User`s guide

2 Importing, Viewing, and Preprocessing Data

2-14

Lowess and Loess: Local Regression Smoothing

The names “lowess” and “loess” are derived from the term “locally weighted

scatter plot smooth,” as both methods use locally weighted linear regression to

smooth data.

The smoothing process is considered local because, like the moving average

method, each smoothed value is determined by neighboring data points defined

within the span. The process is weighted because a regression weight function

is defined for the data points contained within the span. In addition to the

regression weight function, you can use a robust weight function, which makes

the process resistant to outliers. Finally, the methods are differentiated by the

model used in the regression: lowess uses a linear polynomial, while loess uses

a quadratic polynomial.

The local regression smoothing methods used by the Curve Fitting Toolbox

follow these rules:

• The span can be even or odd.

• You can specify the span as a percentage of the total number of data points

in the data set. For example, a span of 0.1 uses 10% of the data points.

The regression smoothing and robust smoothing procedures are described in

detail below.

Local Regression Smoothing Procedure

The local regression smoothing process follows these steps for each data point:

1 Compute the regression weights for each data point in the span. The weights

are given by the tricube function shown below.

x is the predictor value associated with the response value to be smoothed,

are the nearest neighbors of x as defined by the span, and d(x) is the

distance along the abscissa from x to the most distant predictor value within

the span. The weights have these characteristics:

- The data point to be smoothed has the largest weight and the most

influence on the fit.

- Data points outside the span have zero weight and no influence on the fit.

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dx()

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