Interval-Based Composite Indicators

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1 University of Rome Niccolo Cusano Conference of European Statistics Stakeholders 22 November 2014

2 1 Building Composite Indicators 2 (ICI) 3 Constructing ICI 4 Application on real data

3 Composite Indicators Composite indicators measure concepts which can be multi-dimensional and cannot be defined by using a unique single indicator (e.g poverty, competitiveness). They are very useful in order to compare country performances and are very important in policy communication (Saltelli 2007 and Nardo et al. 2005). Various examples are in Tarantola (2010). E-Business Readiness Composite Indicator based on Adoption of ICT and Use of ICT.

4 Composite Indicators They can be used in various ways: for policy evaluation and more in general for decision making. The diffusion of these indicators is a relevant signal of the importance of the statistical tool (Tarantola & Saltelli 2007) There are three determinants in having a good composite indicator (Tarantola 2010): A relevant theoretical framework Good quality of the data A good methodology in the construction of the indicator

5 Construction of a Composite Indicator There are these steps on building composite indicators (Nardo et al. 2005, Tarantola 2010): Step 1. Development of a relevant theoretical framework Step 2. Selection of indicators Step 3. Missing data imputation Step 4. Multivariate analysis Step 5. Data normalisation Step 6. Weighting and aggregation Step 7. Test for Robustness Step 8. Composite indicator decontruction Step 9. Association with other variables Step 10. Presentation and dissemination of the indicator

6 Construction of a Composite Indicator So we have in most of cases: Y c u = Q I q,u w q (1) q=1 With c = 1... C and 0 w q 1 and Q q=1 w q = 1. Differently we can consider a different aggregation function, the geometric one: Y c u = Q (w q I q,u ) 1/Q (2) q=1 where as well in this case c = 1... C and 0 w q 1 and Q q=1 w q = 1 In this sense various authors discuss the different aggregation function which can be used in building composite indicators.

7 (ICI) Aims and structure Aim: building composite indicators by making it more robust, resilient, defensible and useful in the negotiation (Tarantola 2010). In particular an interval-based composite indicator with these characteristics can be based on an interval (for example) and not on a single value. In this sense we can use symbolic data in order to endogenize the concept of variation inside the composite indicator. The concept of variation can be very useful in composite indicators because some inputs can be measured with uncertainty or there is need to take in to account different assumptions on the construction of the composite indicator (robustness assesment).

8 (ICI) Working jointly with classical and interval-based composite indicators We can also explicity taking in to account both interval observations and classical observations. In particular we can work on both the representations. We can simply transform the intervals in classical data by considering the centre of the interval (for example). However we can to consider the variabiity associated. In fact the classical data based on a single value can be considered cases of an intervals with no variation. Intervals has a their own algebra so we can to consider statistical methodologies based on intervals (and other symbolic data). See in this sense Gioia Lauro (2005) and Moore (1966)

9 ICI, Interval and Symbolic Data Following Bock and Diday (2000) Brito (2014) classical data can be represented in n p matrices. Each n statistical unit has one single value for each p variable (in column) Table: Classical Data Matrix Country Variable 1 Variable 2 Variable 3 Variable 4 Australia Austria Belgium Canada

10 ICI, Interval and Symbolic Data Symbolic data and interval data, in particular, take in to account the variability of the data. Variability can occur for many reasons. Important reasons can be: imprecise measurement (Gioia Lauro 2005) vagueness (Lauro Palumbo 2005) Variable values are: sets intervals distributions and so on. See: Noirhomme-Fraiture & Brito (2011), Bock and Diday (2000)

11 ICI, Interval and Symbolic Data Interval Data Intervals An interval is an ordered pair, where [x] = [x L, x U ]. x L, x U are the interval bounds such that x L x U. Lower and Upper Bounds [X ] t = [X t,l, X t,u ] where < X t,l X t,u < Center and radius [X ] t = X t,c, X t,r where X t,c = (X t,l + X t,u )/2 and X t,r = (X t,u X t,l )/2

12 An Example of Interval Data Visualization Here we consider an example of interval data: interval time series. An interval time series x t is an ordered sequence of interval-valued variables x t = [x Lt, x Ut ] = x Ct, x Rt for t = 1, 2...n. In particular we use Endesa daily data (years ) transformed in weekly interval data (temporal aggregation). It is important to note in the analysis are used jointly the classical data and the interval data (Drago et al. 2013).

