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Statistical analysis of circular data
Book reviews/Dynamics of Atmospheres and Oceans 21 (1994) 213-225
215
Five reports on marine chemistry follow the descriptive set. As is typical of chemical oceanography, about one-third of this material concerns measurement techniques. Some interesting features in distributions of more or less exotic chemical constituents are described, but not much of novelty about the deep circulation seems to have been inferred from them. Next are three reports on measurements of particulate fluxes. They will be of more significance to investigations of JGOFS-nature than to direct circulation studies. Three reports on acoustics conclude the book. They are unrelated to anything else (or even to the book's title), because they deal entirely with technical details of acoustic methods, and describe nothing about the ocean. In his introductory chapter Professor Teramoto explained that the motivation for the project was the question, "What is the configuration of the deep circulation of the western North Pacific?" The project did not fully answer this question, of course, but it certainly made some incremental gains, as usefully set out in this book. However, he also stated that the program was "naturally expected to give a solution to the question on what the flow configuration under the Kuroshio is," and I did not perceive that this goal was accomplished. Unfortunately, the publisher has printed several of the figures too small to be read. Most egregious is the set of plots showing distributions of trace elements in the Pacific Ocean (p. 85), which could have been quite interesting, but is in fact quite useless, because the scales are undecipherable to the naked eye, and only a little less cryptic to a hand lens. Elsevier is well known for charging so much for its books as to put them beyond the reach of most individuals. The high price of this one (US$148.50), as well as its varied content, will surely also limit sales. But libraries should probably have it available to their patrons. B.A. WARREN (Woods Hole, MA, USA)
Statistical Analysis of Circular Data, N.I. Fisher. Cambridge University Press, Cambridge, UK, 1993, hardcover, xviii + 277 pp., £35, ISBN 0-521-35018-2. This well-written book deals with the statistical treatment of two-dimensional orientations (angles measured in the plane) which are to be found in most fields of science including geology, geophysics, meteorology and oceanography. In general, directional data are modelled as unit vectors. The length of the vector sum can be converted into the concentration parameter K describing lack of scatter of observations about their vector mean. The Von Mises distribution is as important to circular data as the normal distribution to ordinary (linear) data.
216
Book reviews/Dynamics of Atmospheres and Oceans 21 (1994) 213-225
The book is a companion volume to Statistical Analysis of Spherical Data (Fisher et al., 1987) with methods for handling three-dimensional orientations where the Fisher distribution (introduced for the treatment of palaeomagnetic measurements by R.A. Fisher in 1953) is often used for modelling the density function of the unit vectors. A difference in approach between the two books is that in the current volume more use is made of nonparametric smoothing and bootstrap methods. Also, on page 40, the author remarks that, in his experience, many spherical data sets are well-modelled by Fisher distributions with K/> 2 whereas most circular data sets appear to have underlying distributions with K < 2. In this comparison, K = 2 represents approximately the smallest value for which density at the antipode can be regarded as negligibly small for either the Von Mises or the Fisher distribution. The organization of this book is as follows. Chapter 1: Introduction (contains interesting historical notes); Chapter 2: Descriptive methods (emphasis on useful graphical display); Chapter 3: Models (probability distributions on the circle including the uniform, wrapped normal and Von Mises distributions); Chapter 4: Analysis of a single sample of data (includes estimation of median and mean directions, goodness-of-fit test for Von Mises model, statistical analysis of a random sample of unit vectors from a multimodal distribution); Chapter 5: Analysis of two or more samples, and of other experimental layouts (e.g. tests for common mean direction and concentration parameter of two or more von Mises distributions); Chapter 6: Correlation and regression; Chapter 7: Analysis of data with temporal or spatial structure; Chapter 8: Some modern statistical techniques for testing and estimation. At the end of the book there are two appendices containing 13 statistical tables and the 24 data sets used for examples earlier in the book, references and an index. The analysis of directional data has a long history. Although N.I. Fisher's account in Chapter 1 is interesting, I feel that it is biased and offer the following additions: one of the first averages on record was taken by William Borough in 1581 for a set of compass readings; in 1755, T. Simpson discussed "the method practiced by astronomers to diminish the errors arising from the imperfections of instruments and of the organs of sense by taking