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6.1: Introduction to Non-Brownian Motion - Biology


Detailed studies of contemporary evolution have revealed a rich variety of processes that influence how traits evolve through time. Consider the famous studies of Darwin’s finches, Geospiza, in the Galapagos islands carried out by Peter and Rosemary Grant, among others (e.g. Grant and Grant 2011). These studies have documented the action of natural selection on traits from one generation to the next. One can see very clearly how changes in climate – especially the amount of rainfall – affect the availability of different types of seeds (Grant and Grant 2002). These changing resources in turn affect which individuals survive within the population. When natural selection acts on traits that can be inherited from parents to offspring, those traits evolve.

One can obtain a dataset of morphological traits, including measurements of body and beak size and shape, along with a phylogenetic tree for several species of Darwin’s finches. Imagine that you have the goal of analyzing the tempo and mode of morphological evolution across these species of finch. We can start by fitting a Brownian motion model to these data. However, a Brownian model (which, as we learned in Chapter 3, corresponds to a few simple scenarios of trait evolution) hardly seems realistic for a group of finches known to be under strong and predictable directional selection.

Brownian motion is very commonly used in comparative biology: in fact, a large number of comparative methods that researchers use for continuous traits assumes that traits evolve under a Brownian motion model. The scope of other models beyond Brownian motion that we can use to model continuous trait data on trees is somewhat limited. However, more and more methods are being developed that break free of this limitation, moving the field beyond Brownian motion. In this chapter I will discuss these approaches and what they can tell us about evolution. I will also describe how moving beyond Brownian motion can point the way forward for statistical comparative methods.

In this chapter, I will consider four ways that comparative methods can move beyond simple Brownian motion models: by transforming the variance-covariance matrix describing trait covariation among species, by incorporating variation in rates of evolution, by accounting for evolutionary constraints, and by modeling adaptive radiation and ecological opportunity. It should be apparent that the models listed here do not span the complete range of possibilities, and so my list is not meant to be comprehensive. Instead, I hope that readers will view these as examples, and that future researchers will add to this list and enrich the set of models that we can fit to comparative data.


6.1: Introduction to Non-Brownian Motion - Biology

The process of diffusion is the most elementary stochastic transport process. Brownian motion, the representative model of diffusion, played a important role in the advancement of scientific fields such as physics, chemistry, biology and finance. However, in recent decades, non-diffusive transport processes with non-Brownian statistics were observed experimentally in a multitude of scientific fields. Examples include human travel, in-cell dynamics, the motion of bright points on the solar surface, the transport of charge carriers in amorphous semiconductors, the propagation of contaminants in groundwater, the search patterns of foraging animals and the transport of energetic particles in turbulent plasmas. These examples showed that the assumptions of the classical diffusion paradigm, assuming an underlying uncorrelated (Markovian), Gaussian stochastic process, need to be relaxed to describe transport processes exhibiting a non-local character and exhibiting long-range correlations. This article does not aim at presenting a complete review of non-diffusive transport, but rather an introduction for readers not familiar with the topic. For more in depth reviews, we recommend some references in the following. First, we recall the basics of the classical diffusion model and then we present two approaches of possible generalizations of this model: the Continuous-Time-Random-Walk (CTRW) and the fractional Lévy motion (fLm).


Contents

Within the scientific literature the term 1/f noise is sometimes used loosely to refer to any noise with a power spectral density of the form

where f is frequency, and 0 < α < 2, with exponent α usually close to 1. The canonical case with α = 1 is called pink noise. [3] General 1/f α -like noises occur widely in nature and are a source of considerable interest in many fields. The distinction between the noises with α near 1 and those with a broad range of α approximately corresponds to a much more basic distinction. The former (narrow sense) generally come from condensed-matter systems in quasi-equilibrium, as discussed below. [4] The latter (broader sense) generally correspond to a wide range of non-equilibrium driven dynamical systems.

Pink noise sources include flicker noise in electronic devices. In their study of fractional Brownian motion, [5] Mandelbrot and Van Ness proposed the name fractional noise (sometimes since called fractal noise) to describe 1/f α noises for which the exponent α is not an even integer, [6] or that are fractional derivatives of Brownian (1/f 2 ) noise.

