Plotting in MATLAB. Trevor Spiteri

Transcription

1 Functions and Special Department of Communications and Computer Engineering Faculty of Information and Communication Technology University of Malta 29 February, 2008

2 Functions and Special Outline Multiple Interacting with Functions and Plotting Functions Function Discovery Special Different 3-D

3 Functions and Special Multiple Interacting with Outline Multiple Interacting with Functions and Plotting Functions Function Discovery Special Different 3-D

4 Functions and Special Multiple Interacting with The elements of a plot Plot title Legend Height of a Falling Object Versus Time Model Measurement 400 Data symbol Axis label Height (m) Grid Tick-mark label Tick mark Time (s)

5 Functions and Special Multiple Interacting with The elements of a plot explained The scale is the range and spacing of the numbers. The most-used scale is the rectilinear scale, also known as the linear scale. Another kind of scale is the logarithmic scale. Tick marks are placed on the axis to help visualize the numbers. A function is always plotted as a line, with no data symbols. When plotting data, data symbols are used, except if they are too dense. When plotting more than one curve, they must be distinguished. This distinction can be made using a legend or labels.

6 Functions and Special Multiple Interacting with A correct plot Each axis must be labelled with the name and its units. Each axis should have regularly spaced tick marks. These should not be too sparse or too dense. When plotting data points, use a data symbol such as a circle. If there are many symbols, use a dot. When plotting points generated by using a function, data symbols should not be used.

7 Functions and Special Multiple Interacting with Plotting x-y data To plot x data against y data, type: >> plot(x, y) To change the line specification, type: >> plot(x, y, s) s contains the line style specifier, the marker specifier, and/or the colour specifier. For example, for a solid line, a plus data marker, and the red colour, the specifier would be '- + r'

8 Functions and Special Multiple Interacting with Line specification string syntax Line styles Markers Colours Solid line - Plus sign (+) + Red r Dashed line -- Circle ( ) o Green g Dotted line : Asterisk ( ) * Blue b Dash-dot line -. Point ( ). Cyan c Cross ( ) x Magenta m Square ( ) s Yellow y Diamond ( ) d Black k Upward triangle ( ) ^ White w Downward triangle ( ) v Right triangle ( ) > Left triangle ( ) < Five-pointed star p Six-pointed star h

9 Functions and Special Multiple Interacting with Labels, titles, and grids The xlabel function sets the label for the x-axis. The ylabel function sets the label for the y-axis. The title function sets the plot title. The grid on command displays the grid. The grid off command hides the grid.

10 Functions and Special Multiple Interacting with function sets the axis of the plot. To make the plot a square, type: >> axis square To make the x-axis and y-axis have the same scale, type: >> axis equal 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]) To return to automatic axis, type: >> axis auto

11 Functions and Special Multiple Interacting with Scales The most commonly-used scale is the rectilinear scale, also known as the linear scale. The plot command uses the linear scale. The loglog command is the same as the plot command, except that logarithmic scales are used for both the x-axis and the y-axis. The semilogx command is the same as the plot command, except that a logarithmic scale is used on the x-axis. The semilogy command is the same as the plot command, except that a logarithmic scale is used on the y-axis.

12 Functions and Special Multiple Interacting with More than one line on the same axis To plot more than one line on the same axis, one way is to type: >> plot(x1, y1, s1, x2, y2, s2,...) Another way is to use the hold command. >> plot(x1, y1, s1) >> hold on >> plot(x2, y2, s2)... >> hold off If x and y are matrices, to plot the columns of y against the columns of x, type: >> plot(x, y)

13 Functions and Special Multiple Interacting with The legend The legend can be used to distinguish between different lines. To display a legend, use the legend command. >> legend(string 1, string 2,...) The specified strings will be used as labels. The default legend location is the top right corner. To change the location of the legend: >> legend(string 1,..., 'location', specifier) The specifier for a location inside the axis can be 'North', 'NorthEast', 'East', and so on. The default location is 'NorthEast'. The specifier for a location outside the axis can be 'NorthOutside', 'NorthEastOutside', and so on. For MATLAB to choose a place automatically, the specifier can be 'Best' or 'BestOutside'.

