SigmaPlot provides seven different data smoothing
algorithms that should satisfy most smoothing needs – negative
exponential, loess, running average, running median,
bisquare, inverse square and inverse distance. Each
smoother contains options that make them very flexible. For
example, unequally spaced data that occurs in clumps
is better analyzed using the nearest neighbor rather
than a fixed bandwidth method. Also, outlier rejection
is available in some smoothers.
Two Dimensional Smoothing
Smoothing is used to elicit trends from noisy
data. The three examples in Tukey´s book Exploratory
Data Analysis (AddisonWesley, 1977) show the need
for smoothing beautifully. The trends in the U.S.
gold production from 1872 to 1956, Figure 1A, are
fairly clear.
The peaks and valleys in the U.S. wheat production,
Figure 1B, are less clear. I challenge you to
visually find the trends in the annual New York
City precipitation data shown in Figure 1C. The
loess algorithm will be used to smooth these data
sets. “loess” means locally weighted regression. Each
point along the smooth curve is obtained from a
regression of data points close to the curve point
with the closest points more heavily weighted. The
amount of smoothing, which affects the number of
points in the regression is determined by the user. A
weighted regression is performed for each point
along the smooth curve.
Figure 1. Data with trends that are increasingly
more difficult to visualize
loess smoothed curves for the three examples in
Figure 1 are shown in Figure 2. The smoothed curves
in Figure 2A and 2B make the trends in the gold
and wheat data very clear. It is still difficult
to visualize in the raw data the precipitation
trend shown in Figure 2C. To confirm the results
of the loess smoothed curve the histogram of average
rainfall in ten year intervals was computed and
superimposed on the smooth curve. There is a good
comparison between the histogram and the loess
smooth.
The loess smoothing parameters were varied to
achieve the best visualization. A polynomial degree
of one was used in all cases. A 0.1 sampling proportion
was used in Figure 2A and B and 0.3 in Figure 2C. Since
the data was unequally spaced along the x axis
the nearest neighbor bandwidth method was used. The
default number of intervals (100) for generation
of the smooth curve was found to be the best. This
generates a line using straight lines between curve
points. Sometimes this leads to sharp corners
in the smooth so the spline interpolation line
type (Smoothed (spline)) was used.
Figure 2. Smoothed curves for data in Figure 1.
A ten year average rainfall histogram is also shown
in C.
Several of the smoothing methods, including loess,
are based on local polynomial regression and the
polynomial order is selectable. Increasing the
order tends to include more high frequency components
in the smooth. The effect of increasing the order
from 1 (local linear regressions) to 2 (local quadratic
regressions) is shown in Figure 3. The effect
is to increase peak height magnitude and introduce
additional high frequency components (wiggles)
in B. A subsequent increase of the sampling proportion
in C results in a smooth very much like the original
for order 1 in A.
Figure 3. Effect of increasing the regression
polynomial order. The order is 1 and sampling
proportion is 0.1 in A. The order is increased
to 2 in B and then the sampling proportion is increased
to 0.2 in C.
Three Dimensional Smoothing
Visualizing spatial relationships in a three dimensional
scatter plot can be very difficult. The strongest
three dimensional cue is provided by an animated
rotation of the data. Since this is not possible
in paper publications we must resort to using drop
lines, enclosing the graph with additional axes,
etc. Figure 4 shows that a smooth surface also
helps. This data describes the reaction characteristics
on an isomer of hexane. The smooth surface B clearly
shows the trends with respect to temperature and
reaction rate whereas visualizing this in the scatterplot
A is difficult.
Figure 4. The data trend in A is easily visualized
with a loess smoothed surface, B.
This data is relatively sparse so a large sampling
proportion 0.6 was required to avoid oscillations
and spikes in the loess surface. A polynomial
degree of 1 and the nearest neighbor bandwidth
method were used. The Preview feature allows a
quick comparison of smoothing methods on a given
data set. For this data essentially equivalent
smooth surfaces were obtained with the negative
exponential and bisquare methods.
The bandwidth method option is also very useful. The
nearest neighbor method works well for unequally
spaced data. The data in Figure 3 is unequally
spaced in both X and Y. Compare the smoothing
results using the nearest neighbor and fixed methods
shown in Figure 5. The result for the fixed method
is about the best that could be obtained by varying
the sampling proportion with a value of 0.8 shown.
Figure 5. Comparison of bandwidth methods for
unequally spaced data. Nearest neighbor on the
left and fixed on the right.
Additional Computational Details
Smoothers is a generic name for a variety
of techniques that can be used to either smooth a
data set by removing undesired highfrequency components
(locations of rapid variation, such as noise contamination),
or to resample dependent variable values
to other independent variable locations using the
values of the data at nearby points. The smoothing
methods provided in SigmaPlot operate by weighting
the data in a neighborhood of the smoothing location
and applying linear or nonlinear methods to combine
the weighted values to produce a smoothed value. These
nonparametric smoothing techniques provide a good
complement to the parameterized curve/surface fitting
facility (Regression Wizard) in SigmaPlot. For
data subjected to measurement errors, noise, etc.,
either method can be used to predict behavior or
to estimate true values.
The kernel used in the smoothing computation and
the smoothing method are given in the following
table.
Algorithm 
Weighting Kernel 
Method to Compute Smoothed Value 
Negative Exponential 
Gaussian 
Polynomial or Loess Fitting 
Loess 
Tricube 
Polynomial or Loess Fitting 
Running Average 
Uniform 
Mean 
Running Median 
Uniform 
Median 
Bisquare 
Biweight 
Polynomial Fitting 
Inverse Square 
Cauchy 
Mean 
Inverse Distance 
Inverse Distance 
Mean 
The equations used for each kernel are:
Kernel 
Kernel Formula (y=0 for 2D Smoothers) 
Uniform 
1 
Biweight 
(1  x^{2} – y^{2})^{2} 
Tricube 
(1  sqrt(x^{2}+y^{2})^{3})^{3} 
Gaussian 
exp(x^{2}y^{2}) 
Cauchy 
1/(1+x^{2}+y^{2}) 
Inverse Distance (3D only) 
1/sqrt(x^{2}+y^{2}) 
