1. Chapter 11 LOcally-WEighted Scatterplot Smoother (Lowess)
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 equation or statistical test.
Lowess Smooth
No equation or statistical test.
Shows General Trend.

3. LOcally-WEighted Scatterplot Smoother (Lowess)
Purpose
Summarizes general location or trend
Not constrained to be linear, or monotonic
Highlights patterns for two or more groups
Draws eye towards the main point
Validates using a model with a specific form
Straight line
Exponential decay
Etc.

4. LOcally-WEighted Scatterplot Smoother (Lowess)
Disadvantages
No tests of significance, only exploratory
No single equation for smooth
Method
Iterative reweighed regression
Weights decrease as point gets further away from both X and Y direction

5. Weighted Least Squares
Find slope and intercept that minimizes the Sum of Weighted Squared Residuals (SWSE)

6. How Lowess Works
Lowess calculates a new smoothed y-value for each x-value.
A fraction (default f = 0.5) of all points, using the data closest in x-value on either side of the (x,y) point.

7. How Lowess Works

8. Tri-cube Weight Function
How Lowess Works
Tri-cube Weight Function

9. How Lowess Works

10. How Lowess Works

11. How Lowess Works

12. How Lowess Works

13. Why Use Lowess?
Let the data tell you what is going on instead of assuming a model from the start.

14. Plot Lowess for Each Group

15. p-values Do Not Tell the Whole Story!
Not a Good Fit!
Slope = p < 0.001
Lowess Validates
a log Model

16. p-values Do Not Tell the Whole Story!
A log model fits much better than a linear model.
P-value is also <

17. ... > Data View > Smoother
Minitab Lowess
... > Data View > Smoother
Use to fit a lowess smoother to the following plots:
Scatter plot
Matrix plot
Histogram
Time series plot
After creating a graph, you can:
Change the lowess smoothing parameters, add groups, and change the fitted line attributes .
Add a lowess smoother.
Smoother Dialog box items:
None: Choose to suppress the lowess smoother.
Lowess: Choose to display the lowess smoother.
Degree of smoothing: Enter a number between 0 and 1 for the fraction (f) of the total number of points used to calculate the fitted values at each x-value. The default is 0.5.
Number of steps: Enter a number from 0 to 10 to specify the number of iterations of smoothing to use to limit the influence of outliers. Each step reduces the weight given to outliers in the next iteration. The default is 2.
Source: Minitab 15

18. Statistical Methods in Water Resources by D.R. Helsel and R.M. Hirsch
Reading Assignment
Chapter 10.5 Smoothing
(pages 285 to 292)
Statistical Methods in Water Resources
by D.R. Helsel and R.M. Hirsch