diagnostics
As for any MRA studies, applying MRA with the present regression model neces- sitates the usage of certain diagnostics that are crucial to come to some impor- tant decisions: if iterative weighting applies, if a particular deterministic ozone predictor should enter the regression equation, which order n of the statistical
model AR(n) is appropriate, and if the regression requirements of parsimony
and independent normally distributed residuals are met. This section assesses if weighted regression is necessary, introduces parsimony tests, discusses the asso- ciated results, and finally defines the diagnostics of residual autocorrelation and distribution. It is important to recall that the regression requirements must hold in order to get valid confidence bands and intervals.
3.4.1
Stationarity
The iteratively weighted MRA results in significantly thinner confidence bands than the equivalent MRA without weighting (not shown), which is a typical effect (Carroll and Ruppert, 1988). For the three latitude bands, the standard deviation ratio of the noisiest and the least noisiest month of the year has a value of around 3.0, 2.8 (January:July) for the tropics, 3.2 (February:September) for the NH mid-latitudes, and 3.2 (September:February) for the SH mid-latitudes. Thus, according toCarroll and Ruppert (1988), the use of weighted least squares is necessary. For the tropics and NH mid-latitudes, the iterative weighting routine converges quickly after a few iterations, but more slowly for the SH mid-latitudes where the monthly standard deviations begin to stabilise only after about ten iterations. In case of suboptimal ozone predictors, the routine may not converge at all.
3.4.2
Parsimony
Under certain circumstances, individual MRA statistical diagnostic tools can mis- lead, particularly in case of nonlinear MRA. Therefore, when examining parsi- mony, several diagnostics should be consulted that rely on different characteristics of a particular MRA result (Draper and Smith, 1998;Milionis and Davies, 1994). Here, the approach byPandit and Wu (1983) as well as the following tests are applied. The statistical significance of a regression parameter reducing the RSS is tested via the F-statistics at the 95% ptobability level. Also, deterministic as well as stochastic parameter confidence intervals should not include zero, except for the sinusoidal phase parameters (Equation 3.4) whose cyclic response does not change sign at zero parameter value. Each of the deterministic and stochastic parameters contributing to the regression equations (Table 3.1) are chosen to meet these criteria.
There is a range of other tests which are not used (i.a. Draper and Smith, 1998; Ryan, 1997). Bates and Watts (1988), for instance, provide a t-test which involves the statistical significance of the ratio parameter estimate divided by the parameter confidence interval width.
Beyond the approach by Pandit and Wu (1983), the pairwise parameter cor-
relation associated with the approximate correlation matrix is assessed. Bates and Watts (1988) argue that the pairwise parameter correlation coefficients of a nonlinear regression model should definetely not exceed the absolute value of 0.99, indicating detrimental overparameterisation. The present MRA considers a
higher amount of data points than the data sets which Bates and Watts (1988)
typically analyse, and the nonlinearity for most of the parameters is not great (Section 3.5.3); hence, some critical value markedly lower than 0.99 is proba- bly more appropriate. The combined regression model of this Chapter appears
to meet this 0.99 requirement: the pairwise correlation coefficients are weaker than 0.01 for the three latitude bands, except for the ODS trend terms at SH mid-latitudes which display a peak pairwise correlation of about 0.18.
Both multicollinearity of several parameters and pairwise collinearity tend to inflate the widths of confidence intervals and bands, and violate the principle of
parsimony (i.a. Ryan, 1997). Awareness is hence important of the correlation
matrix helping detect pairwise collinearity, but not multicollinearity. The sta- tistical diagnostic of variance inflation factors helping to detect multicollinearity (i.a. Ryan, 1997) is beyond the scope of this Chapter, but a useful performance of the test in terms of nonlinear regression is not granted anyways. Using orthog- onalised ozone predictors would indeed circumvene the problem of collinearity, but would also complicate the physical interpretation of the individual predictors (i.a. Draper and Smith, 1998).
3.4.3
Residual autocorrelation and distribution
In order to validate the regression assumptions of independent and normally dis- tributed residuals, checks of the residual autocorrelation and distribution apply. According toPandit and Wu (1983), the null hypothesis of independent residuals at any given lag is rejected at the 95% (99%) probability level if the residual au- tocorrelation for the lag lies outside the confidence limit±1.96/√T (±2.32/√T), whereT is the number of data points; in this context, the residual autocorrelation function is commonly plotted out to a lag of roughly T /5.
The above approach does not account for autocorrelations at successive lags tending to depend on each other. Hence, the large-lag standard error of the autocorrelation function (i.a.Box et al., 1994) gives a more appropriate confidence limit (Figures 3.4a, 3.5a, and 3.5b). If the autocorrelation function at any lag lower thanT /5 exceeds the limit clearly, the regression model should have a more suitable deterministic part, or a higher order stochastic component. Following straightforward considerations there is a 5% (1%) chance of the autocorrelation at any lag exceeding the 95% (99%) confidence limit even if the null hypothesis holds. It is therefore a good idea to seek physical causes behind statistically significant autocorrelation.
In order to give a rough indication of the residual distribution, of the exis- tence of outliers in particular, residuals plots are provided together with levels of standard deviation as a function of time. Outliers are usually defined as those residuals which are at least three to four standard deviations away from zero (i.a.
Ryan, 1997). Distribution plots are not shown, though these have been made and do not reveal severe departures from normality such as double peaks or strong skewness, but tend to indicate a more heavily than normal tailed residual distri- bution.