities": [[32, "domains-and-probabilities"]], "Dropout": [[2, "dropout"], [42, "dropout"], [43, "dropout"]], "ELU function": [[42, "elu-function"], [43, "elu-function"], [44, "elu-function"]], "Economy-size SVD": [[36, "economy-size-svd"]], "Efficient Polynomial Multiplication": [[45, "efficient-polynomial-multiplication"]], "Elements of Probability Theory and Statistical Data Analysis": [[32, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[11, null], [47, "ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods"], [48, "ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods"]], "Entropy and the ID3 algorithm": [[10, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[35, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[2, "evaluate-model-performance-on-test-data"], [43, "evaluate-model-performance-on-test-data"], [44, "evaluate-model-performance-on-test-data"]], "Example 2": [[36, "example-2"]], "Example 3": [[36, "example-3"]], "Example 4": [[36, "example-4"]], "Example Matrix": [[36, "example-matrix"]], "Example code for Bias-Variance tradeoff": [[38, "example-code-for-bias-variance-tradeoff"]], "Example of Usage of Bayes\u2019 theorem": [[37, "example-of-usage-of-bayes-theorem"], [38, "example-of-usage-of-bayes-theorem"]], "Example of own Standard scaling": [[36, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[36, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[3, "example-exponential-decay"], [44, "example-exponential-decay"]], "Example: Population growth": [[3, "example-population-growth"], [44, "example-population-growth"]], "Example: Solving the one dimensional Poisson equation": [[44, "example-solving-the-one-dimensional-poisson-equation"]], "Example: Solving the wave equation with Neural Networks": [[44, "example-solving-the-wave-equation-with-neural-networks"]], "Example: The diffusion equation": [[3, "example-the-diffusion-equation"], [44, "example-the-diffusion-equation"]], "Example: binary classification problem": [[2, "example-binary-classification-problem"], [43, "example-binary-classification-problem"], [44, "example-binary-classification-problem"]], "Examples": [[35, "examples"]], "Examples of CNN setups": [[45, "examples-of-cnn-setups"]], "Examples of XOR, OR and AND gates": [[41, "examples-of-xor-or-and-and-gates"], [42, "examples-of-xor-or-and-and-gates"]], "Examples of likelihood functions used in logistic regression and neural networks": [[8, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Examples of likelihood functions used in logistic regression and nueral networks": [[0, "examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks"], [39, "examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks"]], "Exercise 1": [[23, "exercise-1"]], "Exercise 1 - Understand the feed forward pass": [[24, "exercise-1-understand-the-feed-forward-pass"]], "Exercise 1: Analytical exercises": [[17, "exercise-1-analytical-exercises"], [18, "exercise-1-analytical-exercises"]], "Exercise 1: Expectation values for ordinary least squares expressions": [[19, "exercise-1-expectation-values-for-ordinary-least-squares-expressions"]], "Exercise 1: Including more data": [[42, "exercise-1-including-more-data"]], "Exercise 1: Linear and logistic regression methods": [[25, "exercise-1-linear-and-logistic-regression-methods"]], "Exercise 1: Setting up various Python environments": [[1, "exercise-1-setting-up-various-python-environments"], [16, "exercise-1-setting-up-various-python-environments"], [35, "exercise-1-setting-up-various-python-environments"]], "Exercise 2": [[23, "exercise-2"]], "Exercise 2 - Gradient with one layer using autograd": [[24, "exercise-2-gradient-with-one-layer-using-autograd"]], "Exercise 2: Adding Ridge Regression": [[18, "exercise-2-adding-ridge-regression"]], "Exercise 2: Deep learning": [[25, "exercise-2-deep-learning"]], "Exercise 2: Expectation values for Ridge regression": [[19, "exercise-2-expectation-values-for-ridge-regression"]], "Exercise 2: Extended program": [[42, "exercise-2-extended-program"]], "Exercise 2: making your own data and exploring scikit-learn": [[1, "exercise-2-making-your-own-data-and-exploring-scikit-learn"], [16, "exercise-2-making-your-own-data-and-exploring-scikit-learn"], [17, "exercise-2-making-your-own-data-and-exploring-scikit-learn"], [35, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3": [[23, "exercise-3"]], "Exercise 3 - Gradient with one layer writing backpropagation by hand": [[24, "exercise-3-gradient-with-one-layer-writing-backpropagation-by-hand"]], "Exercise 3: Decision trees and ensemble methods": [[25, "exercise-3-decision-trees-and-ensemble-methods"]], "Exercise 3: Normalizing our data": [[1, "exercise-3-normalizing-our-data"]], "Exercise 3: Split data in test and training data": [[16, "exercise-3-split-data-in-test-and-training-data"], [17, "exercise-3-split-data-in-test-and-training-data"], [35, "exercise-3-split-data-in-test-and-training-data"]], "Exercise 4 - Custom activation for each layer": [[23, "exercise-4-custom-activation-for-each-layer"]], "Exercise 4 - Gradient with two layers writing backpropagation by hand": [[24, "exercise-4-gradient-with-two-layers-writing-backpropagation-by-hand"]], "Exercise 4: Adding Ridge Regression": [[1, "exercise-4-adding-ridge-regression"]], "Exercise 4: Optimization part": [[25, "exercise-4-optimization-part"]], "Exercise 5 - Gradient with any number of layers writing backpropagation by hand": [[24, "exercise-5-gradient-with-any-number-of-layers-writing-backpropagation-by-hand"]], "Exercise 5 - Processing multiple inputs at once": [[23, "exercise-5-processing-multiple-inputs-at-once"]], "Exercise 5: Analysis of results": [[25, "exercise-5-analysis-of-results"]], "Exercise 5: Analytical exercises": [[1, "exercise-5-analytical-exercises"]], "Exercise 6 - Batched inputs": [[24, "exercise-6-batched-inputs"]], "Exercise 6 - Predicting on real data": [[23, "exercise-6-predicting-on-real-data"]], "Exercise 7 - Training": [[24, "exercise-7-training"]], "Exercise 7 - Training on real data (Optional)": [[23, "exercise-7-training-on-real-data-optional"]], "Exercise 8 (Optional) - Object orientation": [[24, "exercise-8-optional-object-orientation"]], "Exercise week 47": [[25, null]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[7, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[7, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[7, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[7, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[7, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[7, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[1, "exercises"], [16, "exercises"], [35, "exercises"]], "Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40": [[0, null]], "Exercises and Projects": [[7, "exercises-and-projects"]], "Exercises and lab session week 43": [[44, "exercises-and-lab-session-week-43"]], "Exercises week 34": [[16, null]], "Exercises week 35": [[17, null]], "Exercises week 36": [[18, null]], "Exercises week 37": [[19, null]], "Exercises week 38": [[20, null]], "Exercises week 39": [[21, null]], "Exercises week 41": [[22, null]], "Exercises week 42": [[23, null]], "Exercises week 43": [[24, null]], "Expectation value and variance": [[37, "expectation-value-and-variance"], [38, "expectation-value-and-variance"]], "Expectation value and variance for \\boldsymbol{\\beta}": [[37, "expectation-value-and-variance-for-boldsymbol-beta"], [38, "expectation-value-and-variance-for-boldsymbol-beta"]], "Expectation values": [[32, "expectation-values"]], "Explicit derivatives": [[42, "explicit-derivatives"], [43, "explicit-derivatives"]], "Exploding gradients": [[42, "exploding-gradients"], [43, "exploding-gradients"]], "Extending to more predictors": [[0, "extending-to-more-predictors"], [39, "extending-to-more-predictors"]], "Extending to more than one variable": [[39, "extending-to-more-than-one-variable"], [40, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[35, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[13, "feed-forward-neural-networks"], [41, "feed-forward-neural-networks"], [42, "feed-forward-neural-networks"]], "Feed-forward pass": [[2, "feed-forward-pass"], [43, "feed-forward-pass"], [44, "feed-forward-pass"]], "Feedback connections": [[46, "feedback-connections"]], "Final back propagating equation": [[13, "final-back-propagating-equation"], [42, "final-back-propagating-equation"], [43, "final-back-propagating-equation"]], "Final derivatives": [[42, "final-derivatives"]], "Final expression": [[42, "final-expression"], [43, "final-expression"]], "Final expressions": [[40, "final-expressions"]], "Final expressions for the biases of the hidden layer": [[42, "final-expressions-for-the-biases-of-the-hidden-layer"], [43, "final-expressions-for-the-biases-of-the-hidden-layer"]], "Final part": [[45, "final-part"], [46, "final-part"]], "Final regressor code": [[47, "final-regressor-code"], [48, "final-regressor-code"]], "Final technicalities I": [[44, "final-technicalities-i"]], "Final technicalities II": [[44, "final-technicalities-ii"]], "Final technicalities III": [[44, "final-technicalities-iii"]], "Final technicalities IV": [[44, "final-technicalities-iv"]], "Final visualization": [[45, "final-visualization"], [46, "final-visualization"]], "Finally, evaluate the model": [[45, "finally-evaluate-the-model"], [46, "finally-evaluate-the-model"]], "Finding the Limit": [[38, "finding-the-limit"]], "Finding the number of parameters": [[45, "finding-the-number-of-parameters"]], "Fine-tuning neural network hyperparameters": [[2, "fine-tuning-neural-network-hyperparameters"], [42, "fine-tuning-neural-network-hyperparameters"], [43, "fine-tuning-neural-network-hyperparameters"]], "First network example, simple percepetron with one input": [[42, "first-network-example-simple-percepetron-with-one-input"], [43, "first-network-example-simple-percepetron-with-one-input"]], "Fitting an Equation of State for Dense Nuclear Matter": [[1, "fitting-an-equation-of-state-for-dense-nuclear-matter"], [35, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[36, "fixing-the-singularity"]], "Flattening Layer": [[45, "flattening-layer"], [46, "flattening-layer"]], "For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week": [[36, "for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week"]], "Forget and input": [[46, "forget-and-input"]], "Format for electronic delivery of report and programs": [[28, "format-for-electronic-delivery-of-report-and-programs"], [29, "format-for-electronic-delivery-of-report-and-programs"], [30, "format-for-electronic-delivery-of-report-and-programs"]], "Forward and reverse modes": [[42, "forward-and-reverse-modes"]], "Four effective ways to learn an RNN and preparing for next week": [[46, "four-effective-ways-to-learn-an-rnn-and-preparing-for-next-week"]], "Fourier series and Toeplitz matrices": [[45, "fourier-series-and-toeplitz-matrices"]], "Frequently used scaling functions": [[36, "frequently-used-scaling-functions"]], "From FFNNs and CNNs to recurrent neural networks (RNNs)": [[46, "from-ffnns-and-cnns-to-recurrent-neural-networks-rnns"]], "From OLS to Ridge and Lasso": [[37, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[13, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Full object-oriented implementation": [[43, "full-object-oriented-implementation"], [44, "full-object-oriented-implementation"]], "Fully Connected Layers": [[45, "fully-connected-layers"], [46, "fully-connected-layers"]], "Functionality in Scikit-Learn": [[36, "functionality-in-scikit-learn"]], "Functions using mathematical functions from Numpy": [[40, "functions-using-mathematical-functions-from-numpy"], [41, "functions-using-mathematical-functions-from-numpy"]], "Further Dimensionality Remarks": [[4, "further-dimensionality-remarks"]], "Further Manipulations": [[36, "further-manipulations"]], "Further example: Computing the Gini index": [[47, "further-example-computing-the-gini-index"], [48, "further-example-computing-the-gini-index"]], "Further properties (important for our analyses later)": [[6, "further-properties-important-for-our-analyses-later"], [36, "further-properties-important-for-our-analyses-later"]], "Further remarks": [[45, "further-remarks"]], "Further simplification": [[45, "further-simplification"]], "Gating mechanism: Long Short Term Memory (LSTM)": [[46, "gating-mechanism-long-short-term-memory-lstm"]], "Gaussian Elimination": [[27, "gaussian-elimination"]], "General Features": [[10, "general-features"], [47, "general-features"]], "General linear models and linear algebra": [[35, "general-linear-models-and-linear-algebra"]], "Generalizing the above one-dimensional case": [[45, "generalizing-the-above-one-dimensional-case"]], "Generalizing the fitting procedure as a linear algebra problem": [[35, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [35, "id1"]], "Generative Adversarial Networks": [[5, "generative-adversarial-networks"]], "Generative Models": [[5, "generative-models"]], "Geometric Interpretation and link with Singular Value Decomposition": [[12, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Getting serious, the back propagation equations for a neural network": [[42, "getting-serious-the-back-propagation-equations-for-a-neural-network"]], "Getting started with Jax, note the way we import numpy": [[22, "getting-started-with-jax-note-the-way-we-import-numpy"], [41, "getting-started-with-jax-note-the-way-we-import-numpy"]], "Gradient Boosting, Classification Example": [[11, "gradient-boosting-classification-example"], [48, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[11, "gradient-boosting-examples-of-regression"], [48, "gradient-boosting-examples-of-regression"]], "Gradient Boosting, algorithm": [[48, "gradient-boosting-algorithm"]], "Gradient Clipping": [[2, "gradient-clipping"], [42, "gradient-clipping"], [43, "gradient-clipping"]], "Gradient Descent Example": [[39, "id1"], [40, "id8"]], "Gradient boosting, making our own code for a regression case": [[48, "gradient-boosting-making-our-own-code-for-a-regression-case"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[11, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"], [48, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[3, "gradient-descent"], [44, "gradient-descent"]], "Gradient descent and Logistic regression": [[0, "gradient-descent-and-logistic-regression"], [41, "gradient-descent-and-logistic-regression"]], "Gradient descent and Ridge": [[39, "gradient-descent-and-ridge"], [40, "gradient-descent-and-ridge"]], "Gradient descent example": [[39, "gradient-descent-example"], [40, "gradient-descent-example"]], "Gradient expressions": [[42, "gradient-expressions"], [43, "gradient-expressions"]], "Gradient method": [[40, "gradient-method"]], "Gradients of loss functions": [[46, "gradients-of-loss-functions"]], "Grading": [[33, "grading"], [35, "grading"]], "Grid Search": [[39, "grid-search"]], "Growing a classification tree": [[47, "growing-a-classification-tree"], [48, "growing-a-classification-tree"]], "Hidden layers": [[42, "hidden-layers"], [43, "hidden-layers"]], "Homogeneous data": [[42, "homogeneous-data"], [43, "homogeneous-data"]], "Housing data, the code": [[1, "housing-data-the-code"]], "How do we set it up?": [[47, "how-do-we-set-it-up"]], "How to do image compression before the era of deep learning": [[45, "how-to-do-image-compression-before-the-era-of-deep-learning"]], "How to set up the cross-validation for Ridge and/or Lasso": [[39, "how-to-set-up-the-cross-validation-for-ridge-and-or-lasso"]], "Hyperplanes and all that": [[9, "hyperplanes-and-all-that"]], "Identifying Terms": [[38, "identifying-terms"]], "Illustration of a single perceptron model and a multi-perceptron model": [[41, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"], [42, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"]], "Implementing a memory cell in a neural network": [[46, "implementing-a-memory-cell-in-a-neural-network"]], "Important Matrix and vector handling packages": [[27, "important-matrix-and-vector-handling-packages"]], "Important observations": [[42, "important-observations"], [43, "important-observations"]], "Important technicalities: More on Rescaling data": [[37, "important-technicalities-more-on-rescaling-data"]], "Importing Keras and Tensorflow": [[45, "importing-keras-and-tensorflow"], [46, "importing-keras-and-tensorflow"]], "Improving gradient descent with momentum": [[40, "improving-gradient-descent-with-momentum"]], "Improving performance": [[2, "improving-performance"], [43, "improving-performance"], [44, "improving-performance"]], "In general not this simple": [[42, "in-general-not-this-simple"]], "In summary": [[33, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[14, "including-stochastic-gradient-descent-with-autograd"], [22, "including-stochastic-gradient-descent-with-autograd"], [40, "including-stochastic-gradient-descent-with-autograd"], [41, "including-stochastic-gradient-descent-with-autograd"]], "Including more classes": [[0, "including-more-classes"], [39, "including-more-classes"]], "Incremental PCA": [[12, "incremental-pca"]], "Independent and Identically Distributed (iid)": [[37, "independent-and-identically-distributed-iid"]], "Independent and Identically Distrubuted (iid)": [[38, "independent-and-identically-distrubuted-iid"]], "Input gate": [[46, "input-gate"]], "Inputs to the activation function": [[42, "inputs-to-the-activation-function"], [43, "inputs-to-the-activation-function"]], "Insights from the paper by Glorot and Bengio": [[42, "insights-from-the-paper-by-glorot-and-bengio"], [43, "insights-from-the-paper-by-glorot-and-bengio"]], "Installing R, C++, cython or Julia": [[35, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[35, "installing-r-c-cython-numba-etc"]], "Instructor information": [[33, "instructor-information"]], "Interpretations and optimizing our parameters": [[35, "interpretations-and-optimizing-our-parameters"], [35, "id2"], [35, "id3"], [36, "interpretations-and-optimizing-our-parameters"], [36, "id1"], [36, "id2"]], "Interpretations of Bayes\u2019 Theorem": [[37, "interpretations-of-bayes-theorem"], [38, "interpretations-of-bayes-theorem"]], "Interpreting the Ridge results": [[36, "interpreting-the-ridge-results"], [37, "interpreting-the-ridge-results"]], "Introducing JAX": [[14, "introducing-jax"], [22, "introducing-jax"], [40, "introducing-jax"], [41, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[12, "introducing-the-covariance-and-correlation-functions"], [36, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[1, "introduction"], [7, "introduction"], [26, "introduction"], [27, "introduction"]], "Introduction to Neural networks": [[41, "introduction-to-neural-networks"], [42, "introduction-to-neural-networks"]], "Introduction to numerical projects": [[28, "introduction-to-numerical-projects"], [29, "introduction-to-numerical-projects"], [30, "introduction-to-numerical-projects"]], "Inverse of Rectangular Matrix": [[37, "inverse-of-rectangular-matrix"]], "Is the Logistic activation function (Sigmoid) our choice?": [[42, "is-the-logistic-activation-function-sigmoid-our-choice"], [43, "is-the-logistic-activation-function-sigmoid-our-choice"]], "Iterative Fitting, Classification and AdaBoost": [[11, "iterative-fitting-classification-and-adaboost"], [47, "iterative-fitting-classification-and-adaboost"], [48, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[11, "iterative-fitting-regression-and-squared-error-cost-function"], [47, "iterative-fitting-regression-and-squared-error-cost-function"], [48, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[12, "kernel-pca"]], "Kernels and non-linearity": [[9, "kernels-and-non-linearity"]], "Key Idea": [[45, "key-idea"], [46, "key-idea"]], "LSTM details": [[46, "lstm-details"]], "LU Decomposition, the inverse of a matrix": [[27, "lu-decomposition-the-inverse-of-a-matrix"]], "Lab sessions on Tuesday and Wednesday": [[45, "lab-sessions-on-tuesday-and-wednesday"]], "Lab session: Material from last week and relevant for the first project": [[39, "lab-session-material-from-last-week-and-relevant-for-the-first-project"]], "Lab sessions Tuesday and Wednesday": [[41, "lab-sessions-tuesday-and-wednesday"]], "Lab sessions week 39": [[40, "lab-sessions-week-39"]], "Lasso Regression": [[37, "lasso-regression"]], "Lasso and Bayes": [[37, "lasso-and-bayes"], [38, "lasso-and-bayes"]], "Lasso case": [[37, "lasso-case"]], "Layers": [[2, "layers"], [43, "layers"], [44, "layers"], [45, "layers"], [46, "layers"]], "Layers of a CNN": [[45, "layers-of-a-cnn"], [46, "layers-of-a-cnn"]], "Layers used to build CNNs": [[4, "layers-used-to-build-cnns"], [45, "layers-used-to-build-cnns"], [46, "layers-used-to-build-cnns"]], "Layout of a neural network with three hidden layers": [[42, "layout-of-a-neural-network-with-three-hidden-layers"]], "Layout of a neural network with three hidden layers (last later = l=L=4, first layer l=0)": [[43, "layout-of-a-neural-network-with-three-hidden-layers-last-later-l-l-4-first-layer-l-0"]], "Layout of a simple neural network with no hidden layer": [[42, "layout-of-a-simple-neural-network-with-no-hidden-layer"], [43, "layout-of-a-simple-neural-network-with-no-hidden-layer"]], "Layout of a simple neural network with one hidden layer": [[42, "layout-of-a-simple-neural-network-with-one-hidden-layer"], [43, "layout-of-a-simple-neural-network-with-one-hidden-layer"]], "Layout of a simple neural network with two input nodes, one hidden layer and one output node": [[42, "layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-and-one-output-node"]], "Layout of a simple neural network with two input nodes, one hidden layer with two hidden noeds and one output node": [[43, "layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-with-two-hidden-noeds-and-one-output-node"]], "Layout of input to first hidden layer l=1 from input layer l=0": [[43, "layout-of-input-to-first-hidden-layer-l-1-from-input-layer-l-0"]], "Learning outcomes": [[26, "learning-outcomes"], [35, "learning-outcomes"]], "Learning rate methods": [[43, "learning-rate-methods"], [44, "learning-rate-methods"]], "Lecture Monday October 21": [[44, "lecture-monday-october-21"]], "Lecture Monday October 7": [[42, "lecture-monday-october-7"]], "Lecture Monday September 23": [[40, "lecture-monday-september-23"]], "Lecture Monday September 23, Optimization, the central part of any Machine Learning algortithm": [[40, "lecture-monday-september-23-optimization-the-central-part-of-any-machine-learning-algortithm"]], "Lecture Monday September 30, 2024": [[41, "lecture-monday-september-30-2024"]], "Lecture October 14, 2024": [[43, "lecture-october-14-2024"]], "Lectures and ComputerLab": [[35, "lectures-and-computerlab"]], "Limitations of NNs": [[42, "limitations-of-nns"], [43, "limitations-of-nns"]], "Limitations of supervised learning with deep networks": [[2, "limitations-of-supervised-learning-with-deep-networks"], [42, "limitations-of-supervised-learning-with-deep-networks"], [43, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[27, null]], "Linear Regression": [[1, null]], "Linear Regression Problems": [[36, "linear-regression-problems"]], "Linear Regression and the SVD": [[37, "linear-regression-and-the-svd"]], "Linear Regression code, Intercept handling first": [[36, "linear-regression-code-intercept-handling-first"]], "Linear Regression, basic elements": [[1, "linear-regression-basic-elements"]], "Linear classifier": [[39, "linear-classifier"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[6, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[6, "linking-the-regression-analysis-with-a-statistical-interpretation"], [37, "linking-the-regression-analysis-with-a-statistical-interpretation"], [38, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with RNNs": [[46, "linking-with-rnns"]], "Linking with the SVD": [[6, "linking-with-the-svd"], [36, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[34, "links-to-relevant-courses-at-the-university-of-oslo"]], "List of contents:": [[45, "list-of-contents"], [46, "list-of-contents"]], "Logistic Regression": [[8, null], [8, "id1"], [39, "logistic-regression"]], "Logistic function as the root of problems": [[42, "logistic-function-as-the-root-of-problems"], [43, "logistic-function-as-the-root-of-problems"]], "MNIST and GANs": [[5, "mnist-and-gans"]], "Machine Learning": [[35, "machine-learning"]], "Machine learning": [[26, "machine-learning"]], "Main textbooks": [[35, "main-textbooks"]], "Making a tree": [[10, "making-a-tree"], [47, "making-a-tree"], [48, "making-a-tree"]], "Making an ADAboost code yourself": [[48, "making-an-adaboost-code-yourself"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[11, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"], [47, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"], [48, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[36, "making-your-own-test-train-splitting"]], "Marginal Probability": [[37, "marginal-probability"], [38, "marginal-probability"]], "Material for Lecture Monday November 4": [[46, "material-for-lecture-monday-november-4"]], "Material for Lecture Monday October 28": [[45, "material-for-lecture-monday-october-28"]], "Material for lecture Monday September 16": [[39, "material-for-lecture-monday-september-16"]], "Material for lecture Monday September 2": [[37, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 9": [[38, "material-for-lecture-monday-september-9"]], "Material for lecture Monday, August 26": [[36, "material-for-lecture-monday-august-26"]], "Material for the active learning sessions Tuesday and Wednesday": [[37, "material-for-the-active-learning-sessions-tuesday-and-wednesday"]], "Material for the active learning sessions on Tuesday and Wednesday": [[42, "material-for-the-active-learning-sessions-on-tuesday-and-wednesday"], [43, "material-for-the-active-learning-sessions-on-tuesday-and-wednesday"]], "Material for the lab sessions": [[38, "material-for-the-lab-sessions"]], "Material for the lab sessions, additional ways to present classification results and other practicalities": [[46, "material-for-the-lab-sessions-additional-ways-to-present-classification-results-and-other-practicalities"]], "Material for the lecture on Monday October 7, 2024": [[42, "material-for-the-lecture-on-monday-october-7-2024"]], "Mathematical Interpretation of Ordinary Least Squares": [[6, "mathematical-interpretation-of-ordinary-least-squares"], [36, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical model": [[41, "mathematical-model"], [41, "id2"], [41, "id3"], [41, "id4"], [41, "id5"]], "Mathematical optimization of convex functions": [[9, "mathematical-optimization-of-convex-functions"]], "Mathematical setup": [[46, "mathematical-setup"]], "Mathematics of CNNs": [[4, "mathematics-of-cnns"], [45, "mathematics-of-cnns"]], "Mathematics of deep learning": [[42, "mathematics-of-deep-learning"], [43, "mathematics-of-deep-learning"], [44, "mathematics-of-deep-learning"]], "Mathematics of deep learning and neural networks": [[42, "mathematics-of-deep-learning-and-neural-networks"]], "Mathematics of the SVD and implications": [[6, "mathematics-of-the-svd-and-implications"], [36, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[35, "matrices-in-python"]], "Matrix multiplication": [[2, "matrix-multiplication"], [43, "matrix-multiplication"], [44, "matrix-multiplication"]], "Matrix multiplications": [[43, "matrix-multiplications"], [44, "matrix-multiplications"]], "Matrix-vector notation": [[41, "matrix-vector-notation"]], "Matrix-vector notation and activation": [[13, "matrix-vector-notation-and-activation"], [41, "matrix-vector-notation-and-activation"]], "Maximum Likelihood Estimation (MLE)": [[37, "maximum-likelihood-estimation-mle"], [38, "maximum-likelihood-estimation-mle"]], "Maximum likelihood": [[39, "maximum-likelihood"]], "Meet the covariance!": [[32, "meet-the-covariance"]], "Meet the Covariance Matrix": [[6, "meet-the-covariance-matrix"], [36, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[36, "meet-the-hessian-matrix"]], "Meet the Pandas": [[35, "meet-the-pandas"]], "Memory considerations": [[45, "memory-considerations"]], "Min-Max Scaling": [[36, "min-max-scaling"]], "Minimization process": [[44, "minimization-process"]], "Minimizing the cost function using gradient descent and automatic differentiation": [[44, "minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation"]], "Minimizing the cross entropy": [[0, "minimizing-the-cross-entropy"], [39, "minimizing-the-cross-entropy"]], "Minor rewrite": [[46, "minor-rewrite"]], "Momentum based GD": [[14, "momentum-based-gd"], [22, "momentum-based-gd"], [40, "momentum-based-gd"], [41, "momentum-based-gd"]], "Momentum parameter": [[40, "momentum-parameter"], [41, "momentum-parameter"]], "More LSTM details": [[46, "more-lstm-details"]], "More autograd": [[40, "more-autograd"], [41, "more-autograd"]], "More bagging": [[47, "more-bagging"], [48, "more-bagging"]], "More basic Statistics and Bayes\u2019 theorem": [[37, "more-basic-statistics-and-bayes-theorem"], [38, "more-basic-statistics-and-bayes-theorem"]], "More classes": [[0, "more-classes"], [39, "more-classes"]], "More complicated Example: The Ising model": [[7, "more-complicated-example-the-ising-model"]], "More complicated function": [[42, "more-complicated-function"]], "More complicated functions using the elements of their arguments directly": [[40, "more-complicated-functions-using-the-elements-of-their-arguments-directly"], [41, "more-complicated-functions-using-the-elements-of-their-arguments-directly"]], "More considerations": [[42, "more-considerations"], [43, "more-considerations"]], "More details": [[44, "more-details"], [44, "id6"]], "More examples on bootstrap and cross-validation and errors": [[38, "more-examples-on-bootstrap-and-cross-validation-and-errors"]], "More interpretations": [[36, "more-interpretations"], [37, "more-interpretations"]], "More limitations": [[42, "more-limitations"], [43, "more-limitations"]], "More on Dimensionalities": [[4, "more-on-dimensionalities"], [45, "more-on-dimensionalities"]], "More on Rescaling data": [[7, "more-on-rescaling-data"]], "More on Steepest descent": [[39, "more-on-steepest-descent"], [40, "more-on-steepest-descent"]], "More on activation functions, output layers": [[42, "more-on-activation-functions-output-layers"], [43, "more-on-activation-functions-output-layers"], [44, "more-on-activation-functions-output-layers"]], "More on convex functions": [[39, "more-on-convex-functions"], [40, "more-on-convex-functions"]], "More on momentum based approaches": [[40, "more-on-momentum-based-approaches"], [41, "more-on-momentum-based-approaches"]], "More on the general approximation theorem": [[42, "more-on-the-general-approximation-theorem"]], "More preprocessing": [[36, "more-preprocessing"]], "More preprocessing examples, two-dimensional example, the Franke function": [[36, "more-preprocessing-examples-two-dimensional-example-the-franke-function"]], "More technicalities": [[44, "more-technicalities"]], "More thinking": [[36, "more-thinking"]], "More top-down perspectives": [[42, "more-top-down-perspectives"], [43, "more-top-down-perspectives"]], "Multiclass classification": [[43, "multiclass-classification"], [44, "multiclass-classification"]], "Multilayer perceptrons": [[13, "multilayer-perceptrons"], [41, "multilayer-perceptrons"], [42, "multilayer-perceptrons"]], "Multivariable functions": [[42, "multivariable-functions"]], "Network requirements": [[3, "network-requirements"], [44, "network-requirements"]], "Neural Networks vs CNNs": [[4, "neural-networks-vs-cnns"], [45, "neural-networks-vs-cnns"], [46, "neural-networks-vs-cnns"]], "Neural network types": [[41, "neural-network-types"], [42, "neural-network-types"]], "Neural networks": [[13, null]], "New expression for the derivative": [[42, "new-expression-for-the-derivative"]], "New image (or volume)": [[45, "new-image-or-volume"]], "New vector": [[45, "new-vector"]], "Note about SVD Calculations": [[36, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[37, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[32, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[27, "numpy-and-arrays"], [35, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[35, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization and Deep learning": [[39, "optimization-and-deep-learning"]], "Optimization, the central part of any Machine Learning algortithm": [[14, null], [39, "optimization-the-central-part-of-any-machine-learning-algortithm"]], "Optimized Convolution2DLayer": [[45, "optimized-convolution2dlayer"], [46, "optimized-convolution2dlayer"]], "Optimizing our parameters": [[35, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[35, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[2, "optimizing-the-cost-function"], [43, "optimizing-the-cost-function"], [44, "optimizing-the-cost-function"]], "Optimizing the parameters": [[42, "optimizing-the-parameters"], [43, "optimizing-the-parameters"]], "Ordinary Differential Equations first": [[44, "ordinary-differential-equations-first"]], "Organizing our data": [[1, "organizing-our-data"], [35, "organizing-our-data"]], "Other Matrix and Vector Operations": [[27, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[5, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[35, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[35, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other ingredients of a neural network": [[42, "other-ingredients-of-a-neural-network"]], "Other measures in classification studies: Cancer Data again": [[0, "other-measures-in-classification-studies-cancer-data-again"], [39, "other-measures-in-classification-studies-cancer-data-again"]], "Other parameters": [[42, "other-parameters"]], "Other popular texts": [[35, "other-popular-texts"]], "Other techniques": [[12, "other-techniques"]], "Other types of networks": [[13, "other-types-of-networks"], [41, "other-types-of-networks"], [42, "other-types-of-networks"]], "Other useful relations": [[36, "other-useful-relations"]], "Other ways of visualizing the trees": [[10, "other-ways-of-visualizing-the-trees"], [47, "other-ways-of-visualizing-the-trees"], [48, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[35, "our-model-for-the-nuclear-binding-energies"]], "Output gate": [[46, "output-gate"]], "Output layer": [[42, "output-layer"], [43, "output-layer"]], "Overarching aims of the exercises this week": [[18, "overarching-aims-of-the-exercises-this-week"], [19, "overarching-aims-of-the-exercises-this-week"], [20, "overarching-aims-of-the-exercises-this-week"], [21, "overarching-aims-of-the-exercises-this-week"], [22, "overarching-aims-of-the-exercises-this-week"], [23, "overarching-aims-of-the-exercises-this-week"], [24, "overarching-aims-of-the-exercises-this-week"], [25, "overarching-aims-of-the-exercises-this-week"]], "Overarching view of a neural network": [[42, "overarching-view-of-a-neural-network"]], "Overview of first week": [[35, "overview-of-first-week"]], "Overview video on Stochastic Gradient Descent": [[40, "overview-video-on-stochastic-gradient-descent"], [41, "overview-video-on-stochastic-gradient-descent"]], "Own code for Ordinary Least Squares": [[35, "own-code-for-ordinary-least-squares"], [36, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[12, "pca-and-scikit-learn"]], "Padding": [[45, "padding"]], "Pandas AI": [[35, "pandas-ai"]], "Parameters of neural networks": [[42, "parameters-of-neural-networks"]], "Parameters to train, common settings": [[45, "parameters-to-train-common-settings"]], "Part a)": [[30, "part-a"]], "Part a) : Ordinary Least Square (OLS) on the Franke function": [[28, "part-a-ordinary-least-square-ols-on-the-franke-function"]], "Part a), setting up the problem": [[30, "part-a-setting-up-the-problem"]], "Part a): Write your own Stochastic Gradient Descent code, first step": [[29, "part-a-write-your-own-stochastic-gradient-descent-code-first-step"]], "Part b)": [[30, "part-b"], [30, "id1"]], "Part b): Adding Ridge regression for the Franke function": [[28, "part-b-adding-ridge-regression-for-the-franke-function"]], "Part b): Writing your own Neural Network code": [[29, "part-b-writing-your-own-neural-network-code"]], "Part c)": [[30, "part-c"]], "Part c) Neural networks": [[30, "part-c-neural-networks"]], "Part c): Adding Lasso for the Franke function": [[28, "part-c-adding-lasso-for-the-franke-function"]], "Part c): Testing different activation functions": [[29, "part-c-testing-different-activation-functions"]], "Part d)": [[30, "part-d"]], "Part d) Neural network complexity": [[30, "part-d-neural-network-complexity"]], "Part d): Classification analysis using neural networks": [[29, "part-d-classification-analysis-using-neural-networks"]], "Part d): Paper and pencil part": [[28, "part-d-paper-and-pencil-part"]], "Part e)": [[30, "part-e"], [30, "id2"]], "Part e): Bias-variance trade-off and resampling techniques": [[28, "part-e-bias-variance-trade-off-and-resampling-techniques"]], "Part e): Write your Logistic Regression code, final step": [[29, "part-e-write-your-logistic-regression-code-final-step"]], "Part f) Critical evaluation of the various algorithms": [[29, "part-f-critical-evaluation-of-the-various-algorithms"]], "Part f): Cross-validation as resampling techniques, adding more complexity": [[28, "part-f-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Part g): Analysis of real data": [[28, "part-g-analysis-of-real-data"]], "Partial Differential Equations": [[3, "partial-differential-equations"], [44, "partial-differential-equations"]], "Paths for project 3": [[30, "paths-for-project-3"]], "Plan for week 39, September 23-27, 2024": [[40, "plan-for-week-39-september-23-27-2024"]], "Plan for week 41, October 7-11": [[42, "plan-for-week-41-october-7-11"]], "Plan for week 44": [[45, "plan-for-week-44"]], "Plan for week 46": [[47, "plan-for-week-46"]], "Plan for week 47": [[48, "plan-for-week-47"]], "Plans for the lab sessions": [[39, "plans-for-the-lab-sessions"]], "Plans for week 35": [[36, "plans-for-week-35"]], "Plans for week 36": [[37, "plans-for-week-36"]], "Plans for week 37, lab sessions": [[38, "plans-for-week-37-lab-sessions"]], "Plans for week 37, lecture Monday": [[38, "plans-for-week-37-lecture-monday"]], "Plans for week 38, lecture Monday September 16": [[39, "plans-for-week-38-lecture-monday-september-16"]], "Plans for week 40": [[41, "plans-for-week-40"]], "Plans for week 43": [[44, "plans-for-week-43"]], "Plans for week 45": [[46, "plans-for-week-45"]], "Plotting the Histogram": [[38, "plotting-the-histogram"]], "Plotting the mean value for each group": [[39, "plotting-the-mean-value-for-each-group"]], "Pooling": [[45, "pooling"]], "Pooling Layer": [[45, "pooling-layer"], [46, "pooling-layer"]], "Pooling arithmetic": [[45, "pooling-arithmetic"]], "Pooling types (From Raschka et al)": [[45, "pooling-types-from-raschka-et-al"]], "Practical tips": [[14, "practical-tips"], [22, "practical-tips"], [40, "practical-tips"], [41, "practical-tips"]], "Practicalities": [[33, "practicalities"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[28, "preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools"]], "Predicting New Points With A Trained Recurrent Neural Network": [[5, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[36, "preprocessing-our-data"]], "Prerequisites": [[35, "prerequisites"]], "Prerequisites and background": [[26, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[4, "prerequisites-collect-and-pre-process-data"], [45, "prerequisites-collect-and-pre-process-data"], [46, "prerequisites-collect-and-pre-process-data"]], "Printing out as text": [[47, "printing-out-as-text"], [48, "printing-out-as-text"]], "Probability Distribution Functions": [[32, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[39, "program-example-for-gradient-descent-with-ridge-regression"], [40, "program-example-for-gradient-descent-with-ridge-regression"]], "Program for stochastic gradient": [[14, "program-for-stochastic-gradient"]], "Project 1 on Machine Learning, deadline October 7 (midnight), 2024": [[28, null]], "Project 2 on 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"rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "RNNs": [[46, "rnns"]], "RNNs in more detail": [[46, "rnns-in-more-detail"]], "RNNs in more detail, part 2": [[46, "rnns-in-more-detail-part-2"]], "RNNs in more detail, part 3": [[46, "rnns-in-more-detail-part-3"]], "RNNs in more detail, part 4": [[46, "rnns-in-more-detail-part-4"]], "RNNs in more detail, part 5": [[46, "rnns-in-more-detail-part-5"]], "RNNs in more detail, part 6": [[46, "rnns-in-more-detail-part-6"]], "RNNs in more detail, part 7": [[46, "rnns-in-more-detail-part-7"]], "Random Forest Algorithm": [[47, "random-forest-algorithm"], [48, "random-forest-algorithm"]], "Random Forests Compared with other Methods on the Cancer Data": [[47, "random-forests-compared-with-other-methods-on-the-cancer-data"], [48, "random-forests-compared-with-other-methods-on-the-cancer-data"]], "Random Numbers": [[32, "random-numbers"]], "Random forests": [[11, "random-forests"], [47, "random-forests"], [48, "random-forests"]], "Randomized Grid Search": [[39, "randomized-grid-search"]], "Randomized PCA": [[12, "randomized-pca"]], "Reading material": [[35, "reading-material"]], "Reading recommendations": [[42, "reading-recommendations"], [43, "reading-recommendations"], [44, "reading-recommendations"]], "Reading recommendations:": [[36, "reading-recommendations"]], "Reading suggestions week 34": [[35, "reading-suggestions-week-34"]], "Recurrent neural networks": [[13, "recurrent-neural-networks"], [41, "recurrent-neural-networks"], [42, "recurrent-neural-networks"]], "Recurrent neural networks (RNNs): Overarching view": [[46, "recurrent-neural-networks-rnns-overarching-view"]], "Recurrent neural networks: Overarching view": [[5, null]], "Reducing the number of degrees of freedom, overarching view": [[1, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [36, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reducing the number of operations": [[42, "reducing-the-number-of-operations"]], "Reformulating the problem": [[3, "reformulating-the-problem"], [44, "reformulating-the-problem"]], "Regression Case": [[11, "regression-case"], [48, "regression-case"]], "Regression analysis and resampling methods": [[28, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[35, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[35, "regression-analysis-overarching-aims-ii"]], "Regression trees": [[47, "regression-trees"], [48, "regression-trees"]], "Regular NNs don\u2019t scale well to full images": [[45, "regular-nns-dont-scale-well-to-full-images"], [46, "regular-nns-dont-scale-well-to-full-images"]], "Regularization": [[2, "regularization"], [43, "regularization"], [44, "regularization"]], "Relevance": [[41, "relevance"], [42, "relevance"], [43, "relevance"]], "Remarks on the speed": [[45, "remarks-on-the-speed"], [46, "remarks-on-the-speed"]], "Reminder on Statistics": 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"resampling-methods-bootstrap-background"]], "Resampling methods: Bootstrap steps": [[38, "resampling-methods-bootstrap-steps"]], "Resampling methods: More Bootstrap background": [[38, "resampling-methods-more-bootstrap-background"]], "Residual Error": [[36, "residual-error"]], "Resources on differential equations and deep learning": [[3, "resources-on-differential-equations-and-deep-learning"], [44, "resources-on-differential-equations-and-deep-learning"]], "Revisiting our Linear Regression Solvers": [[14, "revisiting-our-linear-regression-solvers"]], "Revisiting our Logistic Regression case": [[39, "revisiting-our-logistic-regression-case"], [40, "revisiting-our-logistic-regression-case"]], "Revisiting our first homework": [[39, "revisiting-our-first-homework"], [40, "revisiting-our-first-homework"]], "Rewriting as dot products": [[45, "rewriting-as-dot-products"]], "Rewriting the Covariance and/or Correlation Matrix": [[36, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the \\delta-function": [[38, "rewriting-the-delta-function"]], "Rewriting the fitting procedure as a linear algebra problem": [[35, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[35, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[37, "ridge-regression"]], "Ridge and Bayes": [[37, "ridge-and-bayes"], [38, "ridge-and-bayes"]], "Ridge and LASSO Regression": [[36, "ridge-and-lasso-regression"], [37, "ridge-and-lasso-regression"]], "Ridge and Lasso Regression": [[6, null], [6, "id1"]], "Running with Keras": [[45, "running-with-keras"], [46, "running-with-keras"]], "SGD example": [[40, "sgd-example"], [41, "sgd-example"]], "SVD analysis": [[37, "svd-analysis"]], "Same code but now with momentum gradient descent": [[14, "same-code-but-now-with-momentum-gradient-descent"], [22, "same-code-but-now-with-momentum-gradient-descent"], [22, 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[44, "setting-up-a-multi-layer-perceptron-model-for-classification"]], "Setting up the Back propagation algorithm": [[13, "setting-up-the-back-propagation-algorithm"]], "Setting up the Back propagation algorithm, part 3": [[42, "setting-up-the-back-propagation-algorithm-part-3"], [43, "setting-up-the-back-propagation-algorithm-part-3"], [44, "setting-up-the-back-propagation-algorithm-part-3"]], "Setting up the Matrix to be inverted": [[36, "setting-up-the-matrix-to-be-inverted"]], "Setting up the back propagation algorithm": [[42, "setting-up-the-back-propagation-algorithm"]], "Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations": [[43, "setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations"], [44, "setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations"]], "Setting up the back propagation algorithm, part 1": [[43, "setting-up-the-back-propagation-algorithm-part-1"], 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"setup-of-network"]], "Similar (second order function now) problem but now with AdaGrad": [[14, "similar-second-order-function-now-problem-but-now-with-adagrad"], [22, "similar-second-order-function-now-problem-but-now-with-adagrad"], [40, "similar-second-order-function-now-problem-but-now-with-adagrad"], [41, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[10, "simple-python-code-to-read-in-data-and-perform-classification"], [47, "simple-python-code-to-read-in-data-and-perform-classification"], [48, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple Voting Example, head or tail": [[47, "simple-voting-example-head-or-tail"], [48, "simple-voting-example-head-or-tail"]], "Simple case": [[36, "simple-case"]], "Simple code for solving the above problem": [[37, "simple-code-for-solving-the-above-problem"]], "Simple example": [[39, "simple-example"], [42, "simple-example"]], "Simple 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"simpler-examples-first-and-automatic-differentiation"]], "Slightly different approach": [[40, "slightly-different-approach"], [41, "slightly-different-approach"]], "Smarter way of evaluating the above function": [[42, "smarter-way-of-evaluating-the-above-function"]], "Software and needed installations": [[28, "software-and-needed-installations"], [30, "software-and-needed-installations"], [35, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[3, null]], "Solving differential equations with Deep Learning": [[44, "solving-differential-equations-with-deep-learning"]], "Solving differential equations with RNNs": [[46, "solving-differential-equations-with-rnns"]], "Solving partial differential equations with neural networks": [[30, "solving-partial-differential-equations-with-neural-networks"]], "Solving the equation using Autograd": [[44, "solving-the-equation-using-autograd"]], "Solving the one dimensional Poisson equation": [[3, 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methods, limitations": [[41, "summary-from-last-week-using-gradient-descent-methods-limitations"]], "Summary of RNNs": [[46, "summary-of-rnns"]], "Summary of a typical RNN": [[46, "summary-of-a-typical-rnn"]], "Summing up": [[38, "summing-up"]], "Support Vector Machines, overarching aims": [[9, null]], "Systematic reduction": [[4, "systematic-reduction"], [45, "systematic-reduction"], [46, "systematic-reduction"]], "Teachers": [[35, "teachers"]], "Teachers and Grading": [[33, null]], "Teaching Assistants Fall semester 2023": [[33, "teaching-assistants-fall-semester-2023"]], "Teaching schedule with links to material": [[31, null]], "Technicalities": [[44, "technicalities"]], "Tensorflow": [[43, "tensorflow"], [44, "tensorflow"]], "Test Function for what happens with OLS, Ridge and Lasso": [[37, "test-function-for-what-happens-with-ols-ridge-and-lasso"]], "Testing the Means Squared Error as function of Complexity": [[1, "testing-the-means-squared-error-as-function-of-complexity"], [36, 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"the-cart-algorithm-for-regression"]], "The CIFAR01 data set": [[4, "the-cifar01-data-set"], [45, "the-cifar01-data-set"], [46, "the-cifar01-data-set"]], "The Central Limit Theorem": [[38, "the-central-limit-theorem"]], "The Convolutional Neural Network (CNN)": [[45, "the-convolutional-neural-network-cnn"], [46, "the-convolutional-neural-network-cnn"]], "The Hessian matrix": [[39, "the-hessian-matrix"], [40, "the-hessian-matrix"]], "The Hessian matrix for Ridge Regression": [[39, "the-hessian-matrix-for-ridge-regression"], [40, "the-hessian-matrix-for-ridge-regression"]], "The Jacobian": [[36, "the-jacobian"]], "The MNIST dataset again": [[4, "the-mnist-dataset-again"], [45, "the-mnist-dataset-again"], [46, "the-mnist-dataset-again"]], "The Neural Network": [[43, "the-neural-network"], [44, "the-neural-network"]], "The OLS case": [[37, "the-ols-case"]], "The RELU function family": [[2, "the-relu-function-family"], [42, "the-relu-function-family"], [43, "the-relu-function-family"], [44, "the-relu-function-family"]], "The Ridge case": [[37, "the-ridge-case"]], "The SVD example": [[45, "the-svd-example"]], "The SVD, a Fantastic Algorithm": [[36, "the-svd-a-fantastic-algorithm"]], "The Softmax function": [[2, "the-softmax-function"], [43, "the-softmax-function"], [44, "the-softmax-function"]], "The Squared-Error again! Steepest Descent": [[48, "the-squared-error-again-steepest-descent"]], "The Table": [[47, "the-table"], [48, "the-table"]], "The \\chi^2 function": [[1, "the-chi-2-function"], [35, "the-chi-2-function"], [35, "id4"], [35, "id5"], [35, "id6"], [35, "id7"], [35, "id8"]], "The analytical solution": [[44, "the-analytical-solution"]], "The approximation theorem in words": [[42, "the-approximation-theorem-in-words"]], "The backward pass is linear": [[46, "the-backward-pass-is-linear"]], "The basic structure of your project": [[30, "the-basic-structure-of-your-project"]], "The bias-variance tradeoff": [[7, "the-bias-variance-tradeoff"], [38, "the-bias-variance-tradeoff"]], "The code": [[35, "the-code"]], "The code for solving the ODE": [[3, "the-code-for-solving-the-ode"], [44, "the-code-for-solving-the-ode"]], "The complete code with a simple data set": [[36, "the-complete-code-with-a-simple-data-set"]], "The convolution stage": [[45, "the-convolution-stage"]], "The cost function": [[0, "the-cost-function"]], "The cost function rewritten": [[39, "the-cost-function-rewritten"]], "The cost/loss function": [[36, "the-cost-loss-function"]], "The course has two central parts": [[26, "the-course-has-two-central-parts"]], "The derivative of the Logistic funtion": [[42, "the-derivative-of-the-logistic-funtion"], [43, "the-derivative-of-the-logistic-funtion"]], "The derivative of the cost/loss function": [[39, "the-derivative-of-the-cost-loss-function"], [40, "the-derivative-of-the-cost-loss-function"]], "The derivatives": [[42, "the-derivatives"], [43, "the-derivatives"]], "The equations": [[39, "the-equations"], [40, "the-equations"]], "The equations for ordinary least squares": [[36, "the-equations-for-ordinary-least-squares"]], "The equations to solve": [[39, "the-equations-to-solve"], [40, "the-equations-to-solve"]], "The first Case": [[37, "the-first-case"]], "The forget gate": [[46, "the-forget-gate"]], "The function to solve for": [[44, "the-function-to-solve-for"]], "The gradient step": [[40, "the-gradient-step"], [41, "the-gradient-step"]], "The ideal": [[39, "the-ideal"], [40, "the-ideal"]], "The logistic function": [[0, "the-logistic-function"], [8, "the-logistic-function"], [39, "the-logistic-function"]], "The mean squared error and its derivative": [[36, "the-mean-squared-error-and-its-derivative"]], "The moons example": [[9, "the-moons-example"]], "The multilayer perceptron (MLP)": [[13, "the-multilayer-perceptron-mlp"]], "The network with one input layer, specified number of hidden layers, and one output layer": [[3, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"], [44, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"]], "The optimization problem": [[42, "the-optimization-problem"]], "The ouput layer": [[42, "the-ouput-layer"], [43, "the-ouput-layer"]], "The problem of exploding or vanishing gradients": [[46, "the-problem-of-exploding-or-vanishing-gradients"]], "The problem to solve for": [[44, "the-problem-to-solve-for"]], "The program using Autograd": [[44, "the-program-using-autograd"]], "The same example but now with cross-validation": [[38, "the-same-example-but-now-with-cross-validation"]], "The sensitiveness of the gradient descent": [[39, "the-sensitiveness-of-the-gradient-descent"], [40, "the-sensitiveness-of-the-gradient-descent"]], "The singular value decomposition": [[6, "the-singular-value-decomposition"], [36, "the-singular-value-decomposition"]], "The specific equation to solve for": [[44, "the-specific-equation-to-solve-for"]], "The syntax a.dot(b) when finding the dot product": [[40, "the-syntax-a-dot-b-when-finding-the-dot-product"]], "The training": [[42, "the-training"], [43, "the-training"]], "The trial solution": [[44, "the-trial-solution"], [44, "id4"], [44, "id5"], [44, "id7"]], "The two-dimensional case": [[9, "the-two-dimensional-case"]], "Time decay rate": [[40, "time-decay-rate"], [41, "time-decay-rate"]], "To our real data: nuclear binding energies. 