Société de Calcul Mathématique, SA
Mathematical Modelling Company Corp.

A probability book for Engineers and Researchers:

Probabilistic Information Transfer

We are glad to introduce the new book by Olga Zeydina and Bernard Beauzamy:


Probabilistic Information Transfer


A book you will be proud to have for yourself and to show to your friends !

ISBN: 978-2-9521458-6-2, ISSN : 1767-1175. Size 15,3 x 24 cm. Hardcover, 208 pages.

In real life situations, one rarely has desirably detailed information. It is sometimes incomplete, sometimes corrupted, or with missing or erroneous data. Conversely, some pieces of information do exist. Therefore, there is a natural wish: to try to use the existing information in order to reconstruct some missing items. However, this should be done with two constraints:

  • First, one should not add any artificial information, such as model assumptions (for instance, that some growth is linear, or that some law is gaussian) ;
  • Second, the result should be of probabilistic nature: we do not want a precise value for the reconstruction, but a probability law, which allows estimation of the uncertainties.
This is precisely the topic of this book. We show how to "propagate" the information, from a place where it exists to a place where we want to use it; this propagation deteriorates with the distance, somewhat as a gravitational field decreases with the distance.

The book is organized in three parts: the first part presents the basic rules, accessible with no specific expertise in probabilities; the second presents the applications to real world problems, and the third part gives the theory.

This theory comes from a situation which is rather rare these days: a new mathematical theory, entirely developed by SCM in order to meet a need which was originally expressed by the Industry, namely Framatome ANP (today Areva) in 2003. Since then, the tool was used in many cases: classifying industrial objects (Air Liquide), evaluation of pollutions (Total), estimates for water quality in European rivers (European Environment Agency), applications to the safety of nuclear reactors (Institut de Radioprotection et de Sûreté Nucléaire), and so on.

About the authors :

  • Olga Zeydina, PhD in Probabilistic Methods for Nuclear Safety, has been Research Engineer at the Société de Calcul Mathématique SA, since 2006.
  • Bernard Beauzamy, University Professor (1979-1995), founded SCM SA in 1995 and since then has been Chairman of this company.

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    Table of contents

    I. Introduction 9
    First Part - General Presentation 15
    Chapter I - Presentation of the tool 17
    I. Introduction 17
    II. Construction of the EPH in one dimension 18
    A. Basic construction 18
    B. An example of application of the EPH 21
    III. Construction of the EPH in any dimension 24
    A. Minimal understanding of the situation 24
    B. The distance 25
    C. Changes with respect to the 1-dimensional situation 25
    Chapter II - Taking into account the uncertainties 33
    I. Position of the problem 33
    II. A simple example 34
    III. Precautions to be taken 35
    IV. Taking into account many uncertainties 35
    Chapter III - From local to global estimates 37
    I. Introduction 37
    II. Investigation depending on the dimension 37
    A. Low dimension 37
    B. High dimension 38
    III. A simple example in low dimension 40
    Chapter IV - The EPH in non-homogeneous situations 45
    I. Introduction 45
    II. Notation 45
    III. Construction of the EPH in a one-dimensional space 46
    A. Before the first measurement 46
    B. After the first measurement 47
    C. When several measurements have been made 48
    IV. A simple example 50
    V. Construction of the non-homogeneous EPH in any dimension51
    VI. A simple example 53
    Chapter V - Putting weights upon the parameters 57
    I. Introduction 57
    II. Taking into account previous information 58
    III. Propagating the information 58
    IV. Constructing a non-isotropic EPH 59
    A. The formulas 59
    B. Practical construction 60
    Chapter VI - How to choose the data ? 63
    I. Introduction 63
    II. Different pasts 65
    III. Trends and randomness 66
    IV. Position and speed 69
    Chapter VII - One source, many sensors 73
    I. Presentation 73
    II. Mathematical description of the problem 73
    A. Transfer function 73
    B. Fusion of data 74
    III. A simple example 75
    IV. Other procedures 76

    Second Part - Applications of the EPH 79
    Chapter I - The use of the EPH in Epidemiology 81
    I. Introduction 81
    II. Traditional methods 81
    A. Linear regression 81
    B. Probability law 82
    III. Description of the problem 82
    IV. Building the EPH 83
    A. Specific construction 83
    B. Computing the parameter lambda 84
    C. Fixing the parameters 85
    V. Computation using EPH 85
    VI. Study of increments 87
    Chapter II - Applications to an Industrial Forecast 91
    I. Introduction 91
    II. Difference between statistics and probabilities 91
    III. General comments about the use of statistics by the Industry 92
    A. Regression Method 93
    B. Index Point method 94
    IV. Better methods 96
    A. Variable Weight Method 96
    B. The EPH 97
    Chapter III - Estimating the pollution in a harbour 103
    I. Description of the problem 103
    II. Traditional methods 103
    III. Using the EPH 104
    A. Estimate of the pollution in the first layer 105
    B. Estimates of the pollution for the second layer 107
    C. Where are the risk zones ? 110
    D. Deciding future measurements 112
    Chapter IV - Construction of a proximity index between Industrial Objects 115
    I. Description of the problem 115
    II. A simple example 116
    A. Non-numerical values 117
    B. Normalization of the parameters 117
    C. Taking into account the importance of each parameter 118
    D. The distance between two objects 118
    E. Construction of the proximity index 119
    F. Propagation of the information using the EPH 120
    Chapter V - The EPH and the safety of nuclear reactors 123
    I. Description of the problem 123
    II. Two dimensional problem 125
    A. Step 1: transfer coefficients 126
    B. Step 2: taking uncertainties into account 128
    C. Step 3: introducing the EPH 130
    D. Recombining the collectrons 133
    E. Estimating the uncertainties 133
    III. Three dimensional problem 134

    Third Part - Technical Construction 135
    Chapter I - General construction of the EPH 137
    I. Introduction 137
    II. Notation 138
    III. Building the EPH in the one-dimensional case 138
    A. Before the first measurement 138
    B. After the first measurement 139
    C. Two measurements, one dimension 154
    D. Any number of measurements, one dimension 158
    IV. Building the EPH in the multi-dimensional case 162
    A. Bounds for each parameter 162
    B. Preliminary normalization of the parameters 162
    C. Densities after normalization 163
    D. Case of 2 parameters. 164
    E. General Case: K parameters and N measurements 165
    V. Questions about the model 172
    A. Discrete and continuous entropy 172
    B. Dependence on the discretization 175
    C. Dependence on the min and max values 176
    Chapter II - Looking for dangerous zones 177
    I. Introduction 177
    II. Presentation of the problem 178
    III. Dangerous zones 179
    IV. Search for an extreme point 183
    V. Random search 184
    A. Preliminary warning 184
    B. Uniform distribution on the unit sphere 188
    C. Random search for dangerous zones 190
    Chapter III - Ranking the parameters 193
    I. Introduction 193
    II. Practical approach 194
    Chapter IV - Existing techniques 199
    I. Introduction 199
    II. Deterministic techniques 199
    III. Probabilistic Techniques 200
    A. Kriging Interpolation 200
    B. Use of other information 201
    References 203
    Credits 205


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