`Case 1:14-cv-02396—PGG-MHD Document 148-14 Filed 05/30/19 Page 1 of 2
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`EXHIBIT 4
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`EXHIBIT 4
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`8 OF 8
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`PART
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`80F8
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`Case 1:14-cv-02396-PGG-MHD Document 148-14 Filed 05/30/19 Page 2 of 2
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`260
`UNSUPERVISED LEARNING AND CLUSTERING
`(b) Show that the transfer of a sample i from !Z'i to !1£1 causes JT to change to
`1; = fx - [____!!_!_
`(i - mJ) 1ST1(i - m1) - _..!:.!._l (i - mi) 1ST1(i - mi)] .
`n1 +
`1
`n; -
`(c) Suggest an iterative procedure for minimizing fx.
`17. Use the facts that Sx = Sw + SB, J. = tr Sw, and tr SB =In, llmi - m112
`to derive the equations given in Section 6.9 for the change in J. resulting from trans(cid:173)
`ferring a sample i from cluster !1£, to cluster !1£1•
`18. Let cluster !1£, contain ni samples, and let d ;; be some measure of the distance
`between two clusters fl£, and !1£ 1. In general, one might expect that if !1£; and !Z'; are
`merged to form a new cluster !Z'k, then the distance from !Z'k to some other cluster
`!1£,. is not simply related to d,., and d,.1• However, consider the equation
`d,.k = a.d,.; + a.idM + {3di1 + 'Y ldhi - dMI•
`Show that the following choices for the coefficients °'i, oc;, {J, and y lead to the distance
`functions indicated. (For other cases, see Lance and Williams, 1967.)
`(a) dmin :oci =· <X; = 0.5, {J = 0, y = -0.5.
`(b) dma,x :<X; = <X; = 0.5, {J = 0, 'Y = 0.5.
`n;
`(c) da,vg:rx; = ---
`, {J = y = 0.
`, rx1 = ---
`n; + ni
`n; + n1
`.ni
`n,
`, rx1 -
`, {J -
`n; + n;
`ni + n;
`19. Consider a hierarchical clustering procedure in which clusters are merged so as
`to produce the smallest increase in the sum-of-squared error at each step. If the ith
`cluster contains n; samples with samp le mean m;, show that the smallest increase
`results from merging the pair of clusters for which
`
`•
`2
`(d) drnca,n .a., -
`
`llj
`
`rx,rx,, y - 0.
`
`is minimum.
`20. Consider the representation of the points x1 = (1 0)1, x2 = (0 0)1 and x 3 =
`(0 1)1 by a one-dimensional configuration. To obtain a unique solution, assume that
`the image points satisfy 0 = Yi < Y2 < Y3-
`(a) Show that the criterion function J•• is minimized by the configuration with
`y 2 = (1 + V2)/3 and y3 = 2y 2.
`(b) Show that the criterion function Jtt is minimized by the configuration with
`Y2 = (2 + V2)/4 and y3 = 2y 2•
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