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"zero": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 22, 23, 24, 27, 28, 32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], "zeros_lik": 5, "zeroth": [36, 45], "zfill": 5, "zip": [5, 7, 23, 24, 48], "zm_h": [1, 35], "zn": 1, "zone": 1, "zoom": 35, "zx": [27, 35], "zy": [27, 35], "zz": [27, 35], "\u00f8yvind": [7, 36, 37], "\u03b4": [43, 44]}, "titles": ["Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40", "3. Linear Regression", "14. Building a Feed Forward Neural Network", "15. Solving Differential Equations with Deep Learning", "16. Convolutional Neural Networks", "17. Recurrent neural networks: Overarching view", "4. Ridge and Lasso Regression", "5. Resampling Methods", "6. Logistic Regression", "8. Support Vector Machines, overarching aims", "9. Decision trees, overarching aims", "10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods", "11. Basic ideas of the Principal Component Analysis (PCA)", "13. Neural networks", "7. Optimization, the central part of any Machine Learning algortithm", "12. Clustering and Unsupervised Learning", "Exercises week 34", "Exercises week 35", "Exercises week 36", "Exercises week 37", "Exercises week 38", "Exercises week 39", "Exercises week 41", "Exercises week 42", "Exercises week 43", "Exercise week 47", "Applied Data Analysis and Machine Learning", "2. Linear Algebra, Handling of Arrays and more Python Features", "Project 1 on Machine Learning, deadline October 7 (midnight), 2024", "Project 2 on Machine Learning, deadline November 4 (Midnight)", "Project 3 on Machine Learning, deadline December 9 (midnight), 2024", "Teaching schedule with links to material", "1. Elements of Probability Theory and Statistical Data Analysis", "Teachers and Grading", "Textbooks", "Week 34: Introduction to the course, Logistics and Practicalities", "Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression", "Week 36: Linear Regression and Statistical interpretations", "Week 37: Statistical interpretations and Resampling Methods", "Week 38: Logistic Regression and Optimization", "Week 39: Optimization and Gradient Methods", "Week 40: Gradient descent methods (continued) and start Neural networks", "Week 41 Neural networks and constructing a neural network code", "Week 42 Constructing a Neural Network code with examples", "Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations", "Week 44, Convolutional Neural Networks (CNN)", "Week 45, Convolutional Neural Networks (CCNs) and Recurrent Neural Networks (RNNs)", "Week 46: Decision Trees, Ensemble methods and Random Forests", "Week 47: From Decision Trees to Ensemble 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"a-possible-code-using-scikit-learn"], [49, "a-possible-code-using-scikit-learn"]], "A possible implementation of a neural network": [[45, "a-possible-implementation-of-a-neural-network"]], "A quick Reminder on Lagrangian Multipliers": [[9, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[5, "a-simple-example"], [47, "a-simple-example"]], "A soft classifier": [[9, "a-soft-classifier"]], "A standard BM setup": [[50, "a-standard-bm-setup"]], "A top-down approach, recursive binary splitting": [[48, "a-top-down-approach-recursive-binary-splitting"], [49, "a-top-down-approach-recursive-binary-splitting"]], "A top-down perspective on Neural networks": [[2, "a-top-down-perspective-on-neural-networks"], [43, "a-top-down-perspective-on-neural-networks"], [44, "a-top-down-perspective-on-neural-networks"]], "A typical Decision Tree with its pertinent Jargon, Classification Problem": [[48, "a-typical-decision-tree-with-its-pertinent-jargon-classification-problem"]], "A warm-up example": [[22, "a-warm-up-example"], [42, "a-warm-up-example"]], "A way to Read the Bias-Variance Tradeoff": [[39, "a-way-to-read-the-bias-variance-tradeoff"]], "ADAM algorithm, taken from Goodfellow et al": [[22, "adam-algorithm-taken-from-goodfellow-et-al"], [42, "adam-algorithm-taken-from-goodfellow-et-al"]], "ADAM optimizer": [[14, "adam-optimizer"], [41, "adam-optimizer"], [42, "adam-optimizer"]], "Activation functions": [[13, "activation-functions"], [42, "activation-functions"], [43, "activation-functions"], [44, "activation-functions"], [44, "id3"], [45, "activation-functions"], [45, "id3"], [46, "activation-functions"], [47, "activation-functions"]], "Activation functions, Logistic and Hyperbolic ones": [[42, "activation-functions-logistic-and-hyperbolic-ones"], [43, "activation-functions-logistic-and-hyperbolic-ones"], [44, "activation-functions-logistic-and-hyperbolic-ones"]], "Activation functions, examples": [[45, "activation-functions-examples"]], "AdaBoost Examples": [[48, "adaboost-examples"], [49, "adaboost-examples"], [50, "adaboost-examples"]], "AdaGrad algorithm, taken from Goodfellow et al": [[22, "adagrad-algorithm-taken-from-goodfellow-et-al"], [42, "adagrad-algorithm-taken-from-goodfellow-et-al"]], "Adaptive Boosting, AdaBoost": [[48, "adaptive-boosting-adaboost"], [49, "adaptive-boosting-adaboost"], [50, "adaptive-boosting-adaboost"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[11, "adaptive-boosting-adaboost-basic-algorithm"], [48, "adaptive-boosting-adaboost-basic-algorithm"], [49, "adaptive-boosting-adaboost-basic-algorithm"], [50, "adaptive-boosting-adaboost-basic-algorithm"]], "Add Dense layers on top": [[46, "add-dense-layers-on-top"], [47, "add-dense-layers-on-top"]], "Adding Neural Networks": [[42, "adding-neural-networks"], [43, "adding-neural-networks"]], "Adding a hidden layer": [[43, "adding-a-hidden-layer"], [44, "adding-a-hidden-layer"]], "Adding error analysis and training set up": [[36, "adding-error-analysis-and-training-set-up"], [37, "adding-error-analysis-and-training-set-up"]], "Additional Remarks": [[46, "additional-remarks"], [47, "additional-remarks"]], "Additional courses of interest": [[50, "additional-courses-of-interest"]], "Adjust hyperparameters": [[2, "adjust-hyperparameters"], [44, "adjust-hyperparameters"], [45, "adjust-hyperparameters"]], "Adversarial learning": [[50, "adversarial-learning"]], "Algorithms and codes for Adagrad, RMSprop and Adam": [[22, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"], [41, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"], [42, "algorithms-and-codes-for-adagrad-rmsprop-and-adam"]], "Algorithms for Setting up Decision Trees": [[10, "algorithms-for-setting-up-decision-trees"], [48, "algorithms-for-setting-up-decision-trees"], [49, "algorithms-for-setting-up-decision-trees"]], "Alternative differential equations": [[31, "alternative-differential-equations"]], "An Overview of Ensemble Methods": [[11, "an-overview-of-ensemble-methods"], [48, "an-overview-of-ensemble-methods"], [49, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[5, "an-extrapolation-example"]], "An optimization/minimization problem": [[36, "an-optimization-minimization-problem"]], "Analyzing the last results": [[43, "analyzing-the-last-results"], [44, "analyzing-the-last-results"]], "And with loops": [[41, "and-with-loops"], [42, "and-with-loops"]], "And Logistic Regression": [[41, "and-logistic-regression"], [42, "and-logistic-regression"]], "And a corresponding example using scikit-learn": [[40, "and-a-corresponding-example-using-scikit-learn"], [41, "and-a-corresponding-example-using-scikit-learn"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[37, "and-finally-boldsymbol-x-boldsymbol-x-t"], [38, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And finally ADAM": [[22, "and-finally-adam"], [41, "and-finally-adam"], [42, "and-finally-adam"]], "And what about using neural networks?": [[36, "and-what-about-using-neural-networks"]], "Another Example from Scikit-Learn\u2019s Repository": [[39, "another-example-from-scikit-learn-s-repository"]], "Another Example, now with a polynomial fit": [[38, "another-example-now-with-a-polynomial-fit"]], "Another example, the moons again": [[10, "another-example-the-moons-again"]], "Any other topics, impressions, ideas etc you would like to share with us?": [[26, "any-other-topics-impressions-ideas-etc-you-would-like-to-share-with-us"]], "Applied Data Analysis and Machine Learning": [[27, null]], "Artificial neurons": [[42, "artificial-neurons"], [43, "artificial-neurons"]], "Assumptions made": [[38, "assumptions-made"], [39, "assumptions-made"]], "Autocorrelation function": [[33, "autocorrelation-function"]], "Autoencoders: Overarching view": [[50, "autoencoders-overarching-view"]], "Autograd with more complicated functions": [[41, "autograd-with-more-complicated-functions"], [42, "autograd-with-more-complicated-functions"]], "Automatic differentiation": [[14, "automatic-differentiation"], [41, "automatic-differentiation"], [42, "automatic-differentiation"], [43, "automatic-differentiation"]], "Automatic differentiation through examples": [[43, "automatic-differentiation-through-examples"]], "Back propagation": [[45, "back-propagation"]], "Back propagation and automatic differentiation": [[45, "back-propagation-and-automatic-differentiation"]], "Back propagation in time in equations": [[47, "back-propagation-in-time-in-equations"]], "Back propagation in time through figures, part 1": [[47, "back-propagation-in-time-through-figures-part-1"]], "Back propagation in time, part 2": [[47, "back-propagation-in-time-part-2"]], "Back propagation in time, part 3": [[47, "back-propagation-in-time-part-3"]], "Back propagation in time, part 4": [[47, "back-propagation-in-time-part-4"]], "Back to the Cancer Data": [[12, "back-to-the-cancer-data"]], "Background literature": [[29, "background-literature"], [30, "background-literature"]], "Backpropagation in the convolutional layer": [[46, "backpropagation-in-the-convolutional-layer"], [47, "backpropagation-in-the-convolutional-layer"]], "Backpropagation through time": [[47, "backpropagation-through-time"]], "Bagging": [[11, "bagging"], [48, "bagging"], [49, "bagging"]], "Bagging Examples": [[11, "bagging-examples"]], "Basic Matrix Features": [[28, "basic-matrix-features"]], "Basic Steps of AdaBoost": [[48, "basic-steps-of-adaboost"], [49, "basic-steps-of-adaboost"], [50, "basic-steps-of-adaboost"]], "Basic ideas of the Principal Component Analysis (PCA)": [[12, null]], "Basic layout": [[47, "basic-layout"]], "Basic layout, Figures from Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch": [[47, "basic-layout-figures-from-sebastian-rashcka-et-al-machine-learning-with-sickit-learn-and-pytorch"]], "Basic math of the SVD": [[6, "basic-math-of-the-svd"], [37, "basic-math-of-the-svd"]], "Basics": [[8, "basics"], [40, "basics"]], "Basics of a tree": [[10, "basics-of-a-tree"], [48, "basics-of-a-tree"]], "Basics of an NN": [[43, "basics-of-an-nn"]], "Batch Normalization": [[2, "batch-normalization"], [43, "batch-normalization"], [44, "batch-normalization"]], "Batches and mini-batches": [[41, "batches-and-mini-batches"], [42, "batches-and-mini-batches"]], "Bayesian Machine Learning": [[50, "bayesian-machine-learning"]], "Bayes\u2019 Theorem": [[38, "bayes-theorem"], [39, "bayes-theorem"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[6, "bayes-theorem-and-ridge-and-lasso-regression"], [38, "bayes-theorem-and-ridge-and-lasso-regression"], [39, "bayes-theorem-and-ridge-and-lasso-regression"]], "Best wishes to you all and thanks so much for your heroic efforts this semester": [[50, "best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester"]], "Boltzmann Machines": [[50, "boltzmann-machines"]], "Boltzmann machines (BM)": [[50, "boltzmann-machines-bm"]], "Boosting, a Bird\u2019s Eye View": [[11, "boosting-a-bird-s-eye-view"], [48, "boosting-a-bird-s-eye-view"], [49, "boosting-a-bird-s-eye-view"], [50, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[7, "bootstrap"]], "Brief reminder on Newton-Raphson\u2019s method": [[40, "brief-reminder-on-newton-raphson-s-method"], [41, "brief-reminder-on-newton-raphson-s-method"]], "Bringing it together": [[43, "bringing-it-together"], [44, "bringing-it-together"]], "Bringing it together, first back propagation equation": [[13, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[2, null]], "Building a neural network code": [[44, "building-a-neural-network-code"], [45, "building-a-neural-network-code"]], "Building a tree, regression": [[10, "building-a-tree-regression"], [48, "building-a-tree-regression"], [49, "building-a-tree-regression"]], "Building convolutional neural networks in Tensorflow and Keras": [[46, "building-convolutional-neural-networks-in-tensorflow-and-keras"], [47, "building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Building neural networks in Tensorflow and Keras": [[2, "building-neural-networks-in-tensorflow-and-keras"], [44, "building-neural-networks-in-tensorflow-and-keras"], [45, "building-neural-networks-in-tensorflow-and-keras"]], "Building our own CNN code": [[46, "building-our-own-cnn-code"], [47, "building-our-own-cnn-code"]], "Building up AdaBoost": [[48, "building-up-adaboost"], [49, "building-up-adaboost"], [50, "building-up-adaboost"]], "But noen of these can compete with Newton\u2019s method": [[22, "but-noen-of-these-can-compete-with-newton-s-method"]], "But none of these can compete with Newton\u2019s method": [[41, "but-none-of-these-can-compete-with-newton-s-method"], [42, "but-none-of-these-can-compete-with-newton-s-method"]], "CNNs in brief": [[46, "cnns-in-brief"], [47, "cnns-in-brief"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[4, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, simple example": [[46, "cnns-in-more-detail-simple-example"]], "Cancer Data again now with Decision Trees and other Methods": [[10, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Chain rule": [[43, "chain-rule"]], "Chain rule again": [[47, "chain-rule-again"]], "Chain rule, forward and reverse modes": [[43, "chain-rule-forward-and-reverse-modes"]], "Challenge yourself the coming weekend": [[40, "challenge-yourself-the-coming-weekend"]], "Choose a Model and Algorithm": [[50, "choose-a-model-and-algorithm"]], "Choose cost function and optimizer": [[2, "choose-cost-function-and-optimizer"], [44, "choose-cost-function-and-optimizer"], [45, "choose-cost-function-and-optimizer"]], "Class of functions we can approximate": [[43, "class-of-functions-we-can-approximate"]], "Classical PCA Theorem": [[12, "classical-pca-theorem"]], "Classification and Regression, from linear and logistic regression to neural networks": [[30, "classification-and-regression-from-linear-and-logistic-regression-to-neural-networks"]], "Classification problems": [[40, "classification-problems"]], "Classification tree, how to split nodes": [[48, "classification-tree-how-to-split-nodes"], [49, "classification-tree-how-to-split-nodes"]], "Clustering and Unsupervised Learning": [[15, null]], "Code Example for Cross-validation and k-fold Cross-validation": [[39, "code-example-for-cross-validation-and-k-fold-cross-validation"], [40, "code-example-for-cross-validation-and-k-fold-cross-validation"]], "Code example": [[43, "code-example"], [44, "code-example"]], "Code example for the Bootstrap method": [[39, "code-example-for-the-bootstrap-method"]], "Code examples from week 39 and 40": [[22, "code-examples-from-week-39-and-40"]], "Code for SVD and Inversion of Matrices": [[6, "code-for-svd-and-inversion-of-matrices"], [38, "code-for-svd-and-inversion-of-matrices"]], "Code with a Number of Minibatches which varies": [[41, "code-with-a-number-of-minibatches-which-varies"], [42, "code-with-a-number-of-minibatches-which-varies"]], "Code with a Number of Minibatches which varies, analytical gradient": [[22, "code-with-a-number-of-minibatches-which-varies-analytical-gradient"]], "Codes and Approaches": [[15, "codes-and-approaches"]], "Codes for the SVD": [[6, "codes-for-the-svd"], [37, "codes-for-the-svd"]], "Collect and pre-process data": [[2, "collect-and-pre-process-data"], [44, "collect-and-pre-process-data"], [44, "id2"], [45, "collect-and-pre-process-data"], [45, "id2"]], "Communication channels": [[36, "communication-channels"]], "Commutative process": [[46, "commutative-process"]], "Compact expressions": [[43, "compact-expressions"], [44, "compact-expressions"]], "Compare Bagging on Trees with Random Forests": [[11, "compare-bagging-on-trees-with-random-forests"], [48, "compare-bagging-on-trees-with-random-forests"], [49, "compare-bagging-on-trees-with-random-forests"], [50, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[3, "comparing-with-a-numerical-scheme"], [45, "comparing-with-a-numerical-scheme"]], "Comparison with OLS": [[38, "comparison-with-ols"]], "Compile and train the model": [[46, "compile-and-train-the-model"], [47, "compile-and-train-the-model"]], "Completing the list": [[43, "completing-the-list"], [44, "completing-the-list"]], "Computation of gradients": [[41, "computation-of-gradients"], [42, "computation-of-gradients"]], "Computing a Tree using the Gini Index": [[48, "computing-a-tree-using-the-gini-index"], [49, "computing-a-tree-using-the-gini-index"]], "Computing the Gini Factor": [[48, "computing-the-gini-factor"], [49, "computing-the-gini-factor"]], "Computing the Gini index": [[10, "computing-the-gini-index"]], "Computing the various Gini Indices": [[48, "computing-the-various-gini-indices"], [49, "computing-the-various-gini-indices"]], "Computing the various Gini Indices, Hours slept": [[48, "computing-the-various-gini-indices-hours-slept"]], "Computing the various Gini Indices, Hours studied": [[48, "computing-the-various-gini-indices-hours-studied"]], "Conditional Probability": [[38, "conditional-probability"], [39, "conditional-probability"]], "Conditions on convex functions": [[40, "conditions-on-convex-functions"], [41, "conditions-on-convex-functions"]], "Confidence Intervals": [[39, "confidence-intervals"]], "Conjugate gradient method": [[14, "conjugate-gradient-method"], [41, "conjugate-gradient-method"], [41, "id2"], [41, "id3"], [41, "id4"], [41, "id5"], [41, "id6"], [41, "id7"]], "Conjugate gradient method and iterations": [[41, "conjugate-gradient-method-and-iterations"]], "Convex function": [[40, "convex-function"], [41, "convex-function"]], "Convex functions": [[14, "convex-functions"], [40, "convex-functions"], [41, "convex-functions"]], "Convolution": [[46, "convolution"], [47, "convolution"]], "Convolution Examples: Polynomial multiplication": [[4, "convolution-examples-polynomial-multiplication"], [46, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[4, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolution in the Fourier domain": [[46, "convolution-in-the-fourier-domain"], [47, "convolution-in-the-fourier-domain"]], "Convolution using separable kernels": [[46, "convolution-using-separable-kernels"], [47, "convolution-using-separable-kernels"]], "Convolution2DLayer: convolution in a hidden layer": [[46, "convolution2dlayer-convolution-in-a-hidden-layer"], [47, "convolution2dlayer-convolution-in-a-hidden-layer"]], "Convolutional Neural Network": [[13, "convolutional-neural-network"], [42, "convolutional-neural-network"], [43, "convolutional-neural-network"]], "Convolutional Neural Networks": [[4, null]], "Convolutional Neural Networks (recognizing images)": [[46, "convolutional-neural-networks-recognizing-images"], [47, "convolutional-neural-networks-recognizing-images"]], "Correlation Function and Design/Feature Matrix": [[37, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[12, "correlation-matrix"], [37, "correlation-matrix"]], "Correlation Matrix with Pandas": [[37, "correlation-matrix-with-pandas"]], "Correlation Matrix with Pandas and the Franke function": [[37, "correlation-matrix-with-pandas-and-the-franke-function"]], "Cost complexity pruning": [[48, "cost-complexity-pruning"], [49, "cost-complexity-pruning"]], "Cost functions": [[44, "cost-functions"], [45, "cost-functions"], [46, "cost-functions"], [47, "cost-functions"]], "Counting the number of floating point operations": [[43, "counting-the-number-of-floating-point-operations"]], "Course Format": [[36, "course-format"]], "Covariance Matrix Examples": [[37, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[37, "covariance-and-correlation-matrix"]], "Cross correlation": [[46, "cross-correlation"]], "Cross-validation": [[7, "cross-validation"]], "Cross-validation in brief": [[39, "cross-validation-in-brief"], [40, "cross-validation-in-brief"]], "Deadlines for projects (tentative)": [[36, "deadlines-for-projects-tentative"]], "Decision trees and Regression": [[48, "decision-trees-and-regression"]], "Decision trees, overarching aims": [[10, null], [48, "decision-trees-overarching-aims"]], "Deep learning methods": [[36, "deep-learning-methods"]], "Define model and architecture": [[2, "define-model-and-architecture"], [44, "define-model-and-architecture"], [45, "define-model-and-architecture"]], "Defining different types of RBMs": [[50, "defining-different-types-of-rbms"]], "Defining intermediate operations": [[43, "defining-intermediate-operations"]], "Defining the cost function": [[2, "defining-the-cost-function"], [44, "defining-the-cost-function"], [45, "defining-the-cost-function"]], "Defining the data sets to analyze yourself": [[31, "defining-the-data-sets-to-analyze-yourself"]], "Defining the problem": [[45, "defining-the-problem"]], "Definitions": [[43, "definitions"], [44, "definitions"]], "Demonstration": [[46, "demonstration"], [47, "demonstration"]], "Derivative of the cost function": [[43, "derivative-of-the-cost-function"], [44, "derivative-of-the-cost-function"]], "Derivatives and the chain rule": [[13, "derivatives-and-the-chain-rule"], [43, "derivatives-and-the-chain-rule"], [44, "derivatives-and-the-chain-rule"]], "Derivatives in terms of z_j^L": [[43, "derivatives-in-terms-of-z-j-l"], [44, "derivatives-in-terms-of-z-j-l"]], "Derivatives of the hidden layer": [[43, "derivatives-of-the-hidden-layer"], [44, "derivatives-of-the-hidden-layer"]], "Derivatives, example 1": [[37, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[6, "deriving-ols-from-a-probability-distribution"], [38, "deriving-ols-from-a-probability-distribution"], [39, "deriving-ols-from-a-probability-distribution"]], "Deriving the Lasso Regression Equations": [[37, "deriving-the-lasso-regression-equations"], [38, "deriving-the-lasso-regression-equations"]], "Deriving the Ridge Regression Equations": [[37, "deriving-the-ridge-regression-equations"], [38, "deriving-the-ridge-regression-equations"]], "Deriving the back propagation code for a multilayer perceptron model": [[13, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Description of two-dimensional function": [[29, "description-of-two-dimensional-function"]], "Developing a code for doing neural networks with back propagation": [[2, "developing-a-code-for-doing-neural-networks-with-back-propagation"], [44, "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": [[12, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Did the projects and the teaching material allow you to deepen your insights about Machine Learning methods?": [[26, "did-the-projects-and-the-teaching-material-allow-you-to-deepen-your-insights-about-machine-learning-methods"]], "Different kernels and Mercer\u2019s theorem": [[9, "different-kernels-and-mercer-s-theorem"]], "Differential equations": [[47, "differential-equations"]], "Disadvantages": [[10, "disadvantages"], [48, "disadvantages"], [49, "disadvantages"]], "Discussing the correlation data": [[0, "discussing-the-correlation-data"], [40, "discussing-the-correlation-data"]], "Distributed machine learning": [[50, "distributed-machine-learning"]], "Does Logistic Regression do a better Job?": [[42, "does-logistic-regression-do-a-better-job"], [43, "does-logistic-regression-do-a-better-job"]], "Doing it correctly": [[38, "doing-it-correctly"], [39, "doing-it-correctly"]], "Domains and probabilities": [[33, "domains-and-probabilities"]], "Dropout": [[2, "dropout"], [43, "dropout"], [44, "dropout"]], "Dual learning": [[50, "dual-learning"]], "ELU function": [[43, "elu-function"], [44, "elu-function"], [45, "elu-function"]], "Economy-size SVD": [[37, "economy-size-svd"]], "Efficient Polynomial Multiplication": [[46, "efficient-polynomial-multiplication"]], "Elements of Probability Theory and Statistical Data Analysis": [[33, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[11, null], [48, "ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods"], [49, "ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods"]], "Entropy and the ID3 algorithm": [[10, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[36, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[2, "evaluate-model-performance-on-test-data"], [44, "evaluate-model-performance-on-test-data"], [45, "evaluate-model-performance-on-test-data"]], "Example 2": [[37, "example-2"]], "Example 3": [[37, "example-3"]], "Example 4": [[37, "example-4"]], "Example Matrix": [[37, "example-matrix"]], "Example code for Bias-Variance tradeoff": [[39, "example-code-for-bias-variance-tradeoff"]], "Example of Usage of Bayes\u2019 theorem": [[38, "example-of-usage-of-bayes-theorem"], [39, "example-of-usage-of-bayes-theorem"]], "Example of own Standard scaling": [[37, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[37, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[3, "example-exponential-decay"], [45, "example-exponential-decay"]], "Example: Population growth": [[3, "example-population-growth"], [45, "example-population-growth"]], "Example: Solving the one dimensional Poisson equation": [[45, "example-solving-the-one-dimensional-poisson-equation"]], "Example: Solving the wave equation with Neural Networks": [[45, "example-solving-the-wave-equation-with-neural-networks"]], "Example: The diffusion equation": [[3, "example-the-diffusion-equation"], [45, "example-the-diffusion-equation"]], "Example: binary classification problem": [[2, "example-binary-classification-problem"], [44, "example-binary-classification-problem"], [45, "example-binary-classification-problem"]], "Examples": [[36, "examples"]], "Examples of CNN setups": [[46, "examples-of-cnn-setups"]], "Examples of XOR, OR and AND gates": [[42, "examples-of-xor-or-and-and-gates"], [43, "examples-of-xor-or-and-and-gates"]], "Examples of likelihood functions used in logistic regression and neural networks": [[8, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Examples of likelihood functions used in logistic regression and nueral networks": [[0, "examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks"], [40, "examples-of-likelihood-functions-used-in-logistic-regression-and-nueral-networks"]], "Exercise 1": [[23, "exercise-1"]], "Exercise 1 - Understand the feed forward pass": [[24, "exercise-1-understand-the-feed-forward-pass"]], "Exercise 1: Analytical exercises": [[17, "exercise-1-analytical-exercises"], [18, "exercise-1-analytical-exercises"]], "Exercise 1: Expectation values for ordinary least squares expressions": [[19, "exercise-1-expectation-values-for-ordinary-least-squares-expressions"]], "Exercise 1: Including more data": [[43, "exercise-1-including-more-data"]], "Exercise 1: Linear and logistic regression methods": [[25, "exercise-1-linear-and-logistic-regression-methods"]], "Exercise 1: Setting up various Python environments": [[1, "exercise-1-setting-up-various-python-environments"], [16, "exercise-1-setting-up-various-python-environments"], [36, "exercise-1-setting-up-various-python-environments"]], "Exercise 2": [[23, "exercise-2"]], "Exercise 2 - Gradient with one layer using autograd": [[24, "exercise-2-gradient-with-one-layer-using-autograd"]], "Exercise 2: Adding Ridge Regression": [[18, "exercise-2-adding-ridge-regression"]], "Exercise 2: Deep learning": [[25, "exercise-2-deep-learning"]], "Exercise 2: Expectation values for Ridge regression": [[19, "exercise-2-expectation-values-for-ridge-regression"]], "Exercise 2: Extended program": [[43, "exercise-2-extended-program"]], "Exercise 2: making your own data and exploring scikit-learn": [[1, "exercise-2-making-your-own-data-and-exploring-scikit-learn"], [16, "exercise-2-making-your-own-data-and-exploring-scikit-learn"], [17, "exercise-2-making-your-own-data-and-exploring-scikit-learn"], [36, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3": [[23, "exercise-3"]], "Exercise 3 - Gradient with one layer writing backpropagation by hand": [[24, "exercise-3-gradient-with-one-layer-writing-backpropagation-by-hand"]], "Exercise 3: Decision trees and ensemble methods": [[25, "exercise-3-decision-trees-and-ensemble-methods"]], "Exercise 3: Normalizing our data": [[1, "exercise-3-normalizing-our-data"]], "Exercise 3: Split data in test and training data": [[16, "exercise-3-split-data-in-test-and-training-data"], [17, "exercise-3-split-data-in-test-and-training-data"], [36, "exercise-3-split-data-in-test-and-training-data"]], "Exercise 4 - Custom activation for each layer": [[23, "exercise-4-custom-activation-for-each-layer"]], "Exercise 4 - Gradient with two layers writing backpropagation by hand": [[24, "exercise-4-gradient-with-two-layers-writing-backpropagation-by-hand"]], "Exercise 4: Adding Ridge Regression": [[1, "exercise-4-adding-ridge-regression"]], "Exercise 4: Optimization part": [[25, "exercise-4-optimization-part"]], "Exercise 5 - Gradient with any number of layers writing backpropagation by hand": [[24, "exercise-5-gradient-with-any-number-of-layers-writing-backpropagation-by-hand"]], "Exercise 5 - Processing multiple inputs at once": [[23, "exercise-5-processing-multiple-inputs-at-once"]], "Exercise 5: Analysis of results": [[25, "exercise-5-analysis-of-results"]], "Exercise 5: Analytical exercises": [[1, "exercise-5-analytical-exercises"]], "Exercise 6 - Batched inputs": [[24, "exercise-6-batched-inputs"]], "Exercise 6 - Predicting on real data": [[23, "exercise-6-predicting-on-real-data"]], "Exercise 7 - Training": [[24, "exercise-7-training"]], "Exercise 7 - Training on real data (Optional)": [[23, "exercise-7-training-on-real-data-optional"]], "Exercise 8 (Optional) - Object orientation": [[24, "exercise-8-optional-object-orientation"]], "Exercise week 47": [[25, null]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[7, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[7, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[7, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[7, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[7, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[7, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[1, "exercises"], [16, "exercises"], [36, "exercises"]], "Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40": [[0, null]], "Exercises and Projects": [[7, "exercises-and-projects"]], "Exercises and lab session week 43": [[45, "exercises-and-lab-session-week-43"]], "Exercises week 34": [[16, null]], "Exercises week 35": [[17, null]], "Exercises week 36": [[18, null]], "Exercises week 37": [[19, null]], "Exercises week 38": [[20, null]], "Exercises week 39": [[21, null]], "Exercises week 41": [[22, null]], "Exercises week 42": [[23, null]], "Exercises week 43": [[24, null]], "Exercises week 48": [[26, null]], "Expectation value and variance": [[38, "expectation-value-and-variance"], [39, "expectation-value-and-variance"]], "Expectation value and variance for \\boldsymbol{\\beta}": [[38, "expectation-value-and-variance-for-boldsymbol-beta"], [39, "expectation-value-and-variance-for-boldsymbol-beta"]], "Expectation values": [[33, "expectation-values"]], "Explainable machine learning": [[50, "explainable-machine-learning"]], "Explicit derivatives": [[43, "explicit-derivatives"], [44, "explicit-derivatives"]], "Exploding gradients": [[43, "exploding-gradients"], [44, "exploding-gradients"]], "Extending to more predictors": [[0, "extending-to-more-predictors"], [40, "extending-to-more-predictors"]], "Extending to more than one variable": [[40, "extending-to-more-than-one-variable"], [41, "extending-to-more-than-one-variable"]], "Extremely useful tools, strongly recommended": [[36, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[13, "feed-forward-neural-networks"], [42, "feed-forward-neural-networks"], [43, "feed-forward-neural-networks"]], "Feed-forward pass": [[2, "feed-forward-pass"], [44, "feed-forward-pass"], [45, "feed-forward-pass"]], "Feedback connections": [[47, "feedback-connections"]], "Final back propagating equation": [[13, "final-back-propagating-equation"], [43, "final-back-propagating-equation"], [44, "final-back-propagating-equation"]], "Final derivatives": [[43, "final-derivatives"]], "Final expression": [[43, "final-expression"], [44, "final-expression"]], "Final expressions": [[41, "final-expressions"]], "Final expressions for the biases of the hidden layer": [[43, "final-expressions-for-the-biases-of-the-hidden-layer"], [44, "final-expressions-for-the-biases-of-the-hidden-layer"]], "Final part": [[46, "final-part"], [47, "final-part"]], "Final regressor code": [[48, "final-regressor-code"], [49, "final-regressor-code"]], "Final technicalities I": [[45, "final-technicalities-i"]], "Final technicalities II": [[45, "final-technicalities-ii"]], "Final technicalities III": [[45, "final-technicalities-iii"]], "Final technicalities IV": [[45, "final-technicalities-iv"]], "Final visualization": [[46, "final-visualization"], [47, "final-visualization"]], "Finally, evaluate the model": [[46, "finally-evaluate-the-model"], [47, "finally-evaluate-the-model"]], "Finding the Limit": [[39, "finding-the-limit"]], "Finding the number of parameters": [[46, "finding-the-number-of-parameters"]], "Fine-tuning neural network hyperparameters": [[2, "fine-tuning-neural-network-hyperparameters"], [43, "fine-tuning-neural-network-hyperparameters"], [44, "fine-tuning-neural-network-hyperparameters"]], "First network example, simple percepetron with one input": [[43, "first-network-example-simple-percepetron-with-one-input"], [44, "first-network-example-simple-percepetron-with-one-input"]], "Fitting an Equation of State for Dense Nuclear Matter": [[1, "fitting-an-equation-of-state-for-dense-nuclear-matter"], [36, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[37, "fixing-the-singularity"]], "Flattening Layer": [[46, "flattening-layer"], [47, "flattening-layer"]], "For exercise sessions: Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week": [[37, "for-exercise-sessions-why-linear-regression-aka-ordinary-least-squares-and-family-repeat-from-last-week"]], "Forget and input": [[47, "forget-and-input"]], "Format for electronic delivery of report and programs": [[29, "format-for-electronic-delivery-of-report-and-programs"], [30, "format-for-electronic-delivery-of-report-and-programs"], [31, "format-for-electronic-delivery-of-report-and-programs"]], "Forward and reverse modes": [[43, "forward-and-reverse-modes"]], "Four effective ways to learn an RNN and preparing for next week": [[47, "four-effective-ways-to-learn-an-rnn-and-preparing-for-next-week"]], "Fourier series and Toeplitz matrices": [[46, "fourier-series-and-toeplitz-matrices"]], "Frequently used scaling functions": [[37, "frequently-used-scaling-functions"]], "From FFNNs and CNNs to recurrent neural networks (RNNs)": [[47, "from-ffnns-and-cnns-to-recurrent-neural-networks-rnns"]], "From OLS to Ridge and Lasso": [[38, "from-ols-to-ridge-and-lasso"]], "From one to many layers, the universal approximation theorem": [[13, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Full object-oriented implementation": [[44, "full-object-oriented-implementation"], [45, "full-object-oriented-implementation"]], "Fully Connected Layers": [[46, "fully-connected-layers"], [47, "fully-connected-layers"]], "Functionality in Scikit-Learn": [[37, "functionality-in-scikit-learn"]], "Functions using mathematical functions from Numpy": [[41, "functions-using-mathematical-functions-from-numpy"], [42, "functions-using-mathematical-functions-from-numpy"]], "Further Dimensionality Remarks": [[4, "further-dimensionality-remarks"]], "Further Manipulations": [[37, "further-manipulations"]], "Further example: Computing the Gini index": [[48, "further-example-computing-the-gini-index"], [49, "further-example-computing-the-gini-index"]], "Further properties (important for our analyses later)": [[6, "further-properties-important-for-our-analyses-later"], [37, "further-properties-important-for-our-analyses-later"]], "Further remarks": [[46, "further-remarks"]], "Further simplification": [[46, "further-simplification"]], "Gating mechanism: Long Short Term Memory (LSTM)": [[47, "gating-mechanism-long-short-term-memory-lstm"]], "Gaussian Elimination": [[28, "gaussian-elimination"]], "General Features": [[10, "general-features"], [48, "general-features"]], "General linear models and linear algebra": [[36, "general-linear-models-and-linear-algebra"]], "Generalizing the above one-dimensional case": [[46, "generalizing-the-above-one-dimensional-case"]], "Generalizing the fitting procedure as a linear algebra problem": [[36, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [36, "id1"]], "Generative Adversarial Networks": [[5, "generative-adversarial-networks"]], "Generative Models": [[5, "generative-models"]], "Geometric Interpretation and link with Singular Value Decomposition": [[12, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Getting serious, the back propagation equations for a neural network": [[43, "getting-serious-the-back-propagation-equations-for-a-neural-network"]], "Getting started with Jax, note the way we import numpy": [[22, "getting-started-with-jax-note-the-way-we-import-numpy"], [42, "getting-started-with-jax-note-the-way-we-import-numpy"]], "Goals": [[50, "goals"]], "Gradient Boosting, Classification Example": [[11, "gradient-boosting-classification-example"], [49, "gradient-boosting-classification-example"], [50, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[11, "gradient-boosting-examples-of-regression"], [49, "gradient-boosting-examples-of-regression"], [50, "gradient-boosting-examples-of-regression"]], "Gradient Boosting, algorithm": [[49, "gradient-boosting-algorithm"], [50, "gradient-boosting-algorithm"]], "Gradient Clipping": [[2, "gradient-clipping"], [43, "gradient-clipping"], [44, "gradient-clipping"]], "Gradient Descent Example": [[40, "id1"], [41, "id8"]], "Gradient boosting, making our own code for a regression case": [[49, "gradient-boosting-making-our-own-code-for-a-regression-case"], [50, "gradient-boosting-making-our-own-code-for-a-regression-case"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[11, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"], [49, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"], [50, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[3, "gradient-descent"], [45, "gradient-descent"]], "Gradient descent and Logistic regression": [[0, "gradient-descent-and-logistic-regression"], [42, "gradient-descent-and-logistic-regression"]], "Gradient descent and Ridge": [[40, "gradient-descent-and-ridge"], [41, "gradient-descent-and-ridge"]], "Gradient descent example": [[40, "gradient-descent-example"], [41, "gradient-descent-example"]], "Gradient expressions": [[43, "gradient-expressions"], [44, "gradient-expressions"]], "Gradient method": [[41, "gradient-method"]], "Gradients of loss functions": [[47, "gradients-of-loss-functions"]], "Grading": [[34, "grading"], [36, "grading"]], "Grid Search": [[40, "grid-search"]], "Growing a classification tree": [[48, "growing-a-classification-tree"], [49, "growing-a-classification-tree"]], "Hidden layers": [[43, "hidden-layers"], [44, "hidden-layers"]], "Homogeneous data": [[43, "homogeneous-data"], [44, "homogeneous-data"]], "Housing data, the code": [[1, "housing-data-the-code"]], "How do we set it up?": [[48, "how-do-we-set-it-up"]], "How do you judge your own level of knowledge on machine learning before and after this course?": [[26, "how-do-you-judge-your-own-level-of-knowledge-on-machine-learning-before-and-after-this-course"]], "How to do image compression before the era of deep learning": [[46, "how-to-do-image-compression-before-the-era-of-deep-learning"]], "How to set up the cross-validation for Ridge and/or Lasso": [[40, "how-to-set-up-the-cross-validation-for-ridge-and-or-lasso"]], "How would you improve this course?": [[26, "how-would-you-improve-this-course"]], "Hyperplanes and all that": [[9, "hyperplanes-and-all-that"]], "Identifying Terms": [[39, "identifying-terms"]], "If you did not attend the lectures or the lab sessions, which resources did you use?": [[26, "if-you-did-not-attend-the-lectures-or-the-lab-sessions-which-resources-did-you-use"]], "Illustration of a single perceptron model and a multi-perceptron model": [[42, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"], [43, "illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model"]], "Implementing a memory cell in a neural network": [[47, "implementing-a-memory-cell-in-a-neural-network"]], "Important Matrix and vector handling packages": [[28, "important-matrix-and-vector-handling-packages"]], "Important observations": [[43, "important-observations"], [44, "important-observations"]], "Important technicalities: More on Rescaling data": [[38, "important-technicalities-more-on-rescaling-data"]], "Importing Keras and Tensorflow": [[46, "importing-keras-and-tensorflow"], [47, "importing-keras-and-tensorflow"]], "Improving gradient descent with momentum": [[41, "improving-gradient-descent-with-momentum"]], "Improving performance": [[2, "improving-performance"], [44, "improving-performance"], [45, "improving-performance"]], "In general not this simple": [[43, "in-general-not-this-simple"]], "In summary": [[34, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[14, "including-stochastic-gradient-descent-with-autograd"], [22, "including-stochastic-gradient-descent-with-autograd"], [41, "including-stochastic-gradient-descent-with-autograd"], [42, "including-stochastic-gradient-descent-with-autograd"]], "Including more classes": [[0, "including-more-classes"], [40, "including-more-classes"]], "Incremental PCA": [[12, "incremental-pca"]], "Independent and Identically Distributed (iid)": [[38, "independent-and-identically-distributed-iid"]], "Independent and Identically Distrubuted (iid)": [[39, "independent-and-identically-distrubuted-iid"]], "Input gate": [[47, "input-gate"]], "Inputs to the activation function": [[43, "inputs-to-the-activation-function"], [44, "inputs-to-the-activation-function"]], "Insights from the paper by Glorot and Bengio": [[43, "insights-from-the-paper-by-glorot-and-bengio"], [44, "insights-from-the-paper-by-glorot-and-bengio"]], "Installing R, C++, cython or Julia": [[36, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[36, "installing-r-c-cython-numba-etc"]], "Instructor information": [[34, "instructor-information"]], "Interpretations and optimizing our parameters": [[36, "interpretations-and-optimizing-our-parameters"], [36, "id2"], [36, "id3"], [37, "interpretations-and-optimizing-our-parameters"], [37, "id1"], [37, "id2"]], "Interpretations of Bayes\u2019 Theorem": [[38, "interpretations-of-bayes-theorem"], [39, "interpretations-of-bayes-theorem"]], "Interpreting the Ridge results": [[37, "interpreting-the-ridge-results"], [38, "interpreting-the-ridge-results"]], "Introducing JAX": [[14, "introducing-jax"], [22, "introducing-jax"], [41, "introducing-jax"], [42, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[12, "introducing-the-covariance-and-correlation-functions"], [37, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[1, "introduction"], [7, "introduction"], [27, "introduction"], [28, "introduction"]], "Introduction to Neural networks": [[42, "introduction-to-neural-networks"], [43, "introduction-to-neural-networks"]], "Introduction to numerical projects": [[29, "introduction-to-numerical-projects"], [30, "introduction-to-numerical-projects"], [31, "introduction-to-numerical-projects"]], "Inverse of Rectangular Matrix": [[38, "inverse-of-rectangular-matrix"]], "Is the Logistic activation function (Sigmoid) our choice?": [[43, "is-the-logistic-activation-function-sigmoid-our-choice"], [44, "is-the-logistic-activation-function-sigmoid-our-choice"]], "Iterative Fitting, Classification and AdaBoost": [[11, "iterative-fitting-classification-and-adaboost"], [48, "iterative-fitting-classification-and-adaboost"], [49, "iterative-fitting-classification-and-adaboost"], [50, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[11, "iterative-fitting-regression-and-squared-error-cost-function"], [48, "iterative-fitting-regression-and-squared-error-cost-function"], [49, "iterative-fitting-regression-and-squared-error-cost-function"], [50, "iterative-fitting-regression-and-squared-error-cost-function"]], "Joint distribution": [[50, "joint-distribution"]], "Kernel PCA": [[12, "kernel-pca"]], "Kernels and non-linearity": [[9, "kernels-and-non-linearity"]], "Key Idea": [[46, "key-idea"], [47, "key-idea"]], "LSTM details": [[47, "lstm-details"]], "LU Decomposition, the inverse of a matrix": [[28, "lu-decomposition-the-inverse-of-a-matrix"]], "Lab sessions on Tuesday and Wednesday": [[46, "lab-sessions-on-tuesday-and-wednesday"]], "Lab session: Material from last week and relevant for the first project": [[40, "lab-session-material-from-last-week-and-relevant-for-the-first-project"]], "Lab sessions": [[50, "lab-sessions"]], "Lab sessions Tuesday and Wednesday": [[42, "lab-sessions-tuesday-and-wednesday"]], "Lab sessions and lectures": [[26, "lab-sessions-and-lectures"]], "Lab sessions week 39": [[41, "lab-sessions-week-39"]], "Lasso Regression": [[38, "lasso-regression"]], "Lasso and Bayes": [[38, "lasso-and-bayes"], [39, "lasso-and-bayes"]], "Lasso case": [[38, "lasso-case"]], "Layers": [[2, "layers"], [44, "layers"], [45, "layers"], [46, "layers"], [47, "layers"]], "Layers of a CNN": [[46, "layers-of-a-cnn"], [47, "layers-of-a-cnn"]], "Layers used to build CNNs": [[4, "layers-used-to-build-cnns"], [46, "layers-used-to-build-cnns"], [47, "layers-used-to-build-cnns"]], "Layout of a neural network with three hidden layers": [[43, "layout-of-a-neural-network-with-three-hidden-layers"]], "Layout of a neural network with three hidden layers (last later = l=L=4, first layer l=0)": [[44, "layout-of-a-neural-network-with-three-hidden-layers-last-later-l-l-4-first-layer-l-0"]], "Layout of a simple neural network with no hidden layer": [[43, "layout-of-a-simple-neural-network-with-no-hidden-layer"], [44, "layout-of-a-simple-neural-network-with-no-hidden-layer"]], "Layout of a simple neural network with one hidden layer": [[43, "layout-of-a-simple-neural-network-with-one-hidden-layer"], [44, "layout-of-a-simple-neural-network-with-one-hidden-layer"]], "Layout of a simple neural network with two input nodes, one hidden layer and one output node": [[43, "layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-and-one-output-node"]], "Layout of a simple neural network with two input nodes, one hidden layer with two hidden noeds and one output node": [[44, "layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-with-two-hidden-noeds-and-one-output-node"]], "Layout of input to first hidden layer l=1 from input layer l=0": [[44, "layout-of-input-to-first-hidden-layer-l-1-from-input-layer-l-0"]], "Learning outcomes": [[27, "learning-outcomes"], [36, "learning-outcomes"]], "Learning outcomes and overarching aims of this course": [[50, "learning-outcomes-and-overarching-aims-of-this-course"]], "Learning rate methods": [[44, "learning-rate-methods"], [45, "learning-rate-methods"]], "Lecture Monday October 21": [[45, "lecture-monday-october-21"]], "Lecture Monday October 7": [[43, "lecture-monday-october-7"]], "Lecture Monday September 23": [[41, "lecture-monday-september-23"]], "Lecture Monday September 23, Optimization, the central part of any Machine Learning algortithm": [[41, "lecture-monday-september-23-optimization-the-central-part-of-any-machine-learning-algortithm"]], "Lecture Monday