Summary
Statistical regression methods in environmental chemistry are of vital importance. Regression techniques provide environmental chemical analysts with the ability to calibrate instruments and model large environmental systems.
It has become apparent that ordinary least-squares regression is not well suited to modeling data that contains outliers or strong nonlinearities. In the presence of outlying data robust regression methods prove to be a useful tool, while various non-parametric regression models are useful should the data possess nonlinearities or high levels of noise.
Robust techniques have the ability to detect outliers and dampen their effect on the modeling procedure. Several robust regression methods have been proposed but this article focuses on the least median of squares method and reweighted least squares regression.
The non-parametric models to be discussed include the ACE model, the PI model and the MARS model. Unlike ordinary least squares, these methods have evolved only recently, hence there is only limited documentation available on these methods.
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Abbreviations
- LS:
-
ordinary least squares
- ACE:
-
alternating conditional expectations
- PI:
-
pi implementation
- MARS:
-
multivariate adaptive regression splines
- LMS:
-
least median squares
- RLS:
-
reweighted least squares
- PE gcv :
-
generalized cross-validation estimate of the prediction error
- lo f :
-
lack-of-fit
- mi :
-
maximum level of interaction
- RSS :
-
residual sum of squares
- mad :
-
median absolute deviation
- CV :
-
cross-validating
- NP :
-
number of estimated parameters
- GCV :
-
generalized cross-validation
- MSE :
-
mean square error
- AIC :
-
Akaike’s information criterion
- y :
-
response variable
- X i :
-
i-th predictor variable; i = 1,..., m
- ε :
-
residual component
- β i :
-
i-th regression coefficient
- m :
-
number of predictor variables
- N(·):
-
normal distribution
- n :
-
number of observational units
- Z :
-
sample of data vectors
- Z′:
-
corrupted sample of data vectors
- T :
-
regression estimator
- \((\hat .)\) :
-
estimate of (·)
- ‖(·)‖ :
-
magnitude of (·)
- гn :
-
finite sample breakdown point
- σ 2 :
-
population variance
- σ :
-
population standard deviation
- \({\hat \sigma _{LS}}\) :
-
population standard deviation estimated by LS
- \({\hat \sigma _{LMS}}\) :
-
population standard deviation estimated by LMS
- z :
-
standardized residual component
- |(·)|:
-
modulus of (·)
- w i :
-
weight assigned to the i-th observational unit
- R 2 :
-
coefficient of determination
- R 2cv :
-
cross-validated coefficient of determination
- R 2adj :
-
adjusted coefficient of determination
- s :
-
scatterplot smoother
- c i :
-
i-th cutpoint
- R k :
-
k-th set of observations
- η((·)):
-
neighborhood of (·)
- a :
-
constant
- d ((·)):
-
an even function that decreases with |(·)|
- S0j :
-
weight given to y jin producing a smoothing estimate of the observation x 0
- k :
-
number of observations in a symmetric neighborhood
- r −i :
-
cross-validated residual component
- s −i :
-
fitted smooth with the point xi removed
- \(\hat \xi \) :
-
local error estimate
- K*:
-
initial span
- ϱ(·):
-
truncated power function
- q :
-
spline degree
- t :
-
knot
- θ(·):
-
transformed response
- g(·):
-
transformed predictor
- e 2 :
-
unexplained variance
- θ* (·):
-
optimal transformed response
- g*(·):
-
optimal transformed predictor
- Пj :
-
j-th product in a PI model
- ø(·):
-
cubic spline basis function
- J :
-
number of products in a PI model
- K :
-
number of knots in each cubic spline basis function
- J*:
-
optimal number of products in a PImodel
- U(·):
-
uniform distribution
- B p :
-
p-th multivariate spline basis function
- P :
-
number of multivariate spline basis functions
- V :
-
number of groups the data is divided into for cross-validation
- \((\mathop .\limits^ - )\) :
-
mean value of (·)
- f −v :
-
estimated function when the v-th group has been removed.
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Mallet, Y.L., Coomans, D.H., de Vel, O.Y. (1995). Robust and Non-parametric Methods in Multiple Regressions of Environmental Data. In: Einax, J. (eds) Chemometrics in Environmental Chemistry - Statistical Methods. The Handbook of Environmental Chemistry, vol 2 / 2G. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49148-4_6
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DOI: https://doi.org/10.1007/978-3-540-49148-4_6
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