In pink noise, there is equal energy in all octaves (or similar log bundles) of frequency. In terms of power at a constant bandwidth, pink noise falls off at 3 dB per octave. At high enough frequencies pink noise is never dominant. (White noise has equal energy per frequency interval.)

The human auditory system, which processes frequencies in a roughly logarithmic fashion approximated by the Bark scale, does not perceive different frequencies with equal sensitivity signals around 1–4 kHz sound loudest for a given intensity. However, humans still differentiate between white noise and pink noise with ease.

Graphic equalizers also divide signals into bands logarithmically and report power by octaves audio engineers put pink noise through a system to test whether it has a flat frequency response in the spectrum of interest. Systems that do not have a flat response can be equalized by creating an inverse filter using a graphic equalizer. Because pink noise has a tendency to occur in natural physical systems, it is often useful in audio production. Pink noise can be processed, filtered, and/or effects can be added to produce desired sounds. Pink-noise generators are commercially available.

One parameter of noise, the peak versus average energy contents, or crest factor, is important for testing purposes, such as for audio power amplifier and loudspeaker capabilities because the signal power is a direct function of the crest factor. Various crest factors of pink noise can be used in simulations of various levels of dynamic range compression in music signals. On some digital pink-noise generators the crest factor can be specified.

The power spectrum of pink noise is 1 / f only for one-dimensional signals. For two-dimensional signals (e.g., images) the power spectrum is reciprocal to f 2 In general, in an n-dimensional system, the power spectrum is reciprocal to f n . For higher-dimensional signals it is still true (by definition) that each octave carries an equal amount of noise power. The frequency spectrum of two-dimensional signals, for instance, is also two-dimensional, and the area of the power spectrum covered by succeeding octaves is four times as large.

In the past quarter century, pink noise has been discovered in the statistical fluctuations of an extraordinarily diverse number of physical and biological systems (Press, 1978 [7] see articles in Handel & Chung, 1993, [8] and references therein). Examples of its occurrence include fluctuations in tide and river heights, quasar light emissions, heart beat, firings of single neurons, and resistivity in solid-state electronics resulting in flicker noise.

General 1/f α noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous. [9] In physical systems, they are present in some meteorological data series, the electromagnetic radiation output of some astronomical bodies. In biological systems, they are present in, for example, heart beat rhythms, neural activity, and the statistics of DNA sequences, as a generalized pattern. [10] In financial systems, they are often referred to as a long-term memory effect [ specify ] .

An accessible introduction to the significance of pink noise is one given by Martin Gardner (1978) in his Scientific American column "Mathematical Games". [11] In this column, Gardner asked for the sense in which music imitates nature. Sounds in nature are not musical in that they tend to be either too repetitive (bird song, insect noises) or too chaotic (ocean surf, wind in trees, and so forth). The answer to this question was given in a statistical sense by Voss and Clarke (1975, 1978), who showed that pitch and loudness fluctuations in speech and music are pink noises. [12] [13] So music is like tides not in terms of how tides sound, but in how tide heights vary.

Pink noise describes the statistical structure of many natural images. [14] Recently, it has also been successfully applied to the modeling of mental states in psychology, [15] and used to explain stylistic variations in music from different cultures and historic periods. [16] Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of pitches, will tend towards a pink noise spectrum. [17] Similarly, a generally pink distribution pattern has been observed in film shot length by researcher James E. Cutting of Cornell University, in the study of 150 popular movies released from 1935 to 2005. [18]

Pink noise has also been found to be endemic in human response. Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals. [19] Later, Gilden (1997) and Gilden (2001) found that time series formed from reaction time measurement and from iterated two-alternative forced choice also produced pink noises. [20] [21]

Electronic devices Edit

The principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices. In many cases the specific sources of the fluctuations are known. These include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials. [4] [22] The explanation for the approximately pink spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes. [23] Since the frequency range of the typical noise experiment (e.g., 1 Hz – 1 kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 10 14 Hz), the exponential factors in the Arrhenius equation for the rates are large. Relatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates. In the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because d d f ln ⁡ f = 1 f . >ln f=>.>