September 30, 2024": [[42, "lecture-monday-september-30-2024"]], "Lecture Monday, November 25": [[50, "lecture-monday-november-25"]], "Lecture October 14, 2024": [[44, "lecture-october-14-2024"]], "Lectures and ComputerLab": [[36, "lectures-and-computerlab"]], "Limitations of NNs": [[43, "limitations-of-nns"], [44, "limitations-of-nns"]], "Limitations of supervised learning with deep networks": [[2, "limitations-of-supervised-learning-with-deep-networks"], [43, "limitations-of-supervised-learning-with-deep-networks"], [44, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[28, null]], "Linear Regression": [[1, null]], "Linear Regression Problems": [[37, "linear-regression-problems"]], "Linear Regression and the SVD": [[38, "linear-regression-and-the-svd"]], "Linear Regression code, Intercept handling first": [[37, "linear-regression-code-intercept-handling-first"]], "Linear Regression, basic elements": [[1, "linear-regression-basic-elements"]], "Linear classifier": [[40, "linear-classifier"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[6, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[6, "linking-the-regression-analysis-with-a-statistical-interpretation"], [38, "linking-the-regression-analysis-with-a-statistical-interpretation"], [39, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with RNNs": [[47, "linking-with-rnns"]], "Linking with the SVD": [[6, "linking-with-the-svd"], [37, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[35, "links-to-relevant-courses-at-the-university-of-oslo"]], "List of contents:": [[46, "list-of-contents"], [47, "list-of-contents"]], "Logistic Regression": [[8, null], [8, "id1"], [40, "logistic-regression"]], "Logistic function as the root of problems": [[43, "logistic-function-as-the-root-of-problems"], [44, "logistic-function-as-the-root-of-problems"]], "MNIST and GANs": [[5, "mnist-and-gans"]], "Machine Learning": [[36, "machine-learning"]], "Machine Learning Research": [[50, "machine-learning-research"]], "Machine learning": [[27, "machine-learning"], [50, "machine-learning"]], "Main textbooks": [[36, "main-textbooks"]], "Making a tree": [[10, "making-a-tree"], [48, "making-a-tree"], [49, "making-a-tree"]], "Making an ADAboost code yourself": [[49, "making-an-adaboost-code-yourself"], [50, "making-an-adaboost-code-yourself"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[11, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"], [48, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"], [49, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[37, "making-your-own-test-train-splitting"]], "Marginal Probability": [[38, "marginal-probability"], [39, "marginal-probability"]], "Material for Lecture Monday November 4": [[47, "material-for-lecture-monday-november-4"]], "Material for Lecture Monday October 28": [[46, "material-for-lecture-monday-october-28"]], "Material for lecture Monday September 16": [[40, "material-for-lecture-monday-september-16"]], "Material for lecture Monday September 2": [[38, "material-for-lecture-monday-september-2"]], "Material for lecture Monday September 9": [[39, "material-for-lecture-monday-september-9"]], "Material for lecture Monday, August 26": [[37, "material-for-lecture-monday-august-26"]], "Material for the active learning sessions Tuesday and Wednesday": [[38, "material-for-the-active-learning-sessions-tuesday-and-wednesday"]], "Material for the active learning sessions on Tuesday and Wednesday": [[43, "material-for-the-active-learning-sessions-on-tuesday-and-wednesday"], [44, "material-for-the-active-learning-sessions-on-tuesday-and-wednesday"]], "Material for the lab sessions": [[39, "material-for-the-lab-sessions"]], "Material for the lab sessions, additional ways to present classification results and other practicalities": [[47, "material-for-the-lab-sessions-additional-ways-to-present-classification-results-and-other-practicalities"]], "Material for the lecture on Monday October 7, 2024": [[43, "material-for-the-lecture-on-monday-october-7-2024"]], "Mathematical Interpretation of Ordinary Least Squares": [[6, "mathematical-interpretation-of-ordinary-least-squares"], [37, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical model": [[42, "mathematical-model"], [42, "id2"], [42, "id3"], [42, "id4"], [42, "id5"]], "Mathematical optimization of convex functions": [[9, "mathematical-optimization-of-convex-functions"]], "Mathematical setup": [[47, "mathematical-setup"]], "Mathematics of CNNs": [[4, "mathematics-of-cnns"], [46, "mathematics-of-cnns"]], "Mathematics of deep learning": [[43, "mathematics-of-deep-learning"], [44, "mathematics-of-deep-learning"], [45, "mathematics-of-deep-learning"]], "Mathematics of deep learning and neural networks": [[43, "mathematics-of-deep-learning-and-neural-networks"]], "Mathematics of the SVD and implications": [[6, "mathematics-of-the-svd-and-implications"], [37, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[36, "matrices-in-python"]], "Matrix multiplication": [[2, "matrix-multiplication"], [44, "matrix-multiplication"], [45, "matrix-multiplication"]], "Matrix multiplications": [[44, "matrix-multiplications"], [45, "matrix-multiplications"]], "Matrix-vector notation": [[42, "matrix-vector-notation"]], "Matrix-vector notation and activation": [[13, "matrix-vector-notation-and-activation"], [42, "matrix-vector-notation-and-activation"]], "Maximum Likelihood Estimation (MLE)": [[38, "maximum-likelihood-estimation-mle"], [39, "maximum-likelihood-estimation-mle"]], "Maximum likelihood": [[40, "maximum-likelihood"]], "Meet the covariance!": [[33, "meet-the-covariance"]], "Meet the Covariance Matrix": [[6, "meet-the-covariance-matrix"], [37, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[37, "meet-the-hessian-matrix"]], "Meet the Pandas": [[36, "meet-the-pandas"]], "Memory considerations": [[46, "memory-considerations"]], "Meta learning": [[50, "meta-learning"]], "Min-Max Scaling": [[37, "min-max-scaling"]], "Minimization process": [[45, "minimization-process"]], "Minimizing the cost function using gradient descent and automatic differentiation": [[45, "minimizing-the-cost-function-using-gradient-descent-and-automatic-differentiation"]], "Minimizing the cross entropy": [[0, "minimizing-the-cross-entropy"], [40, "minimizing-the-cross-entropy"]], "Minor rewrite": [[47, "minor-rewrite"]], "Momentum based GD": [[14, "momentum-based-gd"], [22, "momentum-based-gd"], [41, "momentum-based-gd"], [42, "momentum-based-gd"]], "Momentum parameter": [[41, "momentum-parameter"], [42, "momentum-parameter"]], "More LSTM details": [[47, "more-lstm-details"]], "More about RBMs": [[50, "more-about-rbms"]], "More autograd": [[41, "more-autograd"], [42, "more-autograd"]], "More bagging": [[48, "more-bagging"], [49, "more-bagging"]], "More basic Statistics and Bayes\u2019 theorem": [[38, "more-basic-statistics-and-bayes-theorem"], [39, "more-basic-statistics-and-bayes-theorem"]], "More classes": [[0, "more-classes"], [40, "more-classes"]], "More complicated Example: The Ising model": [[7, "more-complicated-example-the-ising-model"]], "More complicated function": [[43, "more-complicated-function"]], "More complicated functions using the elements of their arguments directly": [[41, "more-complicated-functions-using-the-elements-of-their-arguments-directly"], [42, "more-complicated-functions-using-the-elements-of-their-arguments-directly"]], "More considerations": [[43, "more-considerations"], [44, "more-considerations"]], "More details": [[45, "more-details"], [45, "id6"]], "More examples on bootstrap and cross-validation and errors": [[39, "more-examples-on-bootstrap-and-cross-validation-and-errors"]], "More interpretations": [[37, "more-interpretations"], [38, "more-interpretations"]], "More limitations": [[43, "more-limitations"], [44, "more-limitations"]], "More on Dimensionalities": [[4, "more-on-dimensionalities"], [46, "more-on-dimensionalities"]], "More on Rescaling data": [[7, "more-on-rescaling-data"]], "More on Steepest descent": [[40, "more-on-steepest-descent"], [41, "more-on-steepest-descent"]], "More on activation functions, output layers": [[43, "more-on-activation-functions-output-layers"], [44, "more-on-activation-functions-output-layers"], [45, "more-on-activation-functions-output-layers"]], "More on convex functions": [[40, "more-on-convex-functions"], [41, "more-on-convex-functions"]], "More on momentum based approaches": [[41, "more-on-momentum-based-approaches"], [42, "more-on-momentum-based-approaches"]], "More on the general approximation theorem": [[43, "more-on-the-general-approximation-theorem"]], "More preprocessing": [[37, "more-preprocessing"]], "More preprocessing examples, two-dimensional example, the Franke function": [[37, "more-preprocessing-examples-two-dimensional-example-the-franke-function"]], "More technicalities": [[45, "more-technicalities"]], "More thinking": [[37, "more-thinking"]], "More top-down perspectives": [[43, "more-top-down-perspectives"], [44, "more-top-down-perspectives"]], "Multiclass classification": [[44, "multiclass-classification"], [45, "multiclass-classification"]], "Multilayer perceptrons": [[13, "multilayer-perceptrons"], [42, "multilayer-perceptrons"], [43, "multilayer-perceptrons"]], "Multivariable functions": [[43, "multivariable-functions"]], "Network Elements, the energy function": [[50, "network-elements-the-energy-function"]], "Network requirements": [[3, "network-requirements"], [45, "network-requirements"]], "Neural Networks vs CNNs": [[4, "neural-networks-vs-cnns"], [46, "neural-networks-vs-cnns"], [47, "neural-networks-vs-cnns"]], "Neural network types": [[42, "neural-network-types"], [43, "neural-network-types"]], "Neural networks": [[13, null]], "New expression for the derivative": [[43, "new-expression-for-the-derivative"]], "New image (or volume)": [[46, "new-image-or-volume"]], "New vector": [[46, "new-vector"]], "Note about SVD Calculations": [[37, "note-about-svd-calculations"]], "Note on Scikit-Learn": [[38, "note-on-scikit-learn"]], "Numerical experiments and the covariance, central limit theorem": [[33, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[28, "numpy-and-arrays"], [36, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[36, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization Methods and Hyperparameters": [[50, "optimization-methods-and-hyperparameters"]], "Optimization and Deep learning": [[40, "optimization-and-deep-learning"]], "Optimization, the central part of any Machine Learning algortithm": [[14, null], [40, "optimization-the-central-part-of-any-machine-learning-algortithm"]], "Optimized Convolution2DLayer": [[46, "optimized-convolution2dlayer"], [47, "optimized-convolution2dlayer"]], "Optimizing our parameters": [[36, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[36, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[2, "optimizing-the-cost-function"], [44, "optimizing-the-cost-function"], [45, "optimizing-the-cost-function"]], "Optimizing the parameters": [[43, "optimizing-the-parameters"], [44, "optimizing-the-parameters"]], "Ordinary Differential Equations first": [[45, "ordinary-differential-equations-first"]], "Organizing our data": [[1, "organizing-our-data"], [36, "organizing-our-data"]], "Other Matrix and Vector Operations": [[28, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[5, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[36, "other-courses-on-data-science-and-machine-learning-at-uio"], [50, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[36, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other ingredients of a neural network": [[43, "other-ingredients-of-a-neural-network"]], "Other measures in classification studies: Cancer Data again": [[0, "other-measures-in-classification-studies-cancer-data-again"], [40, "other-measures-in-classification-studies-cancer-data-again"]], "Other parameters": [[43, "other-parameters"]], "Other popular texts": [[36, "other-popular-texts"]], "Other techniques": [[12, "other-techniques"]], "Other types of networks": [[13, "other-types-of-networks"], [42, "other-types-of-networks"], [43, "other-types-of-networks"]], "Other useful relations": [[37, "other-useful-relations"]], "Other ways of visualizing the trees": [[10, "other-ways-of-visualizing-the-trees"], [48, "other-ways-of-visualizing-the-trees"], [49, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[36, "our-model-for-the-nuclear-binding-energies"]], "Output gate": [[47, "output-gate"]], "Output layer": [[43, "output-layer"], [44, "output-layer"]], "Overarching aims of the exercises this week": [[18, "overarching-aims-of-the-exercises-this-week"], [19, "overarching-aims-of-the-exercises-this-week"], [20, "overarching-aims-of-the-exercises-this-week"], [21, "overarching-aims-of-the-exercises-this-week"], [22, "overarching-aims-of-the-exercises-this-week"], [23, "overarching-aims-of-the-exercises-this-week"], [24, "overarching-aims-of-the-exercises-this-week"], [25, "overarching-aims-of-the-exercises-this-week"], [26, "overarching-aims-of-the-exercises-this-week"]], "Overarching view of a neural network": [[43, "overarching-view-of-a-neural-network"]], "Overview of first week": [[36, "overview-of-first-week"]], "Overview of week 48": [[50, "overview-of-week-48"]], "Overview video on Stochastic Gradient Descent": [[41, "overview-video-on-stochastic-gradient-descent"], [42, "overview-video-on-stochastic-gradient-descent"]], "Own code for Ordinary Least Squares": [[36, "own-code-for-ordinary-least-squares"], [37, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[12, "pca-and-scikit-learn"]], "Padding": [[46, "padding"]], "Pandas AI": [[36, "pandas-ai"]], "Parameters of neural networks": [[43, "parameters-of-neural-networks"]], "Parameters to train, common settings": [[46, "parameters-to-train-common-settings"]], "Part a)": [[31, "part-a"]], "Part a) : Ordinary Least Square (OLS) on the Franke function": [[29, "part-a-ordinary-least-square-ols-on-the-franke-function"]], "Part a), setting up the problem": [[31, "part-a-setting-up-the-problem"]], "Part a): Write your own Stochastic Gradient Descent code, first step": [[30, "part-a-write-your-own-stochastic-gradient-descent-code-first-step"]], "Part b)": [[31, "part-b"], [31, "id1"]], "Part b): Adding Ridge regression for the Franke function": [[29, "part-b-adding-ridge-regression-for-the-franke-function"]], "Part b): Writing your own Neural Network code": [[30, "part-b-writing-your-own-neural-network-code"]], "Part c)": [[31, "part-c"]], "Part c) Neural networks": [[31, "part-c-neural-networks"]], "Part c): Adding Lasso for the Franke function": [[29, "part-c-adding-lasso-for-the-franke-function"]], "Part c): Testing different activation functions": [[30, "part-c-testing-different-activation-functions"]], "Part d)": [[31, "part-d"]], "Part d) Neural network complexity": [[31, "part-d-neural-network-complexity"]], "Part d): Classification analysis using neural networks": [[30, "part-d-classification-analysis-using-neural-networks"]], "Part d): Paper and pencil part": [[29, "part-d-paper-and-pencil-part"]], "Part e)": [[31, "part-e"], [31, "id2"]], "Part e): Bias-variance trade-off and resampling techniques": [[29, "part-e-bias-variance-trade-off-and-resampling-techniques"]], "Part e): Write your Logistic Regression code, final step": [[30, "part-e-write-your-logistic-regression-code-final-step"]], "Part f) Critical evaluation of the various algorithms": [[30, "part-f-critical-evaluation-of-the-various-algorithms"]], "Part f): Cross-validation as resampling techniques, adding more complexity": [[29, "part-f-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Part g): Analysis of real data": [[29, "part-g-analysis-of-real-data"]], "Partial Differential Equations": [[3, "partial-differential-equations"], [45, "partial-differential-equations"]], "Paths for project 3": [[31, "paths-for-project-3"]], "Perspective on Machine Learning": [[50, "perspective-on-machine-learning"]], "Plan for week 39, September 23-27, 2024": [[41, "plan-for-week-39-september-23-27-2024"]], "Plan for week 41, October 7-11": [[43, "plan-for-week-41-october-7-11"]], "Plan for week 44": [[46, "plan-for-week-44"]], "Plan for week 46": [[48, "plan-for-week-46"]], "Plan for week 47": [[49, "plan-for-week-47"]], "Plans for the lab sessions": [[40, "plans-for-the-lab-sessions"]], "Plans for week 35": [[37, "plans-for-week-35"]], "Plans for week 36": [[38, "plans-for-week-36"]], "Plans for week 37, lab sessions": [[39, "plans-for-week-37-lab-sessions"]], "Plans for week 37, lecture Monday": [[39, "plans-for-week-37-lecture-monday"]], "Plans for week 38, lecture Monday September 16": [[40, "plans-for-week-38-lecture-monday-september-16"]], "Plans for week 40": [[42, "plans-for-week-40"]], "Plans for week 43": [[45, "plans-for-week-43"]], "Plans for week 45": [[47, "plans-for-week-45"]], "Plotting the Histogram": [[39, "plotting-the-histogram"]], "Plotting the mean value for each group": [[40, "plotting-the-mean-value-for-each-group"]], "Pooling": [[46, "pooling"]], "Pooling Layer": [[46, "pooling-layer"], [47, "pooling-layer"]], "Pooling arithmetic": [[46, "pooling-arithmetic"]], "Pooling types (From Raschka et al)": [[46, "pooling-types-from-raschka-et-al"]], "Practical tips": [[14, "practical-tips"], [22, "practical-tips"], [41, "practical-tips"], [42, "practical-tips"]], "Practicalities": [[34, "practicalities"]], "Preamble: Note on writing reports, using reference material, AI and other tools": [[29, "preamble-note-on-writing-reports-using-reference-material-ai-and-other-tools"]], "Predicting New Points With A Trained Recurrent Neural Network": [[5, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preparing Your Data": [[50, "preparing-your-data"]], "Preprocessing our data": [[37, "preprocessing-our-data"]], "Prerequisites": [[36, "prerequisites"]], "Prerequisites and background": [[27, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[4, "prerequisites-collect-and-pre-process-data"], [46, "prerequisites-collect-and-pre-process-data"], [47, "prerequisites-collect-and-pre-process-data"]], "Printing out as text": [[48, "printing-out-as-text"], [49, "printing-out-as-text"]], "Probability Distribution Functions": [[33, "probability-distribution-functions"]], "Program example for gradient descent with Ridge Regression": [[40, "program-example-for-gradient-descent-with-ridge-regression"], [41, "program-example-for-gradient-descent-with-ridge-regression"]], "Program for stochastic gradient": [[14, "program-for-stochastic-gradient"]], "Project 1 on Machine Learning, deadline October 7 (midnight), 2024": [[29, null]], "Project 2 on Machine Learning, deadline November 4 (Midnight)": [[30, null]], "Project 3 on Machine Learning, deadline December 9 (midnight), 2024": [[31, null]], "Project based teaching and active learning": [[26, "project-based-teaching-and-active-learning"]], "Properties of PDFs": [[33, "properties-of-pdfs"]], "Pros and cons of trees, pros": [[10, "pros-and-cons-of-trees-pros"], [48, "pros-and-cons-of-trees-pros"], [49, "pros-and-cons-of-trees-pros"]], "Pruning the tree": [[48, "pruning-the-tree"], [49, "pruning-the-tree"]], "Python installers": [[27, "python-installers"], [36, "python-installers"]], "Quantum deep learning": [[50, "quantum-deep-learning"]], "Quantum machine learning": [[50, "quantum-machine-learning"]], "Quantum machine learning algorithms based on linear algebra": [[50, "quantum-machine-learning-algorithms-based-on-linear-algebra"]], "Quantum reinforcement learning": [[50, "quantum-reinforcement-learning"]], "RMS prop": [[14, "rms-prop"], [41, "rms-prop"], [42, "rms-prop"]], "RMSProp algorithm, taken from Goodfellow et al": [[22, "rmsprop-algorithm-taken-from-goodfellow-et-al"], [42, "rmsprop-algorithm-taken-from-goodfellow-et-al"]], "RMSprop for adaptive learning rate with Stochastic Gradient Descent": [[22, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"], [41, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"], [42, "rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent"]], "RNNs": [[47, "rnns"]], "RNNs in more detail": [[47, "rnns-in-more-detail"]], "RNNs in more detail, part 2": [[47, "rnns-in-more-detail-part-2"]], "RNNs in more detail, part 3": [[47, "rnns-in-more-detail-part-3"]], "RNNs in more detail, part 4": [[47, "rnns-in-more-detail-part-4"]], "RNNs in more detail, part 5": [[47, "rnns-in-more-detail-part-5"]], "RNNs in more detail, part 6": [[47, "rnns-in-more-detail-part-6"]], "RNNs in more detail, part 7": [[47, "rnns-in-more-detail-part-7"]], "Random Forest Algorithm": [[48, "random-forest-algorithm"], [49, "random-forest-algorithm"]], "Random Forest Algorithm, reminder from last week": [[50, "random-forest-algorithm-reminder-from-last-week"]], "Random Forests Compared with other Methods on the Cancer Data": [[48, "random-forests-compared-with-other-methods-on-the-cancer-data"], [49, "random-forests-compared-with-other-methods-on-the-cancer-data"], [50, "random-forests-compared-with-other-methods-on-the-cancer-data"]], "Random Numbers": [[33, "random-numbers"]], "Random forests": [[11, "random-forests"], [48, "random-forests"], [49, "random-forests"]], "Randomized Grid Search": [[40, "randomized-grid-search"]], "Randomized PCA": [[12, "randomized-pca"]], "Reading material": [[36, "reading-material"]], "Reading recommendations": [[43, "reading-recommendations"], [44, "reading-recommendations"], [45, "reading-recommendations"]], "Reading recommendations:": [[37, "reading-recommendations"]], "Reading suggestions week 34": [[36, "reading-suggestions-week-34"]], "Recurrent neural networks": [[13, "recurrent-neural-networks"], [42, "recurrent-neural-networks"], [43, "recurrent-neural-networks"]], "Recurrent neural networks (RNNs): Overarching view": [[47, "recurrent-neural-networks-rnns-overarching-view"]], "Recurrent neural networks: Overarching view": [[5, null]], "Reducing the number of degrees of freedom, overarching view": [[1, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [37, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reducing the number of operations": [[43, "reducing-the-number-of-operations"]], "Reformulating the problem": [[3, "reformulating-the-problem"], [45, "reformulating-the-problem"]], "Regression Case": [[11, "regression-case"], [49, "regression-case"]], "Regression analysis and resampling methods": [[29, "regression-analysis-and-resampling-methods"]], "Regression analysis, overarching aims": [[36, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[36, "regression-analysis-overarching-aims-ii"]], "Regression trees": [[48, "regression-trees"], [49, "regression-trees"]], "Regular NNs don\u2019t scale well to full images": [[46, "regular-nns-dont-scale-well-to-full-images"], [47, "regular-nns-dont-scale-well-to-full-images"]], "Regularization": [[2, "regularization"], [44, "regularization"], [45, "regularization"]], "Reinforcement Learning": [[50, "reinforcement-learning"]], "Relevance": [[42, "relevance"], [43, "relevance"], [44, "relevance"]], "Remarks on the speed": [[46, "remarks-on-the-speed"], [47, "remarks-on-the-speed"]], "Reminder on Statistics": [[7, "reminder-on-statistics"]], "Reminder on books with hands-on material and codes": [[43, "reminder-on-books-with-hands-on-material-and-codes"], [44, "reminder-on-books-with-hands-on-material-and-codes"], [45, "reminder-on-books-with-hands-on-material-and-codes"]], "Reminder on the chain rule and gradients": [[43, "reminder-on-the-chain-rule-and-gradients"]], "Replace or not": [[14, "replace-or-not"], [41, "replace-or-not"], [42, "replace-or-not"]], "Required Technologies": [[27, "required-technologies"]], "Resampling": [[50, "resampling"]], "Resampling Methods": [[7, null]], "Resampling approaches can be computationally expensive": [[39, "resampling-approaches-can-be-computationally-expensive"]], "Resampling methods": [[7, "id1"], [39, "resampling-methods"], [39, "id2"]], "Resampling methods: Bootstrap": [[39, "resampling-methods-bootstrap"]], "Resampling methods: Bootstrap approach": [[39, "resampling-methods-bootstrap-approach"]], "Resampling methods: Bootstrap background": [[39, "resampling-methods-bootstrap-background"]], "Resampling methods: Bootstrap steps": [[39, "resampling-methods-bootstrap-steps"]], "Resampling methods: More Bootstrap background": [[39, "resampling-methods-more-bootstrap-background"]], "Residual Error": [[37, "residual-error"]], "Resources on differential equations and deep learning": [[3, "resources-on-differential-equations-and-deep-learning"], [45, "resources-on-differential-equations-and-deep-learning"]], "Revisiting our Linear Regression Solvers": [[14, "revisiting-our-linear-regression-solvers"]], "Revisiting our Logistic Regression case": [[40, "revisiting-our-logistic-regression-case"], [41, "revisiting-our-logistic-regression-case"]], "Revisiting our first homework": [[40, "revisiting-our-first-homework"], [41, "revisiting-our-first-homework"]], "Rewriting as dot products": [[46, "rewriting-as-dot-products"]], "Rewriting the Covariance and/or Correlation Matrix": [[37, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the \\delta-function": [[39, "rewriting-the-delta-function"]], "Rewriting the fitting procedure as a linear algebra problem": [[36, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[36, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge Regression": [[38, "ridge-regression"]], "Ridge and Bayes": [[38, "ridge-and-bayes"], [39, "ridge-and-bayes"]], "Ridge and LASSO Regression": [[37, "ridge-and-lasso-regression"], [38, "ridge-and-lasso-regression"]], "Ridge and Lasso Regression": [[6, null], [6, "id1"]], "Running with Keras": [[46, "running-with-keras"], [47, "running-with-keras"]], "SGD example": [[41, "sgd-example"], [42, "sgd-example"]], "SVD analysis": [[38, "svd-analysis"]], "Same code but now with momentum gradient descent": [[14, "same-code-but-now-with-momentum-gradient-descent"], [22, "same-code-but-now-with-momentum-gradient-descent"], [22, "id1"], [41, "same-code-but-now-with-momentum-gradient-descent"], [41, "id9"], [41, "id10"], [42, "same-code-but-now-with-momentum-gradient-descent"], [42, "id1"]], "Schedule first week": [[36, "schedule-first-week"]], "Schedulers": [[46, "schedulers"], [47, "schedulers"]], "Schematic Regression Procedure": [[10, "schematic-regression-procedure"], [48, "schematic-regression-procedure"], [49, "schematic-regression-procedure"]], "Searching for Optimal Regularization Parameters \\lambda": [[40, "searching-for-optimal-regularization-parameters-lambda"]], "Second moment of the gradient": [[41, "second-moment-of-the-gradient"], [42, "second-moment-of-the-gradient"]], "Sequential data only?": [[47, "sequential-data-only"]], "Set up the model": [[46, "set-up-the-model"], [47, "set-up-the-model"]], "Setting it up": [[46, "setting-it-up"], [47, "setting-it-up"]], "Setting up a Multi-layer perceptron model for classification": [[44, "setting-up-a-multi-layer-perceptron-model-for-classification"], [45, "setting-up-a-multi-layer-perceptron-model-for-classification"]], "Setting up the Back propagation algorithm": [[13, "setting-up-the-back-propagation-algorithm"]], "Setting up the Back propagation algorithm, part 3": [[43, "setting-up-the-back-propagation-algorithm-part-3"], [44, "setting-up-the-back-propagation-algorithm-part-3"], [45, "setting-up-the-back-propagation-algorithm-part-3"]], "Setting up the Matrix to be inverted": [[37, "setting-up-the-matrix-to-be-inverted"]], "Setting up the back propagation algorithm": [[43, "setting-up-the-back-propagation-algorithm"]], "Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations": [[44, "setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations"], [45, "setting-up-the-back-propagation-algorithm-and-algorithm-for-a-feed-forward-nn-initalizations"]], "Setting up the back propagation algorithm, part 1": [[44, "setting-up-the-back-propagation-algorithm-part-1"], [45, "setting-up-the-back-propagation-algorithm-part-1"]], "Setting up the back propagation algorithm, part 2": [[43, "setting-up-the-back-propagation-algorithm-part-2"], [44, "setting-up-the-back-propagation-algorithm-part-2"], [45, "setting-up-the-back-propagation-algorithm-part-2"]], "Setting up the code": [[45, "setting-up-the-code"]], "Setting up the equations for a neural network": [[43, "setting-up-the-equations-for-a-neural-network"], [44, "setting-up-the-equations-for-a-neural-network"]], "Setting up the network using Autograd": [[45, "setting-up-the-network-using-autograd"]], "Setting up the network using Autograd; The full program": [[3, "setting-up-the-network-using-autograd-the-full-program"], [45, "setting-up-the-network-using-autograd-the-full-program"]], "Setting up the network using Autograd; The trial solution": [[45, "setting-up-the-network-using-autograd-the-trial-solution"]], "Setting up the problem": [[45, "setting-up-the-problem"]], "Setup of Network": [[45, "setup-of-network"]], "Similar (second order function now) problem but now with AdaGrad": [[14, "similar-second-order-function-now-problem-but-now-with-adagrad"], [22, "similar-second-order-function-now-problem-but-now-with-adagrad"], [41, "similar-second-order-function-now-problem-but-now-with-adagrad"], [42, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[10, "simple-python-code-to-read-in-data-and-perform-classification"], [48, "simple-python-code-to-read-in-data-and-perform-classification"], [49, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple Voting Example, head or tail": [[48, "simple-voting-example-head-or-tail"], [49, "simple-voting-example-head-or-tail"]], "Simple case": [[37, "simple-case"]], "Simple code for solving the above problem": [[38, "simple-code-for-solving-the-above-problem"]], "Simple example": [[40, "simple-example"], [43, "simple-example"]], "Simple example code": [[41, "simple-example-code"], [42, "simple-example-code"]], "Simple example to illustrate Ordinary Least Squares, Ridge and Lasso Regression": [[38, "simple-example-to-illustrate-ordinary-least-squares-ridge-and-lasso-regression"]], "Simple geometric interpretation": [[40, "simple-geometric-interpretation"], [41, "simple-geometric-interpretation"]], "Simple implementation of GD for OLS, Ridge and Lasso": [[42, "simple-implementation-of-gd-for-ols-ridge-and-lasso"]], "Simple linear regression model using scikit-learn": [[1, "simple-linear-regression-model-using-scikit-learn"], [36, "simple-linear-regression-model-using-scikit-learn"]], "Simple neural network and the back propagation equations": [[43, "simple-neural-network-and-the-back-propagation-equations"], [44, "simple-neural-network-and-the-back-propagation-equations"]], "Simple program": [[40, "simple-program"], [41, "simple-program"]], "Simpler examples first, and automatic differentiation": [[43, "simpler-examples-first-and-automatic-differentiation"]], "Slightly different approach": [[41, "slightly-different-approach"], [42, "slightly-different-approach"]], "Smarter way of evaluating the above function": [[43, "smarter-way-of-evaluating-the-above-function"]], "Social machine learning": [[50, "social-machine-learning"]], "Software and needed installations": [[29, "software-and-needed-installations"], [31, "software-and-needed-installations"], [36, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[3, null]], "Solving differential equations with Deep Learning": [[45, "solving-differential-equations-with-deep-learning"]], "Solving differential equations with RNNs": [[47, "solving-differential-equations-with-rnns"]], "Solving partial differential equations with neural networks": [[31, "solving-partial-differential-equations-with-neural-networks"]], "Solving the equation using Autograd": [[45, "solving-the-equation-using-autograd"]], "Solving the one dimensional Poisson equation": [[3, "solving-the-one-dimensional-poisson-equation"]], "Solving the wave equation - the full program using Autograd": [[45, "solving-the-wave-equation-the-full-program-using-autograd"]], "Solving the wave equation with Neural Networks": [[3, "solving-the-wave-equation-with-neural-networks"]], "Solving using Newton-Raphson\u2019s method": [[40, "solving-using-newton-raphson-s-method"], [41, "solving-using-newton-raphson-s-method"]], "Some famous Matrices": [[28, "some-famous-matrices"]], "Some parallels from real analysis": [[43, "some-parallels-from-real-analysis"]], "Some selected properties": [[40, "some-selected-properties"]], "Some similarities and differences from DNNs": [[50, "some-similarities-and-differences-from-dnns"]], "Some simple problems": [[14, "some-simple-problems"], [40, "some-simple-problems"], [41, "some-simple-problems"]], "Some useful matrix and vector expressions": [[37, "some-useful-matrix-and-vector-expressions"]], "Splitting our Data in Training and Test data": [[1, "splitting-our-data-in-training-and-test-data"], [36, "splitting-our-data-in-training-and-test-data"], [37, "splitting-our-data-in-training-and-test-data"]], "Squared-Error Example and Iterative Fitting": [[48, "squared-error-example-and-iterative-fitting"], [49, "squared-error-example-and-iterative-fitting"], [50, "squared-error-example-and-iterative-fitting"]], "Standard Approach based on the Normal Distribution": [[39, "standard-approach-based-on-the-normal-distribution"]], "Standard imports first": [[48, "standard-imports-first"], [49, "standard-imports-first"]], "Standard steepest descent": [[14, "standard-steepest-descent"], [41, "standard-steepest-descent"]], "Starting your Machine Learning Project": [[50, "starting-your-machine-learning-project"]], "Statistical analysis": [[39, "statistical-analysis"]], "Statistical analysis and optimization of data": [[27, "statistical-analysis-and-optimization-of-data"], [36, "statistical-analysis-and-optimization-of-data"], [50, "statistical-analysis-and-optimization-of-data"]], "Steepest Descent Example": [[49, "steepest-descent-example"], [50, "steepest-descent-example"]], "Steepest descent": [[14, "steepest-descent"], [40, "steepest-descent"], [41, "steepest-descent"]], "Steepest descent method": [[41, "steepest-descent-method"], [41, "id1"]], "Steepest descent example": [[41, "steepest-descent-example"]], "Still thinking": [[37, "still-thinking"]], "Stochastic Gradient Descent": [[41, "stochastic-gradient-descent"], [42, "stochastic-gradient-descent"]], "Stochastic Gradient Descent (SGD)": [[14, "stochastic-gradient-descent-sgd"], [41, "stochastic-gradient-descent-sgd"], [42, "stochastic-gradient-descent-sgd"]], "Stochastic variables and the main concepts, the discrete case": [[33, "stochastic-variables-and-the-main-concepts-the-discrete-case"]], "Strong correlations": [[46, "strong-correlations"], [47, "strong-correlations"]], "Suggested reading and videos": [[40, "suggested-reading-and-videos"]], "Suggested readings and videos": [[42, "suggested-readings-and-videos"]], "Summarizing: Performing a general discrete convolution (From Raschka et al)": [[46, "summarizing-performing-a-general-discrete-convolution-from-raschka-et-al"]], "Summary from last week, using gradient descent methods, limitations": [[42, "summary-from-last-week-using-gradient-descent-methods-limitations"]], "Summary of RNNs": [[47, "summary-of-rnns"]], "Summary of a typical RNN": [[47, "summary-of-a-typical-rnn"]], "Summary of course": [[50, "summary-of-course"]], "Summing up": [[39, "summing-up"]], "Support Vector Machines, overarching aims": [[9, null]], "Systematic reduction": [[4, "systematic-reduction"], [46, "systematic-reduction"], [47, "systematic-reduction"]], "Teachers": [[36, "teachers"]], "Teachers and Grading": [[34, null]], "Teaching Assistants Fall semester 2023": [[34, "teaching-assistants-fall-semester-2023"]], "Teaching schedule with links to material": [[32, null]], "Technicalities": [[45, "technicalities"]], "Tensorflow": [[44, "tensorflow"], [45, "tensorflow"]], "Test Function for what happens with OLS, Ridge and Lasso": [[38, "test-function-for-what-happens-with-ols-ridge-and-lasso"]], "Testing the Means Squared Error as function of Complexity": [[1, "testing-the-means-squared-error-as-function-of-complexity"], [37, "testing-the-means-squared-error-as-function-of-complexity"]], "Testing the XOR gate and other gates": [[44, "testing-the-xor-gate-and-other-gates"], [45, "testing-the-xor-gate-and-other-gates"]], "Textbooks": [[35, null]], "The back propagation equations for a neural network": [[44, "the-back-propagation-equations-for-a-neural-network"]], "The Algorithm before theorem": [[12, "the-algorithm-before-theorem"]], "The Boston housing data example": [[1, "the-boston-housing-data-example"]], "The Breast Cancer Data, now with Keras": [[2, "the-breast-cancer-data-now-with-keras"], [44, "the-breast-cancer-data-now-with-keras"], [45, "the-breast-cancer-data-now-with-keras"]], "The CART algorithm for Classification": [[10, "the-cart-algorithm-for-classification"], [48, "the-cart-algorithm-for-classification"], [49, "the-cart-algorithm-for-classification"]], "The CART algorithm for Regression": [[10, "the-cart-algorithm-for-regression"], [48, "the-cart-algorithm-for-regression"], [49, "the-cart-algorithm-for-regression"]], "The CIFAR01 data set": [[4, "the-cifar01-data-set"], [46, "the-cifar01-data-set"], [47, "the-cifar01-data-set"]], "The Central Limit Theorem": [[39, "the-central-limit-theorem"]], "The Challenges Facing Machine Learning": [[50, "the-challenges-facing-machine-learning"]], "The Convolutional Neural Network (CNN)": [[46, "the-convolutional-neural-network-cnn"], [47, "the-convolutional-neural-network-cnn"]], "The Hessian matrix": [[40, "the-hessian-matrix"], [41, "the-hessian-matrix"]], "The Hessian matrix for Ridge Regression": [[40, "the-hessian-matrix-for-ridge-regression"], [41, "the-hessian-matrix-for-ridge-regression"]], "The Jacobian": [[37, "the-jacobian"]], "The MNIST dataset again": [[4, "the-mnist-dataset-again"], [46, "the-mnist-dataset-again"], [47, "the-mnist-dataset-again"]], "The Neural Network": [[44, "the-neural-network"], [45, "the-neural-network"]], "The OLS case": [[38, "the-ols-case"]], "The RELU function family": [[2, "the-relu-function-family"], [43, "the-relu-function-family"], [44, "the-relu-function-family"], [45, "the-relu-function-family"]], "The Ridge case": [[38, "the-ridge-case"]], "The SVD example": [[46, "the-svd-example"]], "The SVD, a Fantastic Algorithm": [[37, "the-svd-a-fantastic-algorithm"]], "The Softmax function": [[2, "the-softmax-function"], [44, "the-softmax-function"], [45, "the-softmax-function"]], "The Squared-Error again! Steepest Descent": [[49, "the-squared-error-again-steepest-descent"], [50, "the-squared-error-again-steepest-descent"]], "The Table": [[48, "the-table"], [49, "the-table"]], "The \\chi^2 function": [[1, "the-chi-2-function"], [36, "the-chi-2-function"], [36, "id4"], [36, "id5"], [36, "id6"], [36, "id7"], [36, "id8"]], "The analytical solution": [[45, "the-analytical-solution"]], "The approximation theorem in words": [[43, "the-approximation-theorem-in-words"]], "The backward pass is linear": [[47, "the-backward-pass-is-linear"]], "The basic structure of your project": [[31, "the-basic-structure-of-your-project"]], "The bias-variance tradeoff": [[7, "the-bias-variance-tradeoff"], [39, "the-bias-variance-tradeoff"]], "The code": [[36, "the-code"]], "The code for solving the ODE": [[3, "the-code-for-solving-the-ode"], [45, "the-code-for-solving-the-ode"]], "The complete code with a simple data set": [[37, "the-complete-code-with-a-simple-data-set"]], "The convolution stage": [[46, "the-convolution-stage"]], "The cost function": [[0, "the-cost-function"]], "The cost function rewritten": [[40, "the-cost-function-rewritten"]], "The cost/loss function": [[37, "the-cost-loss-function"]], "The course has two central parts": [[27, "the-course-has-two-central-parts"]], "The derivative of the Logistic funtion": [[43, "the-derivative-of-the-logistic-funtion"], [44, "the-derivative-of-the-logistic-funtion"]], "The derivative of the cost/loss function": [[40, "the-derivative-of-the-cost-loss-function"], [41, "the-derivative-of-the-cost-loss-function"]], "The derivatives": [[43, "the-derivatives"], [44, "the-derivatives"]], "The equations": [[40, "the-equations"], [41, "the-equations"]], "The equations for ordinary least squares": [[37, "the-equations-for-ordinary-least-squares"]], "The equations to solve": [[40, "the-equations-to-solve"], [41, "the-equations-to-solve"]], "The first Case": [[38, "the-first-case"]], "The forget gate": [[47, "the-forget-gate"]], "The function to solve for": [[45, "the-function-to-solve-for"]], "The gradient step": [[41, "the-gradient-step"], [42, "the-gradient-step"]], "The ideal": [[40, "the-ideal"], [41, "the-ideal"]], "The last words?": [[50, "the-last-words"]], "The logistic function": [[0, "the-logistic-function"], [8, "the-logistic-function"], [40, "the-logistic-function"]], "The mean squared error and its derivative": [[37, "the-mean-squared-error-and-its-derivative"]], "The moons example": [[9, "the-moons-example"]], "The multilayer perceptron (MLP)": [[13, "the-multilayer-perceptron-mlp"]], "The network": [[50, "the-network"]], "The network with one input layer, specified number of hidden layers, and one output layer": [[3, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"], [45, "the-network-with-one-input-layer-specified-number-of-hidden-layers-and-one-output-layer"]], "The optimization problem": [[43, "the-optimization-problem"]], "The ouput layer": [[43, "the-ouput-layer"], [44, "the-ouput-layer"]], "The problem of exploding or vanishing gradients": [[47, "the-problem-of-exploding-or-vanishing-gradients"]], "The problem to solve for": [[45, "the-problem-to-solve-for"]], "The program using Autograd": [[45, "the-program-using-autograd"]], "The same example but now with cross-validation": [[39, "the-same-example-but-now-with-cross-validation"]], "The sensitiveness of the gradient descent": [[40, "the-sensitiveness-of-the-gradient-descent"], [41, "the-sensitiveness-of-the-gradient-descent"]], "The singular value decomposition": [[6, "the-singular-value-decomposition"], [37, "the-singular-value-decomposition"]], "The specific equation to solve for": [[45, "the-specific-equation-to-solve-for"]], "The structure of the RBM network": [[50, "the-structure-of-the-rbm-network"]], "The syntax a.dot(b) when finding the dot product": [[41, "the-syntax-a-dot-b-when-finding-the-dot-product"]], "The training": [[43, "the-training"], [44, "the-training"]], "The trial solution": [[45, "the-trial-solution"], [45, "id4"], [45, "id5"], [45, "id7"]], "The two-dimensional case": [[9, "the-two-dimensional-case"]], "Then some basic questions": [[26, "then-some-basic-questions"]], "Time decay rate": [[41, "time-decay-rate"], [42, "time-decay-rate"]], "To our real data: nuclear binding energies. Brief reminder on masses and binding energies": [[36, "to-our-real-data-nuclear-binding-energies-brief-reminder-on-masses-and-binding-energies"]], "To think about, first part": [[37, "to-think-about-first-part"]], "Toeplitz matrices": [[46, "toeplitz-matrices"]], "Topics covered in this course: Statistical analysis and optimization of data": [[36, "topics-covered-in-this-course-statistical-analysis-and-optimization-of-data"]], "Topics we have covered this year": [[50, "topics-we-have-covered-this-year"]], "Tossing coins": [[48, "tossing-coins"], [49, "tossing-coins"]], "Towards the PCA theorem": [[12, "towards-the-pca-theorem"]], "Train and test datasets": [[2, "train-and-test-datasets"], [44, "train-and-test-datasets"], [45, "train-and-test-datasets"]], "Transfer learning": [[50, "transfer-learning"]], "Transforming images": [[46, "transforming-images"]], "Two first-order differential equations": [[47, "two-first-order-differential-equations"]], "Two parameters": [[0, "two-parameters"], [40, "two-parameters"]], "Two-dimensional Objects": [[4, "two-dimensional-objects"]], "Two-dimensional objects": [[46, "two-dimensional-objects"]], "Type of problem": [[3, "type-of-problem"], [45, "type-of-problem"]], "Types of Machine Learning": [[36, "types-of-machine-learning"]], "Types of Machine Learning, a repetition": [[50, "types-of-machine-learning-a-repetition"]], "Understanding what happens": [[39, "understanding-what-happens"]], "Universal approximation theorem": [[43, "universal-approximation-theorem"]], "Unsupported functions": [[41, "unsupported-functions"]], "Updating the gradients": [[43, "updating-the-gradients"], [44, "updating-the-gradients"], [45, "updating-the-gradients"]], "Usage of CNN code": [[46, "usage-of-cnn-code"], [47, "usage-of-cnn-code"]], "Usage of activation functions": [[46, "usage-of-activation-functions"], [47, "usage-of-activation-functions"]], "Usage of cost functions": [[46, "usage-of-cost-functions"], [47, "usage-of-cost-functions"]], "Usage of schedulers": [[46, "usage-of-schedulers"], [47, "usage-of-schedulers"]], "Usage of the above learning rate schedulers": [[44, "usage-of-the-above-learning-rate-schedulers"], [45, "usage-of-the-above-learning-rate-schedulers"]], "Useful Python libraries": [[27, "useful-python-libraries"], [36, "useful-python-libraries"]], "Usefulness of the weekly exercises": [[26, "usefulness-of-the-weekly-exercises"]], "Using Autograd": [[14, "using-autograd"]], "Using Autograd with OLS": [[41, "using-autograd-with-ols"], [42, "using-autograd-with-ols"]], "Using Automatic differentation with OLS": [[22, "using-automatic-differentation-with-ols"]], "Using Automatic differentiation": [[45, "using-automatic-differentiation"]], "Using Keras": [[44, "using-keras"], [45, "using-keras"]], "Using autograd": [[41, "using-autograd"], [42, "using-autograd"]], "Using forward Euler to solve the ODE": [[3, "using-forward-euler-to-solve-the-ode"], [45, "using-forward-euler-to-solve-the-ode"]], "Using gradient descent methods, limitations": [[14, "using-gradient-descent-methods-limitations"], [40, "using-gradient-descent-methods-limitations"], [41, "using-gradient-descent-methods-limitations"]], "Using recursion": [[41, "using-recursion"], [42, "using-recursion"]], "Using the Voting Classifier": [[48, "using-the-voting-classifier"], [49, "using-the-voting-classifier"]], "Using the chain rule and summing over all k entries": [[43, "using-the-chain-rule-and-summing-over-all-k-entries"], [44, "using-the-chain-rule-and-summing-over-all-k-entries"]], "Using the correlation matrix": [[0, "using-the-correlation-matrix"], [40, "using-the-correlation-matrix"]], "Vanishing gradients": [[43, "vanishing-gradients"], [44, "vanishing-gradients"], [47, "vanishing-gradients"]], "Various steps in cross-validation": [[39, "various-steps-in-cross-validation"], [40, "various-steps-in-cross-validation"]], "Velocity only": [[47, "velocity-only"]], "Verifying the data set": [[46, "verifying-the-data-set"], [47, "verifying-the-data-set"]], "Visualization": [[2, "visualization"], [2, "id1"], [44, "visualization"], [44, "id1"], [45, "visualization"], [45, "id1"]], "Visualizing the Tree, Classification": [[10, "visualizing-the-tree-classification"], [48, "visualizing-the-tree-classification"], [49, "visualizing-the-tree-classification"]], "Visualizing the Tree, The Moons": [[48, "visualizing-the-tree-the-moons"], [49, "visualizing-the-tree-the-moons"]], "Voting and Bagging": [[48, "voting-and-bagging"], [49, "voting-and-bagging"]], "Was it easy to access the course material?": [[26, "was-it-easy-to-access-the-course-material"]], "Was the weekly update with plans etc useful?": [[26, "was-the-weekly-update-with-plans-etc-useful"]], "Week 34: Introduction to the course, Logistics and Practicalities": [[36, null]], "Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression": [[37, null]], "Week 36: Linear Regression and 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Jupyter-notebooks, GitHub, the various textbooks we have recommended, etc etc": [[26, "which-resources-and-tools-did-you-use-jupyter-notebooks-github-the-various-textbooks-we-have-recommended-etc-etc"]], "Why Boltzmann machines?": [[50, "why-boltzmann-machines"]], "Why CNNS for images, sound files, medical images from CT scans etc?": [[46, "why-cnns-for-images-sound-files-medical-images-from-ct-scans-etc"], [47, "why-cnns-for-images-sound-files-medical-images-from-ct-scans-etc"]], "Why Linear Regression (aka Ordinary Least Squares and family)": [[36, "why-linear-regression-aka-ordinary-least-squares-and-family"]], "Why Voting?": [[48, "why-voting"], [49, "why-voting"]], "Why binary splits?": [[48, "why-binary-splits"], [49, "why-binary-splits"]], "Why did you choose this course?": [[26, "why-did-you-choose-this-course"]], "Why multilayer perceptrons?": [[42, "why-multilayer-perceptrons"], [43, "why-multilayer-perceptrons"]], "Why resampling methods": [[39, "why-resampling-methods"]], 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Linear Regression", "14. Building a Feed Forward Neural Network", "15. Solving Differential Equations with Deep Learning", "16. Convolutional Neural Networks", "17. Recurrent neural networks: Overarching view", "4. Ridge and Lasso Regression", "5. Resampling Methods", "6. Logistic Regression", "8. Support Vector Machines, overarching aims", "9. Decision trees, overarching aims", "10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods", "11. Basic ideas of the Principal Component Analysis (PCA)", "13. Neural networks", "7. Optimization, the central part of any Machine Learning algortithm", "12. Clustering and Unsupervised Learning", "Exercises week 34", "Exercises week 35", "Exercises week 36", "Exercises week 37", "Exercises week 38", "Exercises week 39", "Exercises week 41", "Exercises week 42", "Exercises week 43", "Exercise week 47", "Exercises week 48", "Applied Data Analysis and Machine Learning", "2. Linear Algebra, Handling of Arrays and more Python Features", "Project 1 on Machine Learning, deadline October 7 (midnight), 2024", "Project 2 on Machine Learning, deadline November 4 (Midnight)", "Project 3 on Machine Learning, deadline December 9 (midnight), 2024", "Teaching schedule with links to material", "1. Elements of Probability Theory and Statistical Data Analysis", "Teachers and Grading", "Textbooks", "Week 34: Introduction to the course, Logistics and Practicalities", "Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression", "Week 36: Linear Regression and Statistical interpretations", "Week 37: Statistical interpretations and Resampling Methods", "Week 38: Logistic Regression and Optimization", "Week 39: Optimization and Gradient Methods", "Week 40: Gradient descent methods (continued) and start Neural networks", "Week 41 Neural networks and constructing a neural network code", "Week 42 Constructing a Neural Network code with examples", "Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations", "Week 44, Convolutional Neural Networks (CNN)", "Week 45, Convolutional Neural Networks (CCNs) and Recurrent Neural Networks (RNNs)", "Week 46: Decision Trees, Ensemble methods and Random Forests", "Week 47: From Decision Trees to Ensemble 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    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

