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<Paper uid="N06-4008">
  <Title>Automating the Creation of Interactive Glyph-supplemented Scatterplots for Visualizing Algorithm Results</Title>
  <Section position="4" start_page="282" end_page="282" type="metho">
    <SectionTitle>
3 Embedding into Three Dimensions
</SectionTitle>
    <Paragraph position="0"> When visualizing the results of algorithms, users may not have a three-dimensional embedding already available. However, algorithms have been proposed to produce such embeddings, and we now describe some of those available in Ndaona. Ndaona also implements basic dimensionality reduction algorithms such as Principal Components Analysis, Laplacian Eigenmaps, and Isomap.</Paragraph>
    <Section position="1" start_page="282" end_page="282" type="sub_section">
      <SectionTitle>
3.1 Classification Probabilities
</SectionTitle>
      <Paragraph position="0"> If users have a N xK matrix S of prediction probabilities from a K-class classification algorithm, with S(n,k) having the probability (estimated by the algorithm) that the n-th point is in class k, then this can be supplied instead.</Paragraph>
      <Paragraph position="1"> Ndaona uses the Parametric Embedding algorithm (Iwata et al., 2004) to find a low-dimensional embedding of the N points so that pairs of points that were given similar predictions by the classification algorithm (i.e. have low Kullback-Leibler distance between their prediction probability distributions) are closer together.</Paragraph>
    </Section>
    <Section position="2" start_page="282" end_page="282" type="sub_section">
      <SectionTitle>
3.2 Kernel Matrices
</SectionTitle>
      <Paragraph position="0"> Support vector machines (SVMs) and related methods depend on pairwise similarities of points, in the form of a kernel matrix whos (i,j)-th entry represents the similarity of the i-th and j-th points.</Paragraph>
      <Paragraph position="1"> Shawe-Taylor and Christianini (2004) suggest using the eigenvectors corresponding to the three smallest positive eigenvalues of the Laplacian of the N xN kernel matrix to define a N x 3 positions matrix.</Paragraph>
      <Paragraph position="2"> Ndaona implements an alternative that, in our experience, works better -- using the normalized Laplacian of the kernel matrix (with negative entries replaced by zero).</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>