Cevian

2013-07-09T11:01:51Z

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'''DFFITS''' is a diagnostic meant to show how influential a point is in a [[statistical regression]]. It was proposed in 1980.<ref>{{cite book |last=Belsley |first=David A. |last2=Kuh |first2=Edwin |last3=Welsh |first3=Roy E. | year=1980 |title=Regression diagnostics: identifying influential data and sources of collinearity |publisher=[[John Wiley & Sons]] |location=New York |series=Wiley series in probability and mathematical statistics |isbn=0-471-05856-4}}</ref> It is defined as the change ("DFFIT"), in the predicted value for a point, obtained when that point is left out of the regression, "Studentized" by dividing by the estimated standard deviation of the fit at that point:

:<math>\text{DFFITS} = {\widehat{y_i} - \widehat{y_{i(i)}} \over s_{(i)} \sqrt{h_{ii}}}</math>

where <math>\widehat{y_i}</math> and <math>\widehat{y_{i(i)}}</math> are the prediction for point i with and without point i included in the regression,
<math>s_{(i)}</math> is the standard error estimated without the point in question, and <math>h_{ii}</math> is the [[leverage (statistics)|leverage]] for the point.

DFFITS is very similar to the externally [[Studentized residual]], and is in fact equal to the latter times <math>\sqrt{h_{ii}/(1-h_{ii})}</math>.<ref>{{cite book |last=Montogomery |first=Douglas C. |last2=Peck |first2=Elizabeth A. |last3=Vining |first3=G. Geoffrey |title=Introduction to Linear Regression Analysis |edition=5th |year=2012 |publisher=Wiley |isbn=978-0-470-54281-1 |page=[http://books.google.com/books?id=0yR4KUL4VDkC&lpg=PP1&dq=Introduction%20to%20Linear%20Regression%20Analysis%202nd%20edition&pg=PA218#v=onepage&q&f=false 218] |quote=Thus, ''DFFITSi'' is the value of ''R''-student multiplied by the leverage of the ''i''th observation [hii/(1-hii)]1/2. |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470542810.html |accessdate=22 February 2013}}</ref>

Since when the errors are [[Gaussian]] the externally Studentized residual is distributed as [[Student's t]] (with a number of [[Degrees of freedom (statistics)|degrees of freedom]] equal to the number of residual degrees of freedom minus one), DFFITS for a particular point will be distributed according to this same Student's t distribution multiplied by the [[Leverage (statistics)|leverage factor]] <math>\sqrt{h_{ii}/(1-h_{ii})}</math> for that particular point. Thus, for low leverage points, DFFITS is expected to be small, whereas as the leverage goes to 1 the distribution of the DFFITS value widens infinitely.

For a perfectly balanced experimental design (such as a [[factorial design]] or balanced partial factorial design), the leverage for each point is p/n, the number of parameters divided by the number of points. This means that the DFFITS values will be distributed (in the Gaussian case) as <math>\sqrt{p \over n-p} \approx \sqrt{p \over n}</math> times a t variate. Therefore, the authors suggest investigating those points with DFFITS greater than <math>2\sqrt{p \over n}</math>.

Although the raw values resulting from the equations are different, [[Cook's distance]] and DFFITS are conceptually identical and there is a closed-form formula to convert one value to the other (Cohen, Cohen, West & Aiken, 2003).

== Development ==
Previously when assessing a dataset before running a linear regression, the possibility of outliers would be assessed using histograms and scatterplots. Both methods of assessing data points were subjective and there was little way of knowing how much leverage each potential outlier had on the results data. This led to a variety of quantitative measures, including DFFIT, DFBETA.

==References==
<references/>

[[Category:Regression diagnostics]]

Ptolemy's theorem

2013-07-09T10:55:33Z

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{{About|the physical paradox|the Lewis Carroll dialogue|What the Tortoise Said to Achilles }}
In [[physics]], '''Carroll's paradox''' arises when considering the motion of a falling rigid rod that is specially constrained. Considered one way, the [[angular momentum]] stays constant; considered in a different way, it changes. It is named after Michael M. Carroll who first published it in 1984.

== Explanation ==

Consider two [[concentric circles]] of radius <math>r_1</math> and <math>r_2</math> as might be drawn on the face of a wall clock. Suppose a uniform rigid heavy rod of length <math>l=|r_2-r_1|</math> is somehow constrained between these two circles so that one end of the rod remains on the inner circle and the other remains on the outer circle. Motion of the rod along these circles, acting as guides, is frictionless. The rod is held in the three [[o'clock position]] so that it is horizontal, then released.

Now consider the angular momentum about the centre of the rod:

# After release, the rod falls. Being constrained, it must rotate as it moves. When it gets to a vertical six o'clock position, it has lost [[potential energy]] and, because the motion is frictionless, will have gained [[kinetic energy]]. It therefore possesses angular momentum.
# The reaction force on the rod from either circular guide is frictionless, so it must be directed along the rod; there can be no component of the reaction force perpendicular to the rod. Taking [[moment (physics)|moments]] about the center of the rod, there can be no moment acting on the rod, so its angular momentum remains constant. Because the rod starts with zero angular momentum, it must continue to have zero angular momentum for all time.

An apparent resolution of this paradox is that the physical situation cannot occur. To maintain the rod in a radial position the circles have to exert an infinite force. In real life it would not be possible to construct guides that do not exert a significant reaction force perpendicular to the rod. Victor Namias, however, disputed that infinite forces occur, and argued that a finitely thick rod experiences torque about its center of mass even in the limit as it approaches zero width.

==References==
<references/>
*{{cite journal
| last = Carroll
| first = Michael M.
| date = November 1984
| title = Singular constraints in rigid-body dynamics
| journal = [[American Journal of Physics]]
| volume = 52
| issue = 11
| pages = 1010-1012
| doi = 10.1119/1.13777
}}
*{{cite journal
| last = Namias
| first = Victor
| date = May 1986
| title = On an apparent paradox in the motion of a smoothly constrained rod
| journal = [[American Journal of Physics]]
| volume = 54
| issue = 5
| pages = 440-445
| doi = 10.1119/1.14610
}}
*{{cite journal
| last = Felszeghy
| first = Stephen F.
| date = 1986
| title = On so-called singular constraints in rigid-body dynamics
| journal = [[American Journal of Physics]]
| volume = 54
| pages = 585-586
| doi = 10.1119/1.14533
}}

==External links==

[[Category:Mechanics]]
[[Category:Physical paradoxes]]

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Ptolemy's theorem