plotDiagnostics

Plot diagnostics of nonlinear regression model

Syntax

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

Description

plotDiagnostics(mdl) plots diagnostics from the mdl linear model using leverage as the plot type.

plotDiagnostics(mdl,plottype) plots diagnostics in a plot of type plottype.

h = plotDiagnostics(...) returns handles to the lines in the plot.

h = plotDiagnostics(mdl,plottype,Name,Value) plots with additional options specified by one or more Name,Value pair arguments.

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 Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Note

The plot property name-value pairs apply to the first returned handle h(1).

'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. Thus, 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
`'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'`

'LineStyle'

Type of line, a Line Properties specification. For details, see linespec.

'LineWidth'

Width of the line or edges of filled area, in points, a positive scalar. One point is 1/72 inch.

Default: 0.5

'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 pair 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 pair argument.

'MarkerSize'

Size of the marker in points, a strictly positive scalar. One point is 1/72 inch.

Output Arguments

`h`	Vector of handles to lines or patches in the plot.

Examples

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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

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Hat Matrix

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

$J_{i, j} = {\frac{\partial f}{\partial β_{j}} |}_{x_{i}, β}$

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

$\begin{array}{l} 0 \leq h_{i i} \leq 1 \\ \sum_{i = 1}^{n} h_{i i} = p, \end{array}$

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

$D_{i} = \frac{\sum_{j = 1}^{n} {({\hat{y}}_{j} - {\hat{y}}_{j (i)})}^{2}}{p M S E},$

where

${\hat{y}}_{j}$ is the jth fitted response value.
${\hat{y}}_{j (i)}$ 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:

$D_{i} = \frac{r_{i}^{2}}{p M S E} (\frac{h_{i i}}{{(1 - h_{i i})}^{2}}),$

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.

Documentation