k-Means (Concurrency)

Parameters

• add cluster attribute

  If true, a new Attribute called 'cluster' with the cluster_id for each Example is generated. By default the Attribute is created with the special role 'Cluster', except the parameter add as label is true. If so the new Attribute is called 'label' and has the special role 'Label'. The Attribute can also be generated later by using the Apply Model Operator.

• add as label

  If true the new Attribute with the cluster_id is called 'label' and has the special role 'Label'. If the parameter add cluster attribute is false, no new Attribute is created.

• remove unlabeled

  If set to true, Examples which cannot be assigned to a cluster are removed from the output ExampleSet.

• k

  This parameter specifies the number of clusters to determine.

• max runs

  This parameter specifies the maximal number of runs of k-Means with random initialization of the start points that are performed.

• determine good start values

  If true the k start points are determined using the k-means++ heuristic, descriped in "k-means++: The Advantages of Careful Seeding" by David Arthur and Sergei Vassilvitskii 2007.

• max optimization steps

  This parameter specifies the maximal number of iterations performed for one run of k-Means.

• measure types

  This parameter is used for selecting the type of measure to be used for finding the nearest neighbors. The following options are available:

  • Mixed Measures: Mixed Measures are used to calculate distances in case of both nominal and numerical Attributes.
  • Nominal Measures: In case of only nominal Attributes different distance metrices can be used to calculate distances on this nominal Attributes.
  • Numerical Measures: In case of only numerical Attributes different distance metrices can be used to calculate distances on this numerical Attributes.
  • Bregman Divergences: Bregman divergences are more generic "closeness" measure types with does not satisfy the triangle inequality or symmetry. For more details see the parameter divergence.
• mixed measure

  The only available option for mixed measure is the 'Mixed Euclidean Distance'. For numerical values the euclidean distance is calculated. For nomimal values, a distance of 0 is taken if both values are the same and a distance of one is taken otherwise. This parameter is available when the measure type parameter is set to 'mixed measures'.

• nominal measure

  This parameters defines how to calculate distances for only nominal Attributes in the input ExampleSet, in case the measure type is set to nominal measure. In case of using a similarity as a distance measure, the actual distance is calculated as the negative similarity. For the different similarities the following variables are defined:

  e: number of Attribute for which both Examples have equal and non-zero values

  u: number of Attribute for which both Examples have not equal values

  z: number of Attribute for which both Examples have zero values

  • Nominal Distance: Distance of two values is 0 if both values are the same and 1 otherwise.
  • Dice Similarity: With the above mentioned definitions the DiceSimilarity is: 2*e/(2*e+u)
  • Jaccard Similarity: With the above mentioned definitions the JaccardSimilarity is: e/(e+u)
  • Kulczynski Similarity: With the above mentioned definitions the KulczynskiSimilarity is: e/u
  • Rogers Tanimoto Similarity: With the above mentioned definitions the RogersTanimotoSimilarity is: (e+z)/(e+2*u+z)
  • Russell Rao Similarity: With the above mentioned definitions the RussellRaoSimilarity is: e/(e+u+z)
  • Simple Matching Similarity: With the above mentioned definitions the SimpleMatchingSimilarity is: (e+z)/(e+u+z)
• numerical measure

  This parameters defines how to calculate distances for only numerical Attributes in the input ExampleSet, in case the measure type is set to numerical measure. For the different distance measures the following variable is defined:

  y(i,j) : Value of the j.th Attribute of the i.th Example. Hence y(1,3) - y(2,3) is the difference of the values of the third Attribute of the first and second Example.

  In case of using a similarity as a distance measure, the actual distance is calculated as the negative similarity.

