In this section, we return to the opening questions of this chapter concerning some real data sets. In our calculation, we use the Gaussian kernel throughout.
Example 3.7.1. We return to the data set in Example 3.1.1. Previous analyses, such as Fan and Zhang (1999), have ignored the weather effect. The omission of the weather effect seems reasonable from the point of view linear regression, such as model (3.2). However, as we shall see, the weather has an important role to play.
The daily admissions shown in Figure 3.8 (a) suggest
non-stationarity, which is, however, not discernible in the
explanatory variables. This kind of trend was also observed by
Smith, Davis, and Speckman (1999) in their study of the effect of
particulates on human health. They conjecture that the trend is
due to the epidemic effect. We therefore estimate the time
dependence by a simple kernel method and the result is show in
Figure 3.8 (a). Another factor is the day-of-the-week effect,
presumably due to the booking system. The effect of the
day-of-the-week effect can be estimated by a simple regression
method using dummy variables. To better assess the effect of
pollutants, we remove these two factors first. By an abuse of
notation, we shall continue to use the to denote the
`filtered' data, now shown in Figure 3.8 (b).
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As the different pollutant-based and weather-based covariates may
affect the circulatory and respiratory system after different time
delay, we first use the method of Yao and Tong (1994) to select a
suitable lag for each covariate within the model framework
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Now, using the MAVE method and with the bandwidth , we
have
.
Therefore, the number of e.d.r. directions is 4. The corresponding
directions are
Figures 3.9 (a)-(d) show plotted against the e.d.r.
directions. Figures 3.9 (a
)-(d
) are the estimated
regression function of
on the e.d.r. directions and
pointwise 95% confidence bands. See for example Fan and Gibbers (1996).
It suggests that along these directions, there are discernible
changes in the function value.
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We may draw some preliminary conclusions about which
weather-pollutant conditions are more likely to produce adverse
effects on the human circulatory and respiratory system. We have
identified four conditions, which we list in descending order of
importance as follows. (i) The main covariates in
are nitrogen dioxide (
), variation of
temperature (
) and temperature
, with coefficients
0.5394, -0.4652 and -0.6435 respectively. From Figures
3.9 (a) and (a
), the first e.d.r. direction
suggests that continuous cool days with high nitrogen dioxide
level constitute the most important condition. This kind of
weather is very common in the winter in Hong Kong. (ii) The main
covariates in
are ozone (
) and
humidity (
), with coefficients 0.6045 and -0.7544
respectively. Figures 3.10 (b) and (b1) suggest that
dry days with high ozone level constitute the second most
important condition. Dry days are very common in the autumn time
in Hong Kong. (iii) The main covariates in
are nitrogen dioxide (
) and the variation
of the temperature (
), with coefficients 0.7568 and 0.5232
respectively. Figures 3.9 (c) and (c
) suggest that
rapid temperature variation with high level of nitrogen dioxide
constitutes the third important condition. Rapid temperature
variations can take place at almost any time in Hong Kong. (iv)
The main covariates in
are the respirable
suspended particulates
, the variation of temperature
and the level of temperature
, with coefficients
,
and
respectively. Figures 3.9 (d)
and (d
) suggest that high particulate level with rapid
temperature variation in the winter constitutes the fourth
important condition.
Although individually the levels of major pollutants may be considered to be below the acceptable threshold values according to the National Ambient Quality Standard (NAQS) of U.S.A. as shown in Figure 3.10, there is evidence to suggest that give certain weather conditions which exist in Hong Kong, current levels of nitrogen dioxide, ozone and particulates in the territory already pose a considerable health risk to its citizens. Our results have points of contact with the analysis of Smith, Davis, and Speckman (1999), which focused on the effect of particulates on human health.
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Example 3.7.2. We continue with Example 3.1.2.
The first model was fitted by Maran (1953):
Now, using the dimension reduction method, we obtain the e.d.r.
directions as
and
. No dimensional reduction is needed
and the number of e.d.r. is 2. It is interesting to see that the
first e.d.r. direction practically coincides with that of model
(3.46). (Note that
).
Without the benefit of the second direction, the inadequacy of
model (3.46) is not surprising. Next, the sum of squared
(SS) of the residuals listed in Table 3.5 suggests that
an additive model can achieve almost the same SS of the residuals.
Therefore, we may entertain an additive model of the form
Based on the above observation, it further seems reasonable to fit
a generalized partially linear single-index model of the form
(3.41). Using the method described in Example 3.6.3, we have
fitted the model
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