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-TITLE: Introduction to Applied Data Analysis and Machine Learning
-AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
-DATE: today
-
-
-
-===== Introduction =====
+======= Introduction to Applied Data Analysis and Machine Learning =======
 
 During the last two decades there has been a swift and amazing
 development of Machine Learning techniques and algorithms that impact
@@ -147,8 +141,6 @@ insights and knowledge we have about a specific system, play a central
 role when we develop a specific machine learning algorithm. 
 
 
-===== Machine Learning, an extremely rich field  =====
-
 Machine learning is an extremely rich field, in spite of its young
 age. The increases we have seen during the last  decades in
 computational capabilities have been followed by developments of
@@ -164,8 +156,6 @@ communities of developers in the thousands or more. And the number of
 code developers and contributors keeps increasing.
 
 
-=====  A multidisciplinary approach =====
-
 Not all the
 algorithms and methods can be given a rigorous mathematical
 justification (for example decision trees and random forests), opening up thereby large rooms for experimenting and
@@ -176,12 +166,6 @@ Carlo methods are central elements in a proper understanding of many
 of the algorithms and methods we will discuss.
 
 
-
-
-
-===== Types of Machine Learning =====
-
-
 The approaches to machine learning are many, but are often split into
 two main categories.  In *supervised learning* we know the answer to a
 problem, and let the computer deduce the logic behind it. On the other
@@ -195,35 +179,24 @@ solely from rewards and punishment.
 Another way to categorize machine learning tasks is to consider the
 desired output of a system.  Some of the most common tasks are:
 
-!bpop
 * 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 often supervised learning.
 
 * 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.
 
 * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand.  It is thus a form of unsupervised learning.
-!epop
 
 
-
-
-===== Essential elements of ML =====
-
 The methods we cover have three main topics in common, irrespective of
 whether we deal with supervised or unsupervised learning.
-!bpop
+
 * The first ingredient is normally our data set (which can be subdivided into training, validation  and test data). Many find the most difficult part of using Machine Learning to be the set up of your data in a meaningful way. 
 
 * The second item is a model which is normally a function of some parameters.  The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. 
 
 * The last ingredient is a so-called _cost/loss_ function (or error function) which allows us to present an estimate on how good our model is in reproducing the data it is supposed to train.  
-!epop
 
 
 
-
-
-===== An optimization/minimization problem =====
-
 At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called _gradient methods_.
 
 
@@ -279,10 +252,6 @@ is that if we are not specific about what we mean by a *correct* model, there
 could easily be many different models that fit the given data set *equally well*.
 
 
-
-===== What is a good model? Can we define it?  =====
-
-
 The central question is this: what leads us to say that a model is correct or
 optimal for a given data set? To make the model inference problem well posed, i.e.,
 to guarantee that there is a unique optimal model for the given data, we need to