File Name: pattern recognition and machine learning springer 2006 .zip
- Pattern Recognition and Machine Learning
- Pattern Recognition and Machine Learning, by Christopher M. Bishop
- Machine Learning Lecture Notes Pdf
Pattern Recognition and Machine Learning
Springer has provided excellent support throughout the final stages of preparation of this book, and I would like to thank my commissioning editor John Kimmel for his support and professionalism, as well as Joseph Piliero for his help in designing the cover and the text format and MaryAnn Brickner for her numerous contributions during the production phase. The inspiration for the cover design came from a discussion with Antonio Criminisi. I also wish to thank Oxford University Press for permission to reproduce excerpts from an earlier textbook, Neural Networks for Pattern Recognition Bishop, a.
I would also like to thank Asela Gunawardana for plotting the spectrogram in Figure 13 IntroductionThe problem of searching for patterns in data is a fundamental one and has a long and successful history.
For instance, the extensive astronomical observations of Tycho Brahe in the 16 th century allowed Johannes Kepler to discover the empirical laws of planetary motion, which in turn provided a springboard for the development of classical mechanics. Similarly, the discovery of regularities in atomic spectra played a key role in the development and verification of quantum physics in the early twentieth century.
The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories.
Consider the example of recognizing handwritten digits, illustrated in Figure 1. The goal is to build a machine that will take such a vector x as input and that will produce the identity of the digit 0,. This is a nontrivial problem due to the wide variability of handwriting. It could be tackled using handcrafted rules or heuristics for distinguishing the digits based on the shapes of the strokes, but in practice such an approach leads to a proliferation of rules and of exceptions to the rules and so on, and invariably gives poor results.
The categories of the digits in the training set are known in advance, typically by inspecting them individually and hand-labelling them. We can express the category of a digit using target vector t, which represents the identity of the corresponding digit. Suitable techniques for representing categories in terms of vectors will be discussed later.
Note that there is one such target vector t for each digit image x. The precise form of the function y x is determined during the training phase, also known as the learning phase, on the basis of the training data. Once the model is trained it can then determine the identity of new digit images, which are said to comprise a test set. The ability to categorize correctly new examples that differ from those used for training is known as generalization. In practical applications, the variability of the input vectors will be such that the training data can comprise only a tiny fraction of all possible input vectors, and so generalization is a central goal in pattern recognition.
For most practical applications, the original input variables are typically preprocessed to transform them into some new space of variables where, it is hoped, the pattern recognition problem will be easier to solve. For instance, in the digit recognition problem, the images of the digits are typically translated and scaled so that each digit is contained within a box of a fixed size. This greatly reduces the variability within each digit class, because the location and scale of all the digits are now the same, which makes it much easier for a subsequent pattern recognition algorithm to distinguish between the different classes.
This pre-processing stage is sometimes also called feature extraction. Note that new test data must be pre-processed using the same steps as the training data. Pre-processing might also be performed in order to speed up computation. For example, if the goal is real-time face detection in a high-resolution video stream, the computer must handle huge numbers of pixels per second, and presenting these directly to a complex pattern recognition algorithm may be computationally infeasible.
These features are then used as the inputs to the pattern recognition algorithm. For instance, the average value of the image intensity over a rectangular subregion can be evaluated extremely efficiently Viola and Jones, , and a set of such features can prove very effective in fast face detection. Because the number of such features is smaller than the number of pixels, this kind of pre-processing represents a form of dimensionality reduction.
Care must be taken during pre-processing because often information is discarded, and if this information is important to the solution of the problem then the overall accuracy of the system can suffer. Applications in which the training data comprises examples of the input vectors along with their corresponding target vectors are known as supervised learning problems. Cases such as the digit recognition example, in which the aim is to assign each input vector to one of a finite number of discrete categories, are called classification problems.
If the desired output consists of one or more continuous variables, then the task is called regression. An example of a regression problem would be the prediction of the yield in a chemical manufacturing process in which the inputs consist of the concentrations of reactants, the temperature, and the pressure.
In other pattern recognition problems, the training data consists of a set of input vectors x without any corresponding target values. The goal in such unsupervised learning problems may be to discover groups of similar examples within the data, where it is called clustering, or to determine the distribution of data within the input space, known as density estimation, or to project the data from a high-dimensional space down to two or three dimensions for the purpose of visualization.
Finally, the technique of reinforcement learning Sutton and Barto, is concerned with the problem of finding suitable actions to take in a given situation in order to maximize a reward.
Here the learning algorithm is not given examples of optimal outputs, in contrast to supervised learning, but must instead discover them by a process of trial and error. Typically there is a sequence of states and actions in which the learning algorithm is interacting with its environment. In many cases, the current action not only affects the immediate reward but also has an impact on the reward at all subsequent time steps.
