R help - Sep 2009 - A stat related question

Can I ask a small stat. related question here?

Suppose I have two predictors for a time series processes and accuracy of
predictor is measured from MSEs. My question is, if two predictors give same
MSE then, necessarily they have to be identical? Can anyone provide me any
counter example?

Thanks.
-- 
View this message in context:
http://www.nabble.com/A-stat-related-question-tp25505618p25505618.html
Sent from the R help mailing list archive at Nabble.com.

Let's assume you have just three observations, and x-- = 1:3 for your
observations. 

Predictor 1:	y = x^2
Predictor 2: 	y = 1 if x=1
			y = 4 if x=2
			y = 9 if x=3
			y = 0 elsewhere

These predictors are obviously not the same, but will give the same Mean
Squared Error for your data (whatever your observed y-values are). This
should suffice as a counter example. Or did I misunderstand your
question?

HTH, Michael

> -----Original Message-----
> From: r-help-bounces at r-project.org 
> [mailto:r-help-bounces at r-project.org] On Behalf Of RON70
> Sent: Freitag, 18. September 2009 11:23
> To: r-help at r-project.org
> Subject: [R] A stat related question
> 
> 
> Can I ask a small stat. related question here?
> 
> Suppose I have two predictors for a time series processes and 
> accuracy of predictor is measured from MSEs. My question is, 
> if two predictors give same MSE then, necessarily they have 
> to be identical? Can anyone provide me any counter example?
> 
> Thanks.
> --
> View this message in context: 
> http://www.nabble.com/A-stat-related-question-tp25505618p25505618.html
> Sent from the R help mailing list archive at Nabble.com.
> 
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide 
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

On 18/09/2009, at 9:23 PM, RON70 wrote:
>
> Can I ask a small stat. related question here?
>
> Suppose I have two predictors for a time series processes and  
> accuracy of
> predictor is measured from MSEs. My question is, if two predictors  
> give same
> MSE then, necessarily they have to be identical? Can anyone provide  
> me any
> counter example?
Counter example:

xmpl.df <- structure(list(y = c(-0.367234642740975,  
0.185230564865609, 0.581823727365507,
1.39973682729268, -0.727292059474465, 1.30254263204414,  
0.335848119752074,
1.03850609869762, 0.920728568290646, 0.720878162866862,  
-1.04311893856785,
-0.0901863866107067, 0.623518161999544, -0.953523357772344,  
-0.542828814573857,
0.580996497681682, 0.768178737834591, 0.463767588540167,  
-0.88577629740968,
-1.09978089864786), x1 = c(0.206067430466075, -0.132238579133420,
0.0299230903476012, 0.0770661103560109, 0.0371133529511250,  
-0.0520909837658339,
0.230634542906874, -0.0500870952845974, 0.319228715708252,  
-0.0445038917047473,
0.194516706231773, 0.366107384673495, -0.276282276770058,  
-0.0822685230586955,
-0.0568443308533714, 0.0776057819874248, -0.0832235252633287,
-0.497827207484688, -0.460077637514818, 0.197180935204927), x2 = c 
(0.0933724365258708,
0.290885869560421, -0.0537456615562362, -0.245617952924438,  
-0.375140161451431,
-0.0161691421541291, 0.156173578334144, 0.216101027538157,  
0.0175689640482125,
0.0199243858378162, -0.0866770708194298, 0.00756428018151888,
-0.514631477389958, -0.00411244710635592, -0.203127938586995,
0.337864750427246, 0.0317949224635923, -0.115158146496248,  
0.434123920996512,
0.00900586257173104)), .Names = c("y", "x1",
"x2"), row.names = c(NA,
-20L), class = "data.frame")

The predictors x1 and x2 are *orthogonal* to each other, yet yield  
exactly
the same model when y is regressed on each of them.

To construct such an example think in terms of geometry and linear  
algebra.

Let ``o'' be the constant n-vector all of whose entries are 1.
Take an n-vector y and a unit n-vector x1 which is orthogonal to
``o'' (i.e. which has mean 0).  Construct a unit vector x2 which is
in the othocomplement of V_1 = <o,x1> = the span of o and x1, and  
which has
the same inner product with y as has x1.

To do the latter --- choose any two unit vectors, u1 and u2 in the  
orthocomplement
of V_1, let x2 = a*u1 + b*u2 and choose a and b so that a^2 + b^2 = 1  
and
(y,x2) = (y,x1).  Note that ``(v1,v2)'' means the inner (dot) product  
of v1 and v2.

``Choosing'' a and b involves solving a quadratic equation.

To get things in orthocomplements of things, use the Gramm-Schmidt  
orthonormalization
algorithm.

	cheers,

		Rolf Turner

######################################################################
Attention:\ This e-mail message is privileged and confid...{{dropped:9}}