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forget that When the economy is weaker than average, people always ask, Why would the consumer buy big-ticket items The answer, he says, is that people forget the market is a leading indicator: Look at the classic consumer discretionary stocks big-ticket items like luxury goods and cruise lines It s hard for people to realize that when you re late in a bear market, once the bottom comes, consumer discretionary stocks do tremendous Although history gives insights into what could happen in the future, Fisher has some cautionary advice One part is to look at history, and the other part is not to rely on it, he quips History tells you what happened before, but as we move forward, history never does the exact same thing The Great Humilator finds a way to twist it around Fisher is amused when people approach him with predictions of what s going to happen in the future I ask, Has this happened a lot in history What do you base that on And if it never happened before in history, they are making a very strong statement Fisher says if that event happened a lot in the past, then maybe the statement is believable But if it has never happened before, you d better have some powerful evidence to make me believe this event will happen People tell me things all the time that are so improbable because they have happened so rarely in history People who bet on the black swan will get their neck chipped

Figure 96 (a) Four centers chosen by farthest- rst traversal (b) The resulting clusters (a)

Integer roots are reciprocal-of-integer powers Suppose we take some number or quantity a, and raise it to the power 1/p where p is a positive integer We write this as

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a1/p We can surround the exponent with parentheses for clarity If we do that to the above expression, we get a(1/p) In this case, the parentheses are not technically necessary because the whole ratio is written as a superscript anyway When we take a reciprocal-of-integer power of a quantity, the result is often called a root If you have a number and raise it to the power 1/p, it is the same thing as taking the pth root of that number If p is a positive integer, then the pth root of a quantity is something we must multiply by itself p times in order to get that quantity

It s clear that this algorithm returns a valid partition What s more, the resulting diameter is guaranteed to be at most twice optimal Here s the argument Let x X be the point farthest from 1 , , k (in other words the next center we would have chosen, if we wanted k + 1 of them), and let r be its distance to its closest center Then every point in X must be within distance r of its cluster center By the triangle inequality, this means that every cluster has diameter at most 2r But how does r relate to the diameter of the optimal clustering Well, we have identi ed k + 1 points { 1 , 2 , , k , x} that are all at a distance at least r from each other (why ) Any partition into k clusters must put two of these points in the same cluster and must therefore have diameter at least r This algorithm has a certain high-level similarity to our scheme for VERTEX COVER Instead of a maximal matching, we use a different easily computable structure a set of k points that cover all of X within some radius r, while at the same time being mutually separated by a distance of at least r This structure is used both to generate a clustering and to give a lower bound on the optimal clustering We know of no better approximation algorithm for this problem 274

Fisher wrote a book, How to Smell a Rat, about helping people to identify clues in advance when companies or financial advisors are unethical Although he says that fraudulent companies such as WorldCom and Enron are relatively rare, there are some important lessons I don t believe you can have all that much faith in the earnings-per-share number that some company puts out, he notes I don t know if the accounting is that exact to begin with Let me put it another way: The people who worked at Enron who had all of their life savings in Enron were unwise to take a double bet I don t want to see my 401(k) invested in the company that I work for To protect yourself, Fisher suggests that you diversify your investments

The square root If the general formulas above confuse you, it can help if we look at an example We know that

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