For example, by using appropriate reinforcement learning techniques a neural network can learn to play the game of backgammon to a high standard Tesauro, Here the network must learn to take a board position as input, along with the result of a dice throw, and produce a strong move as the output.
This is done by having the network play against a copy of itself for perhaps a million games. A major challenge is that a game of backgammon can involve dozens of moves, and yet it is only at the end of the game that the reward, in the form of victory, is achieved. The reward must then be attributed appropriately to all of the moves that led to it, even though some moves will have been good ones and others less so.
This is an example of a credit assignment problem. A general feature of reinforcement learning is the trade-off between exploration, in which the system tries out new kinds of actions to see how effective they are, and exploitation, in which the system makes use of actions that are known to yield a high reward. Too strong a focus on either exploration or exploitation will yield poor results.
Reinforcement learning continues to be an active area of machine learning research. However, a 10 points, shown as blue circles, each comprising an observation of the input variable x along with the corresponding target variable t. Our goal is to predict the value of t for some new value of x, without knowledge of the green curve. Although each of these tasks needs its own tools and techniques, many of the key ideas that underpin them are common to all such problems. One of the main goals of this chapter is to introduce, in a relatively informal way, several of the most important of these concepts and to illustrate them using simple examples.
Later in the book we shall see these same ideas re-emerge in the context of more sophisticated models that are applicable to real-world pattern recognition applications. This chapter also provides a self-contained introduction to three important tools that will be used throughout the book, namely probability theory, decision theory, and information theory. Although these might sound like daunting topics, they are in fact straightforward, and a clear understanding of them is essential if machine learning techniques are to be used to best effect in practical applications.
Suppose we observe a real-valued input variable x and we wish to use this observation to predict the value of a real-valued target variable t. For the present purposes, it is instructive to consider an artificial example using synthetically generated data because we then know the precise process that generated the data for comparison against any learned model.
Figure 1. By generating data in this way, we are capturing a property of many real data sets, namely that they possess an underlying regularity, which we wish to learn, but that individual observations are corrupted by random noise. This noise might arise from intrinsically stochastic i. Our goal is to exploit this training set in order to make predictions of the value t of the target variable for some new value x of the input variable.
This is intrinsically a difficult problem as we have to generalize from a finite data set. Furthermore the observed data are corrupted with noise, and so for a given x there is uncertainty as to the appropriate value for t.
Probability theory, discussed in Section 1. For the moment, however, we shall proceed rather informally and consider a simple approach based on curve fitting. The polynomial coefficients w 0 ,. Note that, although the polynomial function y x, w is a nonlinear function of x, it is a linear function of the coefficients w. Functions, such as the polynomial, which are linear in the unknown parameters have important properties and are called linear models and will be discussed extensively in Chapters 3 and 4.
The values of the coefficients will be determined by fitting the polynomial to the training data. This can be done by minimizing an error function that measures the misfit between the function y x, w , for any given value of w, and the training set data points. We shall discuss the motivation for this choice of error function later in this chapter. For the moment we simply note that it is a nonnegative quantity that would be zero if, and only if, the We can solve the curve fitting problem by choosing the value of w for which E w is as small as possible.
Because the error function is a quadratic function of the coefficients w, its derivatives with respect to the coefficients will be linear in the elements of w, and so the minimization of the error function has a unique solution, denoted by w , which can be found in closed form.
The resulting polynomial is Exercise 1. This latter behaviour is known as over-fitting. As we have noted earlier, the goal is to achieve good generalization by making accurate predictions for new data. We can obtain some quantitative insight into the dependence of the generalization performance on M by considering a separate test set comprising data points generated using exactly the same procedure used to generate the training set points but with new choices for the random noise values included in the target values.
For each choice of M , we can then evaluate the residual value of E w given by 1. However, the test set error has become very large and, as we saw in Figure 1. This may seem paradoxical because a polynomial of given order contains all lower order polynomials as special cases.
We can gain some insight into the problem by examining the values of the coefficients w obtained from polynomials of various order, as shown in Table 1. We see that, as M increases, the magnitude of the coefficients typically gets larger. Intuitively, what is happening is that the more flexible polynomials with larger values of M are becoming increasingly tuned to the random noise on the target values. It is also interesting to examine the behaviour of a given model as the size of the data set is varied, as shown in Figure 1.
We see that, for a given model complexity, the over-fitting problem become less severe as the size of the data set increases. Another way to say this is that the larger the data set, the more complex in other words more flexible the model that we can afford to fit to the data.
Pattern Recognition and Machine Learning, by Christopher M. Bishop
Linear least-squares regression, logistic regression, regularized least squares, bias-variance tradeoff, Perceptron. Press, Duda, Peter E. Hart and David G. Bertsekas and John N.
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Machine Learning Lecture Notes Pdf
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