Index
Note: Page numbers followed by “f” indicate figures, and “t” indicate tables.
A
afun( ) function 149–150
commands for 5–6
Fundamental Theorem of 158
linear 7
of random variables 41
Algebraic eigenvalue problem 168
Aliased datasets 119
Amplitude 113–114
peak, significance of 284–289
polynomial 240–241
precomputation of function 253
small number 242–245
and Taylor series 240–241
value of 239–240
areaofcircle( ) function 301
Arithmetic, commands for 5–6
Artifacts 225
Artificial neural networks 254–255
elements order in 180
sample order in 179–180
singular values of 171–172
factor analysis and 172
computing by hand 193
convolution and 193–194
cross-correlation function generalized from 195
Fourier transform of, power spectral density and 194
function 189t
interpolation and prior information of 228
of smoothing filters on time series datasets 201–202
matrix 191
time series dataset 192
autofun( ) function 199
Auxiliary parameters in linear models 70
Axis xy command 29
axis( ) function 29
B
Backslash operator 84
Backward in time 144
Bad data 85
Bayes Theorem 52
probability and 52–53
Biconjugate gradient method 106
Binned data 42
Binomial theorem 159
Boxcar filter 203
Boxcar function 215
Bracketing observations 225
Bugs 17
C
CAC dataset See Climate Analysis Center dataset
questions for 275–279
statistics for 276t
Cascades, filter 158–161
Causality 140
Central Limit Theorem 48
Character string 14
Characteristic values 168
Characteristic vectors 168
Chi-squared test 274
Clear all command 296
Clipping vectors 12
complex 123
filter 76
time series datasets and 141
Coherence 189t
notes on 306
Color bar 26
black-and-white 31
Column vectors 7
of its row vectors 74–75
input 167–168
output 167–168
Command Window navigation 4–5
Commands 3
for arithmetic and algebra 5–6
axis xy 29
clear all 296
conditional 11–12
function 107
Complex coefficients 123
Complex conjugation 123
Complex exponentials 122–124
with Fourier series 123
with positive and negative frequencies 122
ordering 124
Complex least squares 299–300
Complicated text files, reading 297–298
Conditional commands 11–12
confusion with 52
Conditional probability density function 95
computing 56
of mean 82
conv( ) function 147
autocorrelation and 193–194
cross-correlation function compared to 195
discrete 145
filtering as 144–145
integral 144
alternative form of 145
Convolution theorem 202–203
Convolve 147
Coordinated Universal Time (UTC) 296–297
as covariance 187–192
of joint probability density function 190–191
Counts 25–26
correlation as 187–192
error propagation and 88
of datasets 189t
posterior 93
prior 93
prior information on model parameters and 100–101
probability and 56–58
scatter plots as estimate of 188f
autocorrelation generalizing to 195
convolution compared to 195
Fourier transform of 195
zero-lag 213
Cross-spectral density 195
Cubic polynomial 226
natural 227
Cumulative sum (running sum) 43
Curve fitting 70–72
polynomials and 70–71
Cut and paste, avoiding errors with 18
D
Damped least squares 99–100
prior information and 147–148
of smoothness 148–149
Darcy’s law 108
interpolation problems with 224
probability and 39
purpose of 67–68
in single software environment 1–2
theory compared to practice for 1
vectors and matrices in 7
Data format conversion scripts 298
Data kernel 68
of its row vectors 74–75
discrete Fourier 302–303
grey-shaded plot of 73f
aliased 119
bad 85
covariance matrix of 189t
examining 21–27
expectations for 22
in Microsoft Excel 2
originality and 21–22
origins of 22
plotting 14–16
overlay for 29
side-by-side 29
populations of 37
prediction of 68–69
reality checks for 23
repeated 290
resampling, with duplications, 235 291
of sinusoidal oscillations 115–122
synthetic 109–110
techniques for 1
saving to 16–17
time series 34–35
autocorrelation 192
