Data Analysis for the Geosciences

Michael W. Liemohn, University of Michigan, USA

• Preface xvAcknowledgments xxiAbout the Companion Website xxiii1 Assessment and Uncertainty: Examples and Introductory Concepts 11.1 Chicken Little, Amateur Meteorologist 21.2 Uncertainty Ascribes Meaning to Values 31.3 Significant Figures 31.4 Types of Uncertainty 71.5 Example: Finding Saturn’s Moons 91.6 Comparing Two Numbers: Are They Measuring the Same Value? 111.6.1 Distributions of Number Sets 121.6.2 The Gaussian Distribution 131.6.3 Testing a Specific Value within a Data Set: The z Test 141.6.4 Comparing Two Values Revisited 181.7 Use and Misuse of Statistics 191.8 Example: Solar Wind Density and Space Weather 201.9 Uncertainty and the Scientific Method 221.10 Further Reading 241.11 Exercises in the Geosciences 262 Plotting Data: Visualizing Sets of Numbers 272.1 Plotting One- Dimensional Data 272.1.1 What Makes a Good Plot? 292.1.2 Exploratory Versus Explanatory Plot Styles 312.2 Example: Earth’s Magnetic Field Strength 332.3 Probability Distributions— The Histogram 352.4 Plotting Two Data Sets Against Each Other 392.4.1 Overlaid Histograms 392.4.2 The Scatterplot 402.4.3 The Box Plot 422.4.4 The Box- and- Whisker Scatterplot 432.4.5 The Running Average Plot 442.5 Example: Temperature and Carbon Dioxide 482.6 Scientific Visualization: A Sampling from the Literature 502.6.1 A Very Brief History of Visualization 512.6.2 Good Modern- Day Example Visualizations 532.7 Visualization Best Practices 582.7.1 Levels of Abstraction 582.7.2 A Process for a Good Graphic 612.7.3 Types of Colorblindness 632.7.4 Color Scales 632.8 Further Reading 652.9 Exercises in the Geosciences 673 Uncertainty Analysis: Techniques for Propagating Uncertainty 693.1 Propagating Uncertainty 693.1.1 Calculating Uncertainty with One Independent Variable 693.1.2 Calculating Uncertainty with Two Independent Variables 703.1.3 Calculating Uncertainty with Many Independent Variables 723.2 Example: Atmospheric Density 723.2.1 The Hydrostatic Equilibrium Approximation 723.2.2 One Independent Variable 733.2.3 Two Independent Variables 743.2.4 Many Independent Variables 743.3 Fractional and Percentage Uncertainties 753.4 Special Cases of Uncertainty Propagation 773.4.1 Addition and Subtraction 773.4.2 Multiplication and Division 783.4.2.1 Multiplication of Two Parameters 783.4.2.2 Uncertainty of Air Pressure 793.4.2.3 Division with Correlated Variables 803.4.2.4 Multiplication and Division with Independent Variables 813.4.3 Power Laws 823.4.4 Exponentials and Logarithms 823.4.4.1 Exponential Functions 833.4.4.2 Logarithmic Functions 843.4.5 Trigonometric Functions 843.5 Stepwise Uncertainty Propagation 853.6 Example: Planetary Equilibrium Temperature 873.7 Multistep Processing 903.8 Final Advice on Uncertainty Propagation 913.9 Further Reading 933.10 Exercises in the Geosciences 934 Centroids and Spreads: Analyzing a Set of Numbers 954.1 Quantitatively Describing a Data Set: The Centroid 954.1.1 Three Versions of Mean 964.1.2 More Centroids: Median and Mode 984.1.3 Histograms and the Arithmetic Mean 994.2 Quantitatively Describing a Data Set: Spread 1004.2.1 Measures of Spread: Standard Deviation and Mean Absolute Difference 1004.2.2 Another Measure of Spread: Quantiles 1024.2.3 Spread Via Full Width at Half Maximum 1064.2.4 Spread as an L- p Norm 1074.2.5 Sample Versus Population 1084.3 Random and Systematic Error of a Data Set 1094.4 Which Centroid and Spread to Use and Other Tidbits of Advice 1114.5 Standard Deviation of the Mean 