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By Francesco Parrella

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3. 3. 4. 23) θi ∈ [0, C] θi = +C The Online Support Vector Regression algorithm main purpose is to verify these conditions after that a new sample is added. 24) (θi = +C ∧ h(xi ) ≤ − )} Remaining Set R = {i| θi = 0 ∧ |h(xi )| ≤ } The goal is to find a way to add a new sample to one of the three sets maintaining KKT conditions consistent. 6. 6. Adding a new sample The equations above can  0 1    1 Qs1 s1   ..  .  1 Qsls s1 37 be rewritten in an equivalent matrix form:     ∆b ··· 1 1           Q · · · Qs1 sls   ∆θs1    = −  s1 c  ∆θc   ..

This is the easiest case: the sample can be added to the remaining samples set without changes of other samples. 8. Samples Moving 41 In this situation θc or h(xc ) value of the new sample changes until it reaches the support or the error set. Notice that the arrow represents the sample movements of θi or h(xi ) components. The problem is that these variations influence other samples: At the end of the updating procedure, if variation is too big, some samples could change their θi and h(xi ) values and exit from their set.

In this case the sample, is neither a support sample, neither an error sample. Note that if KKT conditions aren’t violated, this part of the algorithm isn’t used. 14. 2 48 Invalid Error Samples An error sample becomes invalid when its |h(xi )| becomes < . In this case the sample, is neither an error sample, neither a remaining sample, because |θi | = C. 3 Invalid Remaining Samples A remaining sample becomes invalid when its |h(xi )| becomes > . In this case the sample, is neither a remaining sample, neither an error sample, because θi = 0.

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