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
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