Elements of Linear Algebra

by Cohn; P.M.

• ISBN13: 9780412552809
• ISBN10: 0412552809
• Edition: 1st
• Format: Nonspecific Binding
• Copyright: 1994-07-01
• Publisher: Chapman & Hall/

Summary

This volume presents a thorough discussion of systems of linear equations and their solutions. Vectors and matrices are introduced as required and an account of determinants is given. Great emphasis has been placed on keeping the presentation as simple as possible, with many illustrative examples. While all mathematical assertions are proved, the student is led to view the mathematical content intuitively, as an aid to understanding.The text treats the coordinate geometry of lines, planes and quadrics, provides a natural application for linear algebra and at the same time furnished a geometrical interpretation to illustrate the algebraic concepts.

Table of Contents

List of Tables	xi
List of Figures	xv
Preface	xix
Acknowledgments	xxi
Introduction	1	(22)
Hydrologic Frequency Analysis	1	(1)
General Aspects and Approaches	1	(3)
Other Models	4	(2)
Return Period, Probability, and Plotting Positions	6	(2)
Flood Frequency Models	8	(2)
Hydrologic Risk	10	(1)
Regionalization	11	(1)
Tests on Hydrologic Data	12	(11)
Test for Independence and Stationarity	13	(1)
Homogeneity and Stationarity Tests	14	(2)
Test for Outliers	16	(7)
Selection and Evaluation of Parent Distributions: Conventional Moments	23	(30)
Moments of Distributions and Their Sample Estimates	23	(3)
Moment Ratio Diagrams (MRDs)	26	(10)
Probability Plots	36	(2)
Selection of Distributions	38	(9)
Chi-Square and Psi Tests	41	(2)
Kolmogorov-Smirnov Test	43	(4)
Regional Homogeneity and Regionalization	47	(6)
Selection and Evaluation of Parent Distributions: Probability Weighted Moments and L-Moments	53	(20)
Moments of Distributions and Their Sample Estimates	53	(3)
L-Moment Ratio Diagrams	56	(5)
Tests Based on L-Moments	61	(3)
Goodness-of-Fit Tests	61	(1)
Regional Homogeneity Tests	62	(2)
A Case Study	64	(9)
Data and Preliminary Analysis	64	(1)
Regional Homogeneity	65	(4)
Regional Quantile Estimates	69	(4)
Parameter and Quantile Estimation	73	(10)
Introduction	73	(1)
Parameter Estimation	73	(3)
Method of Moments (MOM)	74	(1)
Method of Maximum Likelihood (MLM)	75	(1)
Method of Probability Weighted Moments (PWM)	75	(1)
Quantile Estimation	76	(1)
Confidence Intervals	77	(6)
Standard Error in the MOM	78	(2)
Standard Error in the MLM	80	(1)
Standard Error in the PWM	81	(2)
Normal and Related Distributions	83	(44)
Normal Distribution	83	(13)
Introduction	83	(2)
Parameter Estimation	85	(1)
Method of Moments	85	(1)
Maximum Likelihood (ML) Method	86	(1)
PWM Method	87	(3)
Quantile Estimates	90	(2)
Standard Error	92	(1)
Method of Moments	92	(1)
ML Method	92	(2)
PWM Method	94	(2)
Two-Parameter Lognormal (LN(2)) Distribution	96	(11)
Introduction	96	(1)
Parameter Estimation	97	(1)
Method of Moments	97	(1)
Maximum Likelihood (ML) Method	98	(1)
PWM Method	99	(2)
Quantile Estimates	101	(2)
Standard Error	103	(1)
Method of Moments	103	(1)
ML Method	104	(3)
Three-Parameter Lognormal (LN(3)) Distribution	107	(20)
Introduction	107	(2)
Parameter Estimation	109	(1)
Method of Moments	109	(1)
The Maximum Likelihood (ML) Method	110	(2)
PWM Method	112	(3)
Quantile Estimates	115	(2)
Standard Error	117	(1)
Method of Moments	117	(2)
ML Method	119	(8)
The Gamma Family	127	(80)
Exponential Distribution	127	(13)
Introduction	127	(1)
Parameter Estimation	128	(1)
Method of Moments	128	(2)
