Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: How to find a bounding line?
Replies: 39   Last Post: Jul 12, 2013 5:39 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
quasi

Posts: 10,305
Registered: 7/15/05
Re: How to find a bounding line?
Posted: Jul 10, 2013 4:50 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

ols6000 wrote:
>quasi wrote:
>>
>> To show that, assume an optimal line L with slope a passes
>> through the point (x_k,y_k).
>>
>> By the point-slope formula, the line has the equation

>...
>> y = a*x + (y_k - a*x_k)
>>
>> Let s = SUM(y_i - y(x_i)).
>>
>> Since L is an optimal bounding line, s is minimal, hence a
>> change in the value of a, if it doesn't break the bounding
>> condition, cannot decrease the sum.
>>
>> But the sum s is at most linear as a function of a.
>>
>> Case (1): s is degree 1 as function of a. If L does not pass
>> through any of the other N-1 points, a sufficiently small
>> positive or negative change in a will decrease the value of s

>
>This is not correct. A small change in a can either increase
>or decrease s, depending on the values x_i.


Since s is degree 1 in a, either a positive change in a will
decrease the value of s, or a negative change in a will decrease
the value of s. One change or the other (not both) will decrease
the value of s.

>Whether L passes through any of the other points is irrelevant.

The relevance is that if L is a lower bounding line which
passes through only one of the N points, say (x_k,y_k), then
the line can be rotated slightly in either direction
(clockwise or counterclockwise) through the point (x_k,y_k)
and will still be a lower bounding line.

Recalling that a denotes the slope of the line through the
point (x_k,y_k), a clockwise rotation will decrease the value
of a, and a counterclockwise rotation will increase the value
of a.

One of those rotations will decrease the value of s.

quasi


Date Subject Author
7/7/13
Read How to find a bounding line?
Woody
7/7/13
Read Re: How to find a bounding line?
Scott Berg
7/7/13
Read Re: How to find a bounding line?
Peter Percival
7/7/13
Read Re: How to find a bounding line?
Woody
7/7/13
Read Re: How to find a bounding line?
quasi
7/7/13
Read Re: How to find a bounding line?
quasi
7/8/13
Read Re: How to find a bounding line?
quasi
7/8/13
Read Re: How to find a bounding line?
Woody
7/8/13
Read Re: How to find a bounding line?
quasi
7/8/13
Read Re: How to find a bounding line?
LudovicoVan
7/8/13
Read Re: How to find a bounding line?
LudovicoVan
7/10/13
Read Re: How to find a bounding line?
Woody
7/10/13
Read Re: How to find a bounding line?
quasi
7/8/13
Read Re: How to find a bounding line?
Leon Aigret
7/8/13
Read Re: How to find a bounding line?
Woody
7/10/13
Read Re: How to find a bounding line?
Leon Aigret
7/10/13
Read Re: How to find a bounding line?
Leon Aigret
7/10/13
Read Re: How to find a bounding line?
Woody
7/10/13
Read Re: How to find a bounding line?
RGVickson@shaw.ca
7/10/13
Read Re: How to find a bounding line?
Woody
7/10/13
Read Re: How to find a bounding line?
quasi
7/7/13
Read Re: How to find a bounding line?
quasi
7/7/13
Read Re: How to find a bounding line?
quasi
7/7/13
Read Re: How to find a bounding line?
quasi
7/8/13
Read Re: How to find a bounding line?
William Elliot
7/8/13
Read Re: How to find a bounding line?
Peter Percival
7/8/13
Read Re: How to find a bounding line?
quasi
7/11/13
Read Re: How to find a bounding line?
Woody
7/11/13
Read Re: How to find a bounding line?
quasi
7/11/13
Read Re: How to find a bounding line?
LudovicoVan
7/11/13
Read Re: How to find a bounding line?
quasi
7/11/13
Read Re: How to find a bounding line?
Leon Aigret
7/11/13
Read Re: How to find a bounding line?
Woody
7/11/13
Read Re: How to find a bounding line?
Leon Aigret
7/12/13
Read Re: How to find a bounding line?
Woody
7/12/13
Read Re: How to find a bounding line?
Leon Aigret
7/11/13
Read Re: How to find a bounding line?
Woody
7/12/13
Read Re: How to find a bounding line?
quasi
7/12/13
Read Re: How to find a bounding line?
Woody
7/12/13
Read Re: How to find a bounding line?
quasi

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.