14 Functions and Special Multiple Interacting with Text labels Another way to distinguish between lines is to label them. To write labels, use the text command: >> text(x, y, 'string') The specified string will be placed at the given coordinates. To place a label using the mouse, use the gtext command: >> gtext('string') Then click on the figure for the label to be displayed.

15 Functions and Special Multiple Interacting with More than one figure You can have more than one figure window. The figure command creates a new figure window. The close command closes the current figure window. The close all command closes all the figure windows. To have more than one plot in the same figure window, use the subplot command. >> subplot(m, n, p) The figure is split into an m by n matrix of panes. The p parameter specifies on which pane subsequent commands will work.

16 Functions and Special Multiple Interacting with Saving and exporting figures To save a plot, use the saveas command: >> saveas(handle, filename) >> saveas(handle, filename, format) The handle refers to a particular figure. The gcf function returns the handle of the current figure. The data can be saved to different formats, e.g., 'pdf'. If the format is not included, it is deduced from the extension of the filename. Some common formats are 'fig' for a MATLAB figure, 'bmp', 'jpg', and 'pdf'. To save the current figure as a MATLAB figure, type: >> saveas(gcf, 'filename.fig')

17 Functions and Special Multiple Interacting with The plot user interface Apart from the command interface, plots can be manipulated using the user interface (UI). Editing labels is easy using the UI. The UI makes it possible to insert other items onto a plot, including rectangles, ellipses, text labels, and arrows. Colours, fonts, and styles can be changed using the UI. Figures can be saved and exported from the UI.

18 Functions and Special Plotting Functions Function Discovery Outline Multiple Interacting with Functions and Plotting Functions Function Discovery Special Different 3-D

19 Functions and Special Plotting Functions Function Discovery Plotting functions Suppose you want to plot the function: y = cos(tan(x)) tan(sin(x)) for x [1,2] This can be done using: >> x = 1 : 0.01 : 2; >> y = cos(tan(x)) - tan(sin(x)); >> plot(x, y) A better way is to use the fplot command. >> fun = 'cos(tan(x)) - tan(sin(x))'; >> fplot(fun, [1 2]) fplot automatically chooses the point spacing. The function can be defined as a string, as above. The function parameter can also be a function handle. For example, to plot sin(x), type: >> fplot(@sin, [-pi pi])

20 Functions and Special Plotting Functions Function Discovery Plotting polynomials Suppose you want to plot the polynomial: y = x 5 25x x x 10 for x [ 5,5] To do this, use the polyval function: >> p = [ ]; >> x = -5 : 0.01 : 5; >> y = polyval(p, x); >> plot(x, y)

21 Functions and Special Plotting Functions Function Discovery Discovering a function Function discovery is the process of finding a function that can describe a set of data. Some functions can be discovered by trying to draw the function as a straight line. 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 plot Linear y(x) = mx + b plot(x, y) Power y(x) = bx m loglog(x, y) Exponential y(x) = be mx semilogy(x, y) Logarithmic y(x) = m ln(x) + b semilogx(x, y)

22 Functions and Special Plotting Functions Function Discovery Regression analysis Suppose we control x, an independent variable, and we measure y, a dependent variable. y is a random variable because of observational errors. The least squares method is commonly used to find the best-fit model. If we have k data points for a linear function y = mx + b, the least square method minimizess the error J given by: k J = (mx i + b y i ) 2 i=1 The polyfit(x, y, n) function finds the coefficients of an n-degree polynomial that fits the data best in a least-squares sense.

23 Functions and Special Plotting Functions Function Discovery Finding the coefficients of a linear function Once we get a straight line, we know the function type. Interpolating values on logarithmic scales is difficult. To find the coefficients b and m, we use the linear scale. The polyfit function helps us to find the coefficients. If we get a straight line using the plot command, we have a linear function. A linear function is of the form: y = mx + b To find the polynomial coefficients of a linear function, type: >> p = polyfit(x, y, 1) m is p(1) and b is p(2).

