Abstract
For regression problems, the general practice is to consider a constant variance of the error term across all data. This aims to simplify an often complicated model and relies on the assumption that this error is independent of the input variables. This property is known as homoscedasticity. On the other hand, in the real world, this is often a naive assumption, as we are rarely able to exhaustively include all true explanatory variables for a regression. While Big Data is bringing new opportunities for regression applications, ignoring this limitation may lead to biased estimators and inaccurate confidence and prediction intervals.
This paper aims to study the treatment of non-constant variance in regression models, also known as heteroscedasticity. We apply two methodologies: integration of conditional variance within the regression model itself; treat the regression model as a black box and use a meta-model that analyzes the error separately. We compare the performance of both approaches using two heteroscedastic data sets.
Although accounting for heteroscedasticity in data increases the complexity of the models used, we show that it can greatly improve the quality of the predictions, and more importantly, it can provide a proper notion of uncertainty or “confidence” associated with those predictions. We also discuss the feasibility of the solutions in a Big Data context.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Bishop CM (2006) Pattern recognition and machine learning. Springer, New York
Boukouvalas A, Barillec R, Cornford D (2012) Gaussian process quantile regression using expectation propagation. In: Proceedings of the 29th international conference on machine learning (ICML-12), pp 1695–1702
Breusch TS, Pagan AR (1979) A simple test for heteroscedasticity and random coefficient variation. Econometrica 47(5):1287–1294
Chen C, Hu J, Meng T, Zhang Y (2011) Short-time traffic flow prediction with ARIMA-GARCH model. In: Intelligent vehicles symposium (IV), IEEE, pp 607–612
Chipman JS (2011) International encyclopedia of statistical science. Springer, Berlin, pp 577–582
Cook RD, Weisberg S (1983) Diagnostics for heteroscedasticity in regression. Biometrika 70(1):1–10
Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4):987–1007
Fargas JA, Ben-Akiva ME, Pereira FC (2014) Prediction interval modeling using gaussian process quantile regression. Master’s Thesis, MIT, pp 1–65
Fox CW, Roberts SJ (2012) A tutorial on variational Bayesian inference. Artif Intell Rev 38(2):85–95
Goldberg P, Williams C, Bishop C (1998) Regression with input-dependent noise: a Gaussian process treatment. Adv Neural Inf Process Syst 10:493–499
Goldfeld SM, Quandt RE (1965) Some tests for homoscedasticity. J Am Stat Assoc 60:539–547
Gredilla LG, Titsias MK (2012) Variational heteroscedastic Gaussian process regression. In: 28th international conference on machine learning
Hensman J, Fusi N, Lawrence ND (2013) Gaussian processes for big data. In: Proceedings of the 29th conference annual conference on uncertainty in artificial intelligence (UAI-13), pp 282–290
Kersting K, Plagemann C, Pfaff P, Burgard W (2007) Most likely heteroscedas-tic Gaussian process regression. In: Proceedings of the International Machine Learning Society, pp 393–400
Khosravi A, Mazloumi E, Nahavandi S, Creighton D, Van Lint JWC (2011) Prediction intervals to account for uncertainties in travel time prediction. IEEE Trans Intell Transp Syst 12(2):537–547
Koenker R, Hallock KF (2001) Quantile regression. J Econ Perspect 15(4):143–156
Lee YS, Scholtes S (2014) Empirical prediction intervals revisited. Int J Forecast 30(2):217–234
Leslie DS, Kohn R, Nott DJ (2007) A general approach to heteroscedastic linear regression. Stat Comput 17(2):131–146
Long JS, Ervin LH (1998) Correcting for heteroscedasticity with heteroscedasticity-consistent standard errors in the linear regression model: small sample considerations, Working Paper, Department of Statistics, Indiana University
MacKinnon JG (2012) Thirty years of heteroskedasticity-robust inference, Working Papers, Queen’s University, Department of Economics
MacKinnon JG, White H (1983) Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties, Working Papers, Queen’s University, Department of Economics
Osborne J, Waters E (2002) Four assumptions of multiple regression that researchers should always test. Pract Assess Res Eval 8(2):1–9
Pereira FC, Antoniou C, Fargas C, Ben-Akiva M (2014) A meta-model for estimating error bounds in real-traffic prediction systems. IEEE Trans Intell Trans Syst 15:1–13
Quinonero-Candela J, Rasmussen CE, Williams CKI (2007) Approximation methods for Gaussian process regression, Large-scale kernel machines, pp 203–223
Rasmussen CE, Williams C (2006) Gaussian processes for machine learning. MIT Press, Cambridge, MA
Robinson PM (1987) Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55(4):875–891
Silverman BW (1985) Some aspect of the spline smoothing approach to non-parametric regression curve fitting. J R Stat Soc 47(1):1–52
Snelson E, Ghahramani Z (2007) Local and global sparse Gaussian process approximations. In: International conference on artificial intelligence and statistics, pp 524–531
Taylor JW, Bunn DW (1999) A quantile regression approach to generating prediction intervals. Manag Sci 45(2):225–237
Tsekeris T, Stathopoulos A (2006) Real-time traffic volatility forecasting in urban arterial networks. Transp Res Rec 1964:146–156
Tzikas DG, Likas AC, Galatsanos NP (2008) The variational approximation for Bayesian inference. IEEE Signal Process Mag 25(6):131–146
White H (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48(4):817–838
Zeileis A, Wien W (2004) Econometric computing with HC and HAC covariance matrix estimators. J Stat Softw 11(10):1–17
Zhou B, He D, Sun Z (2006) Traffic predictability based on ARIMA/GARCH model. In: 2nd conference on next generation internet design and engineering, pp 207–214
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Antunes, F., O’Sullivan, A., Rodrigues, F., Pereira, F. (2017). A Review of Heteroscedasticity Treatment with Gaussian Processes and Quantile Regression Meta-models. In: Thakuriah, P., Tilahun, N., Zellner, M. (eds) Seeing Cities Through Big Data. Springer Geography. Springer, Cham. https://doi.org/10.1007/978-3-319-40902-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-40902-3_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40900-9
Online ISBN: 978-3-319-40902-3
eBook Packages: Earth and Environmental ScienceEarth and Environmental Science (R0)