Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Format......Page 12
Acknowledgements......Page 13
1.1 Introduction......Page 15
1.2 Representation of floating-point numbers......Page 18
1.2.2 Binary to decimal system......Page 21
1.2.3 Decimal to binary system......Page 23
1.2.4 Binary representation of floating-point numbers......Page 24
Overflow and underflow errors......Page 25
Binary significand – limits of precision......Page 28
1.3 Methods used to measure error......Page 30
1.4 Significant digits......Page 32
1.5 Round-off errors generated by floating-point operations......Page 34
1.6 Taylor series and truncation error......Page 40
1.6.1 Order of magnitude estimation of truncation error......Page 42
1.6.2 Convergence of a series......Page 46
1.6.3 Finite difference formulas for numerical differentiation......Page 47
1.7 Criteria for convergence......Page 53
1.9 Problems......Page 54
References......Page 60
2.1 Introduction......Page 61
2.2.1 Vectors and matrices......Page 67
Using MATLAB......Page 70
Transpose operations......Page 71
Using MATLAB......Page 72
Using MATLAB......Page 73
Using MATLAB......Page 74
Matrix–matrix multiplication......Page 75
Using MATLAB......Page 77
Norm of a vector......Page 78
Using MATLAB......Page 79
2.2.4 Linear combinations of vectors......Page 80
2.2.5 Vector spaces and basis vectors......Page 83
Matrix rank......Page 85
Using MATLAB......Page 86
Determinant of a matrix......Page 87
Inverse of a matrix......Page 88
2.3 Matrix representation of a system of linear equations......Page 89
2.4.1 Gaussian elimination without pivoting animalnhip......Page 90
Step 1......Page 92
Step 2......Page 93
Algorithm for Gaussian elimination without pivoting......Page 94
Algorithm for back substitution......Page 95
MATLAB program 2.1......Page 96
Using MATLAB......Page 98
MATLAB program 2.2......Page 99
2.5 LU factorization......Page 101
2.5.1 LU factorization without pivoting......Page 102
2.5.2 LU factorization with pivoting......Page 107
2.5.3 The MATLAB lu function......Page 109
2.6 The MATLAB backslash (\\) operator......Page 110
2.7 Ill-conditioned problems and the condition number......Page 111
2.8 Linear regression hemoglobin–oxygen binding......Page 115
2.9 Curve fitting using linear least-squares approximation......Page 121
2.9.1 The normal equations......Page 123
Geometric method......Page 127
Algebraic Method......Page 128
2.9.2 Coefficient of determination and quality of fit......Page 129
2.10 Linear least-squares approximation of transformed equations......Page 132
2.11 Multivariable linear least-squares regression......Page 137
2.12 The MATLAB function polyfit......Page 138
2.13 End of Chapter 2: key points to consider......Page 139
Solving systems of linear equations......Page 141
References......Page 153
3.1 Introduction......Page 155
3.2 Characterizing a population: descriptive statistics......Page 158
3.2.1 Measures of central tendency......Page 159
3.2.2 Measures of dispersion......Page 160
3.3 Concepts from probability......Page 161
3.3.1 Random sampling and probability......Page 163
3.3.2 Combinatorics: permutations and combinations......Page 168
3.4 Discrete probability distributions......Page 171
3.4.1 Binomial distribution......Page 173
3.4.2 Poisson distribution......Page 177
3.5 Normal distribution......Page 180
3.5.1 Continuous probability distributions......Page 181
3.5.2 Normal probability density......Page 183
3.5.3 Expectations of sample-derived statistics......Page 185
3.5.4 Standard normal distribution and the z statistic......Page 189
3.5.5 Confidence intervals using the z statistic and the t statistic......Page 191
3.5.6 Non-normal samples and the centralimit theorem......Page 197
3.6 Propagation of error......Page 200
3.6.1 Addition/subtraction of random variables......Page 201
3.6.2 Multiplication/division of random variables......Page 202
3.6.3 General functional relationship between two random variables......Page 204
3.7 Linear regression error......Page 205
3.7.1 Error in model parameters......Page 207
3.7.2 Error in model predictions......Page 210
3.8 End of Chapter 3: key points to consider......Page 213
3.9 Problems......Page 216
References......Page 222
4.1 Introduction......Page 223
4.2 Formulating a hypothesis......Page 224
Observational study......Page 225
4.2.2 Null and alternate hypotheses......Page 231
