|
Preface |
6 |
|
|
Contents |
8 |
|
|
1 Introduction |
13 |
|
|
Abstract |
13 |
|
|
1.1 General |
13 |
|
|
1.2 Trend Definition and Analysis |
15 |
|
|
1.2.1 Conceptual and Visual Trends |
16 |
|
|
1.2.2 Mathematical Trend |
19 |
|
|
1.2.3 Statistical Trend |
21 |
|
|
1.3 Trend in Some Disciplines |
23 |
|
|
1.3.1 Atmospheric Sciences |
24 |
|
|
1.3.2 Environmental Sciences |
24 |
|
|
1.3.3 Earth Sciences |
24 |
|
|
1.3.4 Engineering |
25 |
|
|
1.3.5 Global Warming |
25 |
|
|
1.3.6 Climate Change |
26 |
|
|
1.3.7 Social Sciences |
26 |
|
|
1.3.7.1 Economy |
27 |
|
|
1.3.7.2 Business |
27 |
|
|
1.3.7.3 Health |
28 |
|
|
1.4 Pros and Cons of Trend Analysis |
29 |
|
|
1.5 Future Research Directions |
29 |
|
|
1.6 Purpose of This Book |
30 |
|
|
References |
31 |
|
|
2 Uncertainty and Time Series |
33 |
|
|
Abstract |
33 |
|
|
2.1 General |
33 |
|
|
2.2 Random and Randomness |
36 |
|
|
2.3 Empirical Frequency and Distribution Function |
37 |
|
|
2.3.1 Empirical Frequency and Trend |
40 |
|
|
2.4 Theoretical Probability Distribution Function (Pdf) |
42 |
|
|
2.5 Statistical Modeling |
44 |
|
|
2.5.1 Deterministic-Uncertain Model |
46 |
|
|
2.5.2 Probabilistic-Statistical Model |
47 |
|
|
2.5.3 Transitional Probability Model |
48 |
|
|
2.6 Stochastic Models |
49 |
|
|
2.6.1 Homogeneity (Consistency) |
50 |
|
|
2.6.2 Stationarity |
51 |
|
|
2.6.3 Periodicity (Seasonality) |
52 |
|
|
2.6.3.1 Known Period Case |
55 |
|
|
2.7 Time Series Truncation |
57 |
|
|
2.7.1 Statistical Truncations |
59 |
|
|
2.8 Data Smoothing |
61 |
|
|
2.8.1 Moving Averages |
62 |
|
|
2.8.2 Difference Smoothing |
62 |
|
|
2.9 Jump (Shift) |
64 |
|
|
2.10 Correlation Coefficients |
65 |
|
|
2.10.1 Pearson Correlation Coefficient |
66 |
|
|
2.10.2 Kendall Correlation Coefficient |
70 |
|
|
2.10.3 Spearman Correlation Coefficient |
71 |
|
|
2.11 Persistence/Nonrandomness |
73 |
|
|
2.11.1 Short-Memory (Correlation) Components |
73 |
|
|
2.11.2 Long-Memory (Persistence) Component |
74 |
|
|
2.11.2.1 Rescaled Range and Hurst Phenomenon |
75 |
|
|
References |
77 |
|
|
3 Statistical Trend Tests |
79 |
|
|
Abstract |
79 |
|
|
3.1 General |
79 |
|
|
3.2 Nonparametric Tests |
80 |
|
|
3.2.1 Data Ordering (Ranks) |
81 |
|
|
3.3 Statistical Tests |
82 |
|
|
3.3.1 Wald–Wolfowitz |
82 |
|
|
3.3.2 Sign Test |
82 |
|
|
3.3.3 Sign Difference Test |
83 |
|
|
3.3.4 Run Test |
84 |
|
|
3.3.5 Mann–Whitney (MW) Test |
85 |
|
|
3.3.6 Kruskal–Wallis (KW) Test |
91 |
|
|
3.3.7 Nonparametric Correlation Coefficient |
95 |
|
|
3.3.8 Spearman’s Rho Test of Trend |
96 |
|
|
3.3.9 Turning Point Test |
97 |
|
|
3.3.10 Mann–Kendall (MK) Test |
98 |
|
|
3.3.10.1 Mann–Kendall Trend Search |
101 |
|
|
3.3.10.2 Sen Slope Estimator |
102 |
|
|