1124.6 Counting Statistics 1134.7 Example: Galactic Cosmic Rays 1164.8 Further Reading 1194.9 Exercises in the Geosciences 1205 Assessing Normality: Tests for Assessing the Gaussian Nature of a Distribution 1235.1 Histogram Check 1245.2 Comparing Centroid and Spread Measures 1265.3 Skew 1285.4 Kurtosis 1305.5 The Chi- Squared Test 1325.6 The Kolmogorov–Smirnov Test 1375.7 Example: pH in a Lake 1395.8 Asymmetric Uncertainties 1425.9 Outliers— Tests for a Single Data Value 1445.10 Combining Centroid and Spread: The Weighted Average 1465.11 Example: pH in a Lake Redux 1485.12 Further Reading 1495.13 Exercises in the Geosciences 1506 Correlating Two Data Sets: Analyzing Two Sets of Numbers Together 1536.1 Comparing Two Number Sets 1536.1.1 Chi- Squared and Kolmogorov–Smirnov Tests 1546.1.2 The Student’s t Test 1556.1.3 The Welch’s t Test 1566.2 Linear Correlation 1576.2.1 Covariance of Two Data Sets 1586.2.2 Pearson Linear Correlation Coefficient 1616.2.3 Spearman Rank- Order Correlation 1636.2.4 Correlation with Logarithms 1676.3 Example: Atmospheric Ozone and Temperature 1686.4 Uncertainty of R 1726.4.1 The Jackknife Method 1726.4.2 The Bootstrap Method 1736.4.3 Uncertainty of R for the Ozone- Temperature Example 1756.5 Correlation and Causation 1776.6 Further Reading 1786.7 Exercises in the Geosciences 1797 Curve Fitting: Fitting a Line between Two Sets of Numbers 1817.1 Linear Regression 1817.1.1 Obtaining A and B 1817.1.2 Uncertainties on A and B 1857.1.3 The Zero- Intercept Special Case 1867.1.4 Weighted Linear Fitting 1877.2 Testing a Linear Fit 1887.3 Example: Human- Induced Seismicity 1917.4 Nonlinear Fitting 1947.4.1 Polynomial Fitting 1947.4.2 Generalized “Linear Coefficient” Fitting 1967.4.3 Exponential Fitting: Linearizing the Dependence on Coefficients 1977.4.4 Piecewise Linear Fitting 1987.4.5 Advice about Curve Fitting 1997.5 Example: The Ozone Hole 2007.6 Iterative Curve Fitting 2037.6.1 One- Dimensional Iterative Curve Fitting 2037.6.2 Multidimensional Iterative Curve Fitting 2057.6.3 Gradient Descent Curve Fitting 2087.7 Final Thoughts on Curve Fitting 2097.8 Further Reading 2107.9 Exercises in the Geosciences 2108 Data- Model Comparison Basics: Philosophies of Calculating and Categorizing Metrics 2138.1 Example Model: River Flow Rate 2138.2 What Is a Model? 2148.3 Visualizing Observed and Modeled Values Together 2178.3.1 Scatterplots of Data and Model Values 2178.3.2 The 2D Histogram Plot 2198.3.3 Overlaid Histogram Plots 2218.3.4 Cumulative Probability Distribution Plots 2228.3.5 Quantile–Quantile Plots 2248.4 Example: Total Solar Irradiance 2268.5 A Diverse Zoo of Metrics 2298.5.1 The Primary Categories of Metrics 2308.5.2 Skill 2318.5.3 Metrics Categories Based on Subsetting 2348.6 The Concept of Model “Goodness of Fit” 2358.7 Application Usability Levels 2368.8 Designing a Meaningful Data- Model Comparison 2378.9 Further Reading 2398.10 Exercises in the Geosciences 2409 Fit Performance Metrics: Data- Model Comparisons Based on Exact Observed and Modeled Values 2439.1 What Is Fit Performance? 