13 Symbolic Data Analysis Intervals are symbolic data. Different types of data in this framework (Drago 2012).

14 Statistical Methods based on Intervals and Symbolic Data Various statistical techniques was considered in the context of symbolic data and interval data in particular (Bock and Diday 2000, Billard 2008, Brito ): Principal Component Analysis Cluster Analysis Linear Regression Discriminant Analysis Time Series Analysis

15 Uncertain Analysis and Robustness Assesment Following Huergo Saisana 2006 and Nardo et. al. 2005: It is very important to consider sensitivity analysis in order to: determining the most relevant sources of variability on the analysis determining the sources of uncertainty Identify the variance of a single output Infact various assumptions can be done on the index development: indicator selection, imputation, weigthing, aggregation and outlier detection (Huergo Saisana 2006) In this context interval data can help to consider to compare the different methodologies used. In this case the intervals are based on the different results obtained in the composite indicator or on the different ranks obtained considering different assumptions.

16 Uncertainty Analysis and Robustness Assesment Following Di Nardo et al we can see the country rankings by considering the original composite indicator TAI in Countries are ranked by the original TAI values, however the uncertain input factors in the assumptions are considered and the intervals generated.

17 New Approaches on Uncertainty Analysis We start to consider the different outcomes related to a composite indicator obtained by different assumptions in the modeling process, then build the interval. In this sense we endogenize the variations inside the interval and we can to take in to account these variations.

18 Interval Data and Robustness Analysis In general interval data (and symbolic data) can be very useful in the robustness analysis. They allow to take in to account the entire data structure of the underlying results and to represent the variability of the single composite indicator.

19 Interval Data and Robustness Analysis We can have a set of different composite indicators by condering different k assumptions with k = 1, 2,... K: Y c k, Y c k,..., Y c K (3) By considering the different assumptions on the composite indicators Yk c, we can define the obtained interval composite indicator c in this way: I [Y ] c = [Y c k, Y c k] (4)

20 Interval Data and Robustness Analysis Where we have different composite indicators c = 1... C. So we can have for each different intervals one for each different composite indicator c: [Y 1 k, Y 1 k ], [Y 2 k, Y 2 k ],..., [Y C k, Y C k ] (5) where c = 1... C. The original composite indicator (that we have computed originally and can be the starting point of the robustness analysis) can be denoted Y c k. At the same time Y k denote the lower bound where as well the Y k denote the upper bound. The intervals can be defined as:. I [Y ] = [Y, Y ] = {Y R : Y Y Y } (6)

21 Interval Data and Robustness Analysis In particular the interval data show relevant features which can be considered. These features are the center (also defined midpoint) and the radii of the interval. For example we can obtain the center: And the radius of the interval: Y c center,k = 1 2 (Y c k + Y c k) (7) Y c radius,k = 1 2 (Y c k Y c k ) (8)

22 Interval Data and Robustness Analysis The center correspond to a location measure considering all the different composite indicators computed by considering different set of assumptions. The radius as well can be interpreted in this way. The length span (the difference between upper and lower bound) represent a variability measure between the different composite indicators based on the various assumptions.

23 Interval Data and Robustness Analysis The interval data are explicitly related to the different assumptions k of the composite indicators. Each intervals correspond to a different set of assumptions considered. The different interval data can be ranked by considering their centre (Lotfi and Fallahnejad 2010 and Mballo Diday 2005). At the contrary the span length can be considered in order to compare the variability of the intervals.

24 Application on real data Data are related to the Industrial Innovation Index in Freudenberg (2003). BERD (Business Enterprise Expenditure on Research and Development) as a percentage of GDP Business researchers per labour force Number of patents in triadic patent families per million population Share of firms with new or technologically improved products or processes A theoretical question can be: what are the key drivers of the industrial innovation? To explore these problems we consider the construction of an index of competitiveness. We start to obtain a classical composite indicator based on tbe more plausible assumption.

25 Application on real data However there are various methods to impute the missing data. The results give different results of the composite indicators and then we can build the intervals. We want to study the robustness of the results by considering different assumptions. In this sense our final outcome in the modelling process is an interval of values related to the different single outcomes obtained by considering the different assumptions From the intervals we consider a statistical analysis.

26 Results Country Min Max Centre Range 1 Sweden Switzerland Japan Germany United States Finland Netherlands Luxembourg Denmark Ireland United Kingdom Canada Korea Austria France

27 Results Country Min Max Centre Range 16 Norway Iceland Belgium Australia Czech Republic Slovak Republic Italy New Zealand Hungary Poland Greece Mexico Spain Turkey Portugal

28 Results Results using centres tend to confirm the results obtained by the classical data: higher ranks for Japan, Germany, Switzerland, Sweden and US. Intervals give other interesting information: there are higher ranges for some countries highly ranked (Japan) this could be considered a risk factor. The variability is determined by the different methodologies used. At the same time some countries with worst ranks can have a better rank if we consider all the values of the intervals. So there are important reasons to consider the economic meaning of the range of the interval as the capability to recover (Greece and Hungary). Finally countries without a range are robust results of the index considered (France and Norway in the middle of the ranking).

29 Conclusions Interval based composite indicators are a promising new tool in the construction of composite indicators. In particular: They consider the classical composite indicators as a particular case without no variation on the values. So the classical and interval composite indicators can be jointly considered in the analysis. They can be used in order to assess robustness of the classical composite indicators (by considering all the assumptions) They are by their own construction robust and resilient. They can be easily transformed in classical composite indicators or the results can be easily used in a classical framework (working with centres for example), so they are very simply defensibile and useful in the typical negotiation process.

30 References Arroyo, J., Gonzlez-Rivera, G., & Maté, C. (2010). Forecasting with interval and histogram data. Some financial applications. Handbook of empirical economics and finance, Billard, L. (2008). Some analyses of interval data. CIT. Journal of Computing and Information Technology, 16(4), Bock, H. H., & Diday, E. (Eds.). (2000). Analysis of symbolic data: exploratory methods for extracting statistical information from complex data. Springer. Brito P. (2014) Statistical Data Analysis. LIAAD Open Day 2014 Brito P. (2014) 2 Taking Variability in Data into Account: Symbolic Data Analysis. University of Salerno

31 References Cherchye, L., Moesen, W., Rogge, N., & Van Puyenbroeck, T. (2009). Constructing a knowledge economy composite indicator with imprecise data. CES-Discussion paper series 09.15, Drago C. (2012) The Density Valued Data Analysis in a Temporal Framework: The Data Model Approach. Universit di Napoli Federico II. Ph.D Dissertation in Statistics Drago C. Lauro C. Scepi G. (2013) Visualization and Analysis of Large Datasets by Beanplot PCA -SIS Conference Proceedings conference Advances in Latent Variables: methods, models and applications, University of Brescia June (2013), collected in the Electronic Book Advances in Latent Variables, Eds. Brentari E., Carpita M., Vita e Pensiero, Milan, Italy

32 References Drago C. Lumbreras S., Maté C and Scepi G. (2013). Forecasting Interval Time Series of Exchange Rates. Working Paper, mimeo Freudenberg, M. (2003). Composite indicators of country performance: a critical assessment (No. 2003/16). OECD Publishing. Gioia, F., & Lauro, C. N. (2005). Basic statistical methods for interval data. Statistica applicata, 17(1). Huergo, L. Saisana, M. (2006) Robustness Assessment for Composite Indicators Lauro, C. N., & Palumbo, F. (2005). Principal component analysis for non-precise data. In New Developments in Classification and Data Analysis (pp ). Springer Berlin Heidelberg.

33 References Lotfi, F. H., & Fallahnejad, R. (2010) Imprecise Shannon s entropy and multi attribute decision making. Entropy, 12 (1), Mballo, C., & Diday, E. (2005). Decision trees on interval valued variables. The electronic journal of symbolic data analysis, 3 (1), Moore, R. E. (1966). Interval analysis (Vol. 4). Englewood Cliffs: Prentice-Hall. Nardo, M., Saisana, M., Saltelli, A., Tarantola, S., Hoffman, A., & Giovannini, E. (2005). Handbook on constructing composite indicators: methodology and user guide (No. 2005/3). OECD publishing. Noirhomme-Fraiture, M., Brito, P. (2011), Far Beyond the Classical Data Models: Symbolic Data Analysis. Statistical Analysis and Data Mining, 4 (2),

34 References Palumbo, F., & Lauro, C. N. (2003). A PCA for interval-valued data based on midpoints and radii. In New developments in Psychometrics (pp ). Springer Japan. Saltelli A. (2007) Composite indicators between analysis and advocacy, Social Indicators Research, 81: Tarantola S. (2010) Ten steps to build Composite Indicators, QMSS-2 Vienna, September 2-nd 2010 Tarantola, S., & Saltelli, A. (2007). Composite indicators: the art of mixing apples and oranges. Composite Indicators-Boon or Bane. Statistisches Bundesamt, Germany.