En la Edad Media

that were built in the interval between the invention of the motor car and the invention of innovative town planners, then you would probably use a different measure. You would measure the distance between my house and the shop in the ‘north-south’ direction and the distance in the ‘east-west’ direction, and then add the two distances together. This would correspond to the distance I actually had to walk. It is often known as the city-block or Manhattan distance and looks like:

dC= |x1− x2| + |y1− y2|. (7.12) The point of this discussion is to show that there is more than one way to measure a distance, and that they can provide radically different answers. These two different distances can be seen in Figure 7.9. Mathematically, these distance measures are known as metrics.

A metric function or norm takes two inputs and gives a scalar (the distance) back, which is positive, and 0 if and only if the two points are the same, symmetric (so that the distance to the shop is the same as the distance back), and obeys the triangle inequality, which says that the distance from a to b plus the distance from b to c should not be less than the direct distance from a to c.

Most of the data that we are going to have to analyse lives in rather more than two dimensions. Fortunately, the Euclidean distance that we know about generalises very well to higher dimensions (and so does the city-block metric). In fact, these two measures are both instances of a class of metrics that work in any number of dimensions. The general measure is the Minkowski metric and it is written as:

Lk(x, y) =

d

X

i=1

|xi− yi|^k

!k¹

. (7.13)

If we put k = 1 then we get the city-block distance (Equation (7.12)), and k = 2 gives the Euclidean distance (Equation (7.11)). Thus, you might possibly see the Euclidean metric written as the L2 norm and the city-block distance as the L1 norm. These norms have another interesting feature. Remember that we can define different averages of a set of numbers. If we define the average as the point that minimises the sum of the distance to every datapoint, then it turns out that the mean minimises the Euclidean distance (the sum-of-squares distance), and the median minimises the L1metric. We met another distance measure earlier: the Mahalanobis distance in Section 2.4.2.

There are plenty of other possible metrics to choose, depending upon the dataspace. We generally assume that the space is flat (if it isn’t, then none of these techniques work, and

we don’t want to worry about that). However, it can still be beneficial to look at other metrics. Suppose that we want our classifier to be able to recognise images, for example of faces. We take a set of digital photos of faces and use the pixel values as features. Then we use the nearest neighbour algorithm that we’ve just seen to identify each face. Even if we ensure that all of the photos are taken fully face-on, there are still a few things that will get in the way of this method. One is that slight variations in the angle of the head (or the camera) could make a difference; another is that different distances between the face and the camera (scaling) will change the results; and another is that different lighting conditions will make a difference. We can try to fix all of these things in preprocessing, but there is also another alternative: use a different metric that is invariant to these changes, i.e., it does not vary as they do. The idea of invariant metrics is to find measures that ignore changes that you don’t want. So if you want to be able to rotate shapes around and still recognise them, you need a metric that is invariant to rotation.

A common invariant metric in use for images is the tangent distance, which is an approx-imation to the Taylor expansion in first derivatives, and works very well for small rotations and scalings; for example, it was used to halve the final error rate on nearest neighbour classification of a set of handwritten letters. Invariant metrics are an interesting topic for further study, and there is a reference for them in the Further Reading section if you are interested.

FURTHER READING

For more on nearest neighbour methods, see:

• T. Hastie and R. Tibshirani. Discriminant adaptive nearest neighbor classification and regression. In David S. Touretzky, Michael C. Mozer, and Michael E. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 8, pages 409–415.

The MIT Press, 1996.

• N.S. Altman. An introduction to kernel and nearest-neighbor nonparametric regres-sion. The American Statistician, 46:175–185, 1992.

The original description of KD-trees is:

• A. Moore. A tutorial on KD-trees. Extract from PhD Thesis, 1991. Available from http://www.cs.cmu.edu/simawm/papers.html.

A reference on the tangent distance is:

• P.Y. Simard, Y.A. Le Cun, J.S. Denker, and B. Victorri. Transformation invariance in pattern recognition: Tangent distance and propagation. International Journal of Imaging Systems and Technology, 11:181–194, 2001.

Some of the material in the chapter is covered in:

• Section 9.2 of C.M. Bishop. Pattern Recognition and Machine Learning. Springer, Berlin, Germany, 2006.

• Chapter 6 (especially Sections 6.1–6.3) of T. Mitchell. Machine Learning. McGraw-Hill, New York, USA, 1997.

• Section 13.3 of T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, 2nd edition, Springer, Berlin, Germany, 2008.

PRACTICE QUESTIONS

Problem 7.1 Extend the Gaussian Mixture Model algorithm to allow for more than two classes in the data. This is not trivial, since it involves modifying the EM algorithm.

Problem 7.2 Modify the KD-tree algorithm so that it works on spheres in the data, rather than rectangles. Since they no longer cover the space you will have to add some cases that fail to return a leaf at all. However, this means that the algorithm will not return points that are far away, which will make the results more accurate. Now modify it so that it does not use the Euclidean distance, but rather the L₁ distance. Compare the results of using these two methods on the iris dataset.

Problem 7.3 Use the small figures of numbers that are available on the book website in order to compute the tangent distance. You will have to write code that rotates the numbers by small amounts in order to check that you have written it correctly. What happens when you make large rotations (particularly of a 6 or 9)? Compare using nearest neighbours with Euclidean distance and the tangent distance to verify the results claimed in the chapter. Extend the experiment to the MNIST dataset.

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