%0 Book %1 diaconis1988group %A Diaconis, Persi %B Institute of Mathematical Statistics Lecture Notes---Monograph Series, 11 %C Hayward, CA %D 1988 %I Institute of Mathematical Statistics %K discrete_fourier_transform reference representation_theory %P vi+198 %T Group representations in probability and statistics %U http://projecteuclid.org/euclid.lnms/1215467407 %X The purpose of this nice little book is to show how the mathematical theory of group representations can be used to solve very concrete problems in probability and statistics. It is mainly concerned with noncommutative finite groups. Let us describe two of its typical simple examples. The first one is card shuffling. A deck of cards is repeatedly shuffled. For instance, a random card is transposed with the top one, or two cards chosen at random with each hand are transposed (one can also cut about half the pack and riffle the two packets together). How many shuffles are necessary in order to bring the deck of cards close to random? Repeated shuffling is modeled as convolving a fixed probability measure on the symmetric group. The problem is to find the rate of convergence of these convolution powers to the uniform distribution. Group representations turns the analysis of convolutions into the more manageable analysis of products. This leads to very explicit new results (in Chapter 3). The second example comes from classical analysis of variance. Let $X=\1,\cdots,n\\times\1,\cdots,m\$. Consider real data $x_ij$, $(i,j)X$, which are collected in a two-way layout. The rows represent the levels of a factor and the columns the levels of a second factor. The order in which the rows (and the columns) appear is not of any significance. Therefore it is natural to introduce the group $G=X_nS_m$ acting on $X$. Let $L(X)$ be the set of functions on $X$. A set of data is an element of this space. Let $L(X)=V_0V_1V_2V_3$ be the decomposition of $L(X)$ into irreducible components under the action of $G$. Then projection on $V_0$ gives the grand mean, the projection on $V_1$ gives the influence of rows, that a $V_2$ gives the influence of columns, and that a $V_3$ gives what is left over (residuals). We just recover the usual ANOVA. But, when this spectral analysis is applied to less standard kinds of data, it gives in a systematic way a decomposition into factors which is not so obvious to guess. This book contains 9 chapters. Chapters 2 and 7 develop, in a very friendly way and from first principles, all the tools of group representations which are needed. Chapters 3 and 4 deal with random walks on groups and study problems of the same kind as the card shuffling example above. The other chapters give statistical applications (such as data analysis as in the second example above, partially ranked data, data with values in homogeneous spaces, models). The book contains a very useful annotated bibliography. This book is remarkable. On the one hand it is a research book (most of the material appears in book form here for the first time), using tools from one of the main active fields in ``pure mathematics''. On the other hand, it is very clear and self-contained. Both the pure mathematician and the applied statistician will find pleasure and excitement in reading it. The text is full of attractive examples. It contains many open questions and I am sure that it will be the starting point for new research on the subject. %@ 0-940600-14-5

@book{diaconis1988group, abstract = {The purpose of this nice little book is to show how the mathematical theory of group representations can be used to solve very concrete problems in probability and statistics. It is mainly concerned with noncommutative finite groups. Let us describe two of its typical simple examples. The first one is card shuffling. A deck of cards is repeatedly shuffled. For instance, a random card is transposed with the top one, or two cards chosen at random with each hand are transposed (one can also cut about half the pack and riffle the two packets together). How many shuffles are necessary in order to bring the deck of cards close to random? Repeated shuffling is modeled as convolving a fixed probability measure on the symmetric group. The problem is to find the rate of convergence of these convolution powers to the uniform distribution. Group representations turns the analysis of convolutions into the more manageable analysis of products. This leads to very explicit new results (in Chapter 3). The second example comes from classical analysis of variance. Let $X=\{1,\cdots,n\}\times\{1,\cdots,m\}$. Consider real data $x_{ij}$, $(i,j)\in X$, which are collected in a two-way layout. The rows represent the levels of a factor and the columns the levels of a second factor. The order in which the rows (and the columns) appear is not of any significance. Therefore it is natural to introduce the group $G=X_n\times S_m$ acting on $X$. Let $L(X)$ be the set of functions on $X$. A set of data is an element of this space. Let $L(X)=V_0\oplus V_1\oplus V_2\oplus V_3$ be the decomposition of $L(X)$ into irreducible components under the action of $G$. Then projection on $V_0$ gives the grand mean, the projection on $V_1$ gives the influence of rows, that a $V_2$ gives the influence of columns, and that a $V_3$ gives what is left over (residuals). We just recover the usual ANOVA. But, when this spectral analysis is applied to less standard kinds of data, it gives in a systematic way a decomposition into factors which is not so obvious to guess. This book contains 9 chapters. Chapters 2 and 7 develop, in a very friendly way and from first principles, all the tools of group representations which are needed. Chapters 3 and 4 deal with random walks on groups and study problems of the same kind as the card shuffling example above. The other chapters give statistical applications (such as data analysis as in the second example above, partially ranked data, data with values in homogeneous spaces, models). The book contains a very useful annotated bibliography. This book is remarkable. On the one hand it is a research book (most of the material appears in book form here for the first time), using tools from one of the main active fields in ``pure mathematics''. On the other hand, it is very clear and self-contained. Both the pure mathematician and the applied statistician will find pleasure and excitement in reading it. The text is full of attractive examples. It contains many open questions and I am sure that it will be the starting point for new research on the subject. }, added-at = {2010-02-04T21:40:41.000+0100}, address = {Hayward, CA}, author = {Diaconis, Persi}, biburl = {https://www.bibsonomy.org/bibtex/2933838b3e918380eb835383259944baf/peter.ralph}, description = {MR: Publications results for "MR Number=(964069)"}, interhash = {f8e3d552ac7a2a48d97ec0820620e592}, intrahash = {933838b3e918380eb835383259944baf}, isbn = {0-940600-14-5}, keywords = {discrete_fourier_transform reference representation_theory}, mrclass = {60-02 (20C99 62-02)}, mrnumber = {MR964069 (90a:60001)}, mrreviewer = {Philippe Bougerol}, pages = {vi+198}, publisher = {Institute of Mathematical Statistics}, series = {Institute of Mathematical Statistics Lecture Notes---Monograph Series, 11}, timestamp = {2010-02-04T21:40:41.000+0100}, title = {Group representations in probability and statistics}, url = {http://projecteuclid.org/euclid.lnms/1215467407}, year = 1988 }