the mean of several observations" (cf. Agterberg, 1974). These early workers calculated the arithmetic mean instead of the vector mean. N.I. Fisher (p. 54) comments on the fact that in the past large amounts of data, e.g. on wind directions were summarized by arithmetic means and standard deviations as if the data were linear. He refers to such statistics as "faulty". In defense of early workers including numerous geologists who calculated the arithmetic mean of circular data, I would like to stress that I do not know of a single instance of a scientist unable to solve what N.I. Fisher calls the "notorious cross-over problem familiar to meteorologists" (p. 31). He illustrates this problem by asking the reader to determine the mean of a sample of three points: 359 °, 1°, 3 ° I think that it is safe to assume that early workers would have answered 1° which, incidentally, also is the vector mean. Use of the vector mean and circular/spherical statistics became only widespread about 25 years ago. At the beginning of the period of transition from angles to unit
Book reviews/Dynamics of Atmospheres and Oceans 21 (1994) 213-225
217
vectors, Agterberg and Briggs (1963) proposed the rule that linear instead of circular statistics may be used as long as the maximum angle of deviation between a unit vector and its mean does not exceed 57 ° (or K >/4). Then the arithmetic mean /~ and vector mean /~,,M are nearly equal to one another and approximate confidence intervals (e.g. 95% of deviations lie within/~ + 1.96o,) can be based on the ordinary standard deviation o-. N.I. Fisher introduces a circular standard deviation 0"v,~ for angles of deviation from the vector mean. One of his rules is that 95% of circular data do not deviate more than 2.060"vM if K t> 0.80. This new rule is less stringent than the one based on 0" but the earlier rule included replacement of /~vM by/~. As a general criticism of statistical treatment of circular/spherical data as it is practiced today, it could be said that the methods applied to vectorial data and axial data are very different. Undirected lines such as axes of elongation of pebbles in glacial till or poles of fractures usually can be regarded as directed lines of which the direction can be inferred from other data such as regional patterns of ice movement or stress fields. This would imply that many types of undirected lines can also be modelled as unit vectors provided that the scatter is not too large (e.g. K >/2). N.I. Fisher circumvents problems of this type partly by the use of nonparametric methods. Readers interested in regional patterns such as unit vector fields to be constructed from measurements on scattered directional features will like Chapter 7 (Analysis of data with temporal or spatial structure) which contains new methods of nonparametric smoothing. Under the heading of highlighting trends in a section on exploratory data analysis, the author sets x i = c o s 0 i , y i = sin0 i (i = 1 . . . . . n) where 0 i is a circular measurement at a point labelled i. His basic approach is that x i and Yi are smoothed separately but identically, and afterwards recombined into a smoothed trend pattern of circular data. N.I. Fisher (p. 173) adopted this approach on the basis of personal communication from G.S. Watson. Examples of application include lineament interpretation from a remotely sensed image (figs. 7.3, 7.13 to 7.15). I would like to point out that a version of this approach was used much earlier for fitting unit vector fields to orientations of microfolds in structural geology (Agterberg, 1974, pp. 499-504). Highlighting trends in patterns of directional features can be regarded as a special case of fitting vector fields by least squares (Agterberg, 1985, p. 14). For circular data in the plane, the argument is as follows. Suppose that for every unit vector Ui = { x i , Y i } , there exists a vector F i = {m(xi),m(Yi)} which belongs to a continuous vector field (V). Minimizing the sum of squares of the lengths of the difference vectors Ui-Fi yields estimates of ~ which can later be normalized. Various methods of spatial smoothing including those discussed by N.I. Fisher can be used for estimating the vectors V at all points in a study area. This volume has been prepared with great care. It is difficult to find mistakes in it (on p, 55, line 3: r >/1.12 should read K >/0.35). It can be recommended as a handbook for those working with circular data. Additionally, it will be of general
Book reviews/Dynamics of Atraospheres and Oceans 21 (1994) 213-225
218
interest to mathematically-inclined earth scientists as a model for computer-assisted statistical data analysis. F R E D E R I K P. A G T E R B E R G (Ottawa, Ont., Canada)
References Agterherg, F.P., 1974. Geomathematics. Elsevier, Amsterdam, 596 pp. Agterberg, F.P., 1985. Spatial analysis in the earth sciences. In: P.S Glaeser (Editor), The Role of Data in Scientific Progress. Elsevier, Amsterdam, pp. 9-18. Agterberg, F.P., and Briggs, G., 1963. Statistical analysis of ripple marks in Atokan and Desmoinesian rocks in the Arkoma basin of east-central Oklahoma. J. Sed. Pet., 33(2):393-410. Fisher, N.I., Lewis, T., and Embleton, B.J.J., 1987. Statistical Analysis of Spherical Data~ Cambridge University Press, Cambridge, UK, 329 pp.
