Linear regression analysis of coastal data
Linear regression analysis of beach level data is demonstrated here using a set of beach profile measurements carried out at locations along the Lincolnshire coast (UK) by the National Rivers Authority (now the Environment Agency) and its predecessors between 1959 and 1991, as described in Sutherland et al. 2007[1]. A short introduction to linear regression theory is given in Appendix A.
Contents
Use of trend line for prediction
In order to better understand and predict beach level lowering in front of a seawall, beach levels were measured in front of seawalls along the Lincolnshire coast (UK). Figure 1 shows the bed level change measured in front of the seawall at Mablethorpe Convalescent Home.
Straight lines fitted to beach level time series give an indication of the rate of change of elevation and hence of erosion or accretion. The measured rates of change are often used to predict future beach levels by assuming that the best-fit rate from one period will be continued into the future. Alternatively, long-term shoreline change rates can be determined using linear regression on cross-shore position versus time data.
Genz et al. (2007[2]) reviewed methods of fitting trend lines, including using end point rates, the average of rates, ordinary least squares (including variations such as jackknifing, weighted least squares and least absolute deviation, with and without weighting functions). Genz et al. recommended that weighted methods should be used if uncertainties are understood, but not otherwise. The ordinary least squares, jackknifing and least absolute deviation method were preferred (with weighting, if appropriate). If the uncertainties are unknown or not quantified then the least absolute deviation method is preferred.
The following question then arises: how useful is a best-fit linear trend as a predictor of future beach levels? In order to examine this, the thirty years of Lincolnshire data have been divided into sections: from 1960 to 1970, from 1970 to 1980, from 1980 to 1990 and from 1960 to 1990, for most of the stations. In each case a least-squares best-fit straight line was fitted to the data and the rates of change in elevation from the different periods are shown below:
- From 1960 to 1970 the rate of change was -17mm/year;
- From 1970 to 1980 the rate of change was -63mm/year;
- From 1980 to 1990 the rate of change was +47mm/year.
- From 1960 to 1990 the rate of change was -25mm/year.
The data above indicates that 10-year rates provide little predictive capability for estimating the change in elevation for the next 10-years, let alone for the planning horizon that might need to be considered for a coastal engineering scheme. None of the 10-year rates are close to the 30-year rate.
A prediction horizon is defined as the average length of time over which a prediction (here an extrapolated trend) produces a better level of prediction of future beach levels than a simple baseline prediction. Sutherland et al. (2007)[1] devised a method of determining the prediction horizon for an extrapolated trend using the Brier Skill Score, here defined as 1 minus the mean squared error of the prediction divided by the mean squared error of the baseline prediction (Sutherland et al., 2004[3]). The baseline prediction was that future beach levels would be the same as the average of the measured levels used to define the trend. A 10 year trend was found to have a prediction horizon of 4 years at Mablethorpe Convalescent Home (Fig. 2). Similar values have been found at other sites in Lincolnshire.
Uncertainty in an extrapolated linear trend increases with the prediction horizon. More importantly, statistical confidence limits derived from the regression assume that the underlying trend remains unchanged. They therefore do not account for changes in coastal processes that alter the rate of erosion or accretion. For beach level time series, errors are often correlated because of seasonal cycles, storm erosion and recovery, and multi-year fluctuations; the large differences between the decadal rates at Mablethorpe suggest such fluctuations. Confidence limits from ordinary least squares would then be too narrow.
For a discussion of beach lowering in front of seawalls see also the article Seawalls and revetments.
Conditions for application
For a linear trend to provide a meaningful estimate of long-term change, the underlying rate of change should remain approximately constant over the period considered.
Ordinary least-squares fitting further assumes:
- the observation errors have zero mean and have constant variance
- the observation errors are uncorrelated.
A Gaussian distribution of errors is additionally assumed for conventional statistical inference based on the fitted coefficients.
