Detecting trends in environmental time series data Francis Chiew The University of Melbourne Brisbane 6 July Sydney 20 July

Workshop Objectives To discuss reasons for detecting trend/change in environmental time series data. To show some visual tools for exploratory data analysis (EDA). To explain basic concepts in formal statistical testing of trend/change in time series data. To show some examples of the statistical methods. To provide hands-on experience on the use of TREND, a model product in the Modelling Toolkit. The methods presented in this workshop can be applied to any time series data, but the focus of the workshop is mainly on annual streamflow, rainfall and other hydrologic data.

TREND TREND is a model product in the Modelling Toolkit ( TREND is designed to facilitate statistical testing for trend, change and randomness in hydrological and other time series data. TREND has 12 statistical tests, based on the WMO/UNESCO Expert Workshop on Trend/Change Detection and the CRC for Catchment Hydrology publication Hydrological Recipes.

Workshop Program 9:00 – 9:15Welcome and Introduction 9:15 – 10:00Presentation on trend detection, exploratory data analysis and formal statistical testing (basic concepts, and types of tests in TREND) 10:00 – 10.30Demonstration of statistical tests (in Excel) 10:30 – 10:50Morning tea 10:50 – 12:30Demonstration of TREND 12:30 – 1.15Lunch 1:15 – 3:00Hands-on experience with TREND 3:00 – 3.20Afternoon tea 3.20 – 5.00 More statistical tests and TREND 9:00 – 9:15Welcome and Introduction 9:15 – 10:00Presentation on trend detection, exploratory data analysis and formal statistical testing (basic concepts, and types of tests in TREND) 10:00 – 10.30Demonstration of statistical tests (in Excel) 10:30 – 10:50Morning tea 10:50 – 12:30Demonstration of TREND 12:30 – 1.15Lunch 1:15 – 3:00Hands-on experience with TREND 3:00 – 3.20Afternoon tea 3.20 – 5.00 More statistical tests and TREND

Why detect trend/change in environmental time series data? Why detect trend/change in environmental time series data? Most water resources systems have been designed and operated based on the assumption of stationary hydrology. If this assumption of stationarity is not valid, current systems may be under or over designed. Trend/change in environmental time series data can be caused by climate change as a result of increased greenhouse gas concentrations land use change (urbanisation, clearing, afforestation, etc…) change in management practice etc … Trend/change in environmental time series data can be caused by climate change as a result of increased greenhouse gas concentrations land use change (urbanisation, clearing, afforestation, etc…) change in management practice etc …

Climate change and climate variability Climate change defines the difference between long-term mean values of a climate parameter or statistic, where the mean is taken over a specified interval of time, usually several decades. Climate change can occur because of internal changes in external forcing either for natural reasons or because of human activities. It is difficult to make clear attribution between these causes. Climate variability can be regarded as the variability (the extremes and differences of monthly, seasonal and annual values from the climatically expected value) inherent in the stationary process approximating the climate on a scale of a few decades. The inter- annual hydroclimate can vary considerably, resulting in difficulty in detecting a statistically significant change in the climate. Climate change defines the difference between long-term mean values of a climate parameter or statistic, where the mean is taken over a specified interval of time, usually several decades. Climate change can occur because of internal changes in external forcing either for natural reasons or because of human activities. It is difficult to make clear attribution between these causes. Climate variability can be regarded as the variability (the extremes and differences of monthly, seasonal and annual values from the climatically expected value) inherent in the stationary process approximating the climate on a scale of a few decades. The inter- annual hydroclimate can vary considerably, resulting in difficulty in detecting a statistically significant change in the climate.

Climate change signal in streamflow data? Clear jump in mean (climate change?) Multi-jumps in mean (inter-decadal variability?)

