Plotting and Regression

PLOTTING, POLYNOMIALS, AND FIND

Key Functions and Terminology

  • PLOT: Used for creating a 2D line plot of an equation or data.

  • SCATTER: Used for creating a scatter plot of data points.

  • ROOTS: Calculates the roots of a polynomial given its coefficients.

  • POLYVAL: Evaluates a polynomial for given x-values.

  • POLYFIT: Fits a polynomial to a data set using least squares.

  • FIND: Identifies positions of specific values in a vector or matrix.

Page 2: Basic Plotting Syntax

  • Two built-in plotting functions:

    • plot(x,y): For plotting a line plot.

    • scatter(x,y): For creating scatter plots.

  • Requirements:

    • x and y must be vectors with equal length.

Page 3: Help Documentation

  • Access help documentation via:

    • help plot or help scatter for formatting syntax.

  • Remember:

    • Familiarize yourself with using resources instead of memorization.

Page 4: Related Plotting Functions

  • figure: Opens a new figure window.

    • figure(n): Opens a figure window numbered "n".

  • hold on: Allows additional plots on the current figure.

  • hold off: Stops holding the current plot (replacement occurs on next plot command).

Page 5: Formatting Plot Elements

  • axis([xMIN xMAX yMIN yMAX]): Sets the limits for the axes.

  • title('string'): Adds a title to the plot.

  • xlabel('string') and ylabel('string'): Adds labels for x and y axes respectively.

  • legend('String 1', 'String 2', ...): Adds a legend for different data sets displayed.

Page 6: Plot Formatting Essentials

  • Minimum requirements for proper plots:

    • Appropriate type of data (theoretical vs. experimental).

    • Correct axes values for data display.

    • Axes labels including name and units.

    • Legend if multiple data sets are included.

Page 7: Polynomials in MATLAB

  • Polynomials are represented by coefficient vectors in MATLAB.

  • Example polynomial: 8x^4 + 3x^3 + 5x + 29 is represented as:

    • p = [8 3 0 0 5 29].

Page 8: Finding Roots of Polynomials

  • ROOTS syntax: R = roots(P).

    • R is the output vector of roots.

    • P is the coefficients vector.

  • Example:

    • C_VEC = [1 -5 -6]; yields roots R = [3.0000 2.0000].

Page 9: Evaluating Polynomials

  • POLYVAL syntax: Y = polyval(P,x).

    • Y= output y-values for polynomial defined by coefficients P for x-values in x.

    • polyval = polyval function

    • P = vector that contains coefficients in order

    • x = vector of x-values where you are calculating the corresponding y-values

    • Select enough x-values for smooth plotting.

Page 10: Fitting Polynomials to Data

  • POLYFIT syntax: P = polyfit(x,y,n).

    • P holds the coefficients for the fitted polynomial

    • polyfit = polyfit function

    • x= vector containing the independent variable data

    • y = vector containing the dependent variable

    • n is the polynomial order

  • Significance:

    • Allows estimation of values between collected data points.

Page 11: Using POLYFIT

  • Steps to use POLYFIT:

    1. Plot the data initially.

    2. Determine the polynomial order based on data trends.

    3. Apply polyfit to get coefficients.

    4. Plot the polynomial fit to evaluate accuracy.

  • Aim for using the lowest polynomial order to prevent overfitting.

Page 12: Example Polynomial Fits

  • Example outputs for various orders using POLYFIT:

    • First Order: FIRST = polyfit(x,y,1) results in coefficients: 4.4108, 45.6319.

    • Second Order: SECOND = polyfit(x,y,2) results in coefficients: -0.3686, -1.4869, 61.7277.

    • Third Order: THIRD = polyfit(x,y,3) results in coefficients: 0.0594, 1.0559, -1.5462, 0.4738.

Page 13: Finding Elements in Vectors

-Returns the index positions where specific values in a vector or matric occur

-logical operators used are <,>,==,~=,>=,<=

  • FIND syntax:

    • Syntax #1: INDEX=find(X) for linear indices.

    • Syntax #2: [R,C]=find(X) for row and column indices.

  • Examples:

    • A>10, A==10, A>10 & A<20 are logical expressions evaluated to find positions.