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Figure 7.2

The distribution of PCB and log10(PCB) values in a sample of size 30 from a reference area and a sample of size 20 from a possibly contaminated area.

second test is whether the observed mean difference is significantly larger than +0.301, at the 5% level of significance. The test statistic is (d − μdH)/SE(d) = 1.108, with 48 df. The probability of a value this large or larger is 0.14, so the result is not significant. The two one-sided tests are both nonsignificant, and there is therefore no evidence against the hypothesis that the sites are equivalent.

The precautionary principle suggests that, in a situation like this, it is the test of nonequivalence that should be used. It is quite apparent from Gore and Patil’s (1994) full set of data that the mean PCB levels are not the same in the phase 1 and the phase 2 sampling areas. Hence, the nonsignificant result for the test of the null hypothesis of equivalence is simply due to the relatively small sample sizes.

Of course, it can reasonably be argued that this example is not very sensible, because if the mean PCB concentration is lower in the potentially damaged area, then no one would mind. This suggests that one-sided tests are needed rather than the two-sided tests presented here. From this point of view, this example should just be regarded as an illustration of the TOST calculations, rather than what might be done in practice.

7.5  Chapter Summary

•Classical null hypothesis tests may not be appropriate in situations such as deciding whether an impacted site has been reclaimed, because the initial assumption should be that this is not the case. The null hypothesis should be that the site is still impacted.

•The U.S. Environmental Protection Agency recommends that, for a site that has not been declared impacted, the null hypothesis should

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Table 7.4

Sediment Concentrations (mg/kg) in Soils for Six Samples (A) Taken outside an Ammunition Testing Site and 24 (B) Samples Taken inside the Site

8

Time Series Analysis

8.1  Introduction

Time series have played a role in several of the earlier chapters. In particular, environmental monitoring (Chapter 5) usually involves collecting observations over time at some fixed sites, so that there is a time series for each of these sites, and the same is true for impact assessment (Chapter 6). However, the emphasis in the present chapter will be different, because the situations that will be considered are where there is a single time series, which may be reasonably long (say with 50 or more observations), and the primary concern will often be to understand the structure of the series.

There are several reasons why a time series analysis may be important. For example:

•It gives a guide to the underlying mechanism that produces the series.

•It is sometimes necessary to decide whether a time series displays a significant trend, possibly taking into account serial correlation, which, if present, can lead to the appearance of a trend in stretches of a time series, although in reality the long-run mean of the series is constant.

•A series shows seasonal variation through the year that needs to be removed to display the true underlying trend.

•The appropriate management action depends on the future values of a series, so it is desirable to forecast these and understand the likely size of differences between the forecast and true values.

There is a vast amount of literature on the modeling of time series. It is not possible to cover this in any detail here; so this chapter just provides an introduction to some of the more popular types of models and provides references to where more information can be found.

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8.2  Components of Time Series

To illustrate the types of time series that arise, some examples can be considered. The first is Jones et al.’s (1998a, 1998b) temperature reconstructions for the Northern and Southern Hemispheres, 1000–1991 AD These two series were constructed using data on temperature-sensitive proxy variables, including tree rings, ice cores, corals, and historic documents from 17 sites worldwide. They are plotted in Figure 8.1.

The series is characterized by a considerable amount of year-to-year variation, with excursions away from the overall mean for periods up to about 100 years, and with these excursions being more apparent in the Northern Hemisphere series. The excursions are typical of the behavior of series with a fairly high level of serial correlation.

In view of the current interest in global warming, it is interesting to see that the Northern Hemisphere temperatures in the latter part of the present century are warmer than the overall mean, but similar to those seen after 1000 AD, although somewhat less variable. The recent pattern of warm Southern Hemisphere temperatures is not seen earlier in the series.

A second example is a time series of the water temperature of a stream in Dunedin, New Zealand, measured every month from January 1989 to December 1997. The series is plotted in Figure 8.2. In this case, not surpris-

Figure 8.1

Average Northern and Southern Hemisphere temperature series, 1000–1991 AD, calculated using data from temperature-sensitive proxy variables at 17 sites worldwide. The heavy horizontal lines on each plot are the overall mean temperatures.

Figure 8.2

Water temperatures measured on a stream in Dunedin, New Zealand, at monthly intervals from January 1989 to December 1997. The overall mean is the heavy horizontal line.

ingly, there is a very strong seasonal component, with the warmest temperatures in January to March, and the coldest temperatures in about the middle of the year. There is no clear trend, although the highest recorded temperature was in January 1989, and the lowest was in August 1997.

A third example is the estimated number of pairs of the sandwich tern (Sterna sandvicensis) on the Dutch Wadden island of Griend for the years 1964 to 1995, as provided by Schipper and Meelis (1997). The situation is that, in the early 1960s, the number of breeding pairs decreased dramatically because of poisoning by chlorated hydrocarbons. The discharge of these toxicants was stopped in 1964, and estimates of breeding pairs were then made annually to see whether the numbers increased. Figure 8.3 shows the estimates obtained.

The time series in this case is characterized by an upward trend, with substantial year-to-year variation around this trend. Another point to note is that the year-to-year variation increased as the series increased. This is an effect that is frequently observed in series with a strong trend.

