Detection of Random Signals in Dependent Gaussian Noise (eBook)

Antonio F. Gualtierotti (Autor)

eBook Download: PDF

2015 | 1st ed. 2015
XXXIV, 1176 Seiten
Springer International Publishing (Verlag)
978-3-319-22315-5 (ISBN)

Infos

The book presents the necessary mathematical basis to obtain and rigorously use likelihoods for detection problems with Gaussian noise. To facilitate comprehension the text is divided into three broad areas - reproducing kernel Hilbert spaces, Cramér-Hida representations and stochastic calculus - for which a somewhat different approach was used than in their usual stand-alone context.

One main applicable result of the book involves arriving at a general solution to the canonical detection problem for active sonar in a reverberation-limited environment. Nonetheless, the general problems dealt with in the text also provide a useful framework for discussing other current research areas, such as wavelet decompositions, neural networks, and higher order spectral analysis.

The structure of the book, with the exposition presenting as many details as necessary, was chosen to serve both those readers who are chiefly interested in the results and those who want to learn the material from scratch. Hence, the text will be useful for graduate students and researchers alike in the fields of engineering, mathematics and statistics.

Inhalt

Preface 8
Prolog 12
Credits and Comments 14
Chapter 1 14
Chapter 2 15
Chapter 3 15
Chapter 4 16
Chapter 5 16
Chapters 6, 7, and 9 16
Chapter 8 17
Chapter 10 18
Chapters 11, 12, 13, 14, and 15 18
Chapter 16 18
Chapter 17 19
Notation and Terminology 20
General Notation 20
Part I: Reproducing Kernel Hilbert Spaces 23
Part II: Cramér-Hida Representations 24
Part III: Likelihoods 26
Contents 30
Part I Reproducing Kernel Hilbert Spaces 36
1 Reproducing Kernel Hilbert Spaces: The Rudiments 37
1.1 Definition and First Properties 37
1.2 Membership in a Reproducing Kernel Hilbert Space 68
1.3 Covariances and Reproducing Kernel Hilbert Spaces 77
1.4 Triangular Covariances 93
1.5 Separable Reproducing Kernel Hilbert Spaces 108
1.6 Subspaces of Reproducing Kernel Hilbert Spaces and Associated Projections 115
1.7 Operators in Reproducing Kernel Hilbert Spaces 133
1.7.1 Bounded Linear Operators 134
1.7.2 Unitary Operators 141
1.7.3 Hilbert-Schmidt Operators 143
1.7.4 Covariance Operators 147
2 The Functions of a Reproducing Kernel Hilbert Space 158
2.1 Kernels and the Operators They Determine 158
2.2 Reproducing Kernel Hilbert Spaces of Measurable Functions 165
2.3 Representations of Reproducing Kernels 169
2.4 Embeddings of Reproducing Kernel Hilbert Spaces 183
2.5 Reproducing Kernel Hilbert Spaces of Functions with Integrable Power 200
2.6 Reproducing Kernel Hilbert Spaces of Continuous Functions 216
2.7 Spectral Theory: A Vademecum 230
2.8 Reproducing Kernel Hilbert Spaces as Images of Ranges of Square Roots of Linear Operators 236
3 Relations Between Reproducing Kernel Hilbert Spaces 249
3.1 Order for Covariances 249
3.2 Contractive Inclusions of Hilbert Spaces 289
3.2.1 Definition and Properties of Contractive Inclusions 289
3.2.2 Complementary and Overlapping Spaces 296
3.3 Intersections of Reproducing Kernel Hilbert Spaces 311
3.4 Dominated Families of Covariances 318
3.4.1 Spectral Representation of Dominated Covariances 318
Result 1: 323
Result 2: 324
Result 3: 324
Result 4: 325
Result 5: 325
Result 6: 325
Result 7: 326
Result 9: 328
3.4.2 Simultaneous Reduction of Covariances 332
4 Reproducing Kernel Hilbert Spaces and Paths of Stochastic Processes 338
4.1 Random Elements with Values in a Reproducing Kernel Hilbert Space 338
4.2 Paths and Values, in Reproducing Kernel Hilbert Spaces, of Random Elements 341
4.3 Paths and Values, in Reproducing Kernel Hilbert Spaces, of Gaussian Elements 349
4.4 Processes of Second Order with Paths in a Reproducing Kernel Hilbert Space 350
4.5 Dichotomies 357
5 Reproducing Kernel Hilbert Spaces and Discrimination 359
5.1 Context of Discrimination 359
5.2 Atoms and Reduced Measures 375
5.3 Dependence of the Lebesgue Decomposition on Some Related Reproducing Kernel Hilbert Spaces 380
5.3.1 Background 380
5.3.2 Lebesgue Decomposition and ``Sizes''of Reproducing Kernel Hilbert Spaces Intersections 383
5.4 Domination of Probabilities on Sub-manifolds 390
5.4.1 The General Case 390
