PhD Document - Universidad de Las Palmas de Gran Canaria

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CHAPTER 2. THE CASE OF SOCIAL ROBOTICS ples. What about the available knowledge about the solution? A priori knowledge about the solution to the task allows to define the hypothesis space. We may know, for example, that the solution is a polynomial function. Better still, we may know that the solution is actually a quadratic. The more knowledge we have of the form of the solution the smaller the search space in which our learner will look for a candidate solution: H0 ⊃ H1 ⊃ H2 ⊃ . . . ⊃ H ∗ H ∗ being the objective hypothesis (the solution). (2.1) Virtually all "practical" learners employ some sort of complexity penalization tech- nique [Scheffer, 1999], including the method of sieves and Bayesian, Minimum Description Length, Vapnik-Chervonenkis dimension and validation methods [Nowak, 2004]. The basic idea of complexity penalization is to minimize the sum of the error in the training set and a complexity measure of the hypothesis space. The quantity that we are ultimately interested in is the risk: R(α) = 1 |y − f(x, α)|dP (x, y) (2.2) 2 α being the parameters of the learner (to be set). The risk is a measure of discrepancy between the values f(x, α) produced by our learner and the objective values y produced by H ∗ . Theoretical work by Vapnik and others has allowed to obtain upper bounds on this risk [Vapnik, 1995]. With probability 1 − η the following bound holds: R(α) ≤ eemp + h(log(2l/h) + 1) − log(η/4) l (2.3) where eemp is the empirical risk (the error on the training set), and h a non-negative integer called VC dimension. The VC dimension of a set of functions Hi, a measure of its complexity, is the maximum number of training points that can be shattered 4 by Hi. Note that we can in principle obtain an eemp as low as we want, though we would have to do it by making our hypothesis space more complex (i.e. by having a large number of degrees de freedom). Alternatively, we can consider only simple hypotheses, but then we would obtain large eemp. 4 for a 2-class recognition problem and a set of l training points, if the set can be labelled in all possible 2 l ways, and for each labelling a member of the set Hi can be found which correctly assigns those labels, then we say that the set of points is shattered by that set of functions. 20
CHAPTER 2. THE CASE OF SOCIAL ROBOTICS From the definition of VC dimension the hypothesis spaces considered above satisfy: h(H0) ≥ h(H1) ≥ . . . ≥ h(H ∗ ) (2.4) The second term of the right-hand side of Equation (2.3) is monotonically increasing with h (see [Burges, 1998]). Thus, if we call: the following holds: V C(Hi) = h(Hi)(log(2l/h(Hi)) + 1) − log(η/4) l (2.5) V C(H0) ≥ V C(H1) ≥ . . . ≥ V C(H ∗ ) (2.6) In Equation (2.3), eemp is the error obtained in the training set for an individual hypothesis. Let eempMax(Hi) be the maximum training error that can be obtained in Hi (i.e. the maximum training error of all the individual hypotheses in Hi). Then: From (2.1): R(α) ≤ eemp + V C(Hi) ≤ eempMax(Hi) + V C(Hi) (2.7) Let us assume that the error in the training set for the objective hypothesis H ∗ is 0. Card(H0) > Card(H1) > Card(H2) . . . > 1 (2.8) From (2.8) the following holds: eempMax(H0) ≥ eempMax(H1) ≥ . . . ≥ 0 (2.9) From (2.7), (2.6) and (2.9) we can conclude that the risk decreases as we increase the amount of knowledge. With reduced knowledge good performances cannot be guaranteed. Therefore, in our case in which available knowledge is little, we cannot guarantee good performance for samples different than those tested. Thus, if we obtain low error in a test set we could be actually overfitting to that test test! 5 . The involved inductive process 5 In the discussion we have not mentioned the effect of the samples used. Obviously, if the test samples are "representative" then the error measured is applicable to the future samples that the system will encounter. As an example, let us suppose that the domain is made up of a very frequent set of samples (set A) plus a very infrequent set of samples (set B). Thus P (x) is much larger for values of x inside A. Then if we measure low error 21
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CHAPTER 5. PERCEPTION 5.3 Audio-Vis
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CHAPTER 5. PERCEPTION modules, inst
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CHAPTER 5. PERCEPTION each angle we
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CHAPTER 5. PERCEPTION Figure 5.12:
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CHAPTER 5. PERCEPTION hypothesis is
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CHAPTER 5. PERCEPTION M4.- Pattern
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CHAPTER 5. PERCEPTION Figure 5.15:
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CHAPTER 5. PERCEPTION nods and shak
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CHAPTER 5. PERCEPTION effect of thi
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CHAPTER 5. PERCEPTION effects of am
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CHAPTER 5. PERCEPTION image decreas
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CHAPTER 5. PERCEPTION with an incre
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CHAPTER 5. PERCEPTION However, we a
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CHAPTER 5. PERCEPTION where y0 is t
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CHAPTER 5. PERCEPTION be distinguis
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CHAPTER 5. PERCEPTION in the sense
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CHAPTER 5. PERCEPTION a) b) c) d) F
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CHAPTER 5. PERCEPTION the habituati
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Chapter 6 Action "CHRISTOF: We’ve
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CHAPTER 6. ACTION Expression: "Surp
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CHAPTER 6. ACTION 6.1.3 Implementat
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CHAPTER 6. ACTION Figure 6.3: Facia
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CHAPTER 6. ACTION c α error 0 o 0.
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CHAPTER 6. ACTION fear anger sorrow
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CHAPTER 6. ACTION However, we belie
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Chapter 7 Behaviour "A good name, l
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CHAPTER 7. BEHAVIOUR used defines t
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CHAPTER 7. BEHAVIOUR step environme
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CHAPTER 7. BEHAVIOUR Predicate Desc
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CHAPTER 7. BEHAVIOUR Action Descrip
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CHAPTER 7. BEHAVIOUR Name Action to
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CHAPTER 7. BEHAVIOUR ➔ Basic emot
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CHAPTER 7. BEHAVIOUR [Reilly, 1996]
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Chapter 8 Evaluation and Discussion
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CHAPTER 8. EVALUATION AND DISCUSSIO
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Chapter 9 Conclusions and Future Wo
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CHAPTER 9. CONCLUSIONS AND FUTURE W
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Appendix A CASIMIRO’s Phrase File
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APPENDIX A. CASIMIRO’S PHRASE FIL
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Bibliography [Adams et al., 2000] B
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BIBLIOGRAPHY [Bruce et al., 2001] A
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BIBLIOGRAPHY [Driscoll et al., 1998
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BIBLIOGRAPHY [Hernández et al., 20
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BIBLIOGRAPHY [Kjeldsen, 2001] R. Kj
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BIBLIOGRAPHY [Martin et al., 2000]
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BIBLIOGRAPHY [Oviatt, 2000] S. Ovia
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BIBLIOGRAPHY [Schulte et al., 1999]
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BIBLIOGRAPHY [Tyrrell, 1993] T. Tyr
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Index absolute pitch, 16 action sel
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INDEX proprioception, 39 prosopagno
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