Extended Index Matrices - Index Matrices: Towards an Augmented Matrix Calculus - page 64

omitting, if necessary, some rows, so that all criteria corresponding to the rows of B ,
be in

(α, β) (α, β)

-negative consonance with D .
For brevity, we say that these criteria are in consonance.
The important particularity in this case is that elements b D , O 1 ,...,

-positive or

b D , O n are

evaluated hardlier than the rest a -elements of B .
Let us have a new object X with estimations x 1 ,..., x p w.r.t. the criteria

C 1 ,..., C p . Then we can solve the following problem: “Predict the value y of object

X w.r.t. criterion D ”.
To solve the problem, we can use one of the following two algorithms.

First Algorithm

We realize the following steps for each i p :

1.1. Determine the values a C i , O j and a C i , O k so that a C i , O j

, 1 ≤ i ≤

< a C i , O k and a C i , O j ≤

x i a C i , O k and a C i , O j is the highest a C i , O r with this property and a C i , O k is the

lowest a C i , O s with this property (for 1

≤

p ).
1.2. If criteria C i and D are in positive consonance, then calculate the value

≤ r , s ≤

b D , O k − b D , O j

y i = b D , O j + ( x i − a C i , O j ).

a C i , O k − a C i , O j

and if criteria C i and D are in negative consonance, then calculate the value

b D , O j − b D , O k

y i = b D , O j + ( x i − a C i , O j ).

a C i , O j . a C i , O k −

1.3. Determine the values

y min = min y i ,

1 ≤ i ≤ p

1
p

y a v e = y i ,

1 ≤ i ≤ p

y max = max

1

y i .

≤ i ≤ p

Now, the value of y can be y a v e or some other number in interval [ y min , y max ] .

If there is no number a C i , O j such that a C i , O j ≤ x i ,or a C i , O k such that x i ≤ a C i , O k ,

then Step 1.2 is omitted and in Step 1.3, the denominator is p s , where s is the number

of omitted cases (if they are smaller than p ). If in Step 1.1, a C i , O j

−

= =

x i a C i , O k

<

and b D , O j b D , O k , then

1
2 (

y i = b D , O k − b D , O j )