      @@ -938,27 +941,27 @@ uncorrelated.

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+[[ 2.9789621   5.29596775  9.61069934  8.94881093  7.29305818  5.06734969
+   8.35612808  9.50699066  6.58518977  5.96699567]
+ [ 5.29596775  9.41511622 17.08580103 15.90910271 12.96552275  9.00868142
+  14.85543733 16.90142882 11.70708167 10.60806265]
+ [ 9.61069934 17.08580103 31.00594727 28.87056912 23.52879528 16.34823563
+  26.95846133 30.67136331 21.2450769  19.25066495]
+ [ 8.94881093 15.90910271 28.87056912 26.88225436 21.90836824 15.2223334
+  25.10183335 28.55902794 19.78192944 17.92487255]
+ [ 7.29305818 12.96552275 23.52879528 21.90836824 17.85477485 12.40582286
+  20.45736943 23.27489698 16.12178015 14.608325  ]
+ [ 5.06734969  9.00868142 16.34823563 15.2223334  12.40582286  8.61979174
+  14.21415299 16.17182246 11.20170657 10.15013036]
+ [ 8.35612808 14.85543733 26.95846133 25.10183335 20.45736943 14.21415299
+  23.43933023 26.66755363 18.4717654  16.73770203]
+ [ 9.50699066 16.90142882 30.67136331 28.55902794 23.27489698 16.17182246
+  26.66755363 30.34038983 21.01582211 19.04293178]
+ [ 6.58518977 11.70708167 21.2450769  19.78192944 16.12178015 11.20170657
+  18.4717654  21.01582211 14.55699091 13.19043259]
+ [ 5.96699567 10.60806265 19.25066495 17.92487255 14.608325   10.15013036
+  16.73770203 19.04293178 13.19043259 11.95216188]]
      @@ -1226,15 +1229,15 @@ more practically oriented methods like the blocking technique.