filter coefficients and 141
polynomials from 158
power in 135
smoothing filters on, autocorrelation function of 201–202
smoothing filters on, power spectral density of 202
Date numbers 296
Degrees of freedom 84
effective 279
DelaunayTri( ) function 233–234
Derivation of generalized least squares 301
Derived quantities 41
calculating 168
Jacobian 59
Deuterium 39–40
Deviatoric quantities 100
DFT See Discrete Fourier transform
diag( ) function 12
Difference due to random variation 269–270
Directories See Folders
Discharge rate 32
datasets segregated by 33
Discrete convolution 145
Discrete Fourier data kernel 302–303
Diurnal oscillations 25
Division-by-zero error 72
E
Edge effects 200
Effective degrees of freedom 279
Eigenvalues 168
algebraic eigenvalue problem 168
distinct 169
Eigenvectors 168
perpendicular 169
Atlantic Rock dataset order of 180
counts 25–26
in factors 166–167
high variance in 173
squares of 173
terminology of 166
of vectors and matrices 8–9
loop for 11–12
Error vector 77
covariance and behavior of 88
cut and paste avoiding 18
division-by-zero 72
examining 77–79
observational 84
for outliers 77
in prior information 93
logarithm of 79–80f
shape of 88
Euler’s formulas 122
eye( ) function 148
F
Factor analysis 37
Factor loadings 166
Factor matrix, rows of 167
elements in 166–167
high variance in 173
squares of 173
minimum number of 167–171
spiky 172–176
minerals and 173
terminology of 166
Fast Fourier transform (FFT) 122
fclose( ) function 297–298
FFT See Fast Fourier transform
fft( ) function 124
fgetl( ) function 297–298
Figure Window zoom feature 24
filterfun( ) function 149–151
boxcar 203
cascades 158–161
coefficients 76
time series datasets and 141
as convolution 144–145
design of 207–208
inefficiency and 149–150
Infinite Impulse Response (IIR) 204–205
inverse 158–161
Fourier transform and 160
z-transform of 158–159
length-two 159
minimum phase 159
polynomials and 157–158
principle of least squares estimation of 198–200
problems solved with 145–152
recursion and 161–162
three-point 200
on time series datasets, autocorrelation function of 201–202
on time series datasets, power spectral density of, 182 202
Finite Impulse Response (FIR) 161–162
Fisher-Snedecor F-probability density function 273
fliplr( ) function 12
Floating-point placeholders 36
navigating 4–5
fopen( ) function 297–298
For loops 11–12
nested 35
omitting 12
Format string placeholders 35–36
Forward in time 144
Fourier analysis 72
grey-shaded plot of 73f
Fourier data kernel, discrete 302–303
Fourier series 115
complex exponentials with 123
Fourier transform compared to 125–126
linear equation form of 119
of autocorrelation, power spectral density and 194
of cross-correlation function 195
fast 122
Fourier series compared to 125–126
inverse filters and 160
lessons learned from 125–126
manipulation of 126
phase ramp and 129–130
power spectral density and 136
window function before 215–216
F-probability density function 273–274
Frequency 113–114
complex exponentials with positive and negative 122
frequencies higher than 117–119
ordering 123–124
Function command 107
Function handle 107
Functions See also Probability density function
autocorrelation 189t
interpolation and prior information of 228
of smoothing filters on time series datasets 201–202
boxcar 215
autocorrelation generalizing to 195
convolution compared to 195
Fourier transform of 195
zero-lag 213
generalized 127
time arithmetic 296
time-variable 179–184
before Fourier transforms 215–216
optimal 217–221
Fundamental Theorem of Algebra 158
Future, predicting 154–156
G
modified principle of least squares and 92
filling in 93
smoothness information for 102–103
Generalized function 127
Generalized least squares 98–100
derivation of 301
for estimation 223
ginput( ) function 24–25
