plotting in matlab - l-università ta'...

PlotsFunctions and Plots

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Plotting in MATLAB

Trevor [email protected]

http://staff.um.edu.mt/trevor.spiteri

Department of Communications and Computer EngineeringFaculty of Information and Communication Technology

University of Malta

29 February, 2008

Trevor Spiteri Plotting in MATLAB

mailto:[email protected]

http://staff.um.edu.mt/trevor.spiteri

http://www.um.edu.mt/ict/cce

http://www.um.edu.mt/ict

http://www.um.edu.mt/


Special Plots

Outline

PlotsPlot BasicsThe axisMultiple PlotsInteracting with Plots

Functions and PlotsPlotting FunctionsFunction Discovery

Special PlotsDifferent Plots3-D Plots



Special Plots

Plot BasicsThe axisMultiple PlotsInteracting with Plots

Outline






Special Plots


The elements of a plot

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500Height of a Falling Object Versus Time

Time (s)

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ModelMeasurement

Plot titleLegend

Data symbol

Axis label

Grid

Tick markTick-mark label



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The elements of a plot explained

Ï The scale is the range and spacing of the numbers.

Ï The most-used scale is the rectilinear scale, also known as thelinear scale.

Ï Another kind of scale is the logarithmic scale.

Ï Tick marks are placed on the axis to help visualize thenumbers.

Ï A function is always plotted as a line, with no data symbols.

Ï When plotting data, data symbols are used, except if they aretoo dense.

Ï When plotting more than one curve, they must bedistinguished.

Ï This distinction can be made using a legend or labels.



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A correct plot

Ï Each axis must be labelled with the name and its units.

Ï Each axis should have regularly spaced tick marks. Theseshould not be too sparse or too dense.

Ï When plotting data points, use a data symbol such as a circle.If there are many symbols, use a dot.

Ï When plotting points generated by using a function, datasymbols should not be used.



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Plotting x-y data

Ï To plot x data against y data, type: >> plot(x, y)Ï To change the line specification, type: >> plot(x, y, s)Ï s contains the line style specifier, the marker specifier, and/or

the colour specifier.

Ï For example, for a solid line, a plus data marker, and the redcolour, the specifier would be '- + r'



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Line specification string syntax

Line styles Markers ColoursSolid line - Plus sign (+) + Red rDashed line -- Circle (◦) o Green gDotted line : Asterisk (∗) * Blue bDash-dot line -. Point (·) . Cyan c

Cross (×) x Magenta mSquare (ä) s Yellow yDiamond (♦) d Black kUpward triangle (4) ^ White wDownward triangle (O) vRight triangle (.) >Left triangle (/) <Five-pointed star pSix-pointed star h



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Labels, titles, and grids

Ï The xlabel function sets the label for the x-axis.

Ï The ylabel function sets the label for the y-axis.

Ï The title function sets the plot title.

Ï The grid on command displays the grid.

Ï The grid off command hides the grid.



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The axis

Ï The axis function sets the axis of the plot.

Ï To make the plot a square, type: >> axis squareÏ To make the x-axis and y-axis have the same scale, type:

>> axis equalÏ To set the x-axis in the range x1–x2 and the y-axis in the range

y1–y2, type: >> axis([x1 x2 y1 y2])Ï To return to automatic axis, type: >> axis auto



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Scales

Ï The most commonly-used scale is the rectilinear scale, alsoknown as the linear scale.

Ï The plot command uses the linear scale.

Ï The loglog command is the same as the plot command,except that logarithmic scales are used for both the x-axis andthe y-axis.

Ï The semilogx command is the same as the plot command,except that a logarithmic scale is used on the x-axis.

Ï The semilogy command is the same as the plot command,except that a logarithmic scale is used on the y-axis.



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More than one line on the same axis

Ï To plot more than one line on the same axis, one way is to type:>> plot(x1, y1, s1, x2, y2, s2, . . .)

Ï Another way is to use the hold command.>> plot(x1, y1, s1)>> hold on>> plot(x2, y2, s2). . .>> hold off

Ï If x and y are matrices, to plot the columns of y against thecolumns of x, type: >> plot(x, y)



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The legendÏ The legend can be used to distinguish between different lines.Ï To display a legend, use the legend command.

>> legend(string1, string2, . . .)Ï The specified strings will be used as labels.Ï The default legend location is the top right corner.Ï To change the location of the legend:

>> legend(string1, . . ., 'location', specifier)Ï The specifier for a location inside the axis can be 'North',

'NorthEast', 'East', and so on.Ï The default location is 'NorthEast'.Ï The specifier for a location outside the axis can be

'NorthOutside', 'NorthEastOutside', and so on.Ï For MATLAB to choose a place automatically, the specifier can

be 'Best' or 'BestOutside'.