There is no known lower bound to background pink noise in electronics. Measurements made down to 10 −6 Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour. [24]

A pioneering researcher in this field was Aldert van der Ziel. [25]

A pink-noise source is sometimes deliberately included on analog synthesizers (although a white-noise source is more common), both as a useful audio sound source for further processing and as a source of random control voltages for controlling other parts of the synthesizer. [ citation needed ]

In gravitational wave astronomy Edit

1/f α noises with α near 1 are a factor in gravitational-wave astronomy. The noise curve at very low frequencies affect pulsar timing arrays, the European Pulsar Timing Array (EPTA) and the future International Pulsar Timing Array (IPTA) at low frequencies are space-borne detectors, the formerly proposed Laser Interferometer Space Antenna (LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initial Laser Interferometer Gravitational-Wave Observatory (LIGO) and its advanced configuration (aLIGO). The characteristic strain of potential astrophysical sources are also shown. To be detectable the characteristic strain of a signal must be above the noise curve. [26]

Climate change Edit

Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the climate system. [27]

Diffusion processes Edit

Many time-dependent stochastic processes are known to exhibit 1/f α noises with α between 0 and 2. In particular Brownian motion has a power spectral density that equals 4D/f 2 , [28] where D is the diffusion coefficient. This type of spectrum is sometimes referred to as Brownian noise. Interestingly, the analysis of individual Brownian motion trajectories also show 1/f 2 spectrum, albeit with random amplitudes. [29] Fractional Brownian motion with Hurst exponent H also show 1/f α power spectral density with α=2H+1 for subdiffusive processes (H<0.5) and α=2 for superdiffusive processes (0.5<H<1). [30]

There are many theories of the origin of pink noise. Some theories attempt to be universal, while others are applicable to only a certain type of material, such as semiconductors. Universal theories of pink noise remain a matter of current research interest.

A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to the central limit theorem of statistics. [31] The Tweedie convergence theorem [32] describes the convergence of certain statistical processes towards a family of statistical models known as the Tweedie distributions. These distributions are characterized by a variance to mean power law, that have been variously identified in the ecological literature as Taylor's law [33] and in the physics literature as fluctuation scaling. [34] When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa. [31] Both of these effects can be shown to be the consequence of mathematical convergence such as how certain kinds of data will converge towards the normal distribution under the central limit theorem. This hypothesis also provides for an alternative paradigm to explain power law manifestations that have been attributed to self-organized criticality. [35]

There are various mathematical models to create pink noise. Although self-organised criticality has been able to reproduce pink noise in sandpile models, these do not have a Gaussian distribution or other expected statistical qualities. [36] [37] It can be generated on computer, for example, by filtering white noise, [38] [39] [40] inverse Fourier transform, [41] or by multirate variants on standard white noise generation. [13] [11]


Retained and Deleted Syllabus –CHEMISTRY


Unit 1 Some Basic Concepts of Chemistry

RETAINED PORTIONDELETED PORTION
1.1 Importance of Chemistry
1.3 Properties of Matter and their Measurement
1.4 Uncertainty in Measurement
1.7 Atomic and Molecular Masses
1.8 Mole Concept and Molar Masses
1.9 Percentage Composition
1.10 Stoichiometry and Stoichiometric
Calculations
1.2 Nature of matter,
1.5 Laws of chemical combination,
1.6 Dalton’s atomic theory

UNIT 2 STRUCTURE OF ATOM

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2.3 Developments Leading to the Bohr’s
Model of Atom
2.4 Bohr’s Model for Hydrogen Atom
2.5 Towards Quantum Mechanical Model of
the Atom
2.6 Quantum Mechanical Model of Atom
2.1 Discovery of Subatomic Particles
2.2 Atomic Models

UNIT 3 CLASSIFICATION OF ELEMENTS AND PERIODICITY IN PROPERTIES

RETAINED PORTIONDELETED PORTION
3.3 Modern Periodic Law and the present
form of the Periodic Table
3.4 Nomen cloture of Elements with
Atomic Numbers > 100
3.5 Electronic Configurations of Elements
and the Periodic Table
3.6 Electronic Configurations and Types
of Elements: s-, p-, d-, f Blocks
3.7 Periodic Trends in Properties of
Elements
Significance of classification, brief history of
the development of periodic table.