Ocean Energies. Environmental, Economic and Technological Aspects of Alternative Power Sources, R.H. Charlie and J.R. Justup, Elsevier Oceanography Series, Amsterdam 1993, hardcover XIX + 534 pp., ISBN 0-444-88248-0. This very comprehensive book covers all known projectors or suggestions on the extraction of energy from a variety of known ocean sources whether they are tides, currents waves, winds, thermal, geothermal, salinity or biomass. These are treated very thoroughly in the text, figures and photographs. In the preface it is stated that the contributions by tidal energy to the overall ready in energy may be increased and several other ocean sources are possible. For some, e.g. waves, hundreds of patents for energy extraction have been taken out and some (usually pilot) plants have been built. While most wave projects suffer from problems on concentration of wave energy some have tried to circumvent the problem by various means of concentrative measures like funnels. Other focus their interest on large bulky installations, which are difficult to build and operate and consequently become uneconomical. Utilizing the difference in temperatures, of surface and deeper waters is "an old dream", which originated to O T E C projects, thoroughly dealt with in the book including some mini-OTEC projects that proved to be economically feasible for small local schemes, with some Japanese projects moving ahead of other schemes worldwide. For centuries winds have provided power for lifting of water and during recent decades also for power plants comprising a great number of windmills for the production of power. Well known are the Dutch and Danish windmill parks, now followed by Californian ones producing not negligible amounts of power.
215
Five reports on marine chemistry follow the descriptive set. As is typical of chemical oceanography, about one-third of this material concerns measurement techniques. Some interesting features in distributions of more or less exotic chemical constituents are described, but not much of novelty about the deep circulation seems to have been inferred from them. Next are three reports on measurements of particulate fluxes. They will be of more significance to investigations of JGOFS-nature than to direct circulation studies. Three reports on acoustics conclude the book. They are unrelated to anything else (or even to the book's title), because they deal entirely with technical details of acoustic methods, and describe nothing about the ocean. In his introductory chapter Professor Teramoto explained that the motivation for the project was the question, "What is the configuration of the deep circulation of the western North Pacific?" The project did not fully answer this question, of course, but it certainly made some incremental gains, as usefully set out in this book. However, he also stated that the program was "naturally expected to give a solution to the question on what the flow configuration under the Kuroshio is," and I did not perceive that this goal was accomplished. Unfortunately, the publisher has printed several of the figures too small to be read. Most egregious is the set of plots showing distributions of trace elements in the Pacific Ocean (p. 85), which could have been quite interesting, but is in fact quite useless, because the scales are undecipherable to the naked eye, and only a little less cryptic to a hand lens. Elsevier is well known for charging so much for its books as to put them beyond the reach of most individuals. The high price of this one (US$148.50), as well as its varied content, will surely also limit sales. But libraries should probably have it available to their patrons. B.A. WARREN (Woods Hole, MA, USA)
Statistical Analysis of Circular Data, N.I. Fisher. Cambridge University Press, Cambridge, UK, 1993, hardcover, xviii + 277 pp., £35, ISBN 0-521-35018-2. This well-written book deals with the statistical treatment of two-dimensional orientations (angles measured in the plane) which are to be found in most fields of science including geology, geophysics, meteorology and oceanography. In general, directional data are modelled as unit vectors. The length of the vector sum can be converted into the concentration parameter K describing lack of scatter of observations about their vector mean. The Von Mises distribution is as important to circular data as the normal distribution to ordinary (linear) data.