In the example of Mablethorpe beach the distribution of residual (i.e. de-trended) beach levels seems to follow the common assumption of a Gaussian (normal) distribution, as shown in Fig. 3. The important requirement of uncorrelated observation errors can be checked with an autocorrelation plot.
If observation errors are significantly correlated in time or space, ordinary least-squares uncertainty estimates are generally not appropriate and the correlation structure should be taken into account. Methods include generalized least squares and, for spatial data, Kriging.
Appendix A: Introduction to linear regression theory
Although regression analysis is explained in many textbooks, a short mathematical introduction is given below. The object of linear regression is to analyze the correlation of a measurable property [math]h[/math] (the so-called 'dependent variable' or 'target') with a number [math]K-1[/math] measurable factors [math]y_k, \; k=2, .., K[/math] (the 'independent variables' or 'regressors'). Linear regression is based on the assumption that [math]h[/math] can be estimated through a linear relationship:
[math]h = \hat{h} +\hat{\epsilon} \; , \quad \hat{h} = a_1 + \sum_{k=2}^K a_k y_k = \sum_{k=1}^K a_k y_k \; , \qquad (A1)[/math]
where [math]\hat{h}[/math] is the estimator, [math]a_k, \; k=2, .., K[/math] are the regression coefficients, [math]a_1[/math] is the intercept and [math]y_1=1[/math]. The term [math]\hat{\epsilon}[/math] is the 'residual', i.e., the difference between the true value [math]h[/math] and the estimator [math]\hat{h}[/math]. (Note: Linear regression means linear in the regression coefficients; the regressors [math]y_k[/math] can be nonlinear quantities, for example [math]y_3=y_2^2[/math].) Now assume that we have observations of [math]N[/math] different instances of the [math]K-1[/math] regressors, [math]y_{ki}, \; i=1, .., N[/math], and corresponding observations of the dependent variable, [math]h_i[/math]. If [math]N\gt K[/math] these observations can be used to estimate the values of the regression coefficients [math]a_k[/math] by looking for the best solution to the linear system
[math]h_i = \sum_{k=1}^K a_k y_{ki} + \hat{\epsilon_i} \; , \qquad (A2)[/math]
where [math]\hat{\epsilon_i}[/math] are the residuals, representing deviations from the assumed regression relationship due to unresolved variability, model approximation and measurement errors in the dependent variable. The residuals are considered to be uncorrelated stochastic variables with zero mean, [math]E[\hat{\epsilon_i}]=0[/math] and variance [math]\sigma_i^2 \equiv E[\hat{\epsilon_i^2}][/math] (we use the notation [math]E[x][/math]= the mean value from a large number of trials of the stochastic variable [math]x[/math]).
What is 'the best solution'? Different criteria can be used. Least square optimization is the most often used criterion, i.e., the solution for which the sum of the squared residuals [math]\Phi[/math] is minimum,
[math]\Phi = \tfrac{1}{2} \sum_{i=1}^N \hat{\epsilon_i}^2 = \tfrac{1}{2} \sum_{i=1}^N \Big(h_i - \sum_{k=1}^K a_k y_{ki}\Big)^2 . \qquad (A3)[/math]
At minimum, [math]\Phi[/math] increases for any change in one of the coefficients [math]a_k[/math], which implies that the partial derivatives are zero:
[math]\Large\frac{\partial \Phi}{\partial a_k}\normalsize = 0 , \; k=1, ..,K . \qquad (A4)[/math]
This condition yields the set of [math]K[/math] linear equations
[math] - \sum_{i=1}^N y_{ki} h_i + \sum_{i=1}^N \sum_{k'=1}^K y_{ki} y_{k'i} a_{k'} = 0 . \qquad (A5)[/math]
from which the regression coefficients [math]a_k[/math] can be solved.