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Text labels

Ï Another way to distinguish between lines is to label them.

Ï To write labels, use the text command:>> text(x, y, 'string')

Ï The specified string will be placed at the given coordinates.

Ï To place a label using the mouse, use the gtext command:>> gtext('string')

Ï Then click on the figure for the label to be displayed.



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More than one figure

Ï You can have more than one figure window.

Ï The figure command creates a new figure window.

Ï The close command closes the current figure window.

Ï The close all command closes all the figure windows.

Ï To have more than one plot in the same figure window, use thesubplot command.>> subplot(m, n, p)

Ï The figure is split into an m by n matrix of panes.

Ï The p parameter specifies on which pane subsequentcommands will work.



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Saving and exporting figures

Ï To save a plot, use the saveas command:>> saveas(handle, filename)>> saveas(handle, filename, format)

Ï The handle refers to a particular figure.

Ï The gcf function returns the handle of the current figure.

Ï The data can be saved to different formats, e.g., 'pdf'.

Ï If the format is not included, it is deduced from the extensionof the filename.

Ï Some common formats are 'fig' for a MATLAB figure, 'bmp','jpg', and 'pdf'.

Ï To save the current figure as a MATLAB figure, type:>> saveas(gcf, 'filename.fig')



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The plot user interface

Ï Apart from the command interface, plots can be manipulatedusing the user interface (UI).

Ï Editing labels is easy using the UI.

Ï The UI makes it possible to insert other items onto a plot,including rectangles, ellipses, text labels, and arrows.

Ï Colours, fonts, and styles can be changed using the UI.

Ï Figures can be saved and exported from the UI.



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Plotting FunctionsFunction Discovery

Outline






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Plotting functions

Ï Suppose you want to plot the function:y = cos(tan(x))− tan(sin(x)) for x ∈ [1,2]

Ï This can be done using:>> x = 1 : 0.01 : 2;>> y = cos(tan(x)) - tan(sin(x));>> plot(x, y)

Ï A better way is to use the fplot command.>> fun = 'cos(tan(x)) - tan(sin(x))';>> fplot(fun, [1 2])

Ï fplot automatically chooses the point spacing.

Ï The function can be defined as a string, as above.

Ï The function parameter can also be a function handle. Forexample, to plot sin(x), type: >> fplot(@sin, [-pi pi])



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Plotting polynomials

Ï Suppose you want to plot the polynomial:y = x5 −25x3 +10x2 +70x−10 for x ∈ [−5,5]

Ï To do this, use the polyval function:>> p = [1 0 -25 10 70 -10];>> x = -5 : 0.01 : 5;>> y = polyval(p, x);>> plot(x, y)



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Discovering a functionÏ Function discovery is the process of finding a function that can

describe a set of data.Ï Some functions can be discovered by trying to draw the

function as a straight line.Ï To discover these function, try each of the scales; plot(x, y),

loglog(x, y), and so on.

Table : Functions that can be plotted as stright lines

Kind of function Form Straight-line plotLinear y(x) = mx+b plot(x, y)Power y(x) = bxm loglog(x, y)Exponential y(x) = bemx semilogy(x, y)Logarithmic y(x) = m ln(x)+b semilogx(x, y)



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Regression analysis

Ï Suppose we control x, an independent variable, and wemeasure y, a dependent variable.

Ï y is a random variable because of observational errors.

Ï The least squares method is commonly used to find the best-fitmodel.

Ï If we have k data points for a linear function y = mx+b, theleast square method minimizess the error J given by:

J =k∑

i=1(mxi +b−yi)

2

Ï The polyfit(x, y, n) function finds the coefficients of ann-degree polynomial that fits the data best in a least-squaressense.



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Finding the coefficients of a linear function

Ï Once we get a straight line, we know the function type.

Ï Interpolating values on logarithmic scales is difficult.

Ï To find the coefficients b and m, we use the linear scale.

Ï The polyfit function helps us to find the coefficients.

Ï If we get a straight line using the plot command, we have alinear function.

Ï A linear function is of the form:y = mx+b

Ï To find the polynomial coefficients of a linear function, type:>> p = polyfit(x, y, 1)

Ï m is p(1) and b is p(2).



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Finding the coefficients of a power function

Ï If we get a straight line using the loglog command, we have apower function.

