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The robustness of an estimate against a single outlier is measured by its influence function The robustness of an estimate against heavier contamination is measured by its breakdown point, which is the largest proportion of outliers that can occur in a sample without entailing the possibility of arbitrarily large bias See Rousseeuw and Leroy (1987, Section 12) The LS, LAD, and Huber M-estimates are not robust in this regard; they have breakdown points of 0% because of their sensitivity to leverage points The rank-based estimate in 6 has a breakdown point less than 30% The maximum possible breakdown point is 50% This is achieved by the least-median-of-squares (LMS) estimate, which is the estimate that minimizes the median of the squared residuals (or, equivalently, minimizes the median of the absolute residuals leil) See Rousseeuw and Leroy (1987).

popped from the stack, and the left child is the second tree popped We then push a pointer to the result back onto the stack This algorithm is essentially the same as that used in a postfix evaluation, with tree creation replacing the binary operator computation

Summary

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In this chapter we examined two uses of stacks in programming language and compiler design We demonstrated that, even though the stack is a simple structure, it is very powerful Stacks can be used to decide whether a sequence of symbols is well balanced The resulting algorithm requires linear time and, equally important, consists of a single sequential scan of the input Operator precedence parsing is a technique that can be used to parse infix expressions It, too, requires linear time and a single sequential scan Two stacks are used in the operator precedence parsing algorithm Although the stacks store different types of objects, the generic mechanism (templates) allows the use of a single stack implementation for both types of objects

expression tree A tree in which the leaves contain operands and the other nodes contain operators (p 432) infix expression An expression in which a binary operator has arguments to its left and right When there are several operators, precedence and associativity determine how the operators are processed (P 420) lexical analysis The process of recognizing tokens in a stream of symbols (p 41 1) operator precedence parsing An algorithm that converts an infix expression to a postfix expression in order to evaluate the infix expression (p 422) postfix expression An expression that can be evaluated by a postfix machine without using any precedence rules (p 421) postfix machine Machine used to evaluate a postfix expression The algorithm it uses is as follows: Operands are pushed onto a stack and an operator pops its operands and then pushes the result At the end of the evaluation, the stack should contain exactly one element, which represents the result (p 421)

In the plot of a simple regression data set, the LMS line is the middle of the narrowest strip containing half of the data points The LMS estimate is simple to describe and is very robust against outliers For certain types of data sets, however, the estimate is not robust against small changes in centrally located data points LMS estimation involves considerable computation and has low efficiency, but it can be useful in the detection of outliers or as a robust initial estimate for iterative procedures More complicated.

precedence table A table used to decide what is removed from the operator stack Left-associative operators have the operator stack precedence set at 1 higher than the input symbol precedence Rightassociative operators go the other way (p 430) state machine A common technique used to parse symbols; at any point, the machine is in some state, and each input character takes it to a new state Eventually, the state machine reaches a state at which a symbol has been recognized (p 414) tokenization The process of generating the sequence of symbols (tokens) from an input stream (p 41 1 )

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1 In production code, input errors must be handled as carefully as possible Being lax in this regard leads to programming errors 2 For the balanced symbol routine, handling quotes incorrectly is a common error 3 For the infix to postfix algorithm, the precedence table must reflect the correct precedence and associativity

Both application programs are available You should probably download the balancing program; it may help you debug other C++ programs

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Contains the balanced symbol program Contains the Tokenizer class interface for checking C++ programs (Figure 122) Contains the Tokenizer class implementation for checking C++ programs Contains the expression evaluator, instantiated for int

regression estimates have been devised that have breakdown points of 50% and also have efficiencies as high as 0.95 for normally distributed errors, but computation is still a problem. See Jureckova and Portnoy (1987) and Yohai and Zamar (1988).

121 Show the result of running the balanced symbol program on a 1 b ( I c [ [ [

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