Testing for trend in the variability of monthly and seasonal temperature using the range and the standard deviation as measures of variability

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1 Testing for trend in the variability of monthly and seasonal temperature using the range and the standard deviation as measures of variability T. Astatkie, Emmanuel K. Yiridoe and J. Stephen Clark T. Astatkie, Department of Engineering, Nova Scotia Agricultural College, P. O. Box 55, Truro, Nova Scotia, Canada, B2N 5E3 KEY WORDS climatic variability, trend, bootstrap, nonstationarity Abstract In the Climate Change literature, the range is predominantly used as a measure of variability, and the presence of trend is typically done on seasonal and/or annual basis. In this study, we used daily average temperature values at 15 spatially distributed sites across Canada to test the presence of trend in variability (using both range and standard deviation) using the bootstrap method by Woodward et al. (1997). The length of the series varied from site to site, ranging from 3 to 151 years. The analysis was done for each month, each season and the annual data. When calculating the standard deviations, supersmoother estimates of temperature were used as the means to make the results invariant to the presence of trend in the mean. The results from the monthly and seasonal analysis revealed the presence of either increasing or decreasing variability for some months and some seasons. The results for the annual data were not so revealing, especially at the sites where some months have increasing and others have decreasing trends. The significance of trend in variability by the range and the standard deviation agreed in less than 3% of the time across sites and across the monthly, seasonal and annual aggregations. The differences between median significance scores was significant in 2 of the 15 sites, and in 6 of the 17 aggregations. 1. Introduction Policy debates, and adaptation and mitigation strategies to climate change highlight the importance of climatic variability. Bryant et al. (2), for example, noted that adaptation to climate change, and efforts to mitigate the impacts of climate change need to emphasize not only changes in long-term mean weather attributes but, in particular, trends in the variability of climatic variables. Given that climatic conditions vary from one period to another, variability is an integral part of climate change. Consequently, response strategies and adaptations to climatic change at the regional and global levels have to address climatic variability. The Intergovernmental Panel on Climate Change (IPCC) and various other task forces have echoed this viewpoint, noting that trends and variability in climate attributes are critical for understanding regional and global climate change (Bonsal et al., 2). Giorgi (22) is more to the point when he asserted that variability is more important than mean changes in climate for determining impacts on human and natural systems. Risks of climate change and the associated policy responses have prompted various studies on the variability, trends, and changes in climatic variables across Canada (e,.g, Bonsal et al., 2; Clark et al., 2; Zhang et al., 2). In spite of the importance of climatic variations to understanding the risks and impacts of climate change and variability, previous studies have not investigated a very fundamental issue on variability: what is the relevant measure of variability for weather attributes? Several studies have used the range (maximum minimum), and other measures of extremes (e.g. minimum, maximum, diurnal temperature range) to analyse trends in climatic variables (see, e.g., Zhang et al., 2). Other studies have used the range and standard deviation as measures of variability (e.g., Giorgi, 22). However, none of these studies have provided a justification for the use of a particular measure of variability for weather attributes. In addition, it is not clear whether different measures of dispersion in weather attributes could lead to different conclusions on the nature and direction of climatic variability. The objectives of this study are to investigate if the two most common measures of variability used to analyse climatic variation (i.e. range and standard deviation) can lead to similar conclusions, and if the level of aggregation (i.e. monthly, seasonal, or annual) affects conclusions regarding trend in climate variability. We do this using temperature data from 15 weather stations across Canada. 2. Two measures of variability Suppose we have observations on a time series X ijt. The 79