      -
      -0.14555142287587747
-3.471032560330219
--0.36388945426885777
-0.8471097389658617 8.314152725401152 16.209550199940885
-2.467818446250011 2.622158030566299 8.055384838739426
-[[ 0.84710974  2.46781845  2.62215803]
- [ 2.46781845  8.31415273  8.05538484]
- [ 2.62215803  8.05538484 16.2095502 ]]
-[21.82751857  0.10066746  3.44262663]
+
      0.02067725499485532
+4.1599466577177715
+-0.010435324202206745
+1.0573975468113945 11.60638779409259 34.06300749480313
+3.3628139153668575 3.9755893793908617 13.085147027563798
+[[ 1.05739755  3.36281392  3.97558938]
+ [ 3.36281392 11.60638779 13.08514703]
+ [ 3.97558938 13.08514703 34.06300749]]
+[40.7125057   0.07536391  5.93892323]
      @@ -1502,7 +1505,7 @@ these limiting values.

      -_images/2d32a16373526bea9bc6f4e8fe926f0ed4c4e23d38a4de33454551f6599a83cf.png +_images/46a1fad1ccb7c31a75875feb55a5327e9f0ae2b02295543e814d1b2fc23ab459.png

      Since our random numbers, which are typically generated via a linear congruential algorithm, @@ -1564,10 +1567,10 @@ assumption for approximating \(\sigma

      -
      0.028592327979280928 1.0458514144974842
+
      0.0065879787737229524 1.0071299074549438
 
      
 
      
-_images/8dd8c0fbf4c42ac2d98c5a66035e455e4d40a3c6c7114c3d05daba0ad8fdf4ba.png
+_images/467927f86cd8a5304cb2f39dbf480015e72d87541abf390f2a0eb8672b465f1e.png

      As can be seen from the plot, the first point gives back the variance and a value of one. diff --git a/doc/LectureNotes/_build/html/teachers.html b/doc/LectureNotes/_build/html/teachers.html index 4df14c2a0..bf7eb6da0 100644 --- a/doc/LectureNotes/_build/html/teachers.html +++ b/doc/LectureNotes/_build/html/teachers.html @@ -255,6 +255,9 @@

    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

    Projects

    Projects

      @@ -1005,8 +1008,8 @@ developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly he
      -
      [ 0.17506058 -1.14302491  0.63485898  1.1235978   0.38873383 -0.17169218
-  0.88719525 -0.01015838 -0.27342837 -1.80652926]
+
      [ 2.05896588  0.43642532 -0.21096409  0.40779804 -1.13963886 -0.53677418
+  0.83572684  0.58831595  0.30882611 -2.07060477]
      @@ -1231,26 +1234,36 @@ lowercase letters for vectors and uppercase letters for matrices)

      -
      [[0.3418169  0.08757698 0.45405198 0.04340609 0.58148701 0.14855269
-  0.07347117 0.79453819 0.78257332 0.91637778]
- [0.81517637 0.35341707 0.55717092 0.03395449 0.79320614 0.22562101
-  0.65971268 0.64276252 0.49192086 0.91375988]
- [0.10740566 0.77091954 0.10656055 0.36547141 0.83190714 0.25533245
-  0.40632604 0.48183592 0.22261204 0.16509611]
- [0.47804768 0.92388196 0.85585927 0.38675536 0.30896123 0.22930735
-  0.78408961 0.30789787 0.58654863 0.5228503 ]
- [0.35826417 0.13768194 0.41982096 0.32226801 0.39507682 0.76931267
-  0.97369602 0.12082909 0.97147296 0.30115792]
- [0.72544872 0.9789659  0.98184665 0.98596221 0.68564602 0.80794793
-  0.66307293 0.83836165 0.98251519 0.63387948]
- [0.21352978 0.27418384 0.91865691 0.36851668 0.0525856  0.7259156
-  0.83058696 0.99696669 0.83207908 0.32633042]
- [0.10500719 0.64631695 0.2220673  0.5920376  0.85070792 0.84332209
-  0.73691262 0.06028861 0.62436053 0.00758035]
- [0.13963148 0.11686929 0.34936681 0.14068659 0.59525417 0.83232065
-  0.32968695 0.3371494  0.85006441 0.96828809]
- [0.88555932 0.25272278 0.26684239 0.38777146 0.73658394 0.81886022
-  0.01409764 0.23749007 0.26368616 0.10068772]]
+
      [[4.62790954e-01 3.05355909e-01 4.36307837e-01 5.75663043e-01
+  8.47455900e-01 8.88156953e-01 3.41214900e-01 4.80217592e-01
+  3.51294625e-01 4.55894547e-01]
+ [9.36928932e-01 1.34614286e-01 1.68706337e-01 5.12168283e-01
+  1.82057187e-01 5.04384702e-01 3.90807328e-01 7.21325310e-01
+  6.82139912e-04 7.75656839e-01]
+ [7.39975586e-01 7.28371709e-02 4.47008569e-01 1.48689896e-01
+  8.77801149e-01 8.77111122e-01 6.14302755e-01 3.05347541e-01
+  5.22304438e-01 1.19480182e-01]
+ [3.75927527e-01 1.79526117e-01 1.10245050e-01 2.14621443e-02
+  2.18149821e-01 6.60754644e-01 7.14889592e-01 2.00365236e-01
+  1.87081411e-01 8.98532247e-01]
+ [9.76246245e-01 7.27600064e-02 6.40428736e-01 2.44633057e-01
+  2.13315320e-01 9.04923596e-01 1.01278843e-01 1.60040487e-02
+  9.57338433e-01 4.54609970e-01]
+ [9.70716394e-01 8.19560587e-01 8.00696646e-01 8.50286966e-01
+  6.01413231e-01 5.35401167e-01 5.09318595e-01 7.50871033e-01
+  1.77320278e-01 1.41563755e-01]
+ [5.58999323e-01 8.27823274e-02 2.45162718e-01 9.35805174e-01
+  3.97967922e-01 6.41998971e-01 2.85999165e-01 9.42305845e-01
+  6.51639460e-01 2.40037335e-01]
+ [2.07463601e-01 6.36551310e-01 3.74118364e-01 9.48754846e-01
+  6.84043753e-01 8.69630846e-01 2.90701292e-01 3.73794006e-01
+  9.56898405e-01 7.53969725e-01]
+ [8.69421667e-01 9.16907984e-01 2.60498453e-01 1.05394015e-01
+  8.28209931e-01 2.18273724e-01 5.70725653e-01 5.90412777e-01
+  3.15706255e-01 1.00031589e-01]
+ [3.13129360e-02 9.74280470e-01 1.56927535e-01 2.05412497e-01
+  3.29614274e-01 1.32235211e-01 9.66580589e-01 6.25369588e-01
+  5.91506268e-01 3.63969272e-01]]
      @@ -1305,13 +1318,13 @@ covariance matrix through the np.linalg.eig() function.