5.4.2 Domination on the Linear Manifold of Evaluations 398
5.4.3 Domination on the Quadratic Manifoldof Evaluations 400
5.5 Discrimination of Translates 405
5.5.1 Preliminaries 406
5.5.2 The Case of a Signal Lying Outside of the Reproducing Kernel Hilbert Space of Noise 407
5.6 Discrimination of Gaussian Laws 414
5.6.1 Some Properties of Gaussian Laws 414
5.6.2 Quadratic Forms of Normal Random Variables 416
5.6.3 Gaussian Laws Have the Properties Required for Domination 418
5.6.4 The Quadratic Manifold of Evaluations: A Source of Hilbert-Schmidt Operators 424
5.6.5 Discrimination of Gaussian Translates 430
5.6.6 Discrimination of Gaussian Laws 434
5.7 An Extension to Mixtures of Gaussian Laws 445
Part II Cramér-Hida Representations 461
6 Cramér-Hida Representations from ``First Principles'' 462
6.1 Preliminaries 462
6.1.1 Context 462
6.1.2 Functions and Determinism 465
6.1.3 Functions with Orthogonal Increments 467
6.2 Hilbert Space Valued, Countably Additive, Orthogonally Scattered Measures: A Summary of Results 474
6.2.1 General Case 474
Measures 474
The Linear Space of a CAOSM 476
Integration 476
Projection onto the Linear Space of a CAOSM 477
Isomorphisms of Linear Spaces 480
Absolute Continuity 480
Change of Measure 481
An Interchange of Integration Lemma 481
Representation of Functions as Integrals with Respect to a CAOSM 483
6.2.2 Case of Intervals 485
6.3 Boundedness, Limits, and Separability 488
6.4 The Cramér-Hida Representation 494
6.4.1 Canonical Representations 496
6.4.2 Proper Canonical Representations 502
7 Cramér-Hida Representations via Direct Integrals 534
7.1 Direct Integrals 534
7.1.1 Measurable Fields of Hilbert Spaces 535
7.1.2 The Direct Integral: Existence 544
7.1.3 The Direct Integral: Properties 547
7.2 Representations of the Linear Closure of the Range of a Function with Values in a Hilbert Space 550
7.3 Cramér's Representation 554
8 Some Facts About Multiplicity 557
8.1 All Multiplicities May Occur 557
8.2 Invariance of Multiplicity 561
8.2.1 The Case of Projections 561
8.2.2 The Case of Unitary Transformations 565
8.2.3 Other Linear Transformations Which Preserve Multiplicity 568
The First Step: The Finite Sums Case 577
The Second Step: The Limiting Procedure 585
8.3 Smoothness and Multiplicity: Multiplicity One 597
8.3.1 Multiplicity One: Smoothness of Integrands 598
8.3.2 Multiplicity One from the Prediction Process'Behavior 615
8.3.3 Approximation by Processes of Multiplicity One 630
8.4 Smoothness and Multiplicity: Goursat Maps 649
8.4.1 Hilbert Spaces from Matrix Measures 650
8.4.2 Martingales in the Wide Sense 653
8.4.3 Integration with Respect to Cramér-HidaMartingales 667
The Integral with Respect to a Cramér-Hida Martingale 669
Properties of the Integral 670
8.4.4 Multiplicity of Martingales in the Wide Sense 671
8.4.5 Goursat Maps: Definition and Properties 674
8.4.6 Goursat and Markov Maps of Ordern in the Wide Sense 686
8.4.7 Covariance Kernels of Markov Maps of Order n in the Wide Sense 693
8.4.8 Multiplicity One for Goursat Maps 700
Riccati Matrix Differential Equations 720
8.4.9 Goursat Representations with Smooth, Deterministic Part, Are Proper 725
9 Cramér-Hida Representations via the Prediction Process 731
9.1 Introductory Remarks 731
9.2 The Case of Karhunen Representations 734
9.2.1 Notation, Assumptions, and Some Consequences 738
9.2.2 Properties of Knight's Martingales 746
9.2.3 The Cramér-Hida Representationfrom the Prediction Process 764
9.3 Cramér-Hida and Knight's Representations 779
9.3.1 Notation, Modifications to the Assumptions, and Consequences 779
9.3.2 Equalities of ?-Algebras 784
9.3.3 The Index of Stationarity of a Gaussian Process 796
9.3.4 The Case of Finite Multiplicity 814
Part III Likelihoods 820
10 Bench and Tools 822
10.1 Some Terminology, Notation, and Attending Facts 822
10.2 Sets of Measure Zero 824
10.2.1 Adjunction of Sets to ?-Algebras 825
Enlargement with One Set and, Possibly, Its Subsets 825
Enlargement with a Family of Sets and, Possibly, with the Subsets of Those Sets 828
10.2.2 Enlargement with All the Sets of Measure Zero and, Possibly, Their Subsets 834
10.2.3 Restriction to a Subset of the Base Set 835
10.3 Stopping and Change of Times 841
10.3.1 Inverting Monotone Increasing Functions 841
The Case of Distribution Functions 841
The Case of Increasing Paths 848
10.3.2 Change of Time for a Continuous Local Martingale 856