  • Euclidean Distance: Square root of the sum of quadratic differences over all Attributes. Dist = Sqrt ( Sum_(j=1) [y(1,j)-y(2,j)]^2 )
  • Canberra Distance: Sum over all Attributes. The summand is the absolute of the difference of the value, divided by the sum of the absolute values. Dist = Sum_(j=1) |y(1,j)-y(2,j)| / (|y(1,j)|+|y(2,j)|) The CanberraDistance is often used to compare ranked list or for intrusion detection in computer security.
  • Chebychev Distance: Maximum of all differences of all Attributes. Dist = max_(j=1) (|y(1,j)-y(2,j)|)
  • Correlation Similarity: The similarity is calculated as the correlation between the Attribute vectors of the two Examples.
  • Cosine Similarity: Similarity measure measuring the cosine of the angle between the Attribute vectors of the two Examples.
  • Dice Similarity: The DiceSimilarity for numerical Attributes is calculated as 2*Y1Y2/(Y1+Y2). Y1Y2 = Sum over product of values = Sum_(j=1) y(1,j)*y(2,j). Y1 = Sum over values of first Example = Sum_(j=1) y(1,j) Y2 = Sum over values of second Example = Sum_(j=1) y(2,j)
  • Dynamic Time Warping Distance: Dynamic Time Warping is often use in Time Series analysis for measuring the distance between two temporal sequences. Here the distance on an optimal "warping" path from the Attribute vector of the first Example to the second Example is calculated.
  • Inner Product Similarity: The similarity is calculated as the sum of the product of the Attribute vectors of the two Examples. Dist = -Similarity = -Sum_(j=1) y(1,j)*y(2,j)
  • Jaccard Similarity: The JaccardSimilarity is calculated as Y1Y2/(Y1+Y2-Y1Y2). See DiceSimilarity for the definition of Y1Y2, Y1 and Y2.
  • Kernel Euclidean Distance: The distance is calculated by the Euclidean distance of the two Examples, in a transformed space. The transformation is defined by the chosen kernel and configured by the parameters kernel type, gamma, sigma1, sigma2, sigma 3, shift, degree, a, b.
  • Manhattan Distance: Sum of the absolute distances of the Attribute values. Dist = Sum_(j=1) |y(1,j)-y(2,j)|
  • Max Product Similarity: The similarity is the maximum of all products of all Attribute values. If the maximum is less or equal to zero the similarity is not defined. Dist = -Similarity = -max_(j=1) (y(1,j)*y(2,j))
  • Overlap Similarity: The similarity is a variant of simple matching for numerical Attributes and is calculated as minY1Y2 / min(Y1,Y2). See DiceSimilarity for the definition of Y1, Y2. minY1Y2 = Sum over the minimum of values = Sum_(j=1) min [y(1,j),y(2,j)] .
• divergence

  This parameter defines which type of Bregman divergence is used when the measure type parameter is set to 'bregman divergences'. For the different distance measures the following variable is defined:

  y(i,j) : Value of the j.th Attribute of the i.th Example. Hence y(1,3) - y(2,3) is the difference of the values of the third Attribute of the first and second Example.