Maximum Likelihood (ML) Method	130	(1)
PWM Method	131	(2)
Quantile Estimates	133	(2)
Standard Error	135	(1)
Method of Moments	135	(1)
ML Method	136	(1)
PWM Method	137	(3)
Two-Parameter Gamma (G(2)) Distribution	140	(14)
Introduction	140	(1)
Parameter Estimation	141	(1)
Method of Moments	141	(1)
Maximum Likelihood (ML) Method	142	(2)
PWM Method	144	(2)
Quantile Estimates	146	(4)
Standard Errors	150	(1)
Method of Moments	150	(1)
ML Method	150	(4)
Pearson (3) Distribution	154	(16)
Introduction	154	(2)
Parameter Estimation	156	(1)
Method of Moments	156	(1)
Maximum Likelihood (ML) Method	157	(2)
PWM Method	159	(4)
Quantile Estimation	163	(2)
Standard Errors	165	(1)
Method of Moments	165	(1)
ML Method	165	(5)
Log-Pearson (3) Distribution	170	(21)
Introduction	170	(4)
Studies on Skewness Coefficients	174	(1)
Parameter Estimation	175	(1)
Method of Moments	175	(1)
Maximum Likelihood (ML) Method	176	(1)
PWM Method	177	(4)
Quantile Estimation	181	(2)
Standard Error	183	(1)
Method of Moments	183	(3)
ML Method	186	(5)
U.S. Water Resources Council Method (WRCM)	191	(16)
Introduction	191	(1)
Frequency Analysis Procedure by the WRCM	192	(7)
Outlier Tests	199	(1)
Confidence Limits	200	(7)
Extreme Value Distributions	207	(52)
Generalized Extreme Value (GEV) Distribution	207	(22)
Parameter Estimation	208	(1)
Method of Moments	208	(4)
Maximum Likelihood (ML) Method	212	(4)
PWM Method	216	(5)
Quantile Estimates	221	(1)
Standard Error	222	(1)
Method of Moments	222	(1)
ML Method	223	(2)
PWM Method	225	(4)
The Extreme Value Type I EV(1) Distribution	229	(12)
Parameter Estimation	232	(1)
Method of Moments	232	(1)
Maximum Likelihood (ML) Method	232	(2)
PWM Method	234	(2)
Quantile Estimates	236	(2)
Standard Error	238	(1)
Method of Moments	238	(1)
ML Method	239	(1)
PWM Method	240	(1)
Weibull Distribution	241	(18)
Parameter Estimation	243	(1)
Method of Moments	243	(2)
Maximum Likelihood (ML) Method	245	(1)
PWM Method	246	(3)
Quantile Estimates	249	(2)
Standard Errors	251	(1)
Method of Moments	251	(1)
ML Method	251	(8)
The Wakeby Distribution	259	(32)
The Five-Parameter Wakeby Distribution (WAK(5))	259	(8)
Introduction	259	(1)
Parameter Estimation	260	(5)
Quantile Estimation	265	(1)
Standard Error	265	(2)
The Four-Parameter Wakeby Distribution (WAK(4))	267	(4)
Introduction	267	(1)
Parameter Estimation	267	(2)
Quantile Estimation	269	(2)
Standard Error	271	(1)
The Generalized Pareto Distribution	271	(20)
Introduction	271	(1)
Parameter Estimation	272	(1)
Method of Moments	272	(3)
Maximum Likelihood (ML) Method	275	(2)
PWM Method	277	(4)
Quantile Estimation	281	(3)
Standard Error	284	(1)
Method of Moments	284	(1)
ML Method	285	(1)
PWM Method	285	(6)
The Logistic Distribution	291	(32)
Logistic Distribution	291	(11)
Introduction	291	(1)
Parameter Estimation	291	(1)
Method of Moments	291	(1)
Maximum Likelihood (ML) Method	292	(2)
PWM Method	294	(3)
Quantile Estimation	297	(1)
Standard Error	298	(1)
Method of Moments	298	(1)
ML Method	299	(1)
PWM Method	300	(2)
Generalized Logistic Distribution	302	(21)
Introduction	302	(2)
Parameter Estimation	304	(1)
Method of Moments	304	(2)
-10 < CS < 10; -1/3 < k < 1/3	306	(1)
0 < CS < 10; 1/3 < k < 1/2	306	(1)
-10 < CS < 0; -1/2 < k < -1/3	306	(1)
The Maximum Likelihood (ML) Method	307	(4)
PWM Method	311	(3)
Quantile Estimation	314	(2)
Standard Error	316	(1)
Method of Moments	316	(1)
ML Method	317	(6)
Computer Program	323	(4)
Introduction	323	(1)
Description of Program	323	(4)
References	327	(16)
Index	343