G302 - Graphing Techniques Applied to Earth Science
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1 G32 - Graphing Techniques Applied to Earth Science I. Graphing Review A. Purpose of graphing - to plot data and allow general relationships to be visualized. B. Basic X-Y (scatterplot) graph 1. Axis a. Y axis = vertical axis (ordinate) b. X axis = horizontal axis (abscissa) C. Graph Trends (see attached figures) 1. Linear Increase / Decrease 2. Constant 3. Parabolic (curvilinear) Increase / Decrease Linear Increase Constant Parabolic D. Determining Slopes of Lines 1. slope of any line on a graph = rise / run = (Y 2 - Y 1 ) / (X 2 - X 1 ) Positive Slope Negative Slope Zero Slope Undefined Slope 1
2 E. Linear Relationships - represented by uniform change of y relative to x 1. General Equation for Line: Y = mx + B where Y = variable on ordinate axis, X = variable on abcissa, m = slope of line, B = y-intercept (value on y axis where line intercepts it) a. Sloping line downward to right = negative slope b. sloping line downward to left = positive slope F. Relationships other than linear 1. Quadratic Equation a. form: Y = ax 2 + bx + C where y is a function of x, while a,b, and c are constants 2. Polynomial Functions of higher order (expanded quadratic equation) a. e.g. form: Y = ax 4 + bx 3 + cx 2 +dx + e 3. Power Functions a. form: Y = ax b equivalent to: ln Y = b(ln X) + a 4. Log Function a. form: Y = b(ln X) + a 5. Exponential Function a. form: ln Y = bx + a equivalent to: Y = ax bx G. Best-Fit Functions 1. process of fitting functions to data using regression analysis a. regression - curve fitting process that maximizes the trend of the fitted curve with the distribution of data (1) Residuals - difference between fit Y values and actual Y values of data at given X values (2) Coefficient of Determination r 2 = 1 - SSe / (SSe + SSr) where r = coefficient of determination, SSe = sum of squares of all residual values, SSr = sum of squares of the difference between all actual Y values and the fit Y value at each X location where the data point occurs (a) Interpretation: i) r 2 values between.7-1. = good fit of function to data distribution ii) r 2 values between.5-.7 = moderate to poor fit iii) r 2 values <.5 = poor fit to data 2
3 II. More on Logarithms A. logarithms = inverse of exponential functions 1. examples a. Y = 1 3 = 1, then log 1 (1) = 3 b. Y = 1-2 =.1, then log 1 (1-2 ) = -2 c. If Y = 1 n, then log 1 (1 n ) = n d. If Y = X n, then log X (X n ) = n B. Uses for logarithms 1. re-arranging equations containing exponential functions 2. reducing exponential functions / curves to straight lines 3. compressing large data ranges C. Natural Logarithms - logs to the base "e", where e = In-Class Example: Logs are used in the classification of sediment grain sizes by use of the following equation: φ = -log 2 (d) where φ = "phi" (greek letter f), and d = diameter of sediment grain in millimeters If a grain has a diameter of 8 mm, what is it's corresponding "phi" size? show all work. If a grain has a diameter of 4 mm, what is it's corresponding "phi" size? show all work. If a grain has a diameter of.2 mm, what is it's corresponding "phi" size? show all work. III. Overview of Graph Types A. X-Y Scatterplot Graphs 1. linear axes 2. log-linear axes 3. log-log axes B. Bar Graphs C. Triangular Graphs (three end-member composition plots) D. Polar-Azimuthal Graphs (directional scatter plots) E. Rose Plots (directional histograms) 3
4 Example Linear Graph Y = 2X Web Hits Per Month Expected Hits > Example Bar Graph Jan-98 Jul-98 Jan-99 Jul-99 Jan- Jul- Jan-1 4
5 Log Fit Polynomial Fit Linear Fit Example Log Equation Y = 4Ln(X) Plotted on Linear Axes
6 Example Log Equation Y = 4Ln(X) Plotted on Semi-Log Axes Example Log Equation Y = 4Ln(X) Plotted on Log-Log Axes
7 Example Polynomial Equation Y = 2X 4 + 3X 3 + 5X 2 + 4X Example Power Equation Y = 5X.4 1 Plotted on Log-Log Axes
8 Example Power Equation Y = 5X Example linear regression of scatter plot data Linear Regression Results.4.2 Fit 1: Linear Equation Y = * X Number of data points used = 116 Average X = Average Y = Residual sum of squares = Regression sum of squares = Coef of determination, R-squared = Residual mean square, sigma-hat-sq'd =
9 Example Quadratic Equation Y = 2X2 + 4X (Q,F,L) Proportions 8 8 Lithoclasts 4 (16,4,44) 6 (13,34,53) (2,22,76) (12,15,73) (29,48,23) (35,34,31) (45,3,25) (48,21,31) (55,15,3) 6 Feldspar (13,2,85) (74,2,24) Quartz 9
10 Example Rose Diagram Polar Plot Example Polar Plot 1
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