  • Generalized I Divergence: The distance is calculated as Sum1 Sum2. It is not applicable if any Attribute value is less or equal to 0. Sum1 = Sum_(j=1) y(1,j)*ln[y(1,j)/y(2,j)] Sum2 = Sum_(j=1) [y(1,j)-y(2,j)]
  • Itakura Saito Distance: The Itakura Saito Distance can only be calculated for ExampleSets with 1 Attribute and values larger 0. Dist = y(1,1)/y(2,1)-ln[y(1,1)/y(2,1)]-1
  • KL Divergence: The Kullback-Leibler divergence is a measure of how one probability distribution diverges from a second expected probability distribution. Dist = Sum_(j=1) [y(1,j)*log_2(y(1,j)/y(2,j))]
  • Logarithmic Loss: The LogarithmicLoss can only be calculated for ExampleSets with 1 Attribute and values larger 0. Dist = y(1,1)*ln[y(1,1)/y(2,1)]-(y(1,1)-y(2,1))
  • Logistic Loss: The LogisticLoss can only be calculated for ExampleSets with 1 Attribute and values larger 0. Dist = y(1,1)*ln[y(1,1)/y(2,1)]+(1-y(1,1))*ln[(1-y(1,1))/(1-y(2,1))]
  • Mahalanobis Distance: The Mahalanobis distance measures the distance between the two Examples under the assumption they are both random vectors of the same distribution. Therefore the covariance matrix S is calculated on the whole ExampleSet and the Distance is calculated as: Dist = Sqrt [ (vecY1-vecY2) S (vecY1-vecY2) ] vecY1 = Attribute vector of Example 1 vecY2 = Attribute vector of Example 2
  • Squared Euclidean Distance: Sum of quadratic differences over all Attributes. Dist = Sum_(j=1) [y(1,j)-y(2,j)]^2
  • Squared Loss: The SquaredLoss can only be calculated for ExampleSets with 1 Attribute. Dist = [y(1,1)-y(2,1)]^2
• kernel type

  This parameter is available only when the numerical measure parameter is set to 'Kernel Euclidean Distance'. The type of the kernel function is selected through this parameter. Following kernel types are supported:

  • dot: The dot kernel is defined by k(x,y) = x*y i.e. it is the inner product of x and y.
  • radial: The radial kernel is defined by k(x,y) = exp(-g*||x-y||^2) where g is gamma, specified by the kernel gamma parameter. The adjustable parameter gamma plays a major role in the performance of the kernel, and should be carefully tuned to the problem at hand.
  • polynomial: The polynomial kernel is defined by k(x,y) = (x*y+1)^d where d is the degree of the polynomial and is specified by the _kernel degree_ parameter. Polynomial kernels are well suited for problems where all the training data is normalized.
  • sigmoid: This is a hyperbolic tangent sigmoid kernel. The distance is calculated as tanh[a*Y1Y2+b] where Y1Y2 is the inner product of the Attribute vector of the two Examples. a and b can be adjusted using the kernel a and kernel b parameters. A common value for a is 1/N, where N is the data dimension. Note that not all choices of a and b lead to a valid kernel function.
  • anova: The anova kernel is defined by the raised to the power d of summation of exp(-g(x-y)) where g is gamma and d is degree. The two are adjusted by the kernel gamma and kernel degree parameters respectively.
  • Epanechnikov: The Epanechnikov kernel is this function (3/4)(1-u2) for u between -1 and 1 and zero for u outside that range. It has the two adjustable parameters kernel sigma1 and kernel degree.
  • gaussian combination: This is the gaussian combination kernel. It has the adjustable parameters kernel sigma1, kernel sigma2 and kernel sigma3.
  • multiquadric: The multiquadric kernel is defined by the square root of ||x-y||^2+c^2. It has the adjustable parameters kernel sigma1 and kernel sigma shift.
• kernel gamma

  This is the SVM kernel parameter gamma. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to radial or anova.

• kernel sigma1

  This is the SVM kernel parameter sigma1. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to epachnenikov, gaussian combination or multiquadric.

• kernel sigma2

  This is the SVM kernel parameter sigma2. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to gaussian combination.

• kernel sigma3

  This is the SVM kernel parameter sigma3. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to gaussian combination.

• kernel shift

  This is the SVM kernel parameter shift. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to multiquadric.

• kernel degree

  This is the SVM kernel parameter degree. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to polynomial, anova or epachnenikov.

• kernel a

  This is the SVM kernel parameter a. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to neural.

• kernel b

  This is the SVM kernel parameter b. This parameter is only available when the numerical measure parameter is set to 'Kernel Euclidean Distance' and the kernel type parameter is set to neural.

• use local random seed

  This parameter indicates if a local random seed should be used for randomization of the k different starting points of the algorithm.

• local random seed

  If the use local random seed parameter is checked this parameter determines the local random seed.