Global Positioning System (GPS) 296–297
Global variables 151
GPS See Global Positioning System
Gradient method 250–253
Graphics, MatLab 28–32
Grey-shaded matrix 32
in MatLab 78–79
griddata( ) function 232
H
Hermitian transpose 299–300
Hertz 113–114
hist( ) function 25–26
Histogram 26f
computing 25
rate information and 32–34
one-sided 275
with probability density function 272–274
scenario for 274–280
two-sided 275
I
ifft( ) function 124
IIR See Infinite Impulse Response
Ill-conditioned matrix 92
imagesc( ) function 32
of heat flow 146f
Normal curve proportional to 146
problematic 146–147
Infinite Impulse Response (IIR) 161–162
Information See Prior information
Input column vectors 167–168
Installing MatLab 3
Integral convolution 144
alternative form of 145
International Earth Rotation and Reference Systems Service (IERS) 296–297
Interpolant 224
data analysis problems of 224
Kriging 228–231
prior information of autocorrelation function for 228
spline 232
traditional approach to 224
triangular meshes used in 232–233
Inverse discrete Fourier transform 115
Fourier transform and 160
z-transform of 158–159
Iterative least squares 247–248
J
Joint probability density function 53–56
correlation of 190–191
in MatLab 53–54
univariate probability density function from 54f
K
Kernel See Data kernel
Krige, D. G 229–230
Kriging 228–231
L
Laplace's equation 109
Leap seconds 296–297
Least squares See Principle of least squares
Length-two filter 159
Linear algebra 7
auxiliary parameters in 70
examples of 82
simplest 69
weighted averages and 74–77
Local quantities 225
for elements of vectors and matrices 11
for loops 12
omitting 12
nested 11
scatter plots and 35
M
data analysis in 2
functions defined in 301
graphics 28–32
grid search in 77–78
installing 3
joint probability density function in 54
Latin names for 147
mathematical constants in 6
organizing 3–4
practical considerations of 3
purpose of 1–3
syntax of 12
autocorrelation 191–192
covariance 93
of datasets 189t
posterior 93
prior 93
data analysis and 7
elements of 8–9
loop for 11–12
factor, rows of 167
grey-shaded 32
ill-conditioned 92
multiplication of 7–8
reorganizing 302
rotation 173–174
sample, rows of 167
of singular values 170–171
sparse 105–107
biconjugate gradient method solving 106
square 13
Toeplitz 76
unary 174
unwrap 107–108
Matrix inverse 13
max( ) function 25–26
Maximum number of iterations 106
Mean 41–44
confidence intervals of 82
formula for 43–44
random variables and 44
sample 44
of model parameter 69–70
variance of 82
univariate probability density function computing 53–54
Median 41–44
calculating 43
Microsoft Excel 297
dataset in 2
min( ) function 25–26
Minimum phase filters 159
Mistakes See Bugs
of datasets 73–74
model parameters in 74
quantitative models and 74
samples as 165–167
Mode 41–44
calculating 42
deceptiveness of 42
datasets as function of 67–69
estimating 69
observed data compared with 77
in mixtures 74
poorly determined 92
principle of least squares and 81
failure of 92
prior information on 92–94
covariance and 100–101
generalized least squares and 98–100
roughness and 102
smoothness as 102–104
as random variable function 48–50
sample mean of 69–70
weighted averages of 75
m-script 6
Multiplication of vectors and matrices 7–8
Multitaper method 221
Multivariate Normal probability density function 59–60
Multivariate probability density function 58
linear functions of 61–64
Normal distributions of 58–60
N
Named variables 2
Natural cubic splines 227
Nested loops 11
scatter plots and 35
information flow 256–259
training 256–259
Neuse River hydrograph 13
derivative of 32
plot of 14f
prediction error filter for 155f
Newton’s method 247
Normal probability density function measuring 47
as random variable 269
Nonspiky orthogonal vectors 174–175