24 Functions and Special Plotting Functions Function Discovery Finding the coefficients of a power function If we get a straight line using the loglog command, we have a power function. A power function is of the form: y = bx m Taking logs: ln(y) = mln(x) + ln(b) This has the form of a first-degree polynomial. To find the coefficients, we type: >> p = polyfit(log(x), log(y), 1) m is p(1) and ln(b) is p(2), that is, b is exp(p(2)).

25 Functions and Special Plotting Functions Function Discovery Finding the coefficients of an exponential function If we get a straight line using the semilogy command, we have an exponential function. An exponential function is of the form: y = be mx Taking logs: ln(y) = mx + ln(b) This has the form of a first-degree polynomial. To find the coefficients, we type: >> p = polyfit(x, log(y), 1) m is p(1) and ln(b) is p(2), that is, b is exp(p(2)).

26 Functions and Special Plotting Functions Function Discovery Finding the coefficients of a logarithmic function If we get a straight line using the semilogx command, we have a logarithmic function. A logarithmic function is of the form: y = mlnx + b This has the form of a first-degree polynomial. To find the coefficients, we type: >> p = polyfit(log(x), y, 1) m is p(1) and b is p(2).

27 Functions and Special Plotting Functions Function Discovery Interpolation and extrapolation Once a function is discovered from the data, it can be used to predict conditions within the range of the data. This is called interpolation. The function can also be used to predict conditions that lie outside the range of the data. This is called extrapolation. Extrapolation assumes that the mathematical model remains the same beyond the data range.

28 Functions and Special Different 3-D Outline Multiple Interacting with Functions and Plotting Functions Function Discovery Special Different 3-D

29 Functions and Special Different 3-D Plotting complex numbers The plot(x, y) command plots values of y against values of x. The plot(y) command usually plots values of y against their indices, that is, against [ ]. But when y contains complex values, plot(y) plots the imaginary parts against the real parts. This can also be written explicitly as: >> plot(real(y), imag(y)) This is the only condition in which imaginary parts of numbers are not ignored by the plot command.

30 Functions and Special Different 3-D Plotting polar plots Polar plots are 2-D plots made using polar coordinates. Polar coordinates are of the form (θ,r). θ is the angular coordinate of a point in radians. r is the radial coordinate of a point. To plot values of θ and r, type: >> polar(theta, r) To specify the line style, type: >> polar(theta, r, s) You can convert polar coordinates to rectangular (Cartesian) coordinates: >> [x, y] = pol2cart(theta, r) You can also convert rectangular coordinates to polar coordinates: >> [theta, r] = cart2pol(x, y)

31 Functions and Special Different 3-D Other plots stem(x, y) plots the data as stems from the x-axis terminating in a marker. stairs(x, y) draws a stairstep graph through the specified points. bar(x, y) draws a bar graph of the specified values. scatter(x, y, areas) draws markers at the specified locations. The area of each marker can be specified. plotyy(x1, y1, x2, y2) produces two y-axis labels. The y-axis label on the left is used to plot y 1 against x 1, and the y-axis label on the right to plot y 2 against x 2. plotyy(x1, y1, x2, y2, fun1, fun2) lets you specify the plotting functions. To plot y 1 against x 1 using plot and y 2 against x 2 using stem, type: >> plotyy(x1, y1,

32 Functions and Special Different 3-D 3-D line plots To plot a function that varies in x, y, and z, use the plot3 command. Suppose we have a parametric function: x = e 0.07t cost y = e 0.07t sint z = t 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)

33 Functions and Special Different 3-D Meshes and surface plots Suppose we have a function of two variables, z = cos(xy). We want to plot this function for x [ 3,3] and y [0,6]. To generate values of x and y, we use the meshgrid function: >> [x y] = meshgrid(-3:0.1:3, 0:0.1:6); Then calculate z: >> z = cos(x.* y); Then plot using the mesh command: >> mesh(x, y, z) To draw a surface instead of a mesh: >> surf(x, y, z) Selecting a proper spacing (0.1 in our example) is important.

34 Functions and Special Different 3-D Contours Topographic plots show contour lines instead of a surface. To create a contour plot, type: >> contour(x, y, z) You can draw a mesh and a contour on the same axis using: >> meshc(x, y, z) You can draw a surface and a contour on the same axis using: >> surfc(x, y, z)