Support Vector Regression Machines
Chris J.C.
From now on, when we write , we mean the feature
space reprentation and we must determine the f components of
The cond reprentation is a support vector regression (SVR) reprentation that was developed by Vladimir Vapnik (1995):
and b. If we expand the term raid to the p’th power, we find f coefficients that multiply the various powers and cross product
in that they have the same number of
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合作计划书terms. However coefficients while coefficients that must be
For instance, suppo we have a
and but for the feature space reprentation, we
have to determine coefficients while for the SVR reprentation we have to determine
coefficients in
reprent the 2N values of The optimum values for the components
of depend on our definition of the loss function and
in the prence of noi, and the last term is a
is placed in front of the first
and we let
let V be a matrix who i’th row is the i’th training vector reprented in feature space (including the constant term “1” which reprents a bias).V is a matrix where the number of rows is the number of examples (N) and the number of columns is the dimensionality of feature space be the
I
tube (Figure 1) so that if the predicted value is within the tube the loss
is zero, while if the predicted point is outside the tube, the loss is the magnitude of the difference between the predicted value and the radius
+
where if the sample point is inside the tube. If the obrved point is
“above” the tube, and
if the obrved point is below the tube
and in this ca Since an obrved point can not be simultaneously我喜欢我自己教案
on both sides of the tube, either a; or unless the point is within the tube, in which ca, both constants will be zero.
If emphasis is placed on the emphasis is
placed on the
is:
因果启示录
+
where the
b, and
mabang
and
=
constant is that it now appears
and a vectors. The quadratic programming problems can be very cpu and memory intensive. Fortunately, we can devi programs that
make u of
the fact that for problems with few support vectors (in comparison to the sample size), storage space is proportional to the number of support vectors. We u an active t method (Bunch and Kaufman, 1980) to solve this quadratic programming problem.
Although we may find
function of
That is, we can express the predicted values as:
using any procedure discusd here. Suppo
plus noi. We define the prediction error (PE) and the modeling error (ME):
For the three Friedman functions we calculated both the prediction
constant in the feature space reprentation.
constant and obtain the prediction error on the validation t. Now repeat with a different
constant that minimizes the validation t prediction example test t. This experiment was repeated for and validation ts of size 40 but one test t of size
the dimensionality of feature space is 66 while for the last two problems, the dimensionality of feature space was 15 (for驻村工作队
constant U,
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was the optimum choice of power.
For the Boston Housing data, we picked randomly from the 506 cas using a training t of size 401, a validation t of size 80 and a test t of size 25. This was repeated 100 times. The optimum power as picked by the validations t varied between
3. Results of experiments
The first experiments we tried were bagging regression trees versus support regression (Table I).
Table I. Modeling error and prediction
trials).
Rather than report the standard
for the first experiment we tried both SVR and bagging on the same training, validation, and test t. If SVR had a better modeling error on the test t, it counted as a win. Thus