2 subscript i refers to the minimum time interval over which X ijt is observed (in our case one day), j refers to the time over which we would like to measure variability (e.g., one month, one season, or one year), and t refers to the year. The range is defined as: R jt = Max (X ijt ) Min(X ijt ) (1) where Max(X ijt ) is the largest value of X ijt over interval j in year t, Min(X ijt ) is the minimum value of X ijt over interval j in year t, and R jt is range over time interval j in year t. The standard deviation is: Xijt E Xijt S, (2) jt = ( $ ( )) 2 njt where EXijt $ ( ) is the estimated expected value of X ijt over interval j in year t, and n jt is the number of observations over interval j in year t. Given the time series nature of the data, E(X ijt ) is a conditional expectation which depends upon the mean (first moment) of the time series. As a practical problem of measurement, it seems clear that one would prefer to use an estimator of E(X ijt ) that involves as few a priori restrictions as possible, if only for the reason that mistakes made in mean estimation will carry over to the measurement of the standard deviation. As a solution to this problem, we suggest that a method be used to estimate E(X ijt ) that will estimate E(X ijt ) most accurately without explicitly expressing a time series model generating X ijt. We do this using the supersmoother (Friedman and Silverman, 1989) values of the inter-temporal arithmetic means of X ijt. We used the supsmu function of S-PLUS to get the supersmoother estimates of the means. These will be robust against any assumption made regarding the time series properties of the series X ijt, (stationary, trend stationary, difference stationary, etc.). A single annual smoothed mean is used as the value of EXijt $ ( ) in equation (2) when calculating the monthly, seasonal (tri-monthly, beginning with January of each year, to approximate the four seasons), and annual standard deviations. The range is also calculated for these time intervals. 3. Testing trend in variability To determine if the variability measures are changing over time, we view the values of the range and the standard deviation as a time series and test for the presence of a linear trend. We do this using the bootstrap-based test for trend developed by Woodward et al. (1997). The procedure involves identifying ARMA models for each time series using the Box-Jenkins model identification strategy and the AIC, calculating the COB (Cochrane and Orcutt Bootstrap) t-statistic, calculating the COBA (Cochrane and Orcutt Bootstrap Adjusted) t-statistic, and finally calculating the onesided (increasing trend) critical value based on B = 399 bootstrap samples for a desired level of significance (in our case 5% and %). COBA adjusts COB so that it has a better control of the level of significance. Woodward et al. (1997) showed that even if the COBA test gives a level of significance that is closer to the nominal, it has lower power when the characteristic polynomial has a root close to 1 and when n is small (about 5). We also used a recently proposed test for trend by Vogelsang (1998), which is based on partial sums. Although this test is robust against the null hypothesis that the series is either stationary or difference stationary, it is a large sample test only. The 15 sites used in our study had data for 3 (small sample) to 151 (large sample) years, and as expected, the conclusions from the two methods were consistent for the large data sets. In this paper, we report the findings from the bootstrap-based tests. 4. Data Daily temperature (in C) data for 15 weather stations across Canada were obtained from the Atmospheric Environment Service, Environment Canada (1992). Details about the sites, including their province, elevation and annual average temperature are summarized in Table Results and discussion 5.1 Trend at the sites The test for trend was completed for the standard deviation and the range time series of each of the 12 months and the 4 seasons of the year, and the annual series for all the 15 stations. For economy of space, the results for the first three stations, shown in Table 2, are indicated as NS (not significant, when the P-value of the test is >.1), MS (marginally significant, when.5 < P-value <.1), and S (significant, when P-value <.5). In general, the results (whether increasing or decreasing) do not exhibit a geographical pattern, however, a majority of them show an increasing trend in variability. At all sites, the actual magnitude of the standard deviations were lower in April and October, compared to the remaining months (see e.g., Figure 1). This is mainly because the temperature during April and October is relatively close to the yearly average, 8