10.4 Exponential Martingales 860
10.4.1 Positive Local Martingales 860
10.4.2 Relations Between Martingalesand Their Exponentials 862
10.5 Random Elements with Values in the Hilbert Space of Sequences 869
10.5.1 Sequence Valued Martingales and Associated Exponentials 870
10.5.2 Sequence Valued Processes with Independent Increments 873
11 Calculus for Cramér-Hida Processes 878
11.1 Integrators: Cramér-Hida Processes 878
11.1.1 The ?-Algebras Generated by a Cramér-HidaProcess 881
11.1.2 The Reproducing Kernel Hilbert Spaceof a Cramér-Hida Process 884
Some Calculus for Operator Valued Functions 891
Manifolds and Subspaces Generated by Functions Whose Values Are Hilbert-Schmidt Operators 892
The Case of B 894
11.2 Families of Integrands 896
11.3 Some Stochastic Integrals and Their Properties 901
11.3.1 Definition of the Integral 902
11.3.2 Properties of the Integral 903
11.3.3 Stochastic Integrals and Change of Space 911
11.4 Local Martingales of a Cramér-Hida Process 915
12 Sample Spaces 929
12.1 Topologies for Sample Spaces 929
12.1.1 Fréchet Spaces 930
The Fréchet Space of Sequences 931
12.1.2 Norms, Quasinorms, and Distances on SampleSpaces 931
12.2 Measurable Subsets of Sample Spaces 934
12.2.1 Evaluation Maps and Borel Sets in the Fréchet Case 934
12.2.2 Evaluation Maps and Borel Sets in the Banach Case 935
12.3 Measures for Sample Spaces 936
13 Likelihoods for Signal Plus ``White Noise'' Versus ``White Noise'' 952
13.1 A Version of Girsanov's Theorem 952
13.1.1 Framework 952
13.1.2 Tools for Absolutely Continuous Changeof Measure 953
13.1.3 A Girsanov Type Theorem 959
13.2 Decomposition of Processes 963
13.3 Likelihoods with Moment Conditions 968
13.4 Likelihoods with Paths Conditions 971
14 Scope of the Signal Plus ``White Noise'' Model (I) 986
14.1 Weak Solutions of Stochastic Differential Equations 986
14.2 A Signal Plus ``White Noise'' Model Is No Restriction 990
15 Scope of the Signal Plus ``White Noise'' Model (II) 997
15.1 The Case of Real Processes 998
15.1.1 Measurable Versions of Conditional Expectations 998
15.1.2 Innovations for Product Square Integrable Signals 1003
15.1.3 Innovations for Product Integrable Signals 1005
15.1.4 Innovations for Signals That Arein the Reproducing Kernel Hilbert Spaceof the Noise 1006
15.2 The Case of Vector Processes 1006
15.3 Strong Solutions of Stochastic Equations 1009
16 Scope of Signal Plus ``White Noise'' Model (III) 1017
16.1 Separable Families of Sets 1017
16.2 Morphisms and Embeddings for Probability Spaces 1020
16.3 Morphisms and Inclusions for Filtrations 1035
16.4 Multiplicity for Algebras of Sets 1036
16.5 Vershik's Lacunary Isomorphism Theorem 1046
16.6 Girsanov's Theorem for Real, Continuous, Local Martingales 1067
16.7 Ocone Martingales 1071
16.7.1 Definitions, Characterization, and Properties 1071
16.7.2 Ocone Martingales and Exponentials 1079
16.7.3 Ocone Martingales, More Properties,and Some Examples 1082
16.8 The Uniqueness Class of Continuous Local Martingales 1088
17 Likelihoods for Signal Plus Gaussian Noise Versus Gaussian Noise 1110
17.1 An Introductory but Instructive Example 1111
17.2 The Cramér-Hida Maps 1118
17.3 Inverse Cramér-Hida Maps 1128
17.3.1 The Inverse for Square Integrable Paths 1128
17.3.2 The Inverses for Real and Continuous Paths 1136
17.4 Absolute Continuity and Likelihoods for the Signal Plus Gaussian Noise Case 1154
17.5 Scope of the Signal Plus Gaussian Noise Model 1159
17.6 From Theory to Practice: Some Comments 1162
17.6.1 Framework 1163
17.6.2 The Approximation's Law for Noise Only 1164
17.6.3 The Approximation's Law for Signal-Plus-Noise 1167
17.6.4 A Recursive Approximation to the Likelihood 1169
17.6.5 Estimating the Drift Parameter Function 1173
17.6.6 Epilogue or …Le mot de la fin 1182
References 1184
Index 1193

Produkt-Details

• Erscheint lt. Verlag: 15.12.2015
• Zusatzinfo: XXXIV, 1176 p. 5 illus.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Cramér-Hida representations • Cramér-Hida representations • dependent signals with arbitrary laws • Girsanov‘s theory • Girsanov‘s theory • Goursat processes • Information and Communication, Circuits • prediction processes • Reproducing kernel Hilbert spaces • Signal Detection • uniqueness class of continuous local martingales
• ISBN-10: 3-319-22315-1 / 3319223151
• ISBN-13: 978-3-319-22315-5 / 9783319223155