Hydroclimate is always changing over various time scales Seasonal Inter-annual ENSO (3-7 years) Interdecadal (20-30 years) “Climate change”

Exploratory data analysis (EDA) involves using graphs to explore, understand and present data. EDA is an iterative process where graphs are plotted and refined so that important features of the data can be seen clearly. EDA is an essential component of any statistical analysis. EDA allows much greater appreciation of the features in the data than summary statistics or statistical significance levels. This is because the human brain and visual system is very powerful at identifying and interpreting patterns. A well conducted EDA may eliminate the need for a formal statistical analysis. A statistical analysis is incomplete with EDA. Statistical test results can be meaningless without a proper understanding of the data. For example, EDA can identify amongst other things, outliers due to poor data quality, strong data independence/autocorrelation, and change in station location or recording technique. Exploratory data analysis (EDA) involves using graphs to explore, understand and present data. EDA is an iterative process where graphs are plotted and refined so that important features of the data can be seen clearly. EDA is an essential component of any statistical analysis. EDA allows much greater appreciation of the features in the data than summary statistics or statistical significance levels. This is because the human brain and visual system is very powerful at identifying and interpreting patterns. A well conducted EDA may eliminate the need for a formal statistical analysis. A statistical analysis is incomplete with EDA. Statistical test results can be meaningless without a proper understanding of the data. For example, EDA can identify amongst other things, outliers due to poor data quality, strong data independence/autocorrelation, and change in station location or recording technique. Exploratory data analysis (EDA)

Time series plot The time series plot is the most useful visual tool for analysing trend/change. The variable of interest is plotted against time as a scatter or line plot. A trend line can be fitted to the data. Trend lines include moving average, linear regression, quadratic regression, and nonparametric smoothing (e.g., lowess smooth). The Excel spreadsheet can be used to perform most of the plotting and visual trend analysis (see examples in Excel spreadsheet). Multiple time series plots can be used when data from several sites (plot same data on the same scale) or variables are available. It can be informative presenting data from several sites within a region (or different variables for a location) on a single page. For example, a trend/change is more conclusive if it is observed in data from several locations. The time series plot is the most useful visual tool for analysing trend/change. The variable of interest is plotted against time as a scatter or line plot. A trend line can be fitted to the data. Trend lines include moving average, linear regression, quadratic regression, and nonparametric smoothing (e.g., lowess smooth). The Excel spreadsheet can be used to perform most of the plotting and visual trend analysis (see examples in Excel spreadsheet). Multiple time series plots can be used when data from several sites (plot same data on the same scale) or variables are available. It can be informative presenting data from several sites within a region (or different variables for a location) on a single page. For example, a trend/change is more conclusive if it is observed in data from several locations.

Example time series plots Scatter plot Line plot with data points Line plot without data points Annual River Nile flows at Aswan in cumecs

Polynomial regression point moving average Linear regression Example trend fits to time series data Lowess smooth

Some commercial software for visual data analysis and time series data analysis Excel Mathematica MATLAB MINITAB SAS SPlus SPSS Systat Excel Mathematica MATLAB MINITAB SAS SPlus SPSS Systat

Statistical methods for detecting trend/change in time series data Change in a time series can occur steadily (a trend), abruptly (a step- change) or in a more complex form. It may affect the mean, median, variance or other aspect of the data. TREND has various statistical methods for detecting trend, step change, differences in means/medians between two data periods and randomness in hydrological time series data. The statistical methods in TREND are based on the WMO/UNESCO WCP Expert Workshop on “Detecting Trend and Other Changes in Hydrological Data” and the CRCCH “Hydrological Recipes”. Kundzewicz, Z.W. and Robson, A. (Editors) (2000) Detecting Trend and Other Changes in Hydrological Data. World Climate Program – Water, WMO/UNESCO, WCDMP-45, WMO/TD 1013, Geneva, 157 pp. Grayson, R.B., Argent, R.M., Nathan, R.J., McMahon, T.A. and Mein, R. (1996) Hydrological Recipes: Estimation Techniques in Australian Hydrology. Cooperative Research Centre for Catchment Hydrology, Australia, 125 pp. Change in a time series can occur steadily (a trend), abruptly (a step- change) or in a more complex form. It may affect the mean, median, variance or other aspect of the data. TREND has various statistical methods for detecting trend, step change, differences in means/medians between two data periods and randomness in hydrological time series data. The statistical methods in TREND are based on the WMO/UNESCO WCP Expert Workshop on “Detecting Trend and Other Changes in Hydrological Data” and the CRCCH “Hydrological Recipes”. Kundzewicz, Z.W. and Robson, A. (Editors) (2000) Detecting Trend and Other Changes in Hydrological Data. World Climate Program – Water, WMO/UNESCO, WCDMP-45, WMO/TD 1013, Geneva, 157 pp. Grayson, R.B., Argent, R.M., Nathan, R.J., McMahon, T.A. and Mein, R. (1996) Hydrological Recipes: Estimation Techniques in Australian Hydrology. Cooperative Research Centre for Catchment Hydrology, Australia, 125 pp.