Nontransient signals 134–137
power of 134–135
power spectral density of 135
impulse response proportional to 145–146
Central Limit Theorem and 48
limitations of 48
multivariate 59–60
N-dimensional 58–59
noisy data measured with 47
outliers and 48
normcdf( ) function 275
Null Hypothesis 269–270
rejection of 270
scenario for testing 274–280
Nyquist's Sampling Theorem 115–116
O
Objects, vectors compared to 233
Observational error 84
Observed periodicities 136–137
One-sided test 275
Orthogonal vectors, nonspiky 174f
Orthonormality of discrete Fourier data kernel 302–303
Outliers See also Errors
error plots for 77
Normal probability density function and 48
Output column vectors 167–168
Overlay 29
in stratosphere 196–197
tropospheric 196–197
P
Parseval's Theorem 135
behavior sensitive to 139–143
Period 113–114
nomenclature for 113–114
observed 136–137
in two-dimensions 234–235
Phase 114
minimum 159
ramp 129–130
Placeholders 35
floating-point 36
in format string 35–36
overlay for 29
side-by-side 29
cubic 226–228
curve fitting and 70–72
filters and 157–158
from time series datasets 158
Populations of data 37
Posterior covariance matrix 93
Posterior estimate of variance 84
of nontransient signals 134–137
spectral density 120
Fourier transform and 135
Fourier transform of autocorrelation and 194
of nontransient signal 135
of smoothing filters on time series datasets 202
in time series datasets 135
of datasets 113
future 154–156
Principle of least squares 77
complex 299–300
damped 99–100
prior information and 147
prior information of smoothness and 148
failure of 91
model parameters and 92
for filter estimation 198–200
gaps in information and modified 92
filling in 93
generalized 98–100
derivation of 301
for estimation 223
model parameters and 80
simple 37
weighted 101
Prior covariance matrix 93
Prior estimate of variance 84
of autocorrelation function for interpolation 228
damped least squares and 148
of smoothness 148–149
error in 93
of model parameters 92–94
covariance and 100–101
generalized least squares and 98–100
roughness and 102
smoothness as 102–104
probability density function and 93
smallness 148–149
Bayesian inference and 52–53
confusion with 52
covariance and 56–58
data analysis and 39
as histogram 40
methods for representing 39–40
upper-case and lower-case letters for 40–41
behavior of 41
conditional 95
computing 56f
Fisher-Snedecor F- 273
hypothesis testing with 272–274
joint 53–56
correlation of 190–191
univariate probability density function from 54f
multivariate 58
linear functions of 61–64
normal distributions of 58–60
Central Limit Theorem and 48
limitations of, 44 48
multivariate 58–60
noisy data measured with 47
outliers and 48
prior information and 93
spatially variable 55
computing 53–54
from joint probability density function 54f
mean and variance computed to 53–54
Properties 233
Q
Quantitative models 67–69
abstract understanding of 68
mixtures and 73–74
simplest 69
R
Radians per unit distance 113–115
Random variables 39–41
algebra of 41
difference due to 269–270
functions of 48–50
mean and 43–44
model parameters as function of 48–50
noise as 269
in scatter plots 187
random( ) function 110
Rate curve 32–33
Rate information, histograms and 32–34
Reality checks 23
Recursion 161–162
Relative time 139–140
Repeated datasets 290
Response time 140–141
Reynolds Channel water quality dataset 208–215
Riemann sum 125–126
Rotation matrix 174
Roughness information 102
Row vectors 7
data kernel as column vector of its 74
of factor matrix 167
of sample matrix 167
Rule for error propagation 61
Running sum 43
S
Sample matrix 167
Sample mean 44
of model parameter 69
variance of 82
Atlantic Rock dataset order of 179–180
as mixtures 165–167
dataset to text file 16–17
old scripts 17
as covariance estimate 188f
