NonLinear Regression Day 1.notebook

1

October 03, 2016

For each of the following models, predict y when x = 2

1 2

3

NonLinear Regression Day 1.notebook

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October 03, 2016

Nonlinear Models

How can I straighten data?

Steps for Regression

1)

2)

3)

Goals for Re-expressing

NonLinear Regression Day 1.notebook

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October 03, 2016

Suppose you have to fit a linear model to some data and

now take a look at the residuals. For each of the following possible residual

plots, tell whether you would retry another expression. Explain why or why not.

TopicNon-Linear Regression

EQ: How can I straighten data?

Non-Linear Regression

Keywordsnonlinear,

re-express, shift,

regression, model

Study Questions:Study Questions:

Steps for Regression:1) Create a _________________________

2) Plot the ________________________

3)If there is a ___________________ , build another ______________________.

1. Make a distribution more _________________ (check out the _________________)

Goals for re‐expressing

2. Make the ______________ of groups more alike (check out a __________________)

3. Make the form seem more ___________________. 

symmetrichistogram

spread

boxplot

linear

scatterplotresidualspattern model

Re­express to equalize spread.

No, there is no pattern "left behind"

Re­express to straighten.

NonLinear Regression Day 1.notebook

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October 03, 2016

Warning You will NOT be able to linearize all data! It’s just another tool to add to our bag!

Example 2Scientist Robert Boyle examined the relationship between the volume in which a gas is contained and the pressure in its container. Boyle used a cylindrical container with a moveable top that could be raised or lowered to change the volume. (Amazing that this was published in 1662, right?!) Some of his data are listed below.

a) Create a scatterplot in your calculator and discuss the relationship between height and pressure.

b) Find the least-squared regression line. Include the correlation coefficient and R2 value. Comment on these values.

c) Re-express the data and sketch a scatterplot utilizing reciprocals.

d) Predict the pressure (inches of mercury) if the volume of a gas is 41 inches.

e) Find the least-squared regression line of this scatterplot. Include the new correlation coefficient and R2 value.

Height

Pres

sureThe association

between height and pressure is strong, negative, and non-linear.

Height

1/Pr

essu

re

= -0.000077 + 0.000713(Height)

1Pressure is an exact relationship!

Pressure = -2.22(Height) + 123.26

1Pressure

r - corr coeffThere is a strong, negative, linear association between

height and pressure.

r2 - determination87.9% of the variation in

pressure is captured (explained) in the linear model.

If you look at a scatterplot between f/stops and a camera lens...they 

actually have a logarithmic relationship.

Shutter Speed

f-st

ops

log(Speed)lo

g(f-

stop

s)

NonLinear Regression Day 1.notebook

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October 03, 2016

Quick Thought:  Why do think we utilize the linear model? Why not just use a curved best fit model?

Staight lines are easy to understand. We  know how to think about the slope and y-intercept. We also want other benefits (equal spread everywhere). We would have to give that up if we didn't make it linear.

Example 3: Gordon Moore, one of the founders of Intel Corporation, predicted in 1965 that the number of transistors on an integrated circuit chip would double every 18 months. This is Moore’s law, one way to measure the revolution in computing. Here are the data on the dates and number of transistors for Intel microprocessors:

Year (since 1970)Transistors The association 

between years since 1970 and the amount of transistors is strong, nonlinear, and positive.

Year (since 1970)

ln(Transistors)

0

1

­1

The plot shows a distinct pattern from negative error to positive as it moves left to right. The residuals are small relative to the transformed y­values and the scatterplot is more linear. We can be comfortable using this model for predictions.

ln(transistors) = 0.366(year) + 7.065

lne(transistors) = 0.366(50) + 7.065lne(transistors) = 25.3562 e25.3562 = 100 billion

Give the equation of the least‑squares regression line. 

Predict the number of transistors on an Intel computer chip in 2020.