Hypotheses The starting point of a statistical test is to define a null hypothesis (H 0 ) and an alternative hypothesis (H 1 ). For example, to test for trend in the mean of a time series, H 0 would be that there is no change in the mean of the data, and H 1 would be that the mean is either increasing or decreasing with time. Test statistic The test statistic is a means of comparing the H 0 and H 1. It is a numerical value calculated from the data series that is being tested. Significance level The significance level is a means of measuring whether the test statistic is very different from values that would typically occur under H 0. Power and errors There are two possible types of errors. Type I error is when H 0 is incorrectly rejected. Type II error is when H 0 is accepted when H 1 is true. A test with low Type II error is said to be powerful. Hypotheses The starting point of a statistical test is to define a null hypothesis (H 0 ) and an alternative hypothesis (H 1 ). For example, to test for trend in the mean of a time series, H 0 would be that there is no change in the mean of the data, and H 1 would be that the mean is either increasing or decreasing with time. Test statistic The test statistic is a means of comparing the H 0 and H 1. It is a numerical value calculated from the data series that is being tested. Significance level The significance level is a means of measuring whether the test statistic is very different from values that would typically occur under H 0. Power and errors There are two possible types of errors. Type I error is when H 0 is incorrectly rejected. Type II error is when H 0 is accepted when H 1 is true. A test with low Type II error is said to be powerful. Basic concepts of statistical testing

Significance level The significance level (  ) is a means of measuring whether the test statistic is very different from values that would typically occur under H 0. Specifically, the significance level is the probability of a test statistic value as extreme as, or more extreme than the observed value assuming no trend/change (H 0 ). For example, for  = 0.05, the critical test statistic value is the value that would be exceeded by 5% of test statistic values obtained from randomly generated data. If the test statistic value is greater than the critical test statistic value, H 0 is rejected. The significance level is therefore the probability that a test detects a trend/change (reject H 0 ) when none is present (Type I error). A possible interpretation of the significance level might be:  > 0.10little evidence against H <  < 0.10possible evidence against H <  < 0.05strong evidence against H 0  < 0.01very strong evidence against H 0 For most traditional statistical methods, critical test statistic values for various significance levels can be looked up in statistical tables or calculated from simple formulas, provided the test assumptions are satisfied. Where test assumptions are violated, resampling methods can be used to estimate the significance level of a test statistic. For detecting trend/change, the critical test statistic value at  /2 is used (two-sided tail). For detecting an increase (or decrease), the critical test statistic value at  is used (one-sided tail). The significance level (  ) is a means of measuring whether the test statistic is very different from values that would typically occur under H 0. Specifically, the significance level is the probability of a test statistic value as extreme as, or more extreme than the observed value assuming no trend/change (H 0 ). For example, for  = 0.05, the critical test statistic value is the value that would be exceeded by 5% of test statistic values obtained from randomly generated data. If the test statistic value is greater than the critical test statistic value, H 0 is rejected. The significance level is therefore the probability that a test detects a trend/change (reject H 0 ) when none is present (Type I error). A possible interpretation of the significance level might be:  > 0.10little evidence against H <  < 0.10possible evidence against H <  < 0.05strong evidence against H 0  < 0.01very strong evidence against H 0 For most traditional statistical methods, critical test statistic values for various significance levels can be looked up in statistical tables or calculated from simple formulas, provided the test assumptions are satisfied. Where test assumptions are violated, resampling methods can be used to estimate the significance level of a test statistic. For detecting trend/change, the critical test statistic value at  /2 is used (two-sided tail). For detecting an increase (or decrease), the critical test statistic value at  is used (one-sided tail).

