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.