1. Chapter 12 Trend Analysis
BAE 5333 Applied Water Resources Statistics
Biosystems and Agricultural Engineering Department
Division of Agricultural Sciences and Natural Resources
Oklahoma State University
Source
Dr. Dennis R. Helsel & Dr. Edward J. Gilroy
2006 Applied Environmental Statistics Workshop
and
Statistical Methods in Water Resources

2. No Signal, Only Random Noise Over Time
Null Hypothesis
For All Trend Tests
No Signal, Only Random Noise Over Time

3. First Alterative Hypothesis
Linear Trend Over Time

4. Second Alterative Hypothesis Nonlinear (Monotonic) Trend Over Time

5. Third Alterative Hypothesis Step Trend, Shift at One Point in Time

6. Seasonal Variation A Common Pattern in Natural Systems
Seasonality – within each year
Trend – over multiple years

7. How to Find a Trend Visual (graphical)
May be easy to identify and easy to communicate to others
Can not quantify
Statistical
Gives a number
Can identify trends with messy data
Technically difficult
Be vewwy,
vewwy quiet
….. I’m
hunting
for a twend.

8. Trend Topics Tests where time is an explanatory variable
Parametric versus nonparametric trend tests
How to add other variables besides time
How to deal with seasonal variation

9. Is an Explanatory Variable
Tests Where Time
Is an Explanatory Variable
If time is measured continuously (i.e. decimal time )
Regression (parametric)
Kendall-Theil (nonparametric)
If time is a grouped variable (1980s, pre- versus post-)
t-test (parametric)
rank-sum (nonparametric)

10. Trend Tests
Time is Continuous

11. Trend Tests - Time is Continuous (cont.)

12. Nonparametric Trend Tests
With Other Variables
Compute Lowess of Y vs. X
Store residuals R
Test residuals R for trend using Kendall’s tau

13. Parametric Trend Tests
With Other Variables
If both time (T) and X are continuous
Use multiple regression
Y = b0 + b1T + b2X
If time is grouped
Use ANCOVA
Time (T) is a 0 or 1, binary variable

14. Decision Rules for Using Continuous or Grouped Measures of Time

15. Parametric vs. Nonparametric Test
Distribution of residuals? Linear or monotonic?
Kendall-Theil
Slope is Median of All Pairwise Slopes
Kendall –Theil compares the change in value vs. time (slope) for each point on the chart and takes the median slope as the a summary statistic describing the magnitude of the trend.

16. Step Trends

17. Step Trend Tests

18. Nonparametric Step Trend Tests with Other Variables
Compute Lowess of Y vs. X
Store residuals R
Test residuals R for trend using rank-sum test

19. Nonparametric Step Trend Tests with Other Variables
For a continuous X variable and time (T) is a grouped variable
Use ANCOVA
Time (T) is a binary variable, 0 or 1
Y = b0 + b1T + b2X

20. Dealing with Seasonal Variation
Parametric methods
Build a better model
Nonparametric methods
Only compare within the same season

21. Continuous Trend Tests with Seasonal Variation

22. Seasonal Kendall Test
Compare all data within the same season to one another
DO NOT compare data across different seasons
Slope is the median of all within-season pair wise slopes.

23. Seasonal Kendall Test

24. Computing the Seasonal Kendall Test
The test statistic Si for EACH SEASON is the
number of plusses (Pi) minus the number of
Minuses (Mi) comparing data only within that
season.
For season i we have:
Si = Pi − Mi

25. Seasonal Kendall Test Statistic Sk

26. Summary: Nonparametric Trend Tests
Simplest
Y = b + mT (Kendall-Theil)
Adjusted for another variable
R = b + Mt
R = Lowess residuals of Y vs. X
With seasonal variation
Y = b + mT
m is Seasonal-Kendall slope

27. Regression with Sine and Cosine
Two new explanatory variables are created, and added to the regression equation
These are the sine and cosine of 2πT
T is time in decimal years (1997.5)
Resulting in one revolution every year

28. Regression with Sine and Cosine
The term 2πT produces one cycle per year, with a minimum and maximum
Slope coefficients times the sine and cosine terms fit the cycle function to the data
Both variables are required, as there are two unknowns for the year
Amplitude
“phase shift” (day of the peak)

29. Regression with Sine and Cosine
Y = b0 + b1T + b2X + b3sin(2πT) + b4cos(2πT)
Keep both sin and cos, or drop both.
Base decision on b3 and b4.
If either are significantly different than zero, keep both terms.

30. Summary: Parametric Trend Tests
Simplest
Y = b0 + b1T
Adjusted for another variable
Y = b0 + b1T + b2X
With seasonal variation
Y = b0 + b1T + b2X + b3 sin(2πT) + b4 cos(2πT)

31. Results of a Seasonal Model

32. MINITAB Laboratory 11 Reading Assignment Chapter 12 Trend Analysis
(pages 323 to 355)
Statistical Methods in Water Resources
by D.R. Helsel and R.M. Hirsch
MINITAB
Laboratory 11