In the more compact matrix notation we may write: [math]H[/math] is the [math]N[/math]-vector with elements [math]h_i[/math], [math]\; A[/math] is the [math]K[/math]-vector with elements [math]a_k[/math] and [math]Y[/math] is the [math]N\times K[/math] matrix with elements [math]Y_{ik}=y_{ki}[/math] (row i contains the regressor values of observation i).
The linear equations (A5) can then be written
[math]Y^T \, H = Y^T \, Y \, A . \qquad (A6)[/math]
The solution is found by inversion of the [math]K \times K[/math] matrix [math]Y^T \, Y[/math], giving
[math]A = (Y^T \, Y)^{-1} Y^T \, H . \qquad (A7)[/math]
The uncertainty of the regression coefficients follows from the covariance matrix [math]\text{Cov}(A) = s^2 (Y^T Y)^{-1}[/math], where [math]s^2 = \sum_{i=1}^N \hat{\epsilon_i}^2 / (N-K)[/math] is the estimated error variance and [math]\hat{\epsilon_i}[/math] are the residuals of the fit. The standard error of coefficient [math]a_k[/math] is the square root of the k-th diagonal element. This estimate is only valid if the residuals are uncorrelated; with positively correlated residuals it underestimates the true uncertainty.
Under the usual linear-model assumptions, least squares provides the best linear unbiased estimator when the observation errors [math]\epsilon_i[/math] have zero mean, equal variance [math]\sigma_i^2=\sigma^2[/math] and are uncorrelated. Gaussian-distributed errors are additionally assumed for conventional exact statistical inference. This is often not the case in practice. A few other cases are discussed below.
Case 1. If positive data have approximately multiplicative errors, so that error magnitude scales with the measured (positive) value, a logarithmic transformation can often stabilize the variance. Linear regression can then be applied to the log-transformed data. Back-transformed predictions estimate the median rather than the mean.
Case 2. The errors differ among the data ('heteroscedasticity'), [math]\sigma_i \ne \sigma_j , \, i \ne j[/math]. If estimates of the variances are known, one can use weighted least squares instead of least squares, i.e., replace (A3) with [math]\Phi = \tfrac{1}{2} \sum_{i=1}^N \big( \hat{\epsilon_i} / \sigma_i \big)^2[/math].
Case 3. Multicollinearity occurs when two or more regressors are strongly linearly related. Perfect linear dependence makes the regression coefficients indeterminate and requires removal or reformulation of redundant regressors. Strong but imperfect multicollinearity does not prevent fitting, but increases the uncertainty of individual regression coefficients.
Case 4. The errors of the [math]N[/math] observations are correlated, [math]c_{ij} \equiv E[\epsilon_i \, \epsilon_j] \ne 0[/math]. If an estimate of the covariances [math]c_{ij} [/math] is known, the generalized least squares method can be used. The solution (A7) is then replaced by
[math]A = (Y^T \, C^{-1} \, Y)^{-1} Y^T \, C^{-1} H , \qquad (A8)[/math]
where [math]C[/math] is the covariance matrix with elements [math]c_{ij}[/math]. For spatial data, another option is to use Kriging, see the article Data interpolation with Kriging.
Related articles
References
- ↑ 1.0 1.1 1.2 Sutherland, J., Brampton, A.H., Obhrai, C., Motyka, G.M., Vun, P.-L. and Dunn, S.L. 2007. Understanding the lowering of beaches in front of coastal defence structures, Stage 2. Defra/EA Joint Flood and Coastal Erosion Risk Management R&D programme Technical Report FD1927/TR
- ↑ Genz, A.S., Fletcher, C.H., Dunn, R.A., Frazer, L.N. and Rooney, J.J. 2007. The predictive accuracy of shoreline change rate methods and alongshore beach variation on Maui, Hawaii. Journal of Coastal Research 23(1): 87 – 105
- ↑ Sutherland, J., Peet, A.H. and Soulsby, R.L. 2004. Evaluating the performance of morphological models. Coastal Engineering 51, pp. 917-939.
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