Resampling to estimate significance level Resampling is a robust method for estimating the significance level of a test statistic. It is particularly useful when the test assumptions are violated. In resampling, the original time series data are resampled to provide many replicates of time series data of equal length as the original data. The time series data for each replicate are obtained by randomly selecting data value from any year in the original time series continuously until a time series of equal length as the original data is constructed. In TREND, the data are resampled with replacement (bootstrapping method), i.e., the replicate series may contain more than one of some values in the original series and none of other values. The test statistic value of the original time series data is then compared with the test statistic values of the generated data (replicates) to estimate the significance level. For example, if the test statistic value of the original data is the same as the 950 th highest value from 1000 replicates, H 0 is rejected at  = 0.05 (i.e., a trend/change is detected, with a 5% probability that this trend/change is incorrectly detected). Resampling is a robust method for estimating the significance level of a test statistic. It is particularly useful when the test assumptions are violated. In resampling, the original time series data are resampled to provide many replicates of time series data of equal length as the original data. The time series data for each replicate are obtained by randomly selecting data value from any year in the original time series continuously until a time series of equal length as the original data is constructed. In TREND, the data are resampled with replacement (bootstrapping method), i.e., the replicate series may contain more than one of some values in the original series and none of other values. The test statistic value of the original time series data is then compared with the test statistic values of the generated data (replicates) to estimate the significance level. For example, if the test statistic value of the original data is the same as the 950 th highest value from 1000 replicates, H 0 is rejected at  = 0.05 (i.e., a trend/change is detected, with a 5% probability that this trend/change is incorrectly detected).

Parametric and non-parametric tests Most statistical tests assume that the time series data are independent and identically distributed. Parametric tests also assume that the time series data and the errors (deviations from the trend) follow a particular distribution. Most parametric tests assume that the data are normally distributed. Parametric tests are useful as they also quantify the change in the data (e.g., change in mean or gradient of trend). Parametric tests are generally more powerful than non-parametric tests. Non-parametric tests are generally distribution-free. They detect trend/change, but do not quantify the size of the trend/change. They are very useful because most hydrologic time series data are not normally distributed. Most statistical tests assume that the time series data are independent and identically distributed. Parametric tests also assume that the time series data and the errors (deviations from the trend) follow a particular distribution. Most parametric tests assume that the data are normally distributed. Parametric tests are useful as they also quantify the change in the data (e.g., change in mean or gradient of trend). Parametric tests are generally more powerful than non-parametric tests. Non-parametric tests are generally distribution-free. They detect trend/change, but do not quantify the size of the trend/change. They are very useful because most hydrologic time series data are not normally distributed.

Statistical tests in TREND Tests for trend Mann-Kendall (non-parametric) Spearman’s Rho (non-parametric) Linear Regression (parametric) Tests for step change in mean/median Distribution Free CUSUM (non-parametric) Cumulative Deviation (parametric) Worsley Likelihood Ratio (parametric) Tests for difference in mean/median in two different data periods Rank-Sum (non-parametric) Student’s t-test (parametric) Tests for randomness Median Crossing (non-parametric) Turning Points (non-parametric) Rank Difference (non-parametric) Autocorrelation (parametric) Tests for trend Mann-Kendall (non-parametric) Spearman’s Rho (non-parametric) Linear Regression (parametric) Tests for step change in mean/median Distribution Free CUSUM (non-parametric) Cumulative Deviation (parametric) Worsley Likelihood Ratio (parametric) Tests for difference in mean/median in two different data periods Rank-Sum (non-parametric) Student’s t-test (parametric) Tests for randomness Median Crossing (non-parametric) Turning Points (non-parametric) Rank Difference (non-parametric) Autocorrelation (parametric)

Cautionary Words Must have good data and must understand data (via exploratory data analysis). Must understand statistical test and the assumptions. A statistical test provides evidence, not proof. Significance is not the same as importance (e.g., a change may be detected, but the size of the change may be so small that it is of no importance). If H 0 is rejected, the reason for the trend/change must be investigated. Must have good data and must understand data (via exploratory data analysis). Must understand statistical test and the assumptions. A statistical test provides evidence, not proof. Significance is not the same as importance (e.g., a change may be detected, but the size of the change may be so small that it is of no importance). If H 0 is rejected, the reason for the trend/change must be investigated.