      -
      0.12144818928795939
-4.458165594343705
-0.07083732341182088
-[[ 1.01630108  3.10680954  3.23143243]
- [ 3.10680954 10.41130925  9.90126156]
- [ 3.23143243  9.90126156 15.84271739]]
-[24.2481237   0.07228932  2.94991469]
+
      -0.022796599742659046
+4.026047728952776
+-0.08626455960524361
+[[ 1.07037904  3.23910863  3.46675556]
+ [ 3.23910863 10.63080397 10.39964939]
+ [ 3.46675556 10.39964939 18.1421739 ]]
+[26.31853116  0.06730967  3.45751608]
      @@ -1536,7 +1549,7 @@ Name: Aragorn, dtype: object 
      ---------------------------------------------------------------------------
 AttributeError                            Traceback (most recent call last)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58960/1326197715.py in ?()
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23270/1326197715.py in ?()
 ----> 6 new_hobbit = {'First Name': ["Peregrin"],
       7               'Last Name': ["Took"],
       8               'Place of birth': ["Shire"],
diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html
index 312abccdb..0dd7b3a9d 100644
--- a/doc/LectureNotes/_build/html/week35.html
+++ b/doc/LectureNotes/_build/html/week35.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

      @@ -1021,7 +1024,7 @@ Since we are not using Scikit-Learn here we can define our own
      -
      0.9952260624201319
+
      0.9954482269282812
      @@ -1038,7 +1041,7 @@ Since we are not using Scikit-Learn here we can define our own
      -
      0.01138349118139106
+
      0.008936538168866639
      @@ -1053,23 +1056,23 @@ Since we are not using Scikit-Learn here we can define our own
      -
      [0.03126071 0.00179226 0.05225439 0.00372999 0.02734884 0.05333172
- 0.02020016 0.01521008 0.00965948 0.0056274  0.01055702 0.01870768
- 0.07847027 0.00766418 0.00385358 0.01374419 0.04181664 0.00933893
- 0.00312107 0.01294475 0.07413571 0.11552511 0.01637903 0.02176175
- 0.00854766 0.04706879 0.05041282 0.00575302 0.02289663 0.01263793
- 0.03325052 0.01806972 0.0066441  0.05997838 0.02861775 0.04882654
- 0.08480358 0.03542954 0.01143948 0.00619982 0.00278866 0.00995497
- 0.02001832 0.03635157 0.00732822 0.01114894 0.07709391 0.01125655
- 0.00816671 0.01114243 0.00550993 0.06590956 0.04805237 0.04394445
- 0.00825943 0.00935925 0.023291   0.05290093 0.01249864 0.02700809
- 0.00271844 0.01806574 0.01238543 0.03213764 0.02680834 0.02269712
- 0.02734783 0.00461485 0.0352873  0.01616311 0.00968169 0.02333957
- 0.00912667 0.03871805 0.01637128 0.02097868 0.00230726 0.08493636
- 0.05203452 0.0498517  0.01619572 0.02339214 0.03510055 0.04566385
- 0.0024319  0.02956756 0.05981231 0.00394204 0.02607172 0.02627472
- 0.02413198 0.00518272 0.02283931 0.01528494 0.01208287 0.02055319
- 0.01180608 0.01603956 0.02803521 0.07787677]
+
      [0.04030775 0.05734435 0.01740914 0.02044029 0.02486199 0.03573222
+ 0.00752453 0.0573522  0.00102185 0.04459142 0.03221764 0.00342095
+ 0.01898739 0.03160225 0.00450564 0.01979508 0.00152143 0.00857666
+ 0.00762566 0.00703236 0.00490815 0.02171406 0.04265231 0.03063862
+ 0.0273619  0.00668013 0.03013331 0.00788426 0.01275493 0.01698978
+ 0.02715714 0.0164838  0.00424614 0.06575617 0.03937194 0.0459393
+ 0.03321665 0.00857944 0.01458459 0.03839052 0.04106931 0.01962178
+ 0.01758251 0.06598382 0.01262347 0.05373197 0.00508233 0.02902908
+ 0.01284702 0.02266783 0.05481819 0.02857842 0.00817663 0.00478164
+ 0.02844986 0.03430817 0.00845377 0.01544721 0.00867529 0.01881047
+ 0.02900428 0.00525951 0.00516358 0.05597496 0.03021621 0.00458845
+ 0.0134627  0.00244413 0.00194759 0.04527404 0.02602746 0.00545794
+ 0.00747748 0.01109822 0.08180185 0.02539315 0.02321685 0.02859203
+ 0.01136476 0.00341608 0.00341404 0.00651846 0.05339963 0.01042549
+ 0.02637872 0.00639212 0.02177328 0.02384275 0.00133169 0.00699813
+ 0.0205469  0.04806452 0.01573671 0.05430636 0.01207681 0.0160536
+ 0.01157049 0.08333949 0.01405686 0.00212278]
      @@ -1138,15 +1141,15 @@ but now splitting the data into a training set and a test set.

      -
      [ 2.06741397 -0.19826431  5.13801381  0.12625265 -0.15070691]
+
      [ 2.04491327 -0.39593882  6.26462211 -1.4899045   0.59718501]
 Training R2
-0.9959102280411898
+0.9973808396916977
 Training MSE
-0.007840268112476161
+0.005799563292953798
 Test R2
-0.9923171995387728
+0.9950846163116835
 Test MSE
-0.016855069963547485
+0.011719944818178452
      @@ -2095,7 +2098,9 @@ MSE with intercept column 0.00411363461744314 MSE with intercept column from SKL 0.004113634617443147 -Manual intercept: 2.083766322923899 +
      +
      +
      Manual intercept: 2.083766322923899
 Fitted beta (wiothout intercept): [0.19569961 3.97898392]
 Sklearn intercept: 2.0837663229239043
 Sklearn fitted beta (without intercept): [0.19569961 3.97898392]
diff --git a/doc/LectureNotes/_build/html/week36.html b/doc/LectureNotes/_build/html/week36.html
index 8a8f0a36b..3a7c38468 100644
--- a/doc/LectureNotes/_build/html/week36.html
+++ b/doc/LectureNotes/_build/html/week36.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

      @@ -1166,7 +1169,7 @@ theorem.

      Bootstrap Statistics :
 original           bias      std. error
- 100.106  15.2126        100.109        0.153146
+ 100.088  15.1355        100.087        0.151546
      @@ -1190,7 +1193,7 @@ original bias std. error
      -_images/3c2235cc56c92d97d90ee28891636edca95a10ee0aaa7903396bc4ca0722bd51.png +_images/9369a99fab3ad63476a60dda7c34258ee45c7d9980558aabda81b593e0a3d307.png
      @@ -1386,9 +1389,7 @@ Error: 0.08426840630693411 Bias^2: 0.0796891867672603 Var: 0.004579219539673834 0.08426840630693411 >= 0.0796891867672603 + 0.004579219539673834 = 0.08426840630693413 - - -
      Polynomial degree: 2
+Polynomial degree: 2
 Error: 0.10398646080125035
 Bias^2: 0.1007711427354898
 Var: 0.0032153180657605116
@@ -1403,9 +1404,7 @@ Error: 0.06844519414009445
 Bias^2: 0.06453579006728324
 Var: 0.003909404072811226
 0.06844519414009445 >= 0.06453579006728324 + 0.003909404072811226 = 0.06844519414009446
-
      -
      -
      Polynomial degree: 5
+Polynomial degree: 5
 Error: 0.05227921801205686
 Bias^2: 0.0481872773043029
 Var: 0.004091940707753939
@@ -1420,9 +1419,7 @@ Error: 0.02760977349102253
 Bias^2: 0.022999498260366312
 Var: 0.004610275230656212
 0.02760977349102253 >= 0.022999498260366312 + 0.004610275230656212 = 0.027609773491022525
-
      -
      -
      Polynomial degree: 8
+Polynomial degree: 8
 Error: 0.017355848195593347
 Bias^2: 0.010331721306655127
 Var: 0.007024126888938232
@@ -1432,9 +1429,7 @@ Error: 0.02660572763718093
 Bias^2: 0.010018312644137363
 Var: 0.016587414993043573
 0.02660572763718093 >= 0.010018312644137363 + 0.016587414993043573 = 0.026605727637180936
-
      -
      -
      Polynomial degree: 10
+Polynomial degree: 10
 Error: 0.021592704588025025
 Bias^2: 0.010516485576645508
 Var: 0.011076219011379514
@@ -1449,9 +1444,7 @@ Error: 0.11547777218872497
 Bias^2: 0.01628578269596628
 Var: 0.09919198949275869
 0.11547777218872497 >= 0.01628578269596628 + 0.09919198949275869 = 0.11547777218872497
-
      -
      -
      Polynomial degree: 13
+Polynomial degree: 13
 Error: 0.22842468702219465
 Bias^2: 0.01975416527185249
 Var: 0.20867052175034223
@@ -1826,12 +1819,12 @@ Mean squared error on test data: 1.07641937
 Degree of polynomial:  12
 Mean squared error on training data: 0.00805074
 Mean squared error on test data: 0.04295757
-
      -
      -
      Degree of polynomial:  13
+Degree of polynomial:  13
 Mean squared error on training data: 0.00781918
 Mean squared error on test data: 0.56965674
-Degree of polynomial:  14
+
      +
      +
      Degree of polynomial:  14
 Mean squared error on training data: 0.00465099
 Mean squared error on test data: 0.28443039
 Degree of polynomial:  15
@@ -1840,45 +1833,43 @@ Mean squared error on test data: 568.47051432
 Degree of polynomial:  16
 Mean squared error on training data: 0.00325450
 Mean squared error on test data: 48.97630233
-
      -
      -
      Degree of polynomial:  17
+Degree of polynomial:  17
 Mean squared error on training data: 0.00242954
 Mean squared error on test data: 2.52780600
-Degree of polynomial:  18
+
      +
      +
      Degree of polynomial:  18
 Mean squared error on training data: 0.00219195
 Mean squared error on test data: 429.25695398
 Degree of polynomial:  19
 Mean squared error on training data: 0.00154853
 Mean squared error on test data: 239.97065359
-
      -
      -
      Degree of polynomial:  20
+Degree of polynomial:  20
 Mean squared error on training data: 0.00140846
 Mean squared error on test data: 1350.24493666
 Degree of polynomial:  21
 Mean squared error on training data: 0.00119688
 Mean squared error on test data: 1840.50530832
-Degree of polynomial:  22
+
      +
      +
      Degree of polynomial:  22
 Mean squared error on training data: 0.00092898
 Mean squared error on test data: 1184.60929685
 Degree of polynomial:  23
 Mean squared error on training data: 0.00089193
 Mean squared error on test data: 3892.17483760
-
      -
      -
      Degree of polynomial:  24
+Degree of polynomial:  24
 Mean squared error on training data: 0.00083355
 Mean squared error on test data: 1332.46736215
 Degree of polynomial:  25
 Mean squared error on training data: 0.00079904
 Mean squared error on test data: 7577.76690383
-Degree of polynomial:  26
-Mean squared error on training data: 0.00075590
-Mean squared error on test data: 1079.36895644
      -
      Degree of polynomial:  27
+
      Degree of polynomial:  26
+Mean squared error on training data: 0.00075590
+Mean squared error on test data: 1079.36895644
+Degree of polynomial:  27
 Mean squared error on training data: 0.00068091
 Mean squared error on test data: 3207.25343155
 Degree of polynomial:  28
@@ -1889,9 +1880,9 @@ Mean squared error on training data: 0.00063866
 Mean squared error on test data: 3099.60342978
 
      
 
      
-
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58983/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23293/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58983/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23293/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(testerror), label='Test Error')
 
      
 
      
@@ -1976,7 +1967,7 @@ Mean squared error on test data: 3099.60342978
 
      
 
      
 
      
-
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58983/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23293/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
   plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
 
      
 
      
diff --git a/doc/LectureNotes/_build/html/week38.html b/doc/LectureNotes/_build/html/week38.html
index 49970b23b..c0a815d8d 100644
--- a/doc/LectureNotes/_build/html/week38.html
+++ b/doc/LectureNotes/_build/html/week38.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

      @@ -1010,7 +1013,7 @@ which equals

      -
      <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x11ff98dc0>
+
      <mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x126ab3dc0>
 
      
 
      
 _images/6e7ee017368b6108792de2f8f1bc208a3f62fb66ba3ace91d0bb16b8c8313ed3.png
@@ -1068,7 +1071,7 @@ which equals
      -
      [<matplotlib.lines.Line2D at 0x126e8a9d0>]
+
      [<matplotlib.lines.Line2D at 0x126fb8b20>]
 
      
 
      
 _images/87c2ad9509e698434c22f92f294f3ea8a669bfd9e5cce9fc96ab15f997da69ab.png
@@ -1362,14 +1365,14 @@ when \(||\nabla_\beta C(\beta_k) || \
      -
      Eigenvalues of Hessian Matrix:[0.31668291 4.15666977]
-[[3.74118981]
- [3.28649035]]
-[[3.74118981]
- [3.28649035]]
+
      Eigenvalues of Hessian Matrix:[0.34526171 4.42891508]
+[[3.73401546]
+ [3.2913872 ]]
+[[3.73401546]
+ [3.2913872 ]]
 
      
 
      
-_images/ec89b6bb83d134561d17bfc7f7dc31aa6db94db6ece70faf26265ae81d65a613.png
+_images/3430c2e5bb18ad9f457af6ed6748c3fa16381ffd0271ca7c0c248e53b04e0fe6.png
      @@ -1397,9 +1400,9 @@ when \(||\nabla_\beta C(\beta_k) || \
      -
      [[4.23463496]
- [2.74399625]]
-[4.20387361] [2.73436201]
+
      [[3.96383084]
+ [2.94004512]]
+[3.99618085] [2.9822844]
      @@ -1499,14 +1502,14 @@ minimum of this function.

      -
      Eigenvalues of Hessian Matrix:[0.30988581 4.80658849]
-[[4.14058224]
- [2.99719628]]
-[[4.14080023]
- [2.99702696]]
+
      Eigenvalues of Hessian Matrix:[0.31443711 4.41185301]
+[[4.01380409]
+ [2.87614787]]
+[[4.0124208 ]
+ [2.87730544]]
 
      
 
      
-_images/b2b6d80c00707d89a55ad71288bf1803be1c0b2d7ecc12a02a5917e520a9cd6d.png
+_images/b4424ef99a3917f39ceb54b5132d60d97fff9ed86b004003694a9733a9d7e3c2.png
      @@ -1617,7 +1620,7 @@ minimum of this function.

      >29 f([0.00115631]) = 0.00000
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58998/394505933.py:33: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23327/394505933.py:33: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
   print('>%d f(%s) = %.5f' % (i, solution, solution_eval))
 
      
 
      
@@ -1728,7 +1731,7 @@ minimum of this function.
      
 >29 f([6.17748881e-07]) = 0.00000
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_58998/476849792.py:39: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23327/476849792.py:39: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
   print('>%d f(%s) = %.5f' % (i, solution, solution_eval))
 
      
 
      
@@ -2790,8 +2793,12 @@ first example shows results with ordinary leats squares.
      
 [[3.94499279]
  [3.03306538]]
 Eigenvalues of Hessian Matrix:[0.31248425 4.44418124]
-theta from own gd
-[[3.94499279]
+
      +
      +
      theta from own gd
+
      +
      +
      [[3.94499279]
  [3.03306538]]
      @@ -3084,16 +3091,16 @@ beta from own Newton code [[4.0586484] [3.0718316]] Eigenvalues of Hessian Matrix:[0.29860173 3.8931686 ] -theta from own gd + + +
      theta from own gd
 [[4.0586484]
  [3.0718316]]
      _images/df9f7915879c2371a6245aa34d566f5914b9f11786649103613f16a241fdf046.png 
      theta from own sdg
-
      -
      -
      [[4.02496085]
+[[4.02496085]
  [3.12081773]]
      @@ -3182,7 +3189,9 @@ Eigenvalues of Hessian Matrix:[0.27470622 4.24106503] theta from own gd [[3.95906059] [3.02296298]] -theta from own sdg with momentum + + +
      theta from own sdg with momentum
 [[3.95611042]
  [2.99475306]]
      diff --git a/doc/LectureNotes/_build/html/week40.html b/doc/LectureNotes/_build/html/week40.html index 1efcd205d..1e32e3884 100644 --- a/doc/LectureNotes/_build/html/week40.html +++ b/doc/LectureNotes/_build/html/week40.html @@ -257,6 +257,9 @@
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

      @@ -631,17 +634,17 @@ We summarize some of these here for the methods we hvae studied in project one, 
      Parameters for OLS using gradient descent
-[[3.633198  ]
- [4.1182509 ]
- [4.44504789]]
+[[4.18426686]
+ [2.51331698]
+ [5.22403374]]
 Parameters for Ridge using gradient descent
-[[3.81542861]
- [3.39782637]
- [4.80980326]]
+[[4.13677307]
+ [2.53440516]
+ [5.21680493]]
 Parameters for Lasso using gradient descent
-[[3.47633954]
- [4.56277628]
- [4.22765626]]
+[[3.40725426]
+ [4.31893412]
+ [4.43233965]]
      @@ -693,11 +696,11 @@ Parameters for Lasso using gradient descent [[4.] [3.] [5.]] -0 [-25.50604212] [-33.67380616] -1 [-1.07718279e-13] [-1.67721558e-13] -2 [-9.23705556e-16] [-1.2228689e-15] -3 [8.70414851e-16] [1.13386647e-15] -4 [-1.86517468e-15] [-2.57274984e-15] +0 [-21.31398212] [-28.05552428] +1 [-3.03046477e-14] [-1.33496431e-14] +2 [-7.99360578e-16] [-9.75256689e-16] +3 [1.77635684e-16] [2.91496906e-16] +4 [1.77635684e-16] [2.91496906e-16] beta from own Newton code [[4.] [3.] @@ -1039,18 +1042,18 @@ function.

      Own inversion
-[[4.06629035]
- [2.90434687]]
-Eigenvalues of Hessian Matrix:[0.28385837 4.07166128]
+[[3.93518583]
+ [3.05586004]]
+Eigenvalues of Hessian Matrix:[0.31020049 4.44279756]
 theta from own gd
-[[4.06629035]
- [2.90434687]]
+[[3.93518583]
+ [3.05586004]]
 theta from own sdg
-[[4.04098389]
- [2.9248133 ]]
+[[3.92681541]
+ [3.07007326]]
      -_images/2de542effe405ced9195c07c5d2061f8911ca88584548f85cb6d28df1f83d693.png +_images/81f5882581a53e21f2c20be0ef5d63c90956ddfe5d34db4fb8f83012ab7b8c4a.png
      @@ -1761,15 +1764,17 @@ first example shows results with ordinary leats squares.

      Own inversion
-[[4.05658312]
- [2.97504928]]
-Eigenvalues of Hessian Matrix:[0.31549215 4.38384294]
-theta from own gd
-[[4.05658312]
- [2.97504928]]
      -_images/1bf8fa67622cbeaf3f9c469a206d89f5936b2fade360d11439ef4a3e43cf6f31.png +
      [[3.95215017]
+ [2.98138287]]
+Eigenvalues of Hessian Matrix:[0.28889272 3.80178359]
+theta from own gd
+[[3.95215017]
+ [2.98138287]]
+
      +
      +_images/5a1776c87da1ba203336178287b47b42ec33ec260ba18e3e2bb9d9f8ed8c8134.png
      @@ -1837,73 +1842,73 @@ theta from own gd 
      Own inversion
 [[4.]
  [3.]]
-Eigenvalues of Hessian Matrix:[0.32421852 4.40457549]
-0 [-8.91626016] [-10.12979569]
-1 [-0.25100049] [0.20953861]
-2 [-0.23252447] [0.19411458]
-3 [-0.21540847] [0.17982591]
-4 [-0.19955236] [0.16658901]
-5 [-0.18486342] [0.15432648]
-6 [-0.17125572] [0.14296659]
-7 [-0.15864967] [0.13244289]
-8 [-0.14697155] [0.12269384]
-9 [-0.13615305] [0.11366241]
-10 [-0.1261309] [0.10529578]
-11 [-0.11684647] [0.09754501]
-12 [-0.10824546] [0.09036477]
-13 [-0.10027757] [0.08371307]
-14 [-0.09289619] [0.07755099]
-15 [-0.08605815] [0.07184251]
-16 [-0.07972345] [0.06655422]
-17 [-0.07385505] [0.06165519]
-18 [-0.06841862] [0.05711679]
-19 [-0.06338236] [0.05291245]
-20 [-0.05871681] [0.04901759]
-21 [-0.0543947] [0.04540943]
-22 [-0.05039073] [0.04206687]
-23 [-0.0466815] [0.03897035]
-24 [-0.0432453] [0.03610176]
-25 [-0.04006203] [0.03344433]
-26 [-0.03711309] [0.03098251]
-27 [-0.03438121] [0.0287019]
-28 [-0.03185043] [0.02658917]
-29 [-0.02950594] [0.02463195]
+Eigenvalues of Hessian Matrix:[0.29823315 4.72747492]
+0 [-17.30620124] [-21.78309726]
+1 [-0.05759628] [0.04549505]
+2 [-0.05396282] [0.04262499]
+3 [-0.05055857] [0.03993599]
+4 [-0.04736908] [0.03741662]
+5 [-0.0443808] [0.03505619]
+6 [-0.04158103] [0.03284467]
+7 [-0.03895789] [0.03077266]
+8 [-0.03650022] [0.02883136]
+9 [-0.03419761] [0.02701253]
+10 [-0.03204025] [0.02530845]
+11 [-0.03001898] [0.02371186]
+12 [-0.02812523] [0.022216]
+13 [-0.02635095] [0.0208145]
+14 [-0.0246886] [0.01950141]
+15 [-0.02313112] [0.01827117]
+16 [-0.02167189] [0.01711853]
+17 [-0.02030472] [0.0160386]
+18 [-0.01902379] [0.01502681]
+19 [-0.01782367] [0.01407884]
+20 [-0.01669927] [0.01319068]
+21 [-0.01564579] [0.01235854]
+22 [-0.01465878] [0.0115789]
+23 [-0.01373403] [0.01084844]
+24 [-0.01286761] [0.01016407]
+25 [-0.01205586] [0.00952287]
+26 [-0.01129531] [0.00892212]
+27 [-0.01058275] [0.00835927]
+28 [-0.00991513] [0.00783192]
+29 [-0.00928964] [0.00733784]
 theta from own gd
-[[3.9156926 ]
- [3.07038096]]
-0 [-0.02733402] [0.02281881]
-1 [-0.02532198] [0.02113913]
-2 [-0.02285443] [0.01907918]
-3 [-0.02043186] [0.01705679]
-4 [-0.01820111] [0.01519453]
-5 [-0.01619211] [0.01351739]
-6 [-0.01439752] [0.01201924]
-7 [-0.01279935] [0.01068507]
-8 [-0.01137774] [0.00949829]
-9 [-0.01011375] [0.0084431]
-10 [-0.00899009] [0.00750504]
-11 [-0.00799123] [0.00667119]
-12 [-0.00710334] [0.00592997]
-13 [-0.0063141] [0.0052711]
-14 [-0.00561255] [0.00468544]
-15 [-0.00498895] [0.00416484]
-16 [-0.00443464] [0.0037021]
-17 [-0.00394191] [0.00329076]
-18 [-0.00350393] [0.00292513]
-19 [-0.00311462] [0.00260012]
-20 [-0.00276855] [0.00231123]
-21 [-0.00246094] [0.00205443]
-22 [-0.00218751] [0.00182617]
-23 [-0.00194446] [0.00162326]
-24 [-0.00172842] [0.0014429]
-25 [-0.00153637] [0.00128259]
-26 [-0.00136567] [0.00114008]
-27 [-0.00121393] [0.00101341]
-28 [-0.00107905] [0.00090081]
-29 [-0.00095916] [0.00080072]
+[[3.97081612]
+ [3.02305221]]
+0 [-0.0087036] [0.00687493]
+1 [-0.00815453] [0.00644123]
+2 [-0.00747538] [0.00590477]
+3 [-0.00680005] [0.00537133]
+4 [-0.00616847] [0.00487245]
+5 [-0.00558986] [0.0044154]
+6 [-0.00506364] [0.00399975]
+7 [-0.00458633] [0.00362272]
+8 [-0.00415381] [0.00328108]
+9 [-0.00376201] [0.0029716]
+10 [-0.00340714] [0.00269129]
+11 [-0.00308574] [0.00243742]
+12 [-0.00279466] [0.00220749]
+13 [-0.00253103] [0.00199925]
+14 [-0.00229228] [0.00181066]
+15 [-0.00207604] [0.00163985]
+16 [-0.0018802] [0.00148516]
+17 [-0.00170284] [0.00134506]
+18 [-0.0015422] [0.00121818]
+19 [-0.00139672] [0.00110327]
+20 [-0.00126497] [0.00099919]
+21 [-0.00114564] [0.00090494]
+22 [-0.00103757] [0.00081957]
+23 [-0.00093969] [0.00074226]
+24 [-0.00085105] [0.00067224]
+25 [-0.00077077] [0.00060883]
+26 [-0.00069806] [0.00055139]
+27 [-0.00063221] [0.00049938]
+28 [-0.00057257] [0.00045227]
+29 [-0.00051856] [0.00040961]
 theta from own gd wth momentum
-[[3.99737032]
- [3.00219529]]
+[[3.99842525]
+ [3.00124389]]
      @@ -1992,22 +1997,20 @@ theta from own gd wth momentum 
      Own inversion
-[[3.80284827]
- [3.14558948]]
-Eigenvalues of Hessian Matrix:[0.33477228 4.32758606]
+[[3.93505095]
+ [3.03255202]]
+Eigenvalues of Hessian Matrix:[0.35715186 4.00152151]
 
      theta from own gd
+[[3.93505095]
+ [3.03255202]]
      -
      [[3.80284827]
- [3.14558948]]
-
      -
      -_images/0fe401714ec88e67e8834af30a476bb19ae5e8a9677eb38e4161007bae099f84.png +_images/327366ea5051007f61df82fc528a36028c298264f7cb3d1565bb97cd4c20255c.png 
      theta from own sdg
-[[3.8286293 ]
- [3.19901555]]
+[[3.93535203]
+ [3.02339886]]
      @@ -2089,19 +2092,15 @@ Eigenvalues of Hessian Matrix:[0.33477228 4.32758606] 
      Own inversion
-[[4.14809298]
- [2.84983419]]
-Eigenvalues of Hessian Matrix:[0.35213219 4.34352956]
+[[4.00794132]
+ [3.02667095]]
+Eigenvalues of Hessian Matrix:[0.29709966 4.68284888]
 theta from own gd
-[[4.14764332]
- [2.85021125]]
-
      -
      -
      theta from own sdg with momentum
-
      -
      -
      [[4.15364228]
- [2.8846002 ]]
+[[4.0068889 ]
+ [3.02750941]]
+theta from own sdg with momentum
+[[4.01544241]
+ [3.0176573 ]]
      @@ -2170,9 +2169,9 @@ theta from own gd 
      theta from own AdaGrad
-[[2.00000539]
- [2.99996887]
- [4.0000263 ]]
+[[1.99856653]
+ [3.01053812]
+ [3.99013096]]
      @@ -2248,9 +2247,9 @@ theta from own gd 
      theta from own RMSprop
-[[2.00051557]
- [2.99775568]
- [4.00224621]]
+[[2.00002452]
+ [2.99973034]
+ [4.00026406]]
      @@ -2330,9 +2329,9 @@ theta from own gd 
      theta from own ADAM
-[[1.99997454]
- [3.00018485]
- [3.99980356]]
+[[1.99997633]
+ [3.00007035]
+ [3.99994513]]
      @@ -2453,7 +2452,7 @@ It provides composable transformations of Python+NumPy programs: differentiate, return asarray(x, dtype=self.dtype) -
      [<matplotlib.lines.Line2D at 0x1109cc4f0>]
+
      [<matplotlib.lines.Line2D at 0x1323c4070>]
 
      
 
      
 _images/02e94753795fba95a52acc4488a9ab82172ba372cbb7c1cc79a3e88c3ba03d69.png
@@ -2488,7 +2487,7 @@ It provides composable transformations of Python+NumPy programs: differentiate,
      -
      <matplotlib.collections.PathCollection at 0x110957c40>
+
      <matplotlib.collections.PathCollection at 0x133fa12e0>
 
      
 
      
 _images/60f34a975702165b9275820e0dda6259f453a92c5ad587b37191bb47a9c67ae9.png
diff --git a/doc/LectureNotes/_build/html/week41.html b/doc/LectureNotes/_build/html/week41.html
index 2682e5f49..d7ccb38fb 100644
--- a/doc/LectureNotes/_build/html/week41.html
+++ b/doc/LectureNotes/_build/html/week41.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

    Projects

      @@ -2379,7 +2382,7 @@ the Hadamard product, meaning element-wise multiplication.

      Old accuracy on training data: 0.1440501043841336
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2715,7 +2718,7 @@ Lambda =  10.0
 Accuracy score on test set:  0.19166666666666668
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2724,7 +2727,7 @@ Lambda =  1e-05
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2733,7 +2736,7 @@ Lambda =  0.0001
 Accuracy score on test set:  0.08611111111111111
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2742,7 +2745,7 @@ Lambda =  0.001
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2751,7 +2754,7 @@ Lambda =  0.01
 Accuracy score on test set:  0.08888888888888889
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2760,7 +2763,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.08611111111111111
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2769,7 +2772,7 @@ Lambda =  1.0
 Accuracy score on test set:  0.08888888888888889
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2778,11 +2781,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.09166666666666666
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2791,11 +2794,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2804,11 +2807,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2817,11 +2820,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2830,11 +2833,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2843,7 +2846,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -2852,11 +2855,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2865,11 +2868,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2878,11 +2881,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2891,11 +2894,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2904,11 +2907,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2917,11 +2920,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2930,11 +2933,11 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -2943,11 +2946,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -3000,15 +3003,15 @@ Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59019/953065564.py:4: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23356/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -3159,218 +3162,31 @@ Lambda =  0.1
 Accuracy score on test set:  0.8888888888888888
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      Learning rate  =  0.0001
 Lambda =  1.0
-Accuracy score on test set:  0.8805555555555555
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.0001
-Lambda =  10.0
-Accuracy score on test set:  0.8944444444444445
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  1e-05
-Accuracy score on test set:  0.975
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  0.0001
-Accuracy score on test set:  0.9777777777777777
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  0.001
-Accuracy score on test set:  0.9805555555555555
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  0.01
-Accuracy score on test set:  0.9861111111111112
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  0.1
-Accuracy score on test set:  0.9805555555555555
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  1.0
-Accuracy score on test set:  0.9777777777777777
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.001
-Lambda =  10.0
-Accuracy score on test set:  0.9444444444444444
-
      -
      -
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
      -
      -
      Learning rate  =  0.01
-Lambda =  1e-05
-Accuracy score on test set:  0.9861111111111112
-
      -
      -
      Learning rate  =  0.01
-Lambda =  0.0001
-Accuracy score on test set:  0.9888888888888889
-
      -
      -
      Learning rate  =  0.01
-Lambda =  0.001
-Accuracy score on test set:  0.9888888888888889
-
      -
      -
      Learning rate  =  0.01
-Lambda =  0.01
-Accuracy score on test set:  0.9861111111111112
-
      -
      -
      Learning rate  =  0.01
-Lambda =  0.1
-Accuracy score on test set:  0.9888888888888889
-
      -
      -
      Learning rate  =  0.01
-Lambda =  1.0
-Accuracy score on test set:  0.9722222222222222
-
      -
      -
      Learning rate  =  0.01
-Lambda =  10.0
-Accuracy score on test set:  0.9527777777777777
-
      -
      -
      Learning rate  =  0.1
-Lambda =  1e-05
-Accuracy score on test set:  0.9027777777777778
-
-Learning rate  =  0.1
-Lambda =  0.0001
 Accuracy score on test set:  0.8583333333333333
      -
      Learning rate  =  0.1
-Lambda =  0.001
-Accuracy score on test set:  0.8722222222222222
+
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:698: UserWarning: Training interrupted by user.
+  warnings.warn("Training interrupted by user.")
 