3 thereby resulting in lower values for the standard deviation. Table 1. Description of the Canadian weather sites included in the study. Station Name Elev (m) Temp ( C) Data Range Prince Rupert, BC Vancouver, BC Whitehorse, YK Yellowknife, NWT Edmonton, AB Lethbridge, AB Saskatoon, SK Winnipeg, MB Churchill, MB Thunder Bay, ON Sault Ste Marie, ON Toronto, ON Quebec City, QC Charlottetown, PEI St. John s, NF The fluctuations in the range as well as the standard deviation, at all sites, were highest in the winter months (November through March) and lowest in the summer months, as expected. In Western Canada (Table 2), Prince Rupert saw a decreasing trend in the standard deviation as well as in the range in December and January, and this was also reflected in the annual standard deviations. The ranges, however increased in February and April, but decreased in May. Vancouver, with the highest annual temperature, had increasing variability in December and April, and decreasing in May and July. Whitehorse, barely above the 6 parallel, had a decreasing trend in the spring months and increasing in October. Yellowknife, the northern- most station, had mostly increasing, whereas Edmonton and Lethbridge had mostly decreasing variability. In Central Canada, Saskatoon (Figure 1) had decreasing variability in February according to both Table 2. The significance of trend in the standard deviation (SD) and the range (R) at five stations in Western Canada. Month/ Season Prince Rupert, BC (3 years) Vancouver, BC (55 years) Whitehorse, YK (49 years) SD R SD R SD R 1. January NS NS NS NS 2. February NS +S NS NS NS 3. March NS NS NS NS NS NS 4. April NS +MS +S NS NS 5. May NS NS NS -S 6. June NS NS NS +S NS 7. July NS NS NS NS NS 8. August NS NS +S +MS NS +S 9. September NS NS NS NS NS. October NS NS NS NS +S +MS 11. November NS NS NS +MS NS NS 12. December -S +MS NS NS NS 1. Winter NS NS NS NS NS NS 2. Spring NS NS NS +S NS 3. Summer NS NS NS NS NS +MS 4. Fall NS +MS NS NS NS Annual NS NS NS NS NS measures. However, the range showed decreasing variability in May and June as well, and the standard deviation showed increasing variability in November and December. Winnipeg had decreasing trend in March, and increasing in November and December according to both measures. In Churchill, the coldest of all stations, the SD and the range gave conflicting results in January and February. The standard deviation was decreasing while the range was increasing. The standard deviation in Churchill was increasing in May, November and December, and decreasing in October. The trend in October was also decreasing in Thunder Bay according to both measures. Its standard deviation, however, increased in May and July. 81

4 Standard Deviation January February March April May June July August September October November December Winter Spring Summer Fall Year -S +MS -S Range Figure 1. Monthly and seasonal time series plot of the standard deviation (solid) and the range (broken) at Saskatoon, Saskatchewan. Significance of the trend (-S = decreasing, = marginally decreasing, +MS = marginally increasing, +S = increasing) is indicated at the left bottom corner for the standard deviation, and right top for the range. In Eastern Canada, Sault Ste Marie showed decreasing trend in October, January and February, and increasing in March, April and July. Toronto, the most metropolitan, was the only site with all increasing trend. This was also reflected in the annual series. On the other hand, Quebec City showed mainly increasing, and Charlottetown mainly decreasing trend. The trend in St. John s was increasing in all, but in September. Its annual trend was also increasing for both measures. 5.2 Standard Deviation and Range Comparison We coded the significance of the tests as -2 if it is significantly decreasing, -1 if it is marginally decreasing, if it is not significant, 1 if it is marginally increasing, and 2 if it is significantly increasing and computed the correlations between the standard deviation and the range significance values for each site and each month, season and the annual (Table 3). Among the 15 sites, there was a marginally significant agreement (positive correlation) at only 4 sites (27%), namely at Prince Rupert, Saskatoon, Sault Ste Marie and Toronto. The other correlations were not significant at all, suggesting that the range is not a good substitute for the standard deviation for trend analysis. We also performed a Wilcoxon Signed Rank