216
Book reviews/Dynamics of Atmospheres and Oceans 21 (1994) 213-225
The book is a companion volume to Statistical Analysis of Spherical Data (Fisher et al., 1987) with methods for handling three-dimensional orientations where the Fisher distribution (introduced for the treatment of palaeomagnetic measurements by R.A. Fisher in 1953) is often used for modelling the density function of the unit vectors. A difference in approach between the two books is that in the current volume more use is made of nonparametric smoothing and bootstrap methods. Also, on page 40, the author remarks that, in his experience, many spherical data sets are well-modelled by Fisher distributions with K/> 2 whereas most circular data sets appear to have underlying distributions with K < 2. In this comparison, K = 2 represents approximately the smallest value for which density at the antipode can be regarded as negligibly small for either the Von Mises or the Fisher distribution. The organization of this book is as follows. Chapter 1: Introduction (contains interesting historical notes); Chapter 2: Descriptive methods (emphasis on useful graphical display); Chapter 3: Models (probability distributions on the circle including the uniform, wrapped normal and Von Mises distributions); Chapter 4: Analysis of a single sample of data (includes estimation of median and mean directions, goodness-of-fit test for Von Mises model, statistical analysis of a random sample of unit vectors from a multimodal distribution); Chapter 5: Analysis of two or more samples, and of other experimental layouts (e.g. tests for common mean direction and concentration parameter of two or more von Mises distributions); Chapter 6: Correlation and regression; Chapter 7: Analysis of data with temporal or spatial structure; Chapter 8: Some modern statistical techniques for testing and estimation. At the end of the book there are two appendices containing 13 statistical tables and the 24 data sets used for examples earlier in the book, references and an index. The analysis of directional data has a long history. Although N.I. Fisher's account in Chapter 1 is interesting, I feel that it is biased and offer the following additions: one of the first averages on record was taken by William Borough in 1581 for a set of compass readings; in 1755, T. Simpson discussed "the method practiced by astronomers to diminish the errors arising from the imperfections of instruments and of the organs of sense by taking the mean of several observations" (cf. Agterberg, 1974). These early workers calculated the arithmetic mean instead of the vector mean. N.I. Fisher (p. 54) comments on the fact that in the past large amounts of data, e.g. on wind directions were summarized by arithmetic means and standard deviations as if the data were linear. He refers to such statistics as "faulty". In defense of early workers including numerous geologists who calculated the arithmetic mean of circular data, I would like to stress that I do not know of a single instance of a scientist unable to solve what N.I. Fisher calls the "notorious cross-over problem familiar to meteorologists" (p. 31). He illustrates this problem by asking the reader to determine the mean of a sample of three points: 359 °, 1°, 3 ° I think that it is safe to assume that early workers would have answered 1° which, incidentally, also is the vector mean. Use of the vector mean and circular/spherical statistics became only widespread about 25 years ago. At the beginning of the period of transition from angles to unit
Book reviews/Dynamics of Atmospheres and Oceans 21 (1994) 213-225
217
vectors, Agterberg and Briggs (1963) proposed the rule that linear instead of circular statistics may be used as long as the maximum angle of deviation between a unit vector and its mean does not exceed 57 ° (or K >/4). Then the arithmetic mean /~ and vector mean /~,,M are nearly equal to one another and approximate confidence intervals (e.g. 95% of deviations lie within/~ + 1.96o,) can be based on the ordinary standard deviation o-. N.I. Fisher introduces a circular standard deviation 0"v,~ for angles of deviation from the vector mean. One of his rules is that 95% of circular data do not deviate more than 2.060"vM if K t> 0.80. This new rule is less stringent than the one based on 0" but the earlier rule included replacement of /~vM by/~. As a general criticism of statistical treatment of circular/spherical data as it is practiced today, it could be said that the methods applied to vectorial data and axial data are very different. Undirected lines such as axes of elongation of pebbles in glacial till or poles of fractures usually can be regarded as directed lines of which the direction can be inferred from other data such as regional