Unit 4 Chemical Bonding and Molecular Structure

RETAINED PORTIONDELETED PORTION
4.1 Kössel- Lewis Approach to Chemical
Bonding
4.2 Ionic or Electrovalent Bond
4.3 Bond Parameters
4.4 The Valence Shell Electron Pair Repulsion
(VSEPR) Theory
4.5 Valence Bond Theory
4.6 Hybridisation
4.7 Molecular Orbital Theory
4.8 Bonding in Some Homonuclear Diatomic Molecules
4.9 Hydrogen Bonding
Nil

Unit 5 States of Matter

RETAINED PORTIONDELETED PORTION
5.1 Intermolecular Forces
5.2 Thermal Energy
5.3 Intermolecular Forces vs Thermal Interactions
5.4 The Gaseous State
5.5 The Gas Laws
5.6 Ideal Gas Equation
5.8 Kinetic Molecular Theory of Gases
5.9 Behaviour of Real Gases: Deviation from
Ideal Gas Behaviour
liquefaction of gases, critical
temperature, kinetic energy and
molecular speeds (elementary
idea), Liquid State- vapour
pressure, viscosity and surface
tension (qualitative idea only, no
mathematical derivations)

Unit 6 Thermodynamics

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6.1 Thermodynamic Terms
6.2 Applications
6.3 Measurement of ΔUand ΔH: Calorimetry
6.4 Enthalpy Change, ΔrH of a Reaction Reaction Enthalpy
6.5 Enthalpies for Different Types of Reactions
6.6 Spontaneity
Heat capacity and specific heat capacity,
Criteria for equilibrium

Unit 7 Equilibrium

RETAINED PORTIONDELETED PORTION
7.1 Equilibrium in Physical Processes
7.2 Equilibrium in Chemical Processes –
Dynamic Equilibrium
7.3 Law of Chemical Equilibrium and
Equilibrium Constant
7.4 Homogeneous Equilibria
7.5 Heterogeneous Equilibria
7.6 Applications of Equilibrium Constants
7.7 Relationship between Equilibrium Constant K,
Reaction Quotient Q and Gibbs Energy G
7.8 Factors Affecting Equilibria
7.9 Ionic Equilibrium in Solution
7.10 Acids, Bases and Salts
7.11 Ionization of Acids and Bases
7.12 Buffer Solutions
7.13 Solubility Equilibria of Sparingly Soluble
Salts
hydrolysis of salts (elementary idea),
Henderson Equation

Unit 8 Redox Reactions

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8.1 Classical Idea of Redox Reactions,
Oxidation and Reduction Reactions
8.2 Redox Reactions in Terms of Electron
Transfer Reactions
8.3 Oxidation Number
Applications of redox reactions

Unit 9 Hydrogen

RETAINED PORTIONDELETED PORTION
9.1 Position of Hydrogen in the Periodic Table
9.5 Hydrides
9.6 Water
9.8 Heavy Water, DO
9.9 Dihydrogen as a Fuel
Preparation, properties and uses of
hydrogen, hydrogen peroxide-epreparation,
reactions and structure and use

Unit 10 The s Block Elements

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10.1 Group1 Elements: Alkali Metals
10.2 General Characteristics of the Compounds
of the Alkali Metals
10.3 Anomalous Properties of Lithium
10.6 Group 2 Elements : Alkaline Earth Metals
10.7 General Characteristics of Compounds of
the Alkaline Earth Metals
10.8 Anomalous Behaviour of Beryllium
10.9 Some Important Compounds of Calcium Ca (OH)2 , CaSO4
Preparation and Properties of Some
Important Compounds:
Sodium Carbonate, Sodium Chloride,
Sodium Hydroxide and Sodium Hydrogen
carbonate, Biological importance of
Sodium and Potassium. Calcium Oxide
and Calcium Carbonate and their
industrial uses, biological importance of
Magnesium and Calcium.