3.3.10.3 Spearman’s Tau |
103 |
|
|
3.3.10.4 Regression Trend |
103 |
|
|
3.3.11 Two-Sample Wilcoxon Test |
104 |
|
|
3.3.11.1 Signed-Wilcoxon Test |
105 |
|
|
3.3.11.2 Wilcoxon Signed Rank Test |
105 |
|
|
3.3.12 von Neuman Test |
106 |
|
|
3.3.13 Cumulative Departures Test |
107 |
|
|
3.3.13.1 Cumulative Deviations |
107 |
|
|
3.3.14 Bayesian Test |
109 |
|
|
3.3.15 Relative Error Test |
110 |
|
|
3.3.16 t Test |
111 |
|
|
3.3.17 Cramer Test |
114 |
|
|
3.3.18 F Test |
115 |
|
|
3.3.19 Truncation Test |
118 |
|
|
3.3.20 Deviations Test |
119 |
|
|
3.3.21 Subtraction Test |
119 |
|
|
3.3.22 ?en Autorun Test |
120 |
|
|
3.3.23 Seasonal Kendall Test |
123 |
|
|
3.4 Unit Root Model Trend Determination |
124 |
|
|
3.4.1 Integration and Dickey–Fuller (DF) Test |
125 |
|
|
3.4.2 The Kwiatkowski, Phillips, Schmidt, and Shin Test |
126 |
|
|
3.4.3 Critical Values of the KPSS Test |
129 |
|
|
3.4.4 Empirical Power of the KPSS |
130 |
|
|
3.4.5 Example: Comparison of the DF and KPSS Tests for Several Macro-Economic Time Series |
133 |
|
|
3.4.5.1 Test of Stationarity Around Mean |
133 |
|
|
3.4.5.2 Test of Stationarity Around a Linear Trend |
136 |
|
|
3.5 Parametric Tests |
136 |
|
|
3.5.1 Regression Analysis |
138 |
|
|
3.5.2 Regression Line Assumptions |
139 |
|
|
3.5.3 Goodness of Fit (R2) for Regression |
140 |
|
|
3.5.4 Cumulative Sum (CUSUM) Method |
141 |
|
|
References |
142 |
|
|
4 Temporal Trend Analysis |
145 |
|
|
Abstract |
145 |
|
|
4.1 General |
145 |
|
|
4.2 Visual Inspection |
147 |
|
|
4.3 Monotonic Trend Analysis |
149 |
|
|
4.4 Scatter Diagrams and Regression Model |
150 |
|
|
4.5 Linear Regression Model |
153 |
|
|
4.5.1 Statistical Procedure |
154 |
|
|
4.6 Unrestricted Regression Model |
157 |
|
|
4.6.1 Application |
159 |
|
|
4.7 Partial Regression Method (PRM) |
160 |
|
|
4.8 Cluster Regression and Markov Chain |
163 |
|
|
4.8.1 Cluster Regression Model |
164 |
|
|
4.8.2 Application and Discussion |
165 |
|
|
4.9 Trend Over-whitening Procedures |
171 |
|
|
4.9.1 Over-whitening (OW) Process |
172 |
|
|
4.9.2 Simulation |
176 |
|
|
4.9.3 Application |
177 |
|
|
References |
185 |
|
|
5 Innovative Trend Analyses |
187 |
|
|
Abstract |
187 |
|
|
5.1 General |
187 |
|
|
5.2 Probability Distribution-Statistical Parameter Trend Implications |
189 |
|
|
5.3 Innovative Trend Identification Methodologies |
194 |
|
|
5.3.1 Application |
196 |
|
|
5.4 Innovative Trend Simulation |
198 |
|
|
5.4.1 Fundamental Methodology |
200 |
|
|
5.4.1.1 Simulation Methodology |
201 |
|
|
5.4.1.2 Dependent Process Simulation Results |
204 |
|
|
5.5 Innovative Trend Significance Test |
211 |
|
|
5.5.1 Deterministic Basis |
212 |
|
|
5.5.2 Stochastic Basis |
214 |
|
|
5.5.2.1 Normally Distributed Stochastic Time Series |