Figure 8.3

The estimated number of breeding sandwich-tern pairs on the Dutch Wadden Island, Griend, from 1964 to 1995.

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Figure 8.4

Yearly sunspot numbers since 1700 from the Royal Observatory of Belgium. The heavy horizontal line is the overall mean.

Finally, Figure 8.4 shows yearly sunspot numbers from 1700 to the present (Solar Influences Data Analysis Centre 2008). The most obvious characteristic of this series is the cycle of about 11 years, although it is also apparent that the maximum sunspot number varies considerably from cycle to cycle.

The examples demonstrate the types of components that may appear in a time series. These are:

1.a trend component, such that there is a long-term tendency for the values in the series to increase or decrease (as for the sandwich tern);

2.a seasonal component for series with repeated measurements within calendar years, such that observations at certain times of the year tend to be higher or lower than those at certain other times of the year (as for the water temperatures in Dunedin);

3.a cyclic component that is not related to the seasons of the year (as for sunspot numbers);

4.a component of excursions above or below the long-term mean or trend that is not associated with the calendar year (as for global temperatures); and

5.a random component affecting individual observations (as in all the examples).

These components cannot necessarily be separated easily. For example, it may be a question of definition as to whether component 4 is part of the trend in a series or is a deviation from the trend.

8.3  Serial Correlation

Serial correlation coefficients measure the extent to which the observations in a series separated by different time differences tend to be similar. They

are calculated in a similar way to the usual Pearson correlation coefficient between two variables. Given data (x1, y1), (x2, y2), …, (xn, yn) on n pairs of observations for variables X and Y, the sample Pearson correlation is calculated as

where x is the sample mean for X and y is the sample mean for Y.

Equation (8.1) can be applied directly to the values (x1, x2), (x2, x3), …, (xn−1, xn) in a time series to estimate the serial correlation, r1, between terms that are

one time period apart. However, what is usually done is to calculate this using a simpler equation, such as

where x is the mean of the whole series. Similarly, the correlation between xi and xi+k can be estimated by

This is sometimes called the autocorrelation at lag k.

There are some variations on equations (8.2) and (8.3) that are sometimes used, and when using a computer program, it may be necessary to determine what is actually calculated. However, for long time series, the different varieties of equations give almost the same values.

The correlogram, which is also called the autocorrelation function (ACF), is a plot of the serial correlations rk against k. It is a useful diagnostic tool for gaining some understanding of the type of series that is being dealt with. A useful result in this respect is that, if a series is not too short (say n > 40) and consists of independent random values from a single distribution (i.e., there is no autocorrelation), then the statistic rk will be approximately normally distributed with a mean of

The significance of the sample serial correlation rk can therefore be assessed by seeing whether it falls within the limits [−1/(n − 1)] ± 1.96/√n. If it is within these limits, then it is not significantly different from zero at about the 5% level.

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Figure 8.5

Correlograms for Northern and Southern Hemisphere temperatures, 1000–1991 AD The broken horizontal lines indicate the limits within which autocorrelations are expected to lie 95% of the time for random series of this length.

Note that there is a multiple testing problem here, because if r1 to r20 are all tested at the same time, for example, then one of these values can be expected to be significant by chance (Section 4.9). This suggests that the limits [−1/(n − 1)]

± 1.96/√n should be used only as a guide to the importance of serial correlation, with the occasional value outside the limits not being taken too seriously.

Figure 8.5 shows the correlograms for the global temperature time series (Figure 8.1). It is interesting to see that these are quite different for the Northern and Southern Hemisphere temperatures. It appears that, for some reason, the Northern Hemisphere temperatures are significantly correlated, even up to about 70 years apart in time. However, the Southern Hemisphere temperatures show little correlation after they are two years or more apart in time.

Figure 8.6 shows the correlogram for the series of monthly temperatures measured for a Dunedin stream (Figure 8.2). Here the effect of seasonal variation is very apparent, with temperatures showing high but decreasing correlations for time lags of 12, 24, 36, and 48 months.

Figure 8.6

Correlogram for the series of monthly temperatures in a Dunedin stream. The broken horizontal lines indicate the 95% limits on autocorrelations expected for a random series of this length.

Figure 8.7

Logarithms (base 10) of the estimated number of pairs of the sandwich tern at Wadden Island.

The time series of the estimated number of pairs of the sandwich tern on Wadden Island displays increasing variation as the mean increases (Figure 8.3). However, the variation is more constant if the logarithm to base 10 of the estimated number of pairs is considered (Figure 8.7). The correlogram has therefore been calculated for the logarithm series, and this is shown in Figure 8.8. Here the autocorrelation is high for observations 1 year apart, decreases to about −0.4 for observations 22 years apart, and then starts to increase again. This pattern must be largely due to the trend in the series.

Finally, the correlogram for the sunspot numbers series (Figure 8.4) is shown in Figure 8.9. The 11-year cycle shows up very obviously with high but decreasing correlations for 11, 22, 33, and 44 years. The pattern is similar to what is obtained from the Dunedin stream temperature series with a yearly cycle.

If nothing else, these examples demonstrate how different types of time series exhibit different patterns of structure.

Figure 8.8

Correlogram for the series of logarithms of the number of pairs of sandwich terns on Wadden Island. The broken horizontal lines indicate the 95% limits on autocorrelations expected for a random series of this length.