With the full system introduced in Chapter 3, now we are ready to discuss how to evaluate the performance of a pattern recognition system: a task that seems easy at first glance but is in fact quite complex. We introduce core concepts such as error and accuracy rates, under- and overfitting, and parameters and hyperparameters. We pay special attention to imbalanced problems. Finally, we present a brief introduction on how confident we can be of the evaluation outcomes. We establish the fact that errors are inevitable in most pattern recognition systems, and also introduce a decomposition of errors into different terms.

This book intends to be self-contained, and this chapter provides a short recap of (almost) all the necessary mathematical background that is required to understand the rest of this book.

HMM (hidden Markov model) is a key tool to handle sequences (time series data), but it is not the only one. We start this chapter with a very brief introduction to a few tools for such data, then devote the rest of this chapter to HMM. We first illustrate what the Markov property is and why it is so important, then naturally present HMM. Three basic problems are introduced in HMM: evaluation, decoding, and learning. Dynamic programming turns out to be the solution to the first two basic problems, and we also introduce Baum--Welch, an algorithm for learning HMM parameters.

Part II introduces domain-independent feature extraction methods, and this chapter presents principal component analysis (PCA). We start from its motivation, using an example. Then we gradually discover and develop the PCA algorithm: starting from zero dimensions, then one dimension, and finally the complete algorithm. We analyze its errors in ideal and practical conditions, and establish the equivalence between maximum variance and minimum reconstruction error. Two important issues are also discussed: when we can use PCA, and the relationship between PCA and SVD (singular value decomposition).

There is no silver bullet: no model can fit all data. Hence, special data requires special algorithms. In this chapter, we deal with two types of special data: sparse data and sequences that can be aligned to each other. We will not dive deep into sparsity learning, which is very complex. Rather, we introduce key concepts: sparsity inducing loss functions, dictionary learning, and what exactly the word sparsity means. For the second part in this chapter, we introduce dynamic time warping (DTW), which deals with sequences that can be aligned with each other (but there are sequences that cannot be aligned, which we will discuss in the next chapter). We use our old tricks: ideas, visualizations, formalizations, to reach the DTW solution. The key idea behind its success is divide-and-conquer and the key technology is dynamic programming.

The normal distribution is the most widely used continuous distribution, but many of its relevant properties are a little bit advanced for an undergraduate course. Hence, Part IV introduces some of these advanced topics. This chapter devotes itself to properties of normal distributions: single- and multivariate normal distributions, moment and canonical parameterizations, sum and product, geometry and the Mahalanobis distance, and conditional distributions. We also show that with these properties, some algorithms will become much easier to understand. We use parameter estimation and the Kalman filter as two such examples.

We cannot miss deep learning in a modern pattern recognition textbook, and we introduce CNN (convolutional neural networks) in this chapter. Although the mathematical derivation of CNN, especially the back-propagation process and gradient computation, is complex, we use a lot of useful tools to help readers understand what exactlyis going on in a CNN. Hence, this chapter focuses on accessibility rather than completeness. In its exercise problems, we introduce more relevant topics and methods.

Unlike PCA, which is unsupervised, FLD uses labels associated with data points, and no doubt it may get better linear features and accuracy than PCA. We start by illustrating this motivation, and practice the problem-solving framework by gradually developing the correct mathematical formulation behind the relatively simple idea behind Fisher's linear discriminant (FLD). We discuss various practical issues: the solution for the binary case, the scenario where this solution breaks down, and how to generalize from tasks with only two categories to many categories.

This chapter is a succinct introduction to basic probabilistic methods for pattern recognition and machine learning. One focus is to clearly present the exact meanings of different terms, including the taxonomy of different probabilistic methods. We present a basic introduction to maximum likelihood and maximum a posteriori estimation, and a very brief example to showcase the concept of Bayesian estimation. For the nonparametric world, we start from the drawbacks of parametric methods, gradually analyzing the properties preferred for a nonparametric one, and finally reach the kernel density estimation, a typical nonparametric method.