2449.2 Running Example: Dst and the O’Brien Model 2459.3 Accuracy 2509.3.1 The Big Three of Accuracy: MSE, RMSE, and MAE 2519.3.2 Neglecting Degrees of Freedom 2539.3.3 Normalizing the Accuracy Measure 2569.3.4 Percentage Accuracy Metrics 2579.3.5 Choosing the Right Accuracy Metric 2619.4 Bias 2629.4.1 Mean Error 2629.4.2 Percentage Bias 2659.5 Precision 2669.5.1 Modeling Yield 2669.5.2 Definitions of Precision Using Standard Deviation 2689.6 Association 2689.6.1 Correlation Coefficient 2699.6.2 Nonlinear Association Metrics 2709.7 Extremes 2729.7.1 Extremes of the Cumulative Probability Distribution 2729.7.2 Using Skew and Kurtosis for an Extremes Assessment 2769.8 Skill 2789.8.1 Prediction Efficiency 2789.8.2 Other Options for Fit Performance Skill 2799.9 Discrimination 2819.10 Reliability 2839.11 Summarizing the Running Example 2869.12 Summary of Fit Performance Metrics 2879.13 Further Reading 2919.14 Exercises in the Geosciences 29210 Event Detection Metrics: Comparing Observed and Modeled Number Sets When Only Event Status Matters 29510.1 Defining an Event 29610.2 Contingency Tables 29910.3 Data- Model Comparisons with Events 30110.4 Running Example: Will It Rain? 30310.5 Significance of a Contingency Table 30710.6 Accuracy 31010.7 Bias 31110.8 Precision 31310.9 Association 31410.9.1 Odds Ratio 31510.9.2 Odds Ratio Skill Score 31610.9.3 Matthews Correlation Coefficient 31710.10 Extremes 31710.11 Skill 32110.11.1 Heidke Skill Score 32110.11.2 Peirce and Clayton Skill Scores 32310.11.3 Gilbert Skill Score 32410.12 Discrimination 32510.13 Reliability 32610.14 Summarizing the Running Example 32710.15 Summary of Event Detection Metrics 32810.16 Further Reading 33010.17 Exercises in the Geosciences 33111 Sliding Thresholds: Event Detection Metrics with a Variable Event Identification 33311.1 Sliding the Event Identification Thresholds 33411.2 Sweeping the Modeled Threshold 33711.3 Sweeping the Data Threshold 34011.4 Sweeping Both Thresholds Simultaneously 34211.5 Metric- Versus- Metric Curves 34411.5.1 ROC Curves 34411.5.2 Alt- ROC Curves 34611.5.3 STONE Curves 34711.6 Application of Sliding Thresholds to the Geophysical Running Examples 34911.6.1 Event Definitions for the Running Examples 34911.6.2 Metric- Versus- Modeled Threshold Curves for the Running Examples 35211.6.3 Metric- Versus- Observed Threshold Curves for the Running Examples 35511.6.4 Metric- Versus- Simultaneous Threshold Sweep Curves for the Running Examples 35711.6.5 Metric- Versus- Metric Analysis for the Running Examples 35911.7 The Power of Sliding Thresholds 36211.8 Further Reading 36411.9 Exercises in the Geosciences 36512 Applications of Metrics and Uncertainty: Final Advice and Introductions to Advanced Topics 36712.1 Choosing the Right Set of Metrics 36712.1.1 Metrics for Fit Performance Assessment on Gaussian Distributions 36812.1.2 Metrics for Fit Performance Assessment on Non- Gaussian Distributions 36912.1.3 Metrics for Event Detection Assessment 37212.2 Combining Metrics for Robust Data- Model Comparisons 37412.2.1 The Accuracy–Bias–Precision Trifecta 37412.2.2 The Accuracy–Association Connection 37612.2.3 The Association–Extremes Linkage 37712.2.4 Expanding Our Understanding of Skill 37812.2.5 Using Discrimination and Reliability Together 37912.3 Uncertainty on Metrics 38012.4 Uncertainty on Fit Performance Metrics for the Dst Running Example 38112.5 A Recipe for Robust Comparisons 38512.6 Metrics and Decision- Making 38712.6.1 Choice Combination Statistics 38812.6.2 Example: Spacecraft- Charging Model 39012.7 Additional Advanced Topics 39212.7.1 Periodicity Analysis 39212.7.2 Time- Lagged Analysis 39312.7.3 Additional Tests 39412.7.4 Multidimensional Data Analysis 39412.7.5 Multidimensional Data- Model Comparisons 39612.7.6 Uncertainty Quantification 39712.7.7 Design of Experiments 39812.7.8 Geographical Information System (GIS) Analysis 39812.7.9 Machine Learning 39912.8 Uncertainty and the Scientist 40012.9 Further Reading 40212.10 Exercises in Geoscience 406Index 407