      
 
      
-
      Learning rate  =  0.1
-Lambda =  0.01
-Accuracy score on test set:  0.9055555555555556
-
-Learning rate  =  0.1
-Lambda =  0.1
-Accuracy score on test set:  0.8805555555555555
+
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
 
      
 
      
-
      Learning rate  =  0.1
-Lambda =  1.0
-Accuracy score on test set:  0.8722222222222222
-
-Learning rate  =  0.1
+
      Learning rate  =  0.0001
 Lambda =  10.0
-Accuracy score on test set:  0.8666666666666667
+Accuracy score on test set:  0.8361111111111111
 
      
 
      
-
      Learning rate  =  1.0
+
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      
+
      
+
      Learning rate  =  0.001
 Lambda =  1e-05
-Accuracy score on test set:  0.08611111111111111
-
-Learning rate  =  1.0
-Lambda =  0.0001
-Accuracy score on test set:  0.10555555555555556
-
      
-
      
-
      Learning rate  =  1.0
-Lambda =  0.001
-Accuracy score on test set:  0.10555555555555556
-
-Learning rate  =  1.0
-Lambda =  0.01
-Accuracy score on test set:  0.17777777777777778
-
-Learning rate  =  1.0
-Lambda =  0.1
-Accuracy score on test set:  0.08333333333333333
-
      
-
      
-
      Learning rate  =  1.0
-Lambda =  1.0
-Accuracy score on test set:  0.08888888888888889
-
-Learning rate  =  1.0
-Lambda =  10.0
-Accuracy score on test set:  0.09444444444444444
-
      
-
      
-
      Learning rate  =  10.0
-Lambda =  1e-05
-Accuracy score on test set:  0.17222222222222222
-
-Learning rate  =  10.0
-Lambda =  0.0001
-Accuracy score on test set:  0.11666666666666667
-
-Learning rate  =  10.0
-Lambda =  0.001
-Accuracy score on test set:  0.10555555555555556
-
-Learning rate  =  10.0
-Lambda =  0.01
-Accuracy score on test set:  0.1388888888888889
-
      
-
      
-
      Learning rate  =  10.0
-Lambda =  0.1
-Accuracy score on test set:  0.11388888888888889
-
      
-
      
-
      Learning rate  =  10.0
-Lambda =  1.0
-Accuracy score on test set:  0.10555555555555556
-
-Learning rate  =  10.0
-Lambda =  10.0
-Accuracy score on test set:  0.09444444444444444
+Accuracy score on test set:  0.9777777777777777
 
      
 
      
 
      
@@ -3417,10 +3233,6 @@ Accuracy score on test set:  0.09444444444444444
 
      
 
      
 
      
-
      
-_images/511c09db890499149b1572167479e7c928860b0aaed18526447cb972a68957dd.png
-_images/d189a2d67f3b4e9b555480356aed0829649f8bb2c6cfe1d6fbd861f8f33d0c50.png
-
      
 
      
 
 
      
@@ -3459,14 +3271,6 @@ how simple solving a machine learning problem can be.
      -
      -
        Cell In[14], line 1
-    pip3 install tensorflow
-         ^
-SyntaxError: invalid syntax
-
      -
      -

      and/or if you use anaconda, just write (or install from the graphical user interface) (current release of CPU-only TensorFlow)

      diff --git a/doc/LectureNotes/_build/html/week43.html b/doc/LectureNotes/_build/html/week43.html index 479bdb87d..c0791c021 100644 --- a/doc/LectureNotes/_build/html/week43.html +++ b/doc/LectureNotes/_build/html/week43.html @@ -257,6 +257,9 @@
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

      @@ -1403,7 +1406,7 @@ the Hadamard product, meaning element-wise multiplication.

      Old accuracy on training data: 0.1440501043841336
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1739,7 +1742,7 @@ Lambda =  10.0
 Accuracy score on test set:  0.19166666666666668
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1748,7 +1751,7 @@ Lambda =  1e-05
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1757,7 +1760,7 @@ Lambda =  0.0001
 Accuracy score on test set:  0.08611111111111111
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1766,7 +1769,7 @@ Lambda =  0.001
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1775,7 +1778,7 @@ Lambda =  0.01
 Accuracy score on test set:  0.08888888888888889
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1784,7 +1787,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.08611111111111111
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1793,7 +1796,7 @@ Lambda =  1.0
 Accuracy score on test set:  0.08888888888888889
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1802,11 +1805,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.09166666666666666
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1815,11 +1818,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1828,11 +1831,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1841,11 +1844,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1854,11 +1857,11 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1867,7 +1870,7 @@ Lambda =  0.1
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
 
      
 
      
@@ -1876,11 +1879,11 @@ Lambda =  1.0
 Accuracy score on test set:  0.10555555555555556
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1889,11 +1892,11 @@ Lambda =  10.0
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1902,11 +1905,11 @@ Lambda =  1e-05
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1915,11 +1918,11 @@ Lambda =  0.0001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1928,11 +1931,11 @@ Lambda =  0.001
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
@@ -1941,38 +1944,43 @@ Lambda =  0.01
 Accuracy score on test set:  0.07777777777777778
      -
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/953065564.py:4: RuntimeWarning: overflow encountered in exp
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
   return 1/(1 + np.exp(-x))
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
   exp_term = np.exp(self.z_o)
-/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_59025/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
   self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
 
      
 
      
-
      ---------------------------------------------------------------------------
-KeyboardInterrupt                         Traceback (most recent call last)
-Cell In[8], line 11
-      8 for j, lmbd in enumerate(lmbd_vals):
-      9     dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
-     10                         n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
----> 11     dnn.train()
-     13     DNN_numpy[i][j] = dnn
-     15     test_predict = dnn.predict(X_test)
-
-Cell In[6], line 98, in NeuralNetwork.train(self)
-     95 self.X_data = self.X_data_full[chosen_datapoints]
-     96 self.Y_data = self.Y_data_full[chosen_datapoints]
----> 98 self.feed_forward()
-     99 self.backpropagation()
-
-Cell In[6], line 38, in NeuralNetwork.feed_forward(self)
-     36 def feed_forward(self):
-     37     # feed-forward for training
----> 38     self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
-     39     self.a_h = sigmoid(self.z_h)
-     41     self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
-
-KeyboardInterrupt: 
+
      Learning rate  =  10.0
+Lambda =  0.1
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  1.0
+Accuracy score on test set:  0.07777777777777778
+
      
+
      
+
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:43: RuntimeWarning: overflow encountered in exp
+  exp_term = np.exp(self.z_o)
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/1630775253.py:44: RuntimeWarning: invalid value encountered in divide
+  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
+
      
+
      
+
      Learning rate  =  10.0
+Lambda =  10.0
+Accuracy score on test set:  0.07777777777777778
 
      
 
      
 
      
@@ -2018,6 +2026,22 @@ Accuracy score on test set:  0.07777777777777778
      +
      +
      /var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_23401/953065564.py:4: RuntimeWarning: overflow encountered in exp
+  return 1/(1 + np.exp(-x))
+
      +
      +_images/e7be2ceafe348b5d5f0e390d3915e62a6f18c62d65f8255a2252346c88a1a2bf.png +_images/4021319392c5e4b499c0bad3ee2a1e5eda71857601b96f9cf9ff5d7d32397a0c.png +
      @@ -2053,6 +2077,329 @@ performance overall.

      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  1e-05
+Accuracy score on test set:  0.18333333333333332
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  0.0001
+Accuracy score on test set:  0.18611111111111112
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  0.001
+Accuracy score on test set:  0.13055555555555556
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  0.01
+Accuracy score on test set:  0.24444444444444444
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  0.1
+Accuracy score on test set:  0.23333333333333334
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  1.0
+Accuracy score on test set:  0.12777777777777777
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  1e-05
+Lambda =  10.0
+Accuracy score on test set:  0.1527777777777778
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  1e-05
+Accuracy score on test set:  0.9111111111111111
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  0.0001
+Accuracy score on test set:  0.8888888888888888
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  0.001
+Accuracy score on test set:  0.8722222222222222
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  0.01
+Accuracy score on test set:  0.8305555555555556
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  0.1
+Accuracy score on test set:  0.8888888888888888
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  1.0
+Accuracy score on test set:  0.8805555555555555
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.0001
+Lambda =  10.0
+Accuracy score on test set:  0.8944444444444445
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  1e-05
+Accuracy score on test set:  0.975
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  0.0001
+Accuracy score on test set:  0.9777777777777777
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  0.001
+Accuracy score on test set:  0.9805555555555555
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  0.01
+Accuracy score on test set:  0.9861111111111112
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  0.1
+Accuracy score on test set:  0.9805555555555555
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  1.0
+Accuracy score on test set:  0.9777777777777777
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.001
+Lambda =  10.0
+Accuracy score on test set:  0.9444444444444444
+
      +
      +
      /Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/neural_network/_multilayer_perceptron.py:691: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
+  warnings.warn(
+
      +
      +
      Learning rate  =  0.01
+Lambda =  1e-05
+Accuracy score on test set:  0.9861111111111112
+
      +
      +
      Learning rate  =  0.01
+Lambda =  0.0001
+Accuracy score on test set:  0.9888888888888889
+
      +
      +
      Learning rate  =  0.01
+Lambda =  0.001
+Accuracy score on test set:  0.9888888888888889
+
      +
      +
      Learning rate  =  0.01
+Lambda =  0.01
+Accuracy score on test set:  0.9861111111111112
+
      +
      +
      Learning rate  =  0.01
+Lambda =  0.1
+Accuracy score on test set:  0.9888888888888889
+
      +
      +
      Learning rate  =  0.01
+Lambda =  1.0
+Accuracy score on test set:  0.9722222222222222
+
+Learning rate  =  0.01
+Lambda =  10.0
+Accuracy score on test set:  0.9527777777777777
+
      +
      +
      Learning rate  =  0.1
+Lambda =  1e-05
+Accuracy score on test set:  0.9027777777777778
+
+Learning rate  =  0.1
+Lambda =  0.0001
+Accuracy score on test set:  0.8583333333333333
+
      +
      +
      Learning rate  =  0.1
+Lambda =  0.001
+Accuracy score on test set:  0.8722222222222222
+
      +
      +
      Learning rate  =  0.1
+Lambda =  0.01
+Accuracy score on test set:  0.9055555555555556
+
+Learning rate  =  0.1
+Lambda =  0.1
+Accuracy score on test set:  0.8805555555555555
+
      +
      +
      Learning rate  =  0.1
+Lambda =  1.0
+Accuracy score on test set:  0.8722222222222222
+
      +
      +
      Learning rate  =  0.1
+Lambda =  10.0
+Accuracy score on test set:  0.8666666666666667
+
+Learning rate  =  1.0
+Lambda =  1e-05
+Accuracy score on test set:  0.08611111111111111
+
      +
      +
      Learning rate  =  1.0
+Lambda =  0.0001
+Accuracy score on test set:  0.10555555555555556
+
+Learning rate  =  1.0
+Lambda =  0.001
+Accuracy score on test set:  0.10555555555555556
+
      +
      +
      Learning rate  =  1.0
+Lambda =  0.01
+Accuracy score on test set:  0.17777777777777778
+
+Learning rate  =  1.0
+Lambda =  0.1
+Accuracy score on test set:  0.08333333333333333
+
+Learning rate  =  1.0
+Lambda =  1.0
+Accuracy score on test set:  0.08888888888888889
+
      +
      +
      Learning rate  =  1.0
+Lambda =  10.0
+Accuracy score on test set:  0.09444444444444444
+
+Learning rate  =  10.0
+Lambda =  1e-05
+Accuracy score on test set:  0.17222222222222222
+
      +
      +
      Learning rate  =  10.0
+Lambda =  0.0001
+Accuracy score on test set:  0.11666666666666667
+
+Learning rate  =  10.0
+Lambda =  0.001
+Accuracy score on test set:  0.10555555555555556
+
+Learning rate  =  10.0
+Lambda =  0.01
+Accuracy score on test set:  0.1388888888888889
+
+Learning rate  =  10.0
+Lambda =  0.1
+Accuracy score on test set:  0.11388888888888889
+
      +
      +
      Learning rate  =  10.0
+Lambda =  1.0
+Accuracy score on test set:  0.10555555555555556
+
+Learning rate  =  10.0
+Lambda =  10.0
+Accuracy score on test set:  0.09444444444444444
+
      +
      +
      @@ -2096,6 +2443,10 @@ performance overall.

      +
      +_images/511c09db890499149b1572167479e7c928860b0aaed18526447cb972a68957dd.png +_images/d189a2d67f3b4e9b555480356aed0829649f8bb2c6cfe1d6fbd861f8f33d0c50.png +
      @@ -2134,6 +2485,14 @@ how simple solving a machine learning problem can be.

      +
      +
        Cell In[12], line 1
+    pip3 install tensorflow
+         ^
+SyntaxError: invalid syntax
+
      +
      +

      and/or if you use anaconda, just write (or install from the graphical user interface) (current release of CPU-only TensorFlow)

      diff --git a/doc/LectureNotes/_build/html/week44.html b/doc/LectureNotes/_build/html/week44.html index 371cb2f57..f3ea96e35 100644 --- a/doc/LectureNotes/_build/html/week44.html +++ b/doc/LectureNotes/_build/html/week44.html @@ -257,6 +257,9 @@
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
    • Exercise week 47
      • +
    • Week 48: Gradient boosting and summary of course
      • +
    • Exercises week 48
      • +

    Projects

      @@ -1574,40 +1577,24 @@ labels = (n_inputs) = (1797,) super().__init__(activity_regularizer=activity_regularizer, **kwargs) -
       1/12 ━━━━━━━━━━━━━━━━━━━━ 1s 131ms/step - accuracy: 0.0938 - loss: 3.2865
+
       1/12 ━━━━━━━━━━━━━━━━━━━━ 1s 176ms/step - accuracy: 0.0625 - loss: 2.4493
 
      
 
      
 
      
- 9/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.1090 - loss: 3.1259  
+ 8/12 ━━━━━━━━━━━━━━━━━━━━ 0s 8ms/step - accuracy: 0.1149 - loss: 2.4557  
 
      
 
      
 
      
-12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.1140 - loss: 3.1040
+12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 31ms/step - accuracy: 0.1176 - loss: 2.4545
+
      
+
      
+
      
+12/12 ━━━━━━━━━━━━━━━━━━━━ 1s 32ms/step - accuracy: 0.1184 - loss: 2.4540
 
      
 
      
 
      Learning rate =  1e-05
 Lambda =  1e-05
-Test accuracy: 0.125
-
      
-
      
-
       1/12 ━━━━━━━━━━━━━━━━━━━━ 1s 133ms/step - accuracy: 0.2812 - loss: 2.6529
-
      
-
      
-
      
- 8/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.2350 - loss: 2.5504  
-
      
-
      
-
      
-12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.2247 - loss: 2.5413
-
      
-
      
-
      
-12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 10ms/step - accuracy: 0.2230 - loss: 2.5413
-
      
-
      
-
      Learning rate =  1e-05
-Lambda =  0.0001
-Test accuracy: 0.203
+Test accuracy: 0.128
 
      
 
      
 
      ---------------------------------------------------------------------------
diff --git a/doc/LectureNotes/_build/html/week45.html b/doc/LectureNotes/_build/html/week45.html
index 8f9cd60a9..515cb5726 100644
--- a/doc/LectureNotes/_build/html/week45.html
+++ b/doc/LectureNotes/_build/html/week45.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

      @@ -941,40 +944,36 @@ labels = (n_inputs) = (1797,) super().__init__(activity_regularizer=activity_regularizer, **kwargs) -
       1/12 ━━━━━━━━━━━━━━━━━━━━ 2s 214ms/step - accuracy: 0.1250 - loss: 3.1816
+
       1/12 ━━━━━━━━━━━━━━━━━━━━ 2s 203ms/step - accuracy: 0.2812 - loss: 2.7253
 
      
 
      
 
      
- 8/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.1078 - loss: 3.0683  
+ 9/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.1957 - loss: 2.8906  
 
      
 
      
 
      
-12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 10ms/step - accuracy: 0.1005 - loss: 3.0830
-
      
-
      
-
      
-12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 10ms/step - accuracy: 0.0998 - loss: 3.0851
+12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.1826 - loss: 2.9261
 
      
 
      
 
      Learning rate =  1e-05
 Lambda =  1e-05
-Test accuracy: 0.092
+Test accuracy: 0.150
 
      
 
      
-
       1/12 ━━━━━━━━━━━━━━━━━━━━ 1s 128ms/step - accuracy: 0.0312 - loss: 3.1505
+
       1/12 ━━━━━━━━━━━━━━━━━━━━ 1s 126ms/step - accuracy: 0.0938 - loss: 2.8780
 
      
 
      
 
      
- 8/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.0554 - loss: 3.1156  
+ 9/12 ━━━━━━━━━━━━━━━━━━━━ 0s 7ms/step - accuracy: 0.1082 - loss: 2.8294  
 
      
 
      
 
      
-12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.0560 - loss: 3.1184
+12/12 ━━━━━━━━━━━━━━━━━━━━ 0s 9ms/step - accuracy: 0.1134 - loss: 2.8359
 
      
 
      
 
      Learning rate =  1e-05
 Lambda =  0.0001
-Test accuracy: 0.058
+Test accuracy: 0.125
 
      
 
      
 
      ---------------------------------------------------------------------------
diff --git a/doc/LectureNotes/_build/html/week46.html b/doc/LectureNotes/_build/html/week46.html
index 7bd5a23de..5f1ed3788 100644
--- a/doc/LectureNotes/_build/html/week46.html
+++ b/doc/LectureNotes/_build/html/week46.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

      @@ -693,13 +696,13 @@ predicting the target features of query instances is as follows:

      2nd degree coefficients:
-zero power:  1.391131752817723
-first power:  0.023929544236763082
-second power:  -0.00011233101388608391
+zero power:  -9.690066070734556
+first power:  0.2579633987876525
+second power:  -0.0008519856133366335
      -_images/54da4dcf618f83f729a4226ed97b89f907d17d25d6f1e00ebabeaabefc39c127.png -_images/eb8a1d5930cd9fc5960f605b1df68a8d022a1f3d3f06c78dd6848a2fc9de92be.png +_images/8e8566a48272bbcd741d192ea8f10c53172ad3cba8f805ceb38c6695c7dd5c2d.png +_images/eb601133698cb5198be6ed45531c33666c9fa7198b8b8a8e59f3cf4e95ddc8eb.png
      @@ -2184,7 +2187,9 @@ SVC 0.896 
      LogisticRegression 0.864
-RandomForestClassifier 0.896
+
      +
      +
      RandomForestClassifier 0.896
 SVC 0.896
      @@ -2302,7 +2307,9 @@ Error: 0.06380941468319971 Bias^2: 0.05160313473529168 Var: 0.01220627994790804 0.06380941468319971 >= 0.05160313473529168 + 0.01220627994790804 = 0.06380941468319971 -Polynomial degree: 2 +
      + +
      Polynomial degree: 2
 Error: 0.043464037468677004
 Bias^2: 0.02659851591375224
 Var: 0.01686552155492476
@@ -2312,9 +2319,7 @@ Error: 0.020716391693769383
 Bias^2: 0.01159033914386312
 Var: 0.00912605254990626
 0.020716391693769383 >= 0.01159033914386312 + 0.00912605254990626 = 0.02071639169376938
-
      -
      -
      Polynomial degree: 4
+Polynomial degree: 4
 Error: 0.02063627410934057
 Bias^2: 0.0117496656370668
 Var: 0.008886608472273775
@@ -2324,7 +2329,9 @@ Error: 0.02087627881701288
 Bias^2: 0.01349183949256158
 Var: 0.007384439324451296
 0.02087627881701288 >= 0.01349183949256158 + 0.007384439324451296 = 0.020876278817012876
-Polynomial degree: 6
+
      +
      +
      Polynomial degree: 6
 Error: 0.02069601123831537
 Bias^2: 0.013918526350129823
 Var: 0.0067774848881855445
@@ -2463,9 +2470,7 @@ this setting.
      
 
      (426, 30)
 (143, 30)
 Test set accuracy Logistic Regression with scaled data: 0.96
-
      
-
      
-
      Test set accuracy SVM with scaled data: 0.96
+Test set accuracy SVM with scaled data: 0.96
 Test set accuracy with Decision Trees and scaled data: 0.87
 
      
 
      
diff --git a/doc/LectureNotes/_build/html/week47.html b/doc/LectureNotes/_build/html/week47.html
index 1871ef282..8ecd398b5 100644
--- a/doc/LectureNotes/_build/html/week47.html
+++ b/doc/LectureNotes/_build/html/week47.html
@@ -257,6 +257,9 @@
 
    • Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods
      • 
 
    • Exercise week 47
      • 
 
+
    • Week 48: Gradient boosting  and summary of course
      • 
+
    • Exercises week 48
      • 
+