patterns of ice movement or stress fields. This would imply that many types of undirected lines can also be modelled as unit vectors provided that the scatter is not too large (e.g. K >/2). N.I. Fisher circumvents problems of this type partly by the use of nonparametric methods. Readers interested in regional patterns such as unit vector fields to be constructed from measurements on scattered directional features will like Chapter 7 (Analysis of data with temporal or spatial structure) which contains new methods of nonparametric smoothing. Under the heading of highlighting trends in a section on exploratory data analysis, the author sets x i = c o s 0 i , y i = sin0 i (i = 1 . . . . . n) where 0 i is a circular measurement at a point labelled i. His basic approach is that x i and Yi are smoothed separately but identically, and afterwards recombined into a smoothed trend pattern of circular data. N.I. Fisher (p. 173) adopted this approach on the basis of personal communication from G.S. Watson. Examples of application include lineament interpretation from a remotely sensed image (figs. 7.3, 7.13 to 7.15). I would like to point out that a version of this approach was used much earlier for fitting unit vector fields to orientations of microfolds in structural geology (Agterberg, 1974, pp. 499-504). Highlighting trends in patterns of directional features can be regarded as a special case of fitting vector fields by least squares (Agterberg, 1985, p. 14). For circular data in the plane, the argument is as follows. Suppose that for every unit vector Ui = { x i , Y i } , there exists a vector F i = {m(xi),m(Yi)} which belongs to a continuous vector field (V). Minimizing the sum of squares of the lengths of the difference vectors Ui-Fi yields estimates of ~ which can later be normalized. Various methods of spatial smoothing including those discussed by N.I. Fisher can be used for estimating the vectors V at all points in a study area. This volume has been prepared with great care. It is difficult to find mistakes in it (on p, 55, line 3: r >/1.12 should read K >/0.35). It can be recommended as a handbook for those working with circular data. Additionally, it will be of general
Book reviews/Dynamics of Atraospheres and Oceans 21 (1994) 213-225
218
interest to mathematically-inclined earth scientists as a model for computer-assisted statistical data analysis. F R E D E R I K P. A G T E R B E R G (Ottawa, Ont., Canada)
References Agterherg, F.P., 1974. Geomathematics. Elsevier, Amsterdam, 596 pp. Agterberg, F.P., 1985. Spatial analysis in the earth sciences. In: P.S Glaeser (Editor), The Role of Data in Scientific Progress. Elsevier, Amsterdam, pp. 9-18. Agterberg, F.P., and Briggs, G., 1963. Statistical analysis of ripple marks in Atokan and Desmoinesian rocks in the Arkoma basin of east-central Oklahoma. J. Sed. Pet., 33(2):393-410. Fisher, N.I., Lewis, T., and Embleton, B.J.J., 1987. Statistical Analysis of Spherical Data~ Cambridge University Press, Cambridge, UK, 329 pp.
Ocean Energies. Environmental, Economic and Technological Aspects of Alternative Power Sources, R.H. Charlie and J.R. Justup, Elsevier Oceanography Series, Amsterdam 1993, hardcover XIX + 534 pp., ISBN 0-444-88248-0. This very comprehensive book covers all known projectors or suggestions on the extraction of energy from a variety of known ocean sources whether they are tides, currents waves, winds, thermal, geothermal, salinity or biomass. These are treated very thoroughly in the text, figures and photographs. In the preface it is stated that the contributions by tidal energy to the overall ready in energy may be increased and several other ocean sources are possible. For some, e.g. waves, hundreds of patents for energy extraction have been taken out and some (usually pilot) plants have been built. While most wave projects suffer from problems on concentration of wave energy some have tried to circumvent the problem by various means of concentrative measures like funnels. Other focus their interest on large bulky installations, which are difficult to build and operate and consequently become uneconomical. Utilizing the difference in temperatures, of surface and deeper waters is "an old dream", which originated to O T E C projects, thoroughly dealt with in the book including some mini-OTEC projects that proved to be economically feasible for small local schemes, with some Japanese projects moving ahead of other schemes worldwide. For centuries winds have provided power for lifting of water and during recent decades also for power plants comprising a great number of windmills for the production of power. Well known are the Dutch and Danish windmill parks, now followed by Californian ones producing not negligible amounts of power.