plotDiagnostics

Class: NonLinearModel

Plot diagnostics of nonlinear regression model

Syntax

plotDiagnostics(mdl)
plotDiagnostics(mdl,plottype)
plotDiagnostics(mdl,plottype,Name,Value)
h = plotDiagnostics(___)


Description

plotDiagnostics(mdl) creates a leverage plot of the nonlinear regression model (mdl) observations. A dotted line in the plot represents the recommended threshold values.

plotDiagnostics(mdl,plottype) specifies the type of observation diagnostics plottype.

plotDiagnostics(mdl,plottype,Name,Value) specifies the graphical properties of diagnostic data points using one or more name-value arguments. For example, you can specify the marker symbol and size for the data points.

h = plotDiagnostics(___) returns graphics objects for the lines or contour in the plot using any of the input argument combinations in the previous syntaxes. Use h to modify the properties of a specific line or contour after you create the plot. For a list of properties, see Line Properties and Contour Properties.

Input Arguments

mdl	Nonlinear regression model, constructed by fitnlm.
plottype	Character vector or string scalar specifying the type of plot: 'contour' Residual vs. leverage with overlaid Cook's contours 'cookd' Cook's distance 'leverage' Leverage (diagonal of Hat matrix) Default: 'leverage'

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Note

The graphical properties listed here are only a subset. For a complete list, see Line Properties. The specified properties determine the appearance of diagnostic data points.

Color	Color of the line or marker, specified as an RGB triplet, hexadecimal color code, color name, or short name for one of the color options listed in the following table. For a custom color, specify an RGB triplet or a hexadecimal color code. An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range [0,1], for example, [0.4 0.6 0.7]. A hexadecimal color code is a character vector or a string scalar that starts with a hash symbol (#) followed by three or six hexadecimal digits, which can range from 0 to F. The values are not case sensitive. Therefore, the color codes "#FF8800", "#ff8800", "#F80", and "#f80" are equivalent. Alternatively, you can specify some common colors by name. This table lists the named color options, the equivalent RGB triplets, and hexadecimal color codes. Color Name Short Name RGB Triplet Hexadecimal Color Code Appearance "red" "r" [1 0 0] "#FF0000" "green" "g" [0 1 0] "#00FF00" "blue" "b" [0 0 1] "#0000FF" "cyan" "c" [0 1 1] "#00FFFF" "magenta" "m" [1 0 1] "#FF00FF" "yellow" "y" [1 1 0] "#FFFF00" "black" "k" [0 0 0] "#000000" "white" "w" [1 1 1] "#FFFFFF" Here are the RGB triplets and hexadecimal color codes for the default colors MATLAB® uses in many types of plots. RGB Triplet Hexadecimal Color Code Appearance [0 0.4470 0.7410] "#0072BD" [0.8500 0.3250 0.0980] "#D95319" [0.9290 0.6940 0.1250] "#EDB120" [0.4940 0.1840 0.5560] "#7E2F8E" [0.4660 0.6740 0.1880] "#77AC30" [0.3010 0.7450 0.9330] "#4DBEEE" [0.6350 0.0780 0.1840] "#A2142F"
LineWidth	Width of the line or edges of filled area, in points, a positive scalar. One point is 1/72 inch. Default: 0.5
Marker	Marker symbol, specified as one of the values in this table. Marker Description Resulting Marker "o" Circle "+" Plus sign "*" Asterisk "." Point "x" Cross "_" Horizontal line "|" Vertical line "square" Square "diamond" Diamond "^" Upward-pointing triangle "v" Downward-pointing triangle ">" Right-pointing triangle "<" Left-pointing triangle "pentagram" Pentagram "hexagram" Hexagram "none" No markers Not applicable
MarkerEdgeColor	Marker outline color, specified as an RGB triplet, hexadecimal color code, color name, or short name for one of the color options listed in the Color name-value argument.
MarkerFaceColor	Fill color for filled markers, specified as an RGB triplet, hexadecimal color code, color name, or short name for one of the color options listed in the Color name-value argument.
MarkerSize	Size of the marker in points, a strictly positive scalar. One point is 1/72 inch.

Output Arguments

h

Graphics objects corresponding to the lines or contour in the plot, returned as a graphics array. Use dot notation to query and set properties of the graphics objects. For details, see Line Properties and Contour Properties. You can use name-value arguments to specify the appearance of diagnostic data points corresponding to the first graphics object h(1).

Examples

Nonlinear Model Leverage Plot

Create a leverage plot of a fitted nonlinear model, and find the points with high leverage.

Load the reaction data and fit a model of the reaction rate as a function of reactants.

load reaction
mdl = fitnlm(reactants,rate,@hougen,[1 .05 .02 .1 2]);

Create a leverage plot of the fitted model.

plotDiagnostics(mdl)

Use data tips to examine the observation with high leverage. A data tip appears when you hover over a data point.

Alternatively, find the high-leverage observation at the command line.

find(mdl.Diagnostics.Leverage > 0.8)

ans =

     6

More About

Hat Matrix

The hat matrix H is defined in terms of the data matrix X and the Jacobian matrix J:

$$J_{i,j}={\frac{\partial f}{\partial {\beta }_j}|}_{x_i,\beta }$$

Here f is the nonlinear model function, and β is the vector of model coefficients.

The Hat Matrix H is

H = J(J^TJ)^–1J^T.

The diagonal elements H_ii satisfy

$$\begin{array}{l}0\le h_{ii}\le 1\\ {\displaystyle \sum _{i=1}^n{h}_{ii}}=p,\end{array}$$

where n is the number of observations (rows of X), and p is the number of coefficients in the regression model.

Leverage

Leverage is a measure of the effect of a particular observation on the regression predictions due to the position of that observation in the space of the inputs.

The leverage of observation i is the value of the ith diagonal term h_ii of the hat matrix H. Because the sum of the leverage values is p (the number of coefficients in the regression model), an observation i can be considered an outlier if its leverage substantially exceeds p/n, where n is the number of observations.

Cook’s Distance

The Cook’s distance D_i of observation i is

$$D_i=\frac{{\displaystyle \sum _{j=1}^n{\left({\widehat{y}}_j-{\widehat{y}}_{j(i)}\right)}^2}}{pMSE},$$

where

• $${\widehat{y}}_j$$ is the jth fitted response value.

• $${\widehat{y}}_{j(i)}$$ is the jth fitted response value, where the fit does not include observation i.

• MSE is the mean squared error.

• p is the number of coefficients in the regression model.

Cook’s distance is algebraically equivalent to the following expression:

$$D_i=\frac{r_i^2}{pMSE}\left(\frac{h_{ii}}{{\left(1-h_{ii}\right)}^2}\right),$$

where e_i is the ith residual.

Tips

• The data cursor displays the values of the selected plot point in a data tip (small text box located next to the data point). The data tip includes the x-axis and y-axis values for the selected point, along with the observation name or number.

References

[1] Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. Applied Linear Statistical Models, Fourth Edition. Irwin, Chicago, 1996.