    Projects

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"30071d16", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "268eaea0", + "metadata": { + "editable": true + }, + "source": [ + "# Exercises week 48\n", + "**November 25-29, 2024**\n", + "\n", + "Date: **Deadline is Friday November 29 at midnight**" + ] + }, + { + "cell_type": "markdown", + "id": "1e04cbee", + "metadata": { + "editable": true + }, + "source": [ + "# Overarching aims of the exercises this week\n", + "\n", + "The exercise this week is a simple course survey and feedback. This\n", + "is important for us in order to improve our teaching material, the\n", + "active learning format and anything else related to a succesful\n", + "mastering of central machine learning methods and their applications." + ] + }, + { + "cell_type": "markdown", + "id": "78363328", + "metadata": { + "editable": true + }, + "source": [ + "### Why did you choose this course?" + ] + }, + { + "cell_type": "markdown", + "id": "d4950ede", + "metadata": { + "editable": true + }, + "source": [ + "### What was your programming knowledge before you started?\n", + "\n", + "And do you feel this course added to your programming competences and skills?" + ] + }, + { + "cell_type": "markdown", + "id": "95e23f0c", + "metadata": { + "editable": true + }, + "source": [ + "### How do you judge your own level of knowledge on machine learning before and after this course?\n", + "\n", + "Here you can discuss your level of skill/knowledge at the start of course\n", + "and at the end of the course and how these matched the level of\n", + "skill/knowledge needed to complete the projects." + ] + }, + { + "cell_type": "markdown", + "id": "db819721", + "metadata": { + "editable": true + }, + "source": [ + "### Did the projects and the teaching material allow you to deepen your insights about Machine Learning methods?\n", + "\n", + "Feel free to comment here." + ] + }, + { + "cell_type": "markdown", + "id": "61d153a4", + "metadata": { + "editable": true + }, + "source": [ + "### Project based teaching and active learning\n", + "\n", + "This is a project based course and we as teachers would like to keep\n", + "it as it is since we see very clearly that people who attend this\n", + "course have a very good learning outcome. Project based courses are\n", + "however demanding (and expensive seen from the university admin) when\n", + "it comes to proper feedback and evaluations. Feel free to discuss\n", + "whether you found a project-based useful. Feel also free to comment upon things we can improve upon or\n", + "alternative ways to assess whether the learning outcomes have been\n", + "achieved. Would you for example prefer a standard 4 hours written exam\n", + "be something you would prefer? Or other alternatives to projects? We\n", + "would very much value your thoughts here since projects are an\n", + "essential part of this course." + ] + }, + { + "cell_type": "markdown", + "id": "ee1af3ff", + "metadata": { + "editable": true + }, + "source": [ + "### Usefulness of the weekly exercises\n", + "\n", + "Did the weekly exercises help in getting started with the projects?\n", + "How relevant where they for solving the projects? Feel free to\n", + "elaborate" + ] + }, + { + "cell_type": "markdown", + "id": "46a35899", + "metadata": { + "editable": true + }, + "source": [ + "### Lab sessions and lectures\n", + "\n", + "Was there a good link between lectures and lab sessions?\n", + "Feel\n", + "free to comment." + ] + }, + { + "cell_type": "markdown", + "id": "f6524f03", + "metadata": { + "editable": true + }, + "source": [ + "### How would you improve this course?\n", + "\n", + "Are there topics which are missing, topics which could have been\n", + "omitted and/or discussed in more depth? Feel free to add your comments\n", + "here such as how to improve to teaching material and more." + ] + }, + { + "cell_type": "markdown", + "id": "ab89b772", + "metadata": { + "editable": true + }, + "source": [ + "## Then some basic questions" + ] + }, + { + "cell_type": "markdown", + "id": "9811e1b4", + "metadata": { + "editable": true + }, + "source": [ + "### Which is your preferred information chanel, Canvas, Discord, mail or other?" + ] + }, + { + "cell_type": "markdown", + "id": "6a3741fa", + "metadata": { + "editable": true + }, + "source": [ + "### Was the weekly update with plans etc useful?" + ] + }, + { + "cell_type": "markdown", + "id": "a51df0b9", + "metadata": { + "editable": true + }, + "source": [ + "### Was it easy to access the course material?" + ] + }, + { + "cell_type": "markdown", + "id": "762d5153", + "metadata": { + "editable": true + }, + "source": [ + "### Which resources and tools did you use? Jupyter-notebooks, GitHub, the various textbooks we have recommended, etc etc" + ] + }, + { + "cell_type": "markdown", + "id": "2bf11aae", + "metadata": { + "editable": true + }, + "source": [ + "### If you did not attend the lectures or the lab sessions, which resources did you use?" + ] + }, + { + "cell_type": "markdown", + "id": "3f92f5ff", + "metadata": { + "editable": true + }, + "source": [ + "### Any other topics, impressions, ideas etc you would like to share with us?" + ] + } + ], + "metadata": {}, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/doc/LectureNotes/_build/html/_sources/week48.ipynb b/doc/LectureNotes/_build/html/_sources/week48.ipynb new file mode 100644 index 000000000..c0298d422 --- /dev/null +++ b/doc/LectureNotes/_build/html/_sources/week48.ipynb @@ -0,0 +1,2818 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "d7c532d5", + "metadata": { + "editable": true + }, + "source": [ + "\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "d27ab2b4", + "metadata": { + "editable": true + }, + "source": [ + "# Week 48: Gradient boosting and summary of course\n", + "**Morten Hjorth-Jensen**, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway\n", + "\n", + "Date: **Nov 24, 2024**\n", + "\n", + "Copyright 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license" + ] + }, + { + "cell_type": "markdown", + "id": "9f55b05a", + "metadata": { + "editable": true + }, + "source": [ + "## Overview of week 48" + ] + }, + { + "cell_type": "markdown", + "id": "e72578f8", + "metadata": { + "editable": true + }, + "source": [ + "## Lecture Monday, November 25\n", + "**Plans for the lecture Monday 25 November, with video suggestions etc.**\n", + "\n", + "1. Boosting and gradient boosting and ensemble models\n", + "\n", + "2. Summary of course\n", + "\n", + "3. Readings and Videos:\n", + "\n", + "a. These lecture notes at \n", + "\n", + "b. See also lecture notes from week 47 at . The lecture on Monday starts with a repetition on AdaBoost before we move over to gradient boosting with examples\n", + "\n", + "\n", + "\n", + "c. Video on Decision trees \n", + "\n", + "d. Video on boosting methods \n", + "\n", + "e. Video on AdaBoost \n", + "\n", + "f. Video on Gradient boost, part 1, parts 2-4 follow thereafter \n", + "\n", + "g. Decision Trees: Rashcka et al chapter 3 pages 86-98, and chapter 7 on Ensemble methods, Voting and Bagging and Gradient Boosting. See also lecture from STK-IN4300, lecture 7 at ." + ] + }, + { + "cell_type": "markdown", + "id": "51ee8fb3", + "metadata": { + "editable": true + }, + "source": [ + "## Lab sessions\n", + "**Lab sessions on Tuesday and Wednesday.**\n", + "\n", + " * Work and Discussion of project 3\n", + "\n", + " * Last weekly exercise\n", + "\n", + " * Lab sessions at usual times.\n", + "\n", + " * For the week of December 2-6, lab sessions start at 10am and end at 4pm, room FØ434, Tuesday and Wednesday" + ] + }, + { + "cell_type": "markdown", + "id": "5b4ed2be", + "metadata": { + "editable": true + }, + "source": [ + "## Random Forest Algorithm, reminder from last week\n", + "\n", + "The algorithm described here can be applied to both classification and regression problems.\n", + "\n", + "We will grow of forest of say $B$ trees.\n", + "* For $b=1:B$\n", + "\n", + "a. Draw a bootstrap sample from the training data organized in our $\\boldsymbol{X}$ matrix.\n", + "\n", + "b. We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached\n", + "\n", + "1. we select $m \\le p$ variables at random from the $p$ predictors/features\n", + "\n", + "2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node\n", + "\n", + "3. split the node into daughter nodes\n", + "\n", + "Finally we output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem." + ] + }, + { + "cell_type": "markdown", + "id": "3e76e282", + "metadata": { + "editable": true + }, + "source": [ + "## Random Forests Compared with other Methods on the Cancer Data" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "acba62b3", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.model_selection import train_test_split \n", + "from sklearn.datasets import load_breast_cancer\n", + "from sklearn.svm import SVC\n", + "from sklearn.linear_model import LogisticRegression\n", + "from sklearn.tree import DecisionTreeClassifier\n", + "from sklearn.ensemble import BaggingClassifier\n", + "\n", + "# Load the data\n", + "cancer = load_breast_cancer()\n", + "\n", + "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", + "print(X_train.shape)\n", + "print(X_test.shape)\n", + "#define methods\n", + "# Logistic Regression\n", + "logreg = LogisticRegression(solver='lbfgs')\n", + "# Support vector machine\n", + "svm = SVC(gamma='auto', C=100)\n", + "# Decision Trees\n", + "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n", + "#Scale the data\n", + "from sklearn.preprocessing import StandardScaler\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "# Logistic Regression\n", + "logreg.fit(X_train_scaled, y_train)\n", + "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", + "# Support Vector Machine\n", + "svm.fit(X_train_scaled, y_train)\n", + "print(\"Test set accuracy SVM with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", + "# Decision Trees\n", + "deep_tree_clf.fit(X_train_scaled, y_train)\n", + "print(\"Test set accuracy with Decision Trees and scaled data: {:.2f}\".format(deep_tree_clf.score(X_test_scaled,y_test)))\n", + "\n", + "\n", + "from sklearn.ensemble import RandomForestClassifier\n", + "from sklearn.preprocessing import LabelEncoder\n", + "from sklearn.model_selection import cross_validate\n", + "# Data set not specificied\n", + "#Instantiate the model with 500 trees and entropy as splitting criteria\n", + "Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion=\"entropy\")\n", + "Random_Forest_model.fit(X_train_scaled, y_train)\n", + "#Cross validation\n", + "accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']\n", + "print(accuracy)\n", + "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(Random_Forest_model.score(X_test_scaled,y_test)))\n", + "\n", + "\n", + "import scikitplot as skplt\n", + "y_pred = Random_Forest_model.predict(X_test_scaled)\n", + "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", + "plt.show()\n", + "y_probas = Random_Forest_model.predict_proba(X_test_scaled)\n", + "skplt.metrics.plot_roc(y_test, y_probas)\n", + "plt.show()\n", + "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "c5cbe0e8", + "metadata": { + "editable": true + }, + "source": [ + "Recall that the cumulative gains curve shows the percentage of the\n", + "overall number of cases in a given category *gained* by targeting a\n", + "percentage of the total number of cases.\n", + "\n", + "Similarly, the receiver operating characteristic curve, or ROC curve,\n", + "displays the diagnostic ability of a binary classifier system as its\n", + "discrimination threshold is varied. It plots the true positive rate against the false positive rate." + ] + }, + { + "cell_type": "markdown", + "id": "19bd77f2", + "metadata": { + "editable": true + }, + "source": [ + "## Compare Bagging on Trees with Random Forests" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "84f1af2e", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "bag_clf = BaggingClassifier(\n", + " DecisionTreeClassifier(splitter=\"random\", max_leaf_nodes=16, random_state=42),\n", + " n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "6c01d900", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "bag_clf.fit(X_train, y_train)\n", + "y_pred = bag_clf.predict(X_test)\n", + "from sklearn.ensemble import RandomForestClassifier\n", + "rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)\n", + "rnd_clf.fit(X_train, y_train)\n", + "y_pred_rf = rnd_clf.predict(X_test)\n", + "np.sum(y_pred == y_pred_rf) / len(y_pred)" + ] + }, + { + "cell_type": "markdown", + "id": "896072f6", + "metadata": { + "editable": true + }, + "source": [ + "## Boosting, a Bird's Eye View\n", + "\n", + "The basic idea is to combine weak classifiers in order to create a good\n", + "classifier. With a weak classifier we often intend a classifier which\n", + "produces results which are only slightly better than we would get by\n", + "random guesses.\n", + "\n", + "This is done by applying in an iterative way a weak (or a standard\n", + "classifier like decision trees) to modify the data. In each iteration\n", + "we emphasize those observations which are misclassified by weighting\n", + "them with a factor." + ] + }, + { + "cell_type": "markdown", + "id": "babc9c9c", + "metadata": { + "editable": true + }, + "source": [ + "## What is boosting? Additive Modelling/Iterative Fitting\n", + "\n", + "Boosting is a way of fitting an additive expansion in a set of\n", + "elementary basis functions like for example some simple polynomials.\n", + "Assume for example that we have a function" + ] + }, + { + "cell_type": "markdown", + "id": "2ed28584", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5b5b9631", + "metadata": { + "editable": true + }, + "source": [ + "where $\\beta_m$ are the expansion parameters to be determined in a\n", + "minimization process and $b(x;\\gamma_m)$ are some simple functions of\n", + "the multivariable parameter $x$ which is characterized by the\n", + "parameters $\\gamma_m$.\n", + "\n", + "As an example, consider the Sigmoid function we used in logistic\n", + "regression. In that case, we can translate the function\n", + "$b(x;\\gamma_m)$ into the Sigmoid function" + ] + }, + { + "cell_type": "markdown", + "id": "7b6c43ec", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\sigma(t) = \\frac{1}{1+\\exp{(-t)}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3962039f", + "metadata": { + "editable": true + }, + "source": [ + "where $t=\\gamma_0+\\gamma_1 x$ and the parameters $\\gamma_0$ and\n", + "$\\gamma_1$ were determined by the Logistic Regression fitting\n", + "algorithm.\n", + "\n", + "As another example, consider the cost function we defined for linear regression" + ] + }, + { + "cell_type": "markdown", + "id": "ee5f259c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "079d8ee3", + "metadata": { + "editable": true + }, + "source": [ + "In this case the function $f(x)$ was replaced by the design matrix\n", + "$\\boldsymbol{X}$ and the unknown linear regression parameters $\\boldsymbol{\\beta}$,\n", + "that is $\\boldsymbol{f}=\\boldsymbol{X}\\boldsymbol{\\beta}$. In linear regression we can \n", + "simply invert a matrix and obtain the parameters $\\beta$ by" + ] + }, + { + "cell_type": "markdown", + "id": "11f821a2", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\boldsymbol{\\beta}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "60990073", + "metadata": { + "editable": true + }, + "source": [ + "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$." + ] + }, + { + "cell_type": "markdown", + "id": "4d7195ab", + "metadata": { + "editable": true + }, + "source": [ + "## Iterative Fitting, Regression and Squared-error Cost Function\n", + "\n", + "The way we proceed is as follows (here we specialize to the squared-error cost function)\n", + "\n", + "1. Establish a cost function, here ${\\cal C}(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m)$.\n", + "\n", + "2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.\n", + "\n", + "3. For $m=1:M$\n", + "\n", + "a. minimize $\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2$ wrt $\\gamma$ and $\\beta$\n", + "\n", + "b. This gives the optimal values $\\beta_m$ and $\\gamma_m$\n", + "\n", + "c. Determine then the new values $f_m(x)=f_{m-1}(x) +\\beta_m b(x;\\gamma_m)$\n", + "\n", + "We could use any of the algorithms we have discussed till now. If we\n", + "use trees, $\\gamma$ parameterizes the split variables and split points\n", + "at the internal nodes, and the predictions at the terminal nodes." + ] + }, + { + "cell_type": "markdown", + "id": "4bee726a", + "metadata": { + "editable": true + }, + "source": [ + "## Squared-Error Example and Iterative Fitting\n", + "\n", + "To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.\n", + "\n", + "For simplicity we assume also that our functions $b(x;\\gamma)=1+\\gamma x$. \n", + "\n", + "This means that for every iteration $m$, we need to optimize" + ] + }, + { + "cell_type": "markdown", + "id": "1190b9a0", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(\\beta_m,\\gamma_m) = \\mathrm{argmin}_{\\beta,\\lambda}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2=\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta(1+\\gamma x_i))^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2702211b", + "metadata": { + "editable": true + }, + "source": [ + "We start our iteration by simply setting $f_0(x)=0$. \n", + "Taking the derivatives with respect to $\\beta$ and $\\gamma$ we obtain" + ] + }, + { + "cell_type": "markdown", + "id": "1b1d9984", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial {\\cal C}}{\\partial \\beta} = -2\\sum_{i}(1+\\gamma x_i)(y_i-\\beta(1+\\gamma x_i))=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cdf7e56a", + "metadata": { + "editable": true + }, + "source": [ + "and" + ] + }, + { + "cell_type": "markdown", + "id": "ac6f3b86", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial {\\cal C}}{\\partial \\gamma} =-2\\sum_{i}\\beta x_i(y_i-\\beta(1+\\gamma x_i))=0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "faf6fb5f", + "metadata": { + "editable": true + }, + "source": [ + "We can then rewrite these equations as (defining $\\boldsymbol{w}=\\boldsymbol{e}+\\gamma \\boldsymbol{x})$ with $\\boldsymbol{e}$ being the unit vector)" + ] + }, + { + "cell_type": "markdown", + "id": "969f86ed", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\gamma \\boldsymbol{w}^T(\\boldsymbol{y}-\\beta\\gamma \\boldsymbol{w})=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "a98a541c", + "metadata": { + "editable": true + }, + "source": [ + "which gives us $\\beta = \\boldsymbol{w}^T\\boldsymbol{y}/(\\boldsymbol{w}^T\\boldsymbol{w})$. Similarly we have" + ] + }, + { + "cell_type": "markdown", + "id": "d8be5438", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta\\gamma \\boldsymbol{x}^T(\\boldsymbol{y}-\\beta(1+\\gamma \\boldsymbol{x}))=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "932f43c5", + "metadata": { + "editable": true + }, + "source": [ + "which leads to $\\gamma =(\\boldsymbol{x}^T\\boldsymbol{y}-\\beta\\boldsymbol{x}^T\\boldsymbol{e})/(\\beta\\boldsymbol{x}^T\\boldsymbol{x})$. Inserting\n", + "for $\\beta$ gives us an equation for $\\gamma$. This is a non-linear equation in the unknown $\\gamma$ and has to be solved numerically. \n", + "\n", + "The solution to these two equations gives us in turn $\\beta_1$ and $\\gamma_1$ leading to the new expression for $f_1(x)$ as\n", + "$f_1(x) = \\beta_1(1+\\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$." + ] + }, + { + "cell_type": "markdown", + "id": "24630eb9", + "metadata": { + "editable": true + }, + "source": [ + "## Iterative Fitting, Classification and AdaBoost\n", + "\n", + "Let us consider a binary classification problem with two outcomes $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n", + "observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values \n", + "$\\{-1,1\\}$.\n", + "\n", + "The error rate of the training sample is then" + ] + }, + { + "cell_type": "markdown", + "id": "9a7fa4b6", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{\\overline{err}}=\\frac{1}{n} \\sum_{i=0}^{n-1} I(y_i\\ne G(x_i)).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7ead62b2", + "metadata": { + "editable": true + }, + "source": [ + "The iterative procedure starts with defining a weak classifier whose\n", + "error rate is barely better than random guessing. The iterative\n", + "procedure in boosting is to sequentially apply a weak\n", + "classification algorithm to repeatedly modified versions of the data\n", + "producing a sequence of weak classifiers $G_m(x)$.\n", + "\n", + "Here we will express our function $f(x)$ in terms of $G(x)$. That is" + ] + }, + { + "cell_type": "markdown", + "id": "91639c49", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e43865c2", + "metadata": { + "editable": true + }, + "source": [ + "will be a function of" + ] + }, + { + "cell_type": "markdown", + "id": "1884c219", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "G_M(x) = \\mathrm{sign} \\sum_{i=1}^M \\alpha_m G_m(x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "bbb1cdbb", + "metadata": { + "editable": true + }, + "source": [ + "## Adaptive Boosting, AdaBoost\n", + "\n", + "In our iterative procedure we define thus" + ] + }, + { + "cell_type": "markdown", + "id": "7e3590f6", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_m(x) = f_{m-1}(x)+\\beta_mG_m(x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2eb8c86b", + "metadata": { + "editable": true + }, + "source": [ + "The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the\n", + "exponential cost/loss function defined as" + ] + }, + { + "cell_type": "markdown", + "id": "e477d15a", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}\\exp{(-y_i(f_{m-1}(x_i)+\\beta G(x_i))}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "b6f01e59", + "metadata": { + "editable": true + }, + "source": [ + "We optimize $\\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.\n", + "This is normally done in two steps. Let us however first rewrite the cost function as" + ] + }, + { + "cell_type": "markdown", + "id": "71c85471", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}w_i^{m}\\exp{(-y_i\\beta G(x_i))},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2d5be340", + "metadata": { + "editable": true + }, + "source": [ + "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$." + ] + }, + { + "cell_type": "markdown", + "id": "e426517c", + "metadata": { + "editable": true + }, + "source": [ + "## Building up AdaBoost\n", + "\n", + "First, for any $\\beta > 0$, we optimize $G$ by setting" + ] + }, + { + "cell_type": "markdown", + "id": "c0c7e993", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "G_m(x) = \\mathrm{sign} \\sum_{i=0}^{n-1} w_i^m I(y_i \\ne G_(x_i)),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7457a096", + "metadata": { + "editable": true + }, + "source": [ + "which is the classifier that minimizes the weighted error rate in predicting $y$.\n", + "\n", + "We can do this by rewriting" + ] + }, + { + "cell_type": "markdown", + "id": "424d5bc9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\exp{-(\\beta)}\\sum_{y_i=G(x_i)}w_i^m+\\exp{(\\beta)}\\sum_{y_i\\ne G(x_i)}w_i^m,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d1fa658b", + "metadata": { + "editable": true + }, + "source": [ + "which can be rewritten as" + ] + }, + { + "cell_type": "markdown", + "id": "f8fc6b15", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(\\exp{(\\beta)}-\\exp{-(\\beta)})\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i))+\\exp{(-\\beta)}\\sum_{i=0}^{n-1}w_i^m=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9ca82f54", + "metadata": { + "editable": true + }, + "source": [ + "which leads to" + ] + }, + { + "cell_type": "markdown", + "id": "6c028686", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_m = \\frac{1}{2}\\log{\\frac{1-\\mathrm{\\overline{err}}}{\\mathrm{\\overline{err}}}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4007a0ef", + "metadata": { + "editable": true + }, + "source": [ + "where we have redefined the error as" + ] + }, + { + "cell_type": "markdown", + "id": "ba456347", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{\\overline{err}}_m=\\frac{1}{n}\\frac{\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i)}{\\sum_{i=0}^{n-1}w_i^m},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "c8df4ef3", + "metadata": { + "editable": true + }, + "source": [ + "which leads to an update of" + ] + }, + { + "cell_type": "markdown", + "id": "da46f9d9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_m(x) = f_{m-1}(x) +\\beta_m G_m(x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5c60f30d", + "metadata": { + "editable": true + }, + "source": [ + "This leads to the new weights" + ] + }, + { + "cell_type": "markdown", + "id": "a5bc4dfe", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "w_i^{m+1} = w_i^m \\exp{(-y_i\\beta_m G_m(x_i))}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "3987cc1e", + "metadata": { + "editable": true + }, + "source": [ + "## Adaptive boosting: AdaBoost, Basic Algorithm\n", + "\n", + "The algorithm here is rather straightforward. Assume that our weak\n", + "classifier is a decision tree and we consider a binary set of outputs\n", + "with $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n", + "observations. Our design matrix is given in terms of the\n", + "feature/predictor vectors\n", + "$\\boldsymbol{X}=[\\boldsymbol{x}_0\\boldsymbol{x}_1\\dots\\boldsymbol{x}_{p-1}]$. Finally, we define also a\n", + "classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\\boldsymbol{y}$. \n", + "\n", + "We have already defined the misclassification error $\\mathrm{err}$ as" + ] + }, + { + "cell_type": "markdown", + "id": "b35df09e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{err}=\\frac{1}{n}\\sum_{i=0}^{n-1}I(y_i\\ne G(x_i)),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "5b6a209c", + "metadata": { + "editable": true + }, + "source": [ + "where the function $I()$ is one if we misclassify and zero if we classify correctly." + ] + }, + { + "cell_type": "markdown", + "id": "d619a097", + "metadata": { + "editable": true + }, + "source": [ + "## Basic Steps of AdaBoost\n", + "\n", + "With the above definitions we are now ready to set up the algorithm for AdaBoost.\n", + "The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.\n", + "1. We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\\dots n-1$. It is easy to see that we must have $\\sum_{i=0}^{n-1}w_i = 1$.\n", + "\n", + "2. We rewrite the misclassification error as" + ] + }, + { + "cell_type": "markdown", + "id": "b7311abb", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mathrm{\\overline{err}}_m=\\frac{\\sum_{i=0}^{n-1}w_i^m I(y_i\\ne G(x_i))}{\\sum_{i=0}^{n-1}w_i},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4bfb4209", + "metadata": { + "editable": true + }, + "source": [ + "1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree.\n", + "\n", + "a. Fit then a given classifier to the training set using the weights $w_i$.\n", + "\n", + "b. Compute then $\\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.\n", + "\n", + "c. Define a quantity $\\alpha_{m} = \\log{(1-\\mathrm{\\overline{err}}_m)/\\mathrm{\\overline{err}}_m}$\n", + "\n", + "d. Set the new weights to $w_i = w_i\\times \\exp{(\\alpha_m I(y_i\\ne G(x_i)}$.\n", + "\n", + "5. Compute the new classifier $G(x)= \\sum_{i=0}^{n-1}\\alpha_m I(y_i\\ne G(x_i)$.\n", + "\n", + "For the iterations with $m \\le 2$ the weights are modified\n", + "individually at each steps. The observations which were misclassified\n", + "at iteration $m-1$ have a weight which is larger than those which were\n", + "classified properly. As this proceeds, the observations which were\n", + "difficult to classifiy correctly are given a larger influence. Each\n", + "new classification step $m$ is then forced to concentrate on those\n", + "observations that are missed in the previous iterations." + ] + }, + { + "cell_type": "markdown", + "id": "4cf4c1f4", + "metadata": { + "editable": true + }, + "source": [ + "## AdaBoost Examples\n", + "\n", + "Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "453dff54", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "from sklearn.ensemble import AdaBoostClassifier\n", + "\n", + "ada_clf = AdaBoostClassifier(\n", + " DecisionTreeClassifier(max_depth=2), n_estimators=200,\n", + " algorithm=\"SAMME.R\", learning_rate=0.01, random_state=42)\n", + "ada_clf.fit(X_train, y_train)\n", + "y_pred = ada_clf.predict(X_test)\n", + "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", + "plt.show()\n", + "y_probas = ada_clf.predict_proba(X_test)\n", + "skplt.metrics.plot_roc(y_test, y_probas)\n", + "plt.show()\n", + "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "d6470164", + "metadata": { + "editable": true + }, + "source": [ + "## Making an ADAboost code yourself" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "b6e01886", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "\n", + "class DecisionStump:\n", + " def fit(self, X, y, weights):\n", + " m, n = X.shape\n", + " self.alpha = 0\n", + " self.threshold = None\n", + " self.polarity = 1\n", + "\n", + " min_error = float('inf')\n", + "\n", + " for feature in range(n):\n", + " feature_values = np.unique(X[:, feature])\n", + "\n", + " for threshold in feature_values:\n", + " for polarity in [1, -1]:\n", + " predictions = np.ones(m)\n", + " predictions[X[:, feature] < threshold] = -1\n", + " predictions *= polarity\n", + "\n", + " error = sum(weights[predictions != y])\n", + "\n", + " if error < min_error:\n", + " min_error = error\n", + " self.alpha = 0.5 * np.log((1 - error) / (error + 1e-10))\n", + " self.threshold = threshold\n", + " self.feature_index = feature\n", + " self.polarity = polarity\n", + "\n", + " def predict(self, X):\n", + " m = X.shape[0]\n", + " predictions = np.ones(m)\n", + " if self.polarity == 1:\n", + " predictions[X[:, self.feature_index] < self.threshold] = -1\n", + " else:\n", + " predictions[X[:, self.feature_index] >= self.threshold] = -1\n", + " return predictions\n", + "\n", + "class AdaBoost:\n", + " def fit(self, X, y, n_estimators):\n", + " m = X.shape[0]\n", + " self.alphas = []\n", + " self.models = []\n", + "\n", + " weights = np.ones(m) / m\n", + "\n", + " for _ in range(n_estimators):\n", + " stump = DecisionStump()\n", + " stump.fit(X, y, weights)\n", + " predictions = stump.predict(X)\n", + "\n", + " error = sum(weights[predictions != y])\n", + " if error == 0:\n", + " break\n", + "\n", + " self.models.append(stump)\n", + " self.alphas.append(stump.alpha)\n", + "\n", + " weights *= np.exp(-stump.alpha * y * predictions)\n", + " weights /= np.sum(weights)\n", + "\n", + " def predict(self, X):\n", + " final_predictions = np.zeros(X.shape[0])\n", + " for alpha, model in zip(self.alphas, self.models):\n", + " final_predictions += alpha * model.predict(X)\n", + " return np.sign(final_predictions)\n", + "\n", + "# Example dataset (X, y)\n", + "X = np.array([[1], [2], [3], [4], [5], [6], [7], [8], [9], [10]])\n", + "y = np.array([-1, -1, -1, -1, 1, 1, 1, 1, 1, 1]) # Labels must be -1 or 1\n", + "\n", + "# Train AdaBoost\n", + "ada = AdaBoost()\n", + "ada.fit(X, y, n_estimators=10)\n", + "\n", + "# Predictions\n", + "predictions = ada.predict(X)\n", + "print(\"Predictions:\", predictions)" + ] + }, + { + "cell_type": "markdown", + "id": "3cede4c9", + "metadata": { + "editable": true + }, + "source": [ + "## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent\n", + "\n", + "Gradient boosting is again a similar technique to Adaptive boosting,\n", + "it combines so-called weak classifiers or regressors into a strong\n", + "method via a series of iterations.\n", + "\n", + "In order to understand the method, let us illustrate its basics by\n", + "bringing back the essential steps in linear regression, where our cost\n", + "function was the least squares function." + ] + }, + { + "cell_type": "markdown", + "id": "94e3b59f", + "metadata": { + "editable": true + }, + "source": [ + "## The Squared-Error again! Steepest Descent\n", + "\n", + "We start again with our cost function ${\\cal C}(\\boldsymbol{y}m\\boldsymbol{f})=\\sum_{i=0}^{n-1}{\\cal L}(y_i, f(x_i))$ where we want to minimize\n", + "This means that for every iteration, we need to optimize" + ] + }, + { + "cell_type": "markdown", + "id": "2f000f62", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(\\hat{\\boldsymbol{f}}) = \\mathrm{argmin}_{\\boldsymbol{f}}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "014da5e9", + "metadata": { + "editable": true + }, + "source": [ + "We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as" + ] + }, + { + "cell_type": "markdown", + "id": "022aee06", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_M(x) = \\sum_{m=0}^M h_m(x).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d0d3e174", + "metadata": { + "editable": true + }, + "source": [ + "In the steepest decent approach we approximate $h_m(x) = -\\rho_m g_m(x)$, where $\\rho_m$ is a scalar and $g_m(x)$ the gradient defined as" + ] + }, + { + "cell_type": "markdown", + "id": "8af32042", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "g_m(x_i) = \\left[ \\frac{\\partial {\\cal L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{m-1}(x_i)}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "4a384d95", + "metadata": { + "editable": true + }, + "source": [ + "With the new gradient we can update $f_m(x) = f_{m-1}(x) -\\rho_m g_m(x)$. Using the above squared-error function we see that\n", + "the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.\n", + "\n", + "Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have" + ] + }, + { + "cell_type": "markdown", + "id": "bd5b9bb9", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "(\\rho_1) = \\mathrm{argmin}_{\\rho}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i+2\\rho y_i)^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "560b3106", + "metadata": { + "editable": true + }, + "source": [ + "## Steepest Descent Example\n", + "\n", + "Optimizing with respect to $\\rho$ we obtain (taking the derivative) that $\\rho_1 = -1/2$. We have then that" + ] + }, + { + "cell_type": "markdown", + "id": "4742d327", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_1(x) = f_{0}(x) -\\rho_1 g_1(x)=-y_i.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "d160a37f", + "metadata": { + "editable": true + }, + "source": [ + "We can then proceed and compute" + ] + }, + { + "cell_type": "markdown", + "id": "f3cde07e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "g_2(x_i) = \\left[ \\frac{\\partial {\\cal L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "cf95ad5e", + "metadata": { + "editable": true + }, + "source": [ + "and find a new value for $\\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**." + ] + }, + { + "cell_type": "markdown", + "id": "bb614b9d", + "metadata": { + "editable": true + }, + "source": [ + "## Gradient Boosting, algorithm\n", + "\n", + "Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,\n", + "so we do not learn a function that can generalize. However, we can modify the algorithm by\n", + "fitting a weak learner to approximate the negative gradient signal. \n", + "\n", + "Suppose we have a cost function $C(f)=\\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard squared-error function" + ] + }, + { + "cell_type": "markdown", + "id": "19e3e8fa", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{y},\\boldsymbol{f})=\\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "9330b34b", + "metadata": { + "editable": true + }, + "source": [ + "The way we proceed in an iterative fashion is to\n", + "1. Initialize our estimate $f_0(x)$.\n", + "\n", + "2. For $m=1:M$, we\n", + "\n", + "a. compute the negative gradient vector $\\boldsymbol{u}_m = -\\partial C(\\boldsymbol{y},\\boldsymbol{f})/\\partial \\boldsymbol{f}(x)$ at $f(x) = f_{m-1}(x)$;\n", + "\n", + "b. fit the so-called base-learner to the negative gradient $h_m(u_m,x)$;\n", + "\n", + "c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;\n", + "\n", + "4. The final estimate is then $f_M(x) = \\sum_{m=1}^M h_m(u_m,x)$." + ] + }, + { + "cell_type": "markdown", + "id": "e648539a", + "metadata": { + "editable": true + }, + "source": [ + "## Gradient Boosting, Examples of Regression" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "d12810d3", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn.ensemble import GradientBoostingRegressor\n", + "import scikitplot as skplt\n", + "from sklearn.metrics import mean_squared_error\n", + "\n", + "n = 100\n", + "maxdegree = 6\n", + "\n", + "# Make data set.\n", + "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", + "\n", + "error = np.zeros(maxdegree)\n", + "bias = np.zeros(maxdegree)\n", + "variance = np.zeros(maxdegree)\n", + "polydegree = np.zeros(maxdegree)\n", + "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", + "\n", + "for degree in range(1,maxdegree):\n", + " model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0) \n", + " model.fit(X_train,y_train)\n", + " y_pred = model.predict(X_test)\n", + " polydegree[degree] = degree\n", + " error[degree] = np.mean( np.mean((y_test - y_pred)**2) )\n", + " bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )\n", + " variance[degree] = np.mean( np.var(y_pred) )\n", + " print('Max depth:', degree)\n", + " print('Error:', error[degree])\n", + " print('Bias^2:', bias[degree])\n", + " print('Var:', variance[degree])\n", + " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", + "\n", + "plt.xlim(1,maxdegree-1)\n", + "plt.plot(polydegree, error, label='Error')\n", + "plt.plot(polydegree, bias, label='bias')\n", + "plt.plot(polydegree, variance, label='Variance')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "f3e95eca", + "metadata": { + "editable": true + }, + "source": [ + "## Gradient Boosting, Classification Example" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "d6426469", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.model_selection import train_test_split \n", + "from sklearn.datasets import load_breast_cancer\n", + "import scikitplot as skplt\n", + "from sklearn.ensemble import GradientBoostingClassifier\n", + "from sklearn.model_selection import cross_validate\n", + "\n", + "# Load the data\n", + "cancer = load_breast_cancer()\n", + "\n", + "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", + "print(X_train.shape)\n", + "print(X_test.shape)\n", + "#now scale the data\n", + "from sklearn.preprocessing import StandardScaler\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "\n", + "gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0) \n", + "gd_clf.fit(X_train_scaled, y_train)\n", + "#Cross validation\n", + "accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']\n", + "print(accuracy)\n", + "print(\"Test set accuracy with Gradient boosting and scaled data: {:.2f}\".format(gd_clf.score(X_test_scaled,y_test)))\n", + "\n", + "import scikitplot as skplt\n", + "y_pred = gd_clf.predict(X_test_scaled)\n", + "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", + "plt.show()\n", + "y_probas = gd_clf.predict_proba(X_test_scaled)\n", + "skplt.metrics.plot_roc(y_test, y_probas)\n", + "plt.show()\n", + "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "8de56415", + "metadata": { + "editable": true + }, + "source": [ + "## XGBoost: Extreme Gradient Boosting\n", + "\n", + "[XGBoost](https://github.com/dmlc/xgboost) or Extreme Gradient\n", + "Boosting, is an optimized distributed gradient boosting library\n", + "designed to be highly efficient, flexible and portable. It implements\n", + "machine learning algorithms under the Gradient Boosting\n", + "framework. XGBoost provides a parallel tree boosting that solve many\n", + "data science problems in a fast and accurate way. See the [article by Chen and Guestrin](https://arxiv.org/abs/1603.02754).\n", + "\n", + "The authors design and build a highly scalable end-to-end tree\n", + "boosting system. It has a theoretically justified weighted quantile\n", + "sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.\n", + "\n", + "It is now the algorithm which wins essentially all ML competitions!!!" + ] + }, + { + "cell_type": "markdown", + "id": "dfc59b18", + "metadata": { + "editable": true + }, + "source": [ + "## Xgboost on the Cancer Data\n", + "\n", + "As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "79e4cc00", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\n", + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from sklearn.model_selection import train_test_split \n", + "from sklearn.datasets import load_breast_cancer\n", + "from sklearn.preprocessing import LabelEncoder\n", + "from sklearn.model_selection import cross_validate\n", + "import scikitplot as skplt\n", + "import xgboost as xgb\n", + "# Load the data\n", + "cancer = load_breast_cancer()\n", + "\n", + "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", + "print(X_train.shape)\n", + "print(X_test.shape)\n", + "#now scale the data\n", + "from sklearn.preprocessing import StandardScaler\n", + "scaler = StandardScaler()\n", + "scaler.fit(X_train)\n", + "X_train_scaled = scaler.transform(X_train)\n", + "X_test_scaled = scaler.transform(X_test)\n", + "\n", + "xg_clf = xgb.XGBClassifier()\n", + "xg_clf.fit(X_train_scaled,y_train)\n", + "\n", + "y_test = xg_clf.predict(X_test_scaled)\n", + "\n", + "print(\"Test set accuracy with Gradient Boosting and scaled data: {:.2f}\".format(xg_clf.score(X_test_scaled,y_test)))\n", + "\n", + "import scikitplot as skplt\n", + "y_pred = xg_clf.predict(X_test_scaled)\n", + "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", + "plt.show()\n", + "y_probas = xg_clf.predict_proba(X_test_scaled)\n", + "skplt.metrics.plot_roc(y_test, y_probas)\n", + "plt.show()\n", + "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", + "plt.show()\n", + "\n", + "\n", + "xgb.plot_tree(xg_clf,num_trees=0)\n", + "plt.rcParams['figure.figsize'] = [50, 10]\n", + "plt.show()\n", + "\n", + "xgb.plot_importance(xg_clf)\n", + "plt.rcParams['figure.figsize'] = [5, 5]\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "c4700b19", + "metadata": { + "editable": true + }, + "source": [ + "## Gradient boosting, making our own code for a regression case" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "eb0594c5", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "class DecisionTreeRegressor:\n", + " def __init__(self, max_depth=3):\n", + " self.max_depth = max_depth\n", + " self.tree = None\n", + " def fit(self, X, y):\n", + " self.tree = self._grow_tree(X, y)\n", + " def _grow_tree(self, X, y, depth=0):\n", + " n_samples, n_features = X.shape\n", + " if depth < self.max_depth:\n", + " best_feature, best_threshold = self._best_split(X, y)\n", + " if best_feature is not None:\n", + " left_indices = X[:, best_feature] < best_threshold\n", + " right_indices = X[:, best_feature] >= best_threshold\n", + " left_child = self._grow_tree(X[left_indices], y[left_indices], depth + 1)\n", + " right_child = self._grow_tree(X[right_indices], y[right_indices], depth + 1)\n", + " return (best_feature, best_threshold, left_child, right_child)\n", + " return np.mean(y)\n", + " def _best_split(self, X, y):\n", + " best_mse = float('inf')\n", + " best_feature, best_threshold = None, None\n", + " n_samples, n_features = X.shape\n", + " \n", + " for feature in range(n_features):\n", + " thresholds = np.unique(X[:, feature])\n", + " for threshold in thresholds:\n", + " left_indices = X[:, feature] < threshold\n", + " right_indices = X[:, feature] >= threshold\n", + " if len(y[left_indices]) > 0 and len(y[right_indices]) > 0:\n", + " left_mse = np.mean((y[left_indices] - np.mean(y[left_indices])) ** 2)\n", + " right_mse = np.mean((y[right_indices] - np.mean(y[right_indices])) ** 2)\n", + " mse = (len(y[left_indices]) * left_mse + len(y[right_indices]) * right_mse) / n_samples\n", + " \n", + " if mse < best_mse:\n", + " best_mse = mse\n", + " best_feature = feature\n", + " best_threshold = threshold\n", + " return best_feature, best_threshold\n", + " def predict(self, X):\n", + " return np.array([self._predict_sample(sample, self.tree) for sample in X])\n", + " def _predict_sample(self, sample, node):\n", + " if isinstance(node, tuple):\n", + " feature, threshold, left_child, right_child = node\n", + " if sample[feature] < threshold:\n", + " return self._predict_sample(sample, left_child)\n", + " else:\n", + " return self._predict_sample(sample, right_child)\n", + " return node\n", + "class GradientBoostingRegressor:\n", + " def __init__(self, n_estimators=100, learning_rate=0.1, max_depth=3):\n", + " self.n_estimators = n_estimators\n", + " self.learning_rate = learning_rate\n", + " self.max_depth = max_depth\n", + " self.models = []\n", + " def fit(self, X, y):\n", + " y_pred = np.zeros(y.shape)\n", + " for _ in range(self.n_estimators):\n", + " residuals = y - y_pred\n", + " model = DecisionTreeRegressor(max_depth=self.max_depth)\n", + " model.fit(X, residuals)\n", + " y_pred += self.learning_rate * model.predict(X)\n", + " self.models.append(model)\n", + " def predict(self, X):\n", + " y_pred = np.zeros(X.shape[0])\n", + " for model in self.models:\n", + " y_pred += self.learning_rate * model.predict(X)\n", + " return y_pred\n", + "# Example usage\n", + "if __name__ == \"__main__\":\n", + " # Sample data\n", + " X = np.array([[1], [2], [3], [4], [5]])\n", + " y = np.array([1.5, 1.7, 3.5, 3.7, 5.0])\n", + " model = GradientBoostingRegressor(n_estimators=100, learning_rate=0.1, max_depth=2)\n", + " model.fit(X, y)\n", + " predictions = model.predict(X)\n", + " print(\"Predictions:\", predictions)" + ] + }, + { + "cell_type": "markdown", + "id": "bbf154c4", + "metadata": { + "editable": true + }, + "source": [ + "## Summary of course" + ] + }, + { + "cell_type": "markdown", + "id": "84d512fc", + "metadata": { + "editable": true + }, + "source": [ + "## What? Me worry? No final exam in this course!\n", + "\n", + "\n", + "\n", + "

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