5 SUPPLEMENT A 5.1 The Minimal Mosaic Partition algorithm snips closest to the root. We prove here the statement of section 2.3 that the partition con- structed by the Minimal Mosaic algorithm has the desirable addi- tional property that its highest snip is closest to the root among all minimal mosaic partitions of the h-tree. In passing we prove also that the algorithm indeed finds a minimal mosaic partition. Denote by T(v) the subtree rooted at v, by N(v) the minimal num- ber of snips needed to create a mosaic partition of T(v), and let A(v) be the set of annotations such that a A(v) if and only if there is a B mosaic partition of T(v) in which the component of v has annota- tion a. Theorem. Let left and right be the two children of the vertex v. Then N(left)+N(right) N(v) N(left)+N(right)+1. Furthermore, 1. N(v)= N(left)+N(right) iff A(left) A(right) . In this case A(v)= A(left) A(right). Sup Fig. 6. Two minimal mosaic partitions. A) A minimal mosaic parti- 2. N(v)= N(left)+N(right)+1 iff A(left) A(right)= . tion whose highest snip is not as high as possible. B) A minimal mosaic In this case A(v)= A(left) A(right). partition whose highest snip as high as possible, as found by the Minimal- Proof. Observe that a minimal mosaic partition of T(v) induces (not Mosaic algorithm. necessarily minimal) mosaic partitions of T(left) and T(right), im- plying N(left)+N(right) N(v). If there is equality, the two induced 5.2 Snipping to minimize misclassification – Proof of cor- mosaic partitions are in fact minimal, and the annotations assigned rectness. to left and right in these partitions are both equal to the annotation The minimum misclassification algorithm is based on dynamic assigned to v. Thus A(left) A(right) A(v). Conversely, for any programming. Its correctness follows from the following easily a in A(left) A(right) a minimal mosaic partition of T(v), in which verified optimal substructure. v is assigned annotation a, can be constructed by adjoining the Let P be a partition of T(v) with the minimum number of misclassi- minimal mosaic partitions of T(left) and T(right) in which the ver- fied leaves, when node v must be assigned the label l and it is tices left and right are assigned annotation a. This proves that permitted to snip k edges (creating a (k+1) partition). Let left and N(v)= N(left)+N(right) and A(v)= A(left) A(right., right be the two children of v. Then the partitions of T(left) and If on the other hand A(left) A(right)= , so that N(v)= T(right) induced by P are optimal for the corresponding induced parameters. N(left)+N(right)+1, then a minimal mosaic partition of T(v) can be constructed from minimal mosaic partitions of T(left) and 5.3 Snipping to minimize misclassification-Time complexity T(right) by giving v the annotation of either left or right. In order to calculate minSnips(v,l,k), four cases have to be consid- Thus A(v)= A(left) A(right). ered (see figure 3, the recurrence formula). In each case, there are up to k+1 possibilities to apportion the k snips to the two subtrees. Corollary. The mosaic partition found by the MinimalMosaic algo- Consequently the time complexity for calculating minSnips(v,l,k), rithm has the property that the induced mosaic partition of any and minNum(v,k), is O(k). The algorithm computes minSnips(v,l,k) subtree T(v) is minimal for that subtree: minSnips(v)=N(v). More- for each node, label and 0≤k<K, hence the time complexity of the over, the partition snips an edge closest to the root, among all algorithm is O(nLK2), where n is the number of genes (tree leaves), minimal mosaic partitions. L is the total number of possible labels and K is the requested Proof. That minSnips(v)=N(v) is easily proven by induction. For number of snips. the second statement let us inspect the vertices of the tree by le- vels, down from the root. If at vertex v case 1 of the theorem is 5.4 Snipping to minimize misclassification – Traceback applicable, no minimal mosaic partition of T(v) can introduce a Once minNum(root,K-1) is computed, the appropriate snips can be snip between v and its immediate children, for otherwise the total found by a traceback, from the root of the tree down to the leaves. number of snips of the partition would exceed the minimum. Con- Let left and right be the two children of root. sequently, as long as the Minimal Mosaic algorithm does not snip Let l* be a label such that minMis(root,l*,K-1)= minNum(root,K- an edge, no minimal mosaic partition can have a snip there. 1). Then the following cases are considered: Supplementary figure 6 presents an example of a minimal mosaic 1. Case 1: minMis(root,l*,K-1)= minMis(left,l*,r)+ min- partition, which induces a partition for a subtree which is not mi- Mis(right,l*,K-1-r) for some 0≤r≤K-1. nimal for that subtree, and whose highest snip is lower than the one In this case there are no snips between root and either of its found by the MinimalMosaic algorithm. children. The traceback continues recursively from both min- Mis(left,l*,r) and minMis(right,l*,K-1-r). 2. Case 2: minMis(root,l*,K-1)= minNum(left,r)+ min- Mis(right,l*,K-2-r) for some 0≤r≤K-2. Dotan-Cohen et al. In this case there is a snip between root and left. The trace- back searches for a label l' such that minMis(left,l',r)= min- Num(left,r) and continues recursively from both min- Mis(left,l',r) and minMis(right,l*,K-2-r). 3. Case 3: minMis(root,l*,K-1)= minNum(right,r)+ min- Mis(left,l*,K-2-r)) for some 0≤r≤K-2. In this case there is a snip between root and right. The trace- back searches for a label l' such that minMis(right,l',r)= min- Num(left,r) and continues recursively from both min- Mis(right,l',r) and minMis(left,l*,K-2-r). Note that in each of the cases there may be more than one value of r which yields an optimal solution. We chose that partition in which the number of snips allotted to T(left) and T(right) are as close to equal as possible. In other words, in the traceback of min- Mis(v,l,k), r=k/2 is the first index to be examined, while r=0 and r=k are the last. 5.5 Simulation study The purpose of the simulation study was to evaluate the success of the algorithm, as measured by its predictive ability, in settings of variable difficulty determined by various “noise” parameters. To this end we constructed datasets of points in a d-dimensional space. Each dataset consisted of g sets of points, each set generated by a different independent Gaussian, as defined below: g p(x) p(i ) p(x | i), i 1 || xd i ,d ||2 d k 1 2 2 p(x | i) e . d 1 2 Here p(i)=1/g. The g Gaussians had identical standard deviations (5 representative standard deviations were used in our analysis). The means of the Gaussians were positioned randomly in a d-dimensional cube [-1,1]d. The points generated from the first Gaussian will be re- ferred to as the first cluster, and the points generated from all other Gaussians together will be referred to as the second cluster. The points were labeled as follows: Of the n points, 80% were labeled with one of 2 labels, according to the cluster they origi- nated from: points from the first cluster (first Gaussian) were la- beled "1", while points from the second cluster (other g-1 Gaus- sians) were labeled "2". To simulate errors in the labeling of the clusters, 10% of the labeled points were given a wrong label: "2" instead of "1" and vice versa. The true labels of the remaining 20% of the points were withheld from the algorithms, and the success of the algorithms in correctly classifying these points forms the basis for the evaluation of their clustering performance. For each standard deviation s such datasets were generated inde- pendently. Because the means were selected at random, the overlap between the different Gaussian and the location of the first Gaus- sian (first cluster) varies between the different datasets. The classi- fication accuracy is measured as the average over the different datasets. In our simulation we used the following numbers: n=5000, g=10, d=10, s=10.
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