4.3 Testing a hypothesis......Page 233
4.3.1 The p value and assessing statistical significance......Page 234
4.3.2 Type I and type II errors......Page 240
4.3.3 Types of variables......Page 242
4.3.4 Choosing a hypothesis test......Page 244
4.4 Parametric tests and assessing normality......Page 245
4.5.1 One-ample z test......Page 249
4.5.2 Two-sample z test......Page 255
4.6.1 One-sample and paired sample t tests......Page 258
4.6.2 Independent two-sample t test......Page 263
The population variances are not equal......Page 264
4.7 Hypothesis testing for population proportions......Page 265
4.7.1 Hypothesis testing for a single population proportion......Page 270
4.7.2 Hypothesis testing for two population proportions......Page 271
4.8 One-way ANOVA......Page 274
4.9 Chi-square tests for nominal scale data......Page 288
4.9.1 Goodness-of-fit test......Page 290
4.9.2 Test of independence......Page 295
4.9.3 Test of homogeneity......Page 299
4.10 More on non-prametric (distribution-free) tests......Page 302
4.10.1 Sign test......Page 303
4.10.2 Wilcoxon signed-rank test......Page 306
4.10.3 Wilcoxon rank-sum test......Page 310
4.12 Problems......Page 313
References......Page 322
5.1 Introduction......Page 324
5.2 Bisection method......Page 326
5.3 Regula-falsi method......Page 333
5.4 Fixed-point iteration......Page 334
5.5 Newton’s method......Page 341
Convergence is not guaranteed......Page 343
Convergence rate is not always quadratic......Page 344
5.6 Secant method......Page 350
5.7 Solving systems of nonlinear equations......Page 352
5.8 MATLAB function fzero......Page 360
5.9 End of Chapter 5: key points to consider......Page 362
5.10 Problems......Page 363
References......Page 367
6.1 Introduction......Page 368
6.2 Polynomial interpolation......Page 375
6.3 Newton–Cotes formulas......Page 385
6.3.1 Trapezoidal rule......Page 386
6.3.2 Simpson’s 1/3 rule......Page 394
6.3.3 Simpson’s 3/8 rule......Page 398
6.4 Richardson’s extrapolation and Romberg integration......Page 401
6.5 Gaussian quadrature......Page 405
6.6 End of Chapter 6: key points to consider......Page 416
6.7 Problems......Page 417
References......Page 422
7.1 Introduction......Page 423
7.2 Euler’s methods......Page 430
7.2.1 Euler’s forward method......Page 431
Local and global truncation errors......Page 432
Stability issues......Page 437
Coupled ODEs......Page 439
7.2.2 Euler’s backward method......Page 442
7.2.3 Modified Euler’s method......Page 445
7.3.1 Second-order RK methods......Page 448
Modified Euler method (explicit)......Page 450
Midpoint method......Page 451
7.3.2 Fourth-order RK methods......Page 452
7.4 Adaptive step size methods......Page 454
7.5 Multistep ODE solvers......Page 465
7.5.1 Adams methods......Page 466
7.5.2 Predictor–corrector methods......Page 468
7.6 Stability and stiff equations......Page 470
7.7 Shooting method for boundary-value problems......Page 475
7.7.1 Linear ODEs......Page 477
7.7.2 Non-linear ODEs......Page 478
7.8 End of Chapter 7: key points to consider......Page 486
7.9 Problems......Page 487
References......Page 492
8.1 Introduction......Page 494
8.2 Unconstrained single-variable optimization......Page 501
8.2.1 Newton’s method......Page 502
8.2.2 Successive parabolic interpolation......Page 506
8.2.3 Golden section search method......Page 509
8.3 Unconstrained multivariable optimization......Page 514
8.3.1 Steepest descent or gradient method......Page 516
8.3.2 Multidimensional Newton’s method......Page 523
8.3.3 Simplex method......Page 527
8.4 Constrained nonlinear optimization......Page 537
8.5 Nonlinear error analysis......Page 544
8.6 End of Chapter 8: key points to consider......Page 547
8.7 Problems......Page 548
References......Page 552
9.1 Introduction......Page 553
9.2 Sequence alignment and database searches......Page 554
9.3 Phylogenetic trees using distance-based methods......Page 568
9.4 End of Chapter 9: key points to consider......Page 571
References......Page 572
A1.1 Matrix operations......Page 574
A1.2 Programming in MATLAB......Page 576
A1.2.1 Operators......Page 579
Conditional control......Page 581
Loop control......Page 583
A1.3 Two-dimensional plotting......Page 584
A1.4.2 Exporting data......Page 587
Appendix B: Location of nodes for Gauss–Legendre quadrature......Page 590
Index for MATLAB commands......Page 592
Index......Page 593