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Lecture 14_Polyn Reg-2012

Lecture 14_Polyn Reg-2012 - Lecture 13 Polynomial...

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1 Lecture 13: Polynomial Regression Stat 102 Textbook reading Chapter 5.2.1 Description Using JMP Relation to Multiple Regression; tests of coefficients Choosing the order of the polynomial CIs (for the mean of Y and for prediction of an additional observation)
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2 Polynomial Regression A Method for Fitting Curvilinear Relationships Reconsider the simple regression problem of estimating Y x = the conditional mean of Y given x . For many problems, Y x is not linear in x . We have suggested transformations of x (perhaps accompanying transformations of Y ) to address this problem. In some situations these yield data in a form suitable for least squares analysis; but in others they do not work well. Polynomial regression is another least-squares technique for fitting curvilinear data. We’ll look at 3 examples, and then explain the theory.
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3 Example 1: How does rainfall affect yield of corn? Data on annual corn yield and average rainfall in six US states (1890-1927). (See Cornyieldrainfall.jmp) Bivariate Fit of YIELD By RAINFALL 20 25 30 35 40 YIELD 6 7 8 9 10 11 12 13 14 15 16 17 RAINFALL Note the generally curved pattern of points, with a max at about 11. Such a pattern cannot be well fit by transformations of x and/or Y .
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4 Example 2: How do people’s incomes change as they age? Weekly wages and age of 200 randomly chosen adult males from the March 1998 C urrent P opulation S urvey 3 4 5 6 7 8 log wage 20 30 40 50 60 70 age We’ll see that there is also a curved pattern here, with max near 45 Note that we’ve already used log e (Wage) as the Y variable.
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5 Arrival Pattern of Calls to a Financial Call Center Data is for week of 7/15/2002 7/19/2002. Data is from call center of a major US bank. Here’s what a call center looks like:
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6 A Call Center (picture is from England, ~1995)
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7 The number of calls made asking for service by an agent is automatically recorded (to the nearest second). The number of calls in each quarter hour of each weekday were totaled. For each data point x = time (to the quarter-hour). For each data point y = # of calls in that quarter hour . The reason for transforming to the sq rt of the # of calls is explained in Brown, Gans, Mandelbaum, Sakov, Shen, Zeltyn and Zhao, (2005) Statistical analysis of a telephone call center: a queueing science perspective ”, Jour. Amer. Statist. Assoc., 100 , 36-50.
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