test on the difference between the significance codes of the standard deviation and the range. We did this test because the distribution of these values is not normal. As the results in Table 3 show, the difference was significant only at two sites: Edmonton with significantly less than zero (P-value =.23) implying that the standard deviation showed significantly more decreasing trends, and Saskatoon with significantly greater than zero (P-value =.16) suggesting that the standard deviation showed more significantly increasing trends. Panel plots of the standard deviation and the range for the 12 months and the 4 seasons are shown at the site where the standard deviation showed more increasing trends (Saskatoon, Figure 1). Among the 17 temporal aggregations (Table not shown), the two measures marginally agreed in 5 aggregations (29%), namely in February (P-value =.79), March (P-value =.36), May (P-value =.45), October (P-value =.) and Winter (P-value =.32). Test of the difference between the significance of the standard deviation and the range indicated that the standard deviation suggested less decreasing trends in February (P-value =.9), March (P-value =.93), October (P-value =.76), and Winter (P-value =.33), whereas in May (P-value =.) and Fall (Pvalue =.36) the standard deviation suggested more increasing trends. 82

5 6. Conclusions In this paper, we studied the empirical evidence relating to trend in climate variability in selected sites across Canada. We compared and contrasted two measures of variability, range and standard deviation, and conclude that standard deviation is a superior measure of variability that should be used in climate change studies. We applied our analysis to monthly, seasonal and annual measures of variability. Our results indicate that there is little agreement between range and standard deviation regarding conclusions concerning trend in variability. This means that range does not seem to be a good substitute for standard deviation to measure trends in climate change. Table 3. Correlation between the significance of the standard deviations and the ranges for the 15 sites. Last column shows the average of the differences (standard deviation - range) of significance and the P-value (in parenthesis) for testing against zero using the Wilcoxon Signed Rank test. Station Name Correlation (p-value) Average Difference Prince Rupert.491 (.45) -.6 (1.) Vancouver.63 (.8) -.12 (.75) Whitehorse.219 (.398) -.6 (.973) Yellowknife.6 (.982) -.35 (.166) Edmonton (.139) -.71 (.23) Lethbridge.122 (.64) -.24 (.246) Saskatoon.544 (.24).53 (.16) Winnipeg.247 (.338).24 (.541) Churchill -.15 (.955) -.6 (1.) Thunder Bay.162 (.532).6 (.984) Sault Ste Marie.531 (.28) -.24 (.344) Toronto.426 (.88).18 (.563) Quebec City.146 (.575).47 (.156) Charlottetown -.98 (.79) -.35 (.176) St. John s.175 (.51) -.6 (.92) Our results also indicate that annual measures of climate change may mask important trends in monthly or seasonal climate change. Inter-temporally disaggregated measures of climate change provide a much more detailed description of climate change than annual measures. Dis-aggregated measures can be compared against predictions coming from more detailed models of climate change as they become available. REFERENCES Bonsal, B.R., X. Zhang, L. A. Vincent, and W. D. Hogg (2). Characteristics of daily and extreme temperatures of Canada. J. Clim. 14: Bryant, C.R., B. Smit, M. Brklacich, T. R. Johnston, J. Smithers, Q. Chiotti and B. Singh (2). Adaptation in Canadian agriculture to climatic variability and change. Clim. Change. 45: Clark, J. S., E. K. Yiridoe, N. D. Burns and T. Astatkie (2). Regional climate change: trend analysis of temperature and precipitation series at selected Canadian sites. Can. J. Agr. Econ. 48: Environment Canada (1992). Canadian Daily Climate Data. Preliminary Temperature and Precipitation Data. Environment Canada, Atmospheric Environment Service, Ottawa. Friedman, J. H. and B. W. Silverman (1989). Flexible parsimonious smoothing and additive modelling (with discussion). Technometrics. 31:3-39. Giorgi, F. (22). Variability and trends of subcontinental scale surface climate in the twentieth century. Part I: observations. Clim.Dyn.18:DOI:.7/s x Vogelsang,T.J. (1998). Trend function hypothesis testing in the presence of serial correlation. Econometrica. 66: Woodward, W. A., S. Bottone and H. L. Gray (1997). Improved tests for trend in time series data. J. Agr., Biol. Env. Stat.. 2: Zhang, X., L. A. Vincent, W. D. Hogg and A. Niitsoo (2). Temperature and precipitation trends in Canada during the 2 th century. ATMOSPHERE-OCEAN. 38: