Data rarely follow the simple models of mathematical statistics. Often, there will be distinct
subsets of observations so that more than one model may be appropriate. Further, parameters
may gradually change over time. In addition, there are often dispersed or grouped outliers
which, in the context of international trade data, may correspond to fraudulent behavior. All
these issues are present in the datasets that are analyzed on a daily basis by the Joint Research
Centre of the European Commission and can only tackled by using methods which are robust
to deviations to model assumptions (see for example [6]). This distance between
mathematical theory and data reality has led, over the last sixty years, to the development of
a large body of work on robust statistics. In the seventies of last century, it was expected
that in the near future any author of an applied article who did not use the robust alterative
would be asked by the referee for an explanation [9]. Now, a further forty years on, there
does not seem to have been the foreseen breakthrough into the wider scientific universe. In
this talk, we initially sketch what we see as some of the reasons for this failure, suggest a
system of interrogating robust analyses, which we call monitoring [5] and describe a series
of robust and efficient methods to detect model deviations, groups of homogeneous
observations [10], multiple outliers and/or sudden level shifts in time series ([8]). Particular
attention will be given to robust and efficient methods (known as forward search) which
enables to use a flexible level of trimming and understand the effect that each unit (outlier
or not) exerts on the model (see for example [1], [2] [7]). Finally, we discuss the extension
of the above methods to transformations and to the big data context. The Box-Cox power
transformation family for non-negative responses in linear models has a long and interesting
history in both statistical practice and theory. The Yeo-Johnson transformation extends the
family to observations that can be positive or negative. In this talk, we describe an extended
Yeo-Johnson transformation that allows positive and negative responses to have different
power transformations ([4] or [3]). As an illustration of the suggested procedure, we analyse
data on the performance of investment funds, 99 out of 309 of which report a loss. The
problem is to use regression to predict medium term performance from two short term
DSS - Dipartimento di Scienze Statistiche - www.dss.uniroma1.it
indicators. It is clear from scatterplots of the data that the negative responses have a lower
variance than the positive ones and a different relationship with the explanatory variables.
Tests and graphical methods from our robust analysis allow the detection of outliers, the
testing of the values of transformation parameters and the building of a simple regression
model. All the methods described in the talk have been included in the FSDA Matlab toolbox
freely donwloadable as a toolbox from Mathworks file exchange or from github at the web
address https://uniprjrc.github.io/FSDA/
References
[1] Atkinson, A. C. and Riani, M. (2000). Robust Diagnostic Regression Analysis. Springer–
Verlag, New York.
[2] Atkinson, A. C., Riani, M., and Cerioli, A. (2004). Exploring Multivariate Data with the
Forward Search. Springer–Verlag, New York.
[3] Atkinson, A. C., Riani, M., and Corbellini, A. (2020). The analysis of transformations
for profit-and-loss data. Journal of the Royal Statistical Society: Series C (Applied
Statistics), 69(2), 251–275.
[4] Atkinson, A. C., Riani, M., and Corbellini, A. (2021). The BoxCox Transformation:
Review and Extensions. Statistical Science, 36(2), 239 – 255.
[5] Cerioli, A., Riani, M., Atkinson, A. C., and Corbellini, A. (2018). The power of
monitoring: How to make the most of a contaminated multivariate sample (with
discussion). Statistical Methods and Applications, 27, 559–666.
https://doi.org/10.1007/s10260–017–0409–8.
[6] Perrotta, D., Torti, F., Cerasa, A., and Riani, M. (2020). The robust estimation of monthly
prices of goods traded by the European Union. Technical Report EUR 30188 EN,
JRC120407, European Commission, Joint Research Centre, Publications Office of the
European Union, Luxembourg. ISBN 978-92-76-18351-8, doi:10.2760/635844.
[7] Riani, M., Atkinson, A. C., and Cerioli, A. (2009). Finding an unknown number of
multivariate outliers. Journal of the Royal Statistical Society, Series B, 71, 447–466.
[8] Rousseeuw, P., Perrotta, D., Riani, M., and Hubert, M. (2019). Robust monitoring of
time series with application to fraud detection. Econometrics and Sttaistics, 9, 108–121.
[9] Stigler, S. M. (2010). The changing history of robustness. The American Statistician, 64,
277–281.
[10] Torti, F., Perrotta, D., Riani, M., and Cerioli, A. (2018). Assessing trimming
methodologies for clustering linear regression data. Advances in Data Analysis and
Classification, 13, 227–257.
3 Dicembre 2021
Marco Riani
Università degli studi di Parma