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1. Introduction
Contents
Table of Contents
1. Introduction
1.1 Scope
1.2 Accompanying material
1.3 Objectives
2. Measurement Uncertainty
2.1 General comments
2.2 Classification of error sources
2.3 More terminology
2.4 The ASME measurement-uncertainty formulation
2.5 Propagation of uncertainty estimates
2.6 Monte Carlo techniques
3. Probability Distribution Functions
3.1 Introductory comment
3.2 Gaussian or normal distribution
3.3 The binomial distribution
3.4 Poisson distribution
3.5 Student's t distribution
3.6 Confidence intervals
4. The Method of Maximum Likelihood
4.1 Basis
4.2 Applications
4.2.1 Weighted averages
4.2.2 Mean of the Poisson probability distribution function
4.2.3 Mean of the binomial probability distribution function
4.2.4 An example: Fitting to CCN measurements
4.2.5 Maximum-likelihood estimation in cases with correlated errors
4.2.6 Estimating exponential-decay time constants
4.3 Application to experimental design
4.4 Relationship to the method of least squares
5. Least-Squares Methods of Fitting Functions to Data
5.1 General discussion and formulas
5.2 Fitting a straight line to observations
5.3 Fitting functions linear in parameters
5.4 Fitting an arbitrary function, not linear in the parameters
5.5 Fitting subject to constraints
5.6 Fitting to minimize the distance of points from a line
6. Linear Regression Analysis
6.1 Simple linear regression
6.2 Effects of measurement errors
6.3 Linear regression with several independent variables
7. Hypothesis testing
7.1 Purpose
7.2 The Chisquare distribution
7.3 The F-test
7.4 Use of the F test in multiple-regression analysis
7.5 Hypothesis testing by likelihood ratios
8. Spectral Analysis
8.1 Introduction
8.2 The Fourier series
8.3 The Fourier integral
8.4 The variance spectrum
8.5 The autocorrelation and autocovariance functions
8.6 Relationship of autocovariance function to variance spectrum
8.7 Plot conventions
8.8 Examples
8.8.1 Hourly measurements of the pressure
8.8.2 Hourly measurements of temperature
8.8.3 Pure random process
8.8.4 Effect of a filter
8.8.5 The autoregressive process
8.9 Aliasing
8.10 Effect of finite sample length
8.11 Removing the trend and tapering the ends of a time series
8.12 Some statistical characteristics of estimators
8.13 The Fast Fourier Transform (FFT)
8.14 The cospectrum
8.15 The "maximum-entropy" method of spectral estimation
9. Numerical Methods
9.1 Introduction
9.2 The Taylor series
9.3 Finite-difference derivatives
9.4 Interpolation and extrapolation
9.4.1 Finite-difference interpolation formulas
9.4.2 Lagrange interpolation
9.4.3 Whittacker's interpolation formula
9.4.4 The cubic spline
9.5 Roots of equations
9.5.1 Newton's method
9.5.2 Interpolation
9.5.3 Other iterative procedures
9.6 Solution of simultaneous equations
9.6.1 Gauss-Seidel iteration
9.6.2 The Newton-Raphson method
9.7 Techniques for numerical integration
10. Experimental Design
10.1 Introduction
10.2 Components in experimental design
10.3 Exploratory vs confirmatory experiments
10.4 Some model experimental designs
10.4.1 Target-control designs
10.4.2 Randomized target design
10.4.3 Non-parametric tests and rerandomization
10.5 Some dangers in experimental design
10.5.1 Unrecognized causes
10.5.2 Climatology
10.5.3 Type-I and Type-II errors
10.5.4 Effects of natural variability
11. Special topics
11.1 Introduction
11.2 Using units in equations
11.3 Nomograms
11.4 Transforming distribution functions
11.5 Moments and associated uncertainties
11.6 Log-normal distributions
11.7 The Maximum-Entropy Approach to Data Analysis
11.7.1 Fundamentals
11.7.2 Application to spectral analysis
11.7.3 Image reconstruction
12. Four-Dimensional Data Assimilation
12.1 Introduction
12.2 Commonly Used Analysis Techniques from 1950 to the late 1980s
12.2.1 Surface fitting
12.2.2 Empirical linear interpolation
12.3 Recently Developed Analysis Techniques
12.3.1 Adaptive filtering
12.3.2 Bayesian approach
12.3.3 Nudging
13. Dynamic Retrievals and Related Strategies
13.1 Introduction
13.2 Doppler Radar Analysis
13.3 Dynamic Retrieval
13.4 Applications of Dynamic Retrieval
13.4.1 Boundary layer investigations
13.4.2 Dynamics of precipitating convection and downdraft forcing
13.4.3 Dynamics of frontal circulations
13.5 Three-Dimensional Extensions of the Dynamic Retrieval Concept
13.5.1 Combined use of the three equations of motion
13.5.2 Incorporation of the thermodynamic energy equation
13.6 Microphysical retrievals
13.7 Single Doppler Radar Retrieval
NCAR Advanced Study Program
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