214 |
|
|
5.5.2.2 Gamma Distributed Stochastic Time Series |
215 |
|
|
5.5.3 Statistical Innovative Trend Test |
216 |
|
|
5.5.4 Application |
217 |
|
|
5.6 Crossing Trend Analysis Methodology |
222 |
|
|
5.6.1 Rational Concept |
224 |
|
|
5.6.2 Theoretical Background |
224 |
|
|
5.6.3 Monte Carlo Simulations |
227 |
|
|
5.6.4 Application |
227 |
|
|
References |
237 |
|
|
6 Spatial Trend Analysis |
239 |
|
|
Abstract |
239 |
|
|
6.1 General |
239 |
|
|
6.2 Numerical Solution |
242 |
|
|
6.3 Spatial Data Analysis |
244 |
|
|
6.4 Homogeneity and Isotropy |
247 |
|
|
6.5 Spatial Trend Surfaces |
250 |
|
|
6.5.1 Horizontal Plane |
252 |
|
|
6.5.2 Horizontal Planes |
253 |
|
|
6.5.3 Inclined Trend Plane |
253 |
|
|
6.5.4 Inclined Trend Planes |
254 |
|
|
6.5.5 Curved Trend Surface |
255 |
|
|
6.5.6 Random Surface |
255 |
|
|
6.6 Spatial Dependence Function (SDF) |
257 |
|
|
6.6.1 Spatial Correlation Parameter Calculation |
259 |
|
|
6.7 Double Mass Curve Test |
262 |
|
|
6.8 Trend Surface Analysis |
266 |
|
|
6.8.1 Planer Trend Regression Analysis |
266 |
|
|
6.8.2 Polynomial Trend Regression Analysis |
269 |
|
|
6.8.3 Kriging Methodology |
274 |
|
|
6.8.3.1 Simple Kriging (SK) |
277 |
|
|
Methodology |
277 |
|
|
6.9 Triple Diagram Model (TDM) |
283 |
|
|
6.9.1 Parallel-Triple Model |
284 |
|
|
6.9.2 Serial-Triple Model |
288 |
|
|
References |
292 |
|
|
7 Trend Variability Detection |
293 |
|
|
Abstract |
293 |
|
|
7.1 General |
293 |
|
|
7.2 Variability Measures |
295 |
|
|
7.2.1 Range |
295 |
|
|
7.2.2 Standard Deviation |
296 |
|
|
7.2.3 The Interquartile Range (IQR) |
298 |
|
|
7.2.4 Investment Variability |
299 |
|
|
7.3 Trend and Variability Detection by Innovative Methodology |
300 |
|
|
7.3.1 Methodology |
301 |
|
|
7.3.2 Simulation Study |
304 |
|
|
7.3.3 Applications |
306 |
|
|
7.4 Trend Significance Limits |
309 |
|
|
7.5 Trend and Variability Analyses by Innovative and Classical Methodologies |
316 |
|
|
7.5.1 ?en Innovative Trend Analysis |
317 |
|
|
7.6 Application and Interpretations |
318 |
|
|
7.6.1 Probability Distribution Functions (pdf) |
320 |
|
|
7.6.2 Different Trends |
321 |
|
|
7.7 Trend and Variability |
323 |
|
|
7.8 Innovative Trend Template and Significance Limits |
326 |
|
|
References |
329 |
|
|
8 Partial Trend Detection |
332 |
|
|
Abstract |
332 |
|
|
8.1 General |
332 |
|
|
8.2 Qualitative Partial Trend Methodology |
335 |
|
|
8.3 Previous Works |
337 |
|
|
8.4 Innovative Piecewise Trend Analysis |
341 |
|
|
8.5 Innovative Trend Template |
346 |
|
|
8.6 Stochastic Simulation Approach |
348 |
|
|
8.7 Data and the Study Area |
352 |
|
|
8.7.1 Partial Trend Groups |
352 |
|
|
8.7.2 Partial Trend Lines |
353 |
|
|
References |
356 |
|
|
Index |
358 |
|