PSI - Issue 2_A

Pavel Skalny / Procedia Structural Integrity 2 (2016) 3727–3734 Pavel Skalny/ Structural Integrity Procedia 00 (2016) 000 – 000

3731

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Table 1. mean values of box counting dimensions and normal vector characteristics y (n)-statistical significant (insignificant) difference. Area Brittle fracture Ductile fracture

Stat. significance differencee

1.18 1.15

1.08 1.02

y y y n n y y

-component (abs) -componnent (abs)

17.51

15.32

0.32 0.27

0.28 0.28

(abs) (abs)

3.1 2.9

2.3 2.2

Table 1. Correlation between proposed characteristics y (n)-statistical significant (insignificant) correlation. 1 0.6 0.71 0.22 0.15 0.49 0.38 y 1 0.75 0.27 0.31 0.41 0.35 y y 1 0.25 0.28 0.35 0.38 -component (abs) n n n 1 0.15 0.8 0.22 -componnent (abs) n y n n 1 0.23 0.75 (abs) y y y y n 1 0.35 (abs) y y y y y y 1

3. Proposed methods

In this chapter there are presented two methods for analyzing normal vectors characteristics. The usage of Bayesian conditional distribution (section 4.2) was already used in Skalny and Strnadel (2015) and Strnadel et al. (2015). In section 4.1 the cluster k-means algorithm is presented. 3.1. K-means The k-means is well known clustering algorithm. It divides the data set ⊂ ℝ to clusters 1 , … , , so every ∈ belongs to the cluster with the nearest center . K-means solves the problem of minimizing the potential function ϕ = ∑ ∑ ‖ − ‖ ∈ ∈ 2 (3) with respect to . Principally the problem of minimizing the potential function can hardly be solved by finding the best solution from all possible realizations-optimal solution. K-means solve the form 1 in finding suboptimal solution with respect to the choice of initial centers (or clusters). The basic procedure of the algorithm can be described in the following way:  Choose initial centers { 1 , 2 , … , } .  Assign each observation to the cluster ( 1, … , ) with the “nearest” center (often used Euclidean distance).  Set new centre as a mean of every cluster.  Repeat previous two steps until no cluster changes. Every change of the k-means algorithm decreases the potential function until the local minimum of the function is achieved. The general k-means algorithm has many modifications, e.g. instead of mean the median or the mode