Ï A power function is of the form:y = bxm

Ï Taking logs:ln(y) = m ln(x)+ ln(b)

Ï This has the form of a first-degree polynomial.

Ï To find the coefficients, we type:>> p = polyfit(log(x), log(y), 1)

Ï m is p(1) and ln(b) is p(2), that is, b is exp(p(2)).



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Finding the coefficients of an exponential function

Ï If we get a straight line using the semilogy command, we havean exponential function.

Ï An exponential function is of the form:y = bemx

Ï Taking logs:ln(y) = mx+ ln(b)


Ï To find the coefficients, we type:>> p = polyfit(x, log(y), 1)

Ï m is p(1) and ln(b) is p(2), that is, b is exp(p(2)).



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Finding the coefficients of a logarithmic function

Ï If we get a straight line using the semilogx command, we havea logarithmic function.

Ï A logarithmic function is of the form:y = m lnx+b


Ï To find the coefficients, we type:>> p = polyfit(log(x), y, 1)

Ï m is p(1) and b is p(2).



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Interpolation and extrapolation

Ï Once a function is discovered from the data, it can be used topredict conditions within the range of the data.

Ï This is called interpolation.

Ï The function can also be used to predict conditions that lieoutside the range of the data.

Ï This is called extrapolation.

Ï Extrapolation assumes that the mathematical model remainsthe same beyond the data range.



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Different Plots3-D Plots

Outline






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Plotting complex numbers

Ï The plot(x, y) command plots values of y against values ofx.

Ï The plot(y) command usually plots values of y against theirindices, that is, against

[1 2 3 . . .

].

Ï But when y contains complex values, plot(y) plots theimaginary parts against the real parts.

Ï This can also be written explicitly as:>> plot(real(y), imag(y))

Ï This is the only condition in which imaginary parts of numbersare not ignored by the plot command.



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Plotting polar plots

Ï Polar plots are 2-D plots made using polar coordinates.

Ï Polar coordinates are of the form (θ,r).

Ï θ is the angular coordinate of a point in radians.

Ï r is the radial coordinate of a point.

Ï To plot values of θ and r, type: >> polar(theta, r)Ï To specify the line style, type: >> polar(theta, r, s)Ï You can convert polar coordinates to rectangular (Cartesian)

coordinates: >> [x, y] = pol2cart(theta, r)Ï You can also convert rectangular coordinates to polar

coordinates: >> [theta, r] = cart2pol(x, y)



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Other plotsÏ stem(x, y) plots the data as stems from the x-axis

terminating in a marker.Ï stairs(x, y) draws a stairstep graph through the specified

points.Ï bar(x, y) draws a bar graph of the specified values.Ï scatter(x, y, areas) draws markers at the specified

locations. The area of each marker can be specified.Ï plotyy(x1, y1, x2, y2) produces two y-axis labels. The

y-axis label on the left is used to plot y1 against x1, and they-axis label on the right to plot y2 against x2.

Ï plotyy(x1, y1, x2, y2, fun1, fun2) lets you specify theplotting functions. To plot y1 against x1 using plot and y2

against x2 using stem, type:>> plotyy(x1, y1, x2, y2, @plot, @stem).



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3-D line plots

Ï To plot a function that varies in x, y, and z, use the plot3command.

Ï Suppose we have a parametric function:

x = e−0.07t cos t

y = e−0.07t sin t

z = t

Ï To plot this function for t ∈ [0,10π]:>> t = 0 : 0.01 : 10*pi;>> x = exp(-0.07 * t) .* cos(t);>> y = exp(-0.07 * t) .* sin(t);>> z = t;>> plot3(x, y, z)



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Meshes and surface plots

Ï Suppose we have a function of two variables, z = cos(xy).

Ï We want to plot this function for x ∈ [−3,3] and y ∈ [0,6].

Ï To generate values of x and y, we use the meshgrid function:>> [x y] = meshgrid(-3:0.1:3, 0:0.1:6);

Ï Then calculate z:>> z = cos(x .* y);

Ï Then plot using the mesh command:>> mesh(x, y, z)

Ï To draw a surface instead of a mesh:>> surf(x, y, z)

Ï Selecting a proper spacing (0.1 in our example) is important.



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Contours

Ï Topographic plots show contour lines instead of a surface.

Ï To create a contour plot, type:>> contour(x, y, z)

Ï You can draw a mesh and a contour on the same axis using:>> meshc(x, y, z)

Ï You can draw a surface and a contour on the same axis using:>> surfc(x, y, z)


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