Highest weight vector

This representation has a highest weight vector 1, and R is spanned by elements... 

Highest weight vectors are useful in the construction of irreducible subrepresentations, a key element of many proofs. 

Definition 8.8 Suppose su(2 ),V, p) is a finite-dimensional Lie algebra representation. Suppose vq is an eigenvector of plfi) with the property that XpVo = 0. Then vq is a highest weight vector for the representation p. 

For example, in P" the polynomial x" is a highest weight vector. The next proposition shows how highest weight vectors generate irreducible subrepresentations. 

Proof. First we must show that V has at least one highest weight vector. Let v... 

In either case we find that YpY vo e W. fo see that W is also invariant under Xp, we argue by induction on/ that Xp(Y U )) e IT. For the base case (k = 0) we know from the definition of a highest weight vector that XpUo = 0 6 VF. The inductive step is... 

Proof. The main idea is to define an isomorphism by mapping the eigenvectors of pli) to the monomials in P". Choose any highest weight vector vo for the representation p. Because V is in irreducible representation of sm(2), the nontrivial subrepresentation VL constructed in Proposition 8.8 must be all of V. Hence we have an explicit basis for V, namely,... 

Next we use Casimirs to find a vector w that is a highest-weight vector for both... 

Exercise 8.9 Construct a Lie algebra representation (sii(2), V, p) with two highest weight vectors Vq and Vi such that their corresponding eigenvalues Ao Al (respectively) are not equal. 

In the q-boson picture the normalized highest weight vector is ... 

In the hypothetical case study, we take Q as the KM estimate of event-free survival probabilify at month 12 as an example. Based on some clinical judgment, the weight vector was chosen as (0.48, 0.29,0.04, 0.19) with the highest weight for PFS and lowest weight for AEl. As all endpoints are on the same scale, no further transformation was applied (i.e., vfx) = x). 

Other weighting schemes have been defined. For example, to representthe Olympic aggregator (which evaluates the average after eliminating the highest and lowest score) the weighting vector is ... 

Extensive studies by Jurs et. al. C1313 with mass spectra showed that a reflexion of the decision plane yields the highest convergence rate. Other methods of weight vector correction and a method with a small positive feedback upon correct classification were investigated by the same authors but did not give better results-... 

A detailed explanation on the derivation of PCA is available in a review paper by Svante (63). In many ways, PLS is similar to PCA except that it looks for correlations between matrices or a matrix and a vector. Robust calibrations can be created using PLS becau,se the correlation is multivariate in nature, making it less susceptible to random noise. PL/S captures the highest variance in the data set and correlates both data blocks simultaneously. Unlike PCA where independent score. sets for each data block is calculated, a common link or weight loading vector (w) is calculated. A regression coefficient, , is used to predict independent variables. 

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