fitorder=nonnegint

The degree of the polynomial used in each local regression. The default value is $1$.

bandwidth=Range(0, 1)

The proportion of the input data points used in each local regression. The default value depends on fitorder and the number of input data points.

iters=nonnegint

The number of iterations when smoothing data of one independent variable. Each iteration makes the data smoother by eliminating outliers, thus making the computation more robust. This option has no effect when the data has more than one independent variable. The default is $2$.

The Lowess command creates a function whose values represent the input data smoothed with the lowess algorithm.

Suppose the input data set $\mathrm{XY}$ is of $m$ independent variables and has $n$ data points, the lowess smoothed value at $x≔\left(x_1\,x_2\,\dots \,x_m\right)$ is computed as follows.

Take the $n\cdot \mathrm{bandwidth}$ points in $\mathrm{XY}$ that are closest to $x$.

Fit a polynomial $P$ of $m$ variables and degree $\mathrm{fitorder}$ to the points using weighted linear least squares, where the weight for a point $w$ is computed by applying the tri-cube weight function to the distance between $x$ and $w$.

Evaluate $P\left(x\right)$.

Running one or more iterations, as specified by the iters option, will produce a set of weights to reduce the influence of outliers (that is, make the computation more robust). At each iteration, the weight of a point depends on the residual of the Lowess curve at that point in the previous iteration. These weights are combined with the weights given by the distance, as described previously.

L will return unevaluated if the arguments are non-convertible to numerics. But if the first and only argument is a Matrix with number of columns equal to the number of parameters of L, a Vector will be returned where the $i$th element is the L applied with the $i$th row of the Matrix as arguments.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{Sample}\left(\mathrm{Uniform}\left(-2\,2\right)\,200\right)\:$

$Y≔\mathrm{Sample}\left(\mathrm{Uniform}\left(-2\,2\right)\,200\right)\:$

$\mathrm{Zerror}≔\mathrm{Sample}\left(\mathrm{Normal}\left(0\,0.1\right)\,200\right)\:$

$Z≔\mathrm{`~`}\left[\mathrm{`*`}\right]\left(X\,\mathrm{`\; \$`}\,\mathrm{map}\left(\exp \,-\mathrm{`~`}\left[\mathrm{`^`}\right]\left(X\,\mathrm{`\; \$`}\,2\right)-\mathrm{`~`}\left[\mathrm{`^`}\right]\left(Y\,\mathrm{`\; \$`}\,2\right)\right)\right)+\mathrm{Zerror}\:$

$\mathrm{XYZ}≔{\mathrm{Matrix}\left(\left[\left[X\right]\,\left[Y\right]\,\left[Z\right]\right]\,\mathrm{datatype}=\mathrm{float}\left[8\right]\right)}^{\mathrm{\%T}}$

$L≔\mathrm{Lowess}\left(\mathrm{XYZ}\,\mathrm{fitorder}=2\,\mathrm{bandwidth}=0.3\right)\:$

$P≔\mathrm{ScatterPlot3D}\left(\mathrm{XYZ}\right)\:$

$Q≔\mathrm{plot3d}\left(L\,-2..2\,-2..2\,\mathrm{grid}=\left[25\,25\right]\right)\:$

$R≔\mathrm{plots}:-\mathrm{shadebetween}\left(L\left(x\,y\right)\,0.4\,x=-1.5..-0.5\,y=-1..1\,\mathrm{showboundary}=\mathrm{false}\,\mathrm{negativeonly}\right)\:$

$\mathrm{plots}:-\mathrm{display}\left(P\,Q\,R\,\mathrm{orientation}=\left[100\,70\,0\right]\,\mathrm{lightmodel}=\mathrm{none}\right)$

$\mathrm{int}\left(0.4-L\left(x\,y\right)\,x=-1.5..-0.5\,y=-1..1\,\mathrm{numeric}\,\mathrm{\epsilon }=0.01\,\mathrm{method}=\mathrm{\_CubaSuave}\right)$

$1.30321733303695$

$X≔\mathrm{Sample}\left(\mathrm{Uniform}\left(0\,\mathrm{\pi }\right)\,200\right)$

$\mathrm{Yerror}≔\mathrm{Sample}\left(\mathrm{Normal}\left(0\,0.1\right)\,200\right)$

$Y≔\mathrm{map}\left(\sin \,X\right)+\mathrm{Yerror}$

$L≔\mathrm{CurveFitting}:-\mathrm{Lowess}\left(X\,Y\,\mathrm{fitorder}=1\,\mathrm{bandwidth}=0.3\right)\:$

$P≔\mathrm{ScatterPlot}\left(X\,Y\right)$

$Q≔\mathrm{plot}\left(L\left(x\right)\,x=0..\mathrm{\pi }\right)$

$R≔\mathrm{plots}:-\mathrm{shadebetween}\left(L\left(x\right)\,0\,x=\frac{\mathrm{\pi }}{8}..\frac{3\mathrm{\pi }}{8}\,\mathrm{showboundary}=\mathrm{false}\,\mathrm{positiveonly}\right)$

$\mathrm{plots}:-\mathrm{display}\left(P\,Q\,R\right)$

$\mathrm{int}\left(L\,\frac{\mathrm{\pi }}{8}..\frac{3\mathrm{\pi }}{8}\,\mathrm{numeric}\,\mathrm{\epsilon }=0.01\right)$

$0.526726498845341$

$\mathrm{Optimization}:-\mathrm{Maximize}\left(L\,\mathrm{map}\left(\mathrm{unapply}\,\left\{-x\,x-\mathrm{\pi }\right\}\,x\right)\,\mathrm{optimalitytolerance}=0.001\right)$

$\left[0.991236282391190437\,\left[\right]\right]$

The Statistics[Lowess] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.