Multiple Indicators Partial Orderings and Ranking and Prioritization

Multiple Indicators Partial Orderings and Ranking and Prioritization without Combining Indicators

Multiple Indicators Two indicators of a person’s size: Scatter Plot: Weight I 1 = Height I 2 = Weight A B Height Is A bigger than B ?

Combining Indicators Simple Average I 1 = Height I 2 = Weight Size = (I 1 + I 2)/2 Bigger than A A Contour of constant size Smaller than A Height

Combining Indicators Weighted Average I 1 = Height I 2 = Weight Size = w 1 I 1 + w 2 I 2 , w 1 + w 2 = 1 w 1 and w 2 reflect: Units of measurement n Relative importance of the two indicators Weight n Slope determines tradeoff between Height and Weight A Contour of constant size Height

Combining Indicators Non-Linear Combination I 1 = Height I 2 = Weight Size = F(I 1 , I 2) Tradeoff varies A Contour of constant size Height

Partial Ordering I 1 = Height Weight I 2 = Weight Not comparable with A Bigger than A A Smaller than A Not comparable with A Height

UNEP HEI National Land, Air, Water Indicators HEI with revised data: n Land: undomesticated land to total land area n Air: (air indicator 1 + air indicator 2) / 2, where air indicator 1 = renewable energy use to total energy use; air indicator 2 = GDP per unit energy use, based on maximum and minimum concept n Water: (water indicator 1 + water indicator 2) / 2, where water indicator 1 = ratio of water available after annual withdrawals to internal water resources; water indicator 2 = ratio of people with access to an improved water source to total population

UNEP HEI Data Matrix

Hasse Diagram---All Countries (labels are HEI ranks)

Hasse Diagram --- Western Europe

Ranking Partially Ordered Sets – 2 b Linear extension decision tree Poset (Hasse Diagram) e a b c d c e f Jump Size: b a 1 b b b e d d e f f f 3 3 d c d e f d d e c f d a e f d e d f e d a c c f e f e f e 2 3 5 4 3 3 2 4 3 4 4 2 2

Ranking Partially Ordered Sets – 3 a Rank-Frequency Distributions Element a Element b Element c Element d Element e Element f Rank

Ranking Partially Ordered Sets – 3 b 16 The curves are stacked one above the other and the result is a linear ordering of the elements: a > b > c > d > e > f

Cumulative Rank Frequency Operator – 1 An example where must be iterated Original Poset (Hasse Diagram) f a b c e g h d g 2 a a f f e e b b ad ad c g h c h

More Indicators do not necessarily mean More Information and More Discrimination n n Suppose two indicator columns are exactly the same. The second column does not add new information or discriminatory capability to that of the first column. Likewise, there is little new information when there is a strong underlying (rank-)correlation between the two indicators. Landscape ecologists have discovered that some fifty landscape pattern metrics amount to essentially five to ten indicators.

Multiple Indicators can convey Redundant Information Combining Indicators (e. g. , by averaging) typically fails to adjust for the Redundancy

Frontier/ Envelope Target (Virtual)

Urban Mixed Forest Agriculture

QOL Relative Efficiency Planning History Relative Efficiency

Off-Farm Employment Relative Efficiency