limitations of 34–37
nested for loops and 35
random variables in 187
comments 18
cut and paste used sparingly in 18
naming variables 17
saving old script 17
start small 18
testing 18
think before you type 17
syntax of 12
Side-by-side plots 29
Sidelobes 203
Simple least squares 98–100
of Atlantic Rock dataset 171–172
of CAC dataset 181–182
factor analysis and 172
matrix of 170
Sinusoidal oscillations 113–115
models composed of 115–122
covariance 246–247
variance 243–245
Smallness information 148–149
three-point 200
autocorrelation function of 201–202
power spectral density of 202
damped least squares and prior 148–149
for gaps in information 102–103
as prior information on model parameters 102–104
data analysis in single 1–2
scripting language 2
spreadsheet 2
spalloc( ) function 151
Sparse matrices 105–107
biconjugate gradient method solving 106
Spatial cycles 113–114
Spatially variable probability density function 55
cross- 195
power 120
Fourier transform of autocorrelation and 194
of smoothing filters on time series datasets 202
Spectral division 161
Spectral hole 160–161
Spikes 141
time-delayed 130
Spiky factors 172–176
minerals and 173
natural 227
interpolation 232
types of 225
Spreadsheet software environment 2
Square matrices 13
sscanf( ) function 297–298
Start time 296
Statistics 270
for calibration test 276t
Stochastic gradient method 252–253
Storm events 32
Stratosphere, ozone in 196–197
String print formatted 35
sum() function 44
svd( ) function 171–172
Swayze, Patrick 52
Synthetic data 12
T
Tapering process 136–137
Tapers 215–216
Taylor series 88
multiple dimensions 245–246
and polynomial approximations 240–241
tcdf( ) function 278
anomaly 181f
Temporal cycles 113–114
statistics for 276t
Chi-squared test 274
one-sided 275
with probability density function 272–274
scenario for 274–280
two-sided 275
scripting language software environment 17
scripts 17
t-test 274–278
Z-test 274
complicated, reading 297–298
saving to 16–17
Text Import Wizard 297
backward in 144
forward in 144
notes on 297–298
relative 139–140
response 140f
Time arithmetic function 296
autocorrelation 192
coherence in 208–215
filter coefficients and 141
polynomials from 157–158
power in 135
similarity in 211
autocorrelation function of 201–202
power spectral density of 202
Time-delayed spikes 130
Time-shift invariance 140
Time-variable functions 179–184
Toeplitz matrix 76
Tolerance 106
logarithm of 79–80f
shape of 85
Transient signals 134
interpolation uses of 233f
TriScatteredInterp( ) function 232
Tropospheric ozone 196–197
Two-sided test 275
U
Ultraviolet light (UV) 196–197
Unary matrix 174
unidrnd( ) function 291
computing 54f
Unit impulse 141
from joint probability density function 54f
mean and variance computed to 53–54
Unwrap matrices 107–108
UTC See Coordinated Universal Time
UV See Ultraviolet light
V
Variables 6
naming 17
persistence of 295–296
random 39–41
algebra of 41
difference due to 269–270
noise as 269
in scatter plots 166
Variance 45–46
disadvantage of 45–46
high, elements of factors having 173
posterior estimate of 84
prior estimate of 84
of sample mean 82
univariate probability density function computing 53–56
characteristic 168
clipping 12
column 7
input 167–168
output 167–168
data analysis and 7
eigenvectors 168
perpendicular 169
elements of 8–9
loop for 11–12
multiplication of 7–8
nonspiky orthogonal 174–175
objects compared to 233
row 7
W
Wavelength 113–114
Wavenumber 113–114
causal filter as 76
linear models and 74–77
of model parameters 75
three-point 75
Weighted least squares 101
before Fourier transforms 215–216
optimal 217–221
X
Z
Zero-lag cross-correlation function 213
zeros ( ) function 27
Z-test 274
z-transform 158
of inverse filter 158–161
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