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

No equation or statistical test. Lowess Smooth No equation or statistical test. Shows General Trend.

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.

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

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

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.

How Lowess Works

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

How Lowess Works

How Lowess Works

How Lowess Works

How Lowess Works

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

Plot Lowess for Each Group

p-values Do Not Tell the Whole Story! Not a Good Fit! Slope = -0.66 p < 0.001 Lowess Validates a log Model

p-values Do Not Tell the Whole Story! A log model fits much better than a linear model. P-value is also < 0.001.

... > 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

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