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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: How to find a bounding line?
Posted: Jul 10, 2013 4:50 PM

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 Woody
7/7/13 Scott Berg
7/7/13 Peter Percival
7/7/13 Woody
7/7/13 quasi
7/7/13 quasi
7/8/13 quasi
7/8/13 Woody
7/8/13 quasi
7/8/13 LudovicoVan
7/8/13 LudovicoVan
7/10/13 Woody
7/10/13 quasi
7/8/13 Leon Aigret
7/8/13 Woody
7/10/13 Leon Aigret
7/10/13 Leon Aigret
7/10/13 Woody
7/10/13 RGVickson@shaw.ca
7/10/13 Woody
7/10/13 quasi
7/7/13 quasi
7/7/13 quasi
7/7/13 quasi
7/8/13 William Elliot
7/8/13 Peter Percival
7/8/13 quasi
7/11/13 Woody
7/11/13 quasi
7/11/13 LudovicoVan
7/11/13 quasi
7/11/13 Leon Aigret
7/11/13 Woody
7/11/13 Leon Aigret
7/12/13 Woody
7/12/13 Leon Aigret
7/11/13 Woody
7/12/13 quasi
7/12/13 Woody
7/12/13 quasi