Unit 11 The p Block Elements

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11.1 Group 13 Elements: The Boron Family
11.2 Important Trends and Anomalous
Properties of Boron
11.4 Uses of Boron and Aluminium and their
Compounds
11.5 Group 14 Elements: The Carbon Family
11.6 Important Trends and Anomalous
Behaviour of Carbon
11.7 Allotropes of Carbon
Some important compounds: Borax, Boric
acid, Boron Hydrides,
Aluminium: Reactions with acids and
alkalies, uses.
Carbon: uses of some important
compounds: oxides. Important
compounds of Silicon and a few uses:
Silicon Tetrachloride,
Silicones, Silicates and Zeolites, their uses.

Unit 12 Organic Chemistry – Some Basic Principles and Techniques

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12.1 General Introduction 12.2 Tetravalence of Carbon: Shapes of Organic
Compounds
12.3 Structural Representations of Organic
Compounds
12.4 Classification of Organic Compounds
12.5 Nomenclature of Organic Compounds
12.6 Isomerism
12.7 Fundamental Concepts in Organic Reaction
Mechanism
methods of purification, qualitative and quantitative analysis

Unit 13 Hydrocarbons

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13.1 Classification
13.2 Alkanes
13.3 Alkenes
13.4 Alkynes
13.5 Aromatic Hydrocarbon
13.6 Carcinogenicity and Toxicity
Free radical mechanism of halogenation, combustion and pyrolysis.

Unit 14 Environmental Chemistry

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NilEntire Unit delete


A Physical Introduction to Suspension Dynamics

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  • Publisher: Cambridge University Press
  • Online publication date: June 2012
  • Print publication year: 2011
  • Online ISBN: 9780511894671
  • DOI: https://doi.org/10.1017/CBO9780511894671
  • Subjects: Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Physics and Astronomy, Mathematical Physics, Fluid Dynamics and Solid Mechanics, Nonlinear Science and Fluid Dynamics
  • Series: Cambridge Texts in Applied Mathematics (45)

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Book description

Understanding the behaviour of particles suspended in a fluid has many important applications across a range of fields, including engineering and geophysics. Comprising two main parts, this book begins with the well-developed theory of particles in viscous fluids, i.e. microhydrodynamics, particularly for single- and pair-body dynamics. Part II considers many-body dynamics, covering shear flows and sedimentation, bulk flow properties and collective phenomena. An interlude between the two parts provides the basic statistical techniques needed to employ the results of the first (microscopic) in the second (macroscopic). The authors introduce theoretical, mathematical concepts through concrete examples, making the material accessible to non-mathematicians. They also include some of the many open questions in the field to encourage further study. Consequently, this is an ideal introduction for students and researchers from other disciplines who are approaching suspension dynamics for the first time.

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Phylogenetic signal in primate behaviour, ecology and life history

Examining biological diversity in an explicitly evolutionary context has been the subject of research for several decades, yet relatively recent advances in analytical techniques and the increasing availability of species-level phylogenies, have enabled scientists to ask new questions. One such approach is to quantify phylogenetic signal to determine how trait variation is correlated with the phylogenetic relatedness of species. When phylogenetic signal is high, closely related species exhibit similar traits, and this biological similarity decreases as the evolutionary distance between species increases. Here, we first review the concept of phylogenetic signal and suggest how to measure and interpret phylogenetic signal in species traits. Second, we quantified phylogenetic signal in primates for 31 variables, including body mass, brain size, life-history, sexual selection, social organization, diet, activity budget, ranging patterns and climatic variables. We found that phylogenetic signal varies extensively across and even within trait categories. The highest values are exhibited by brain size and body mass, moderate values are found in the degree of territoriality and canine size dimorphism, while low values are displayed by most of the remaining variables. Our results have important implications for the evolution of behaviour and ecology in primates and other vertebrates.

1. Introduction

Closely related species tend to exhibit similarities in a range of traits, including morphological, behavioural, life-history and ecological characteristics, because they inherited them from their common ancestors [1]. These similarities form one of the most fundamental patterns in evolutionary biology. Species’ traits that are more similar in close relatives than distant relatives are often said to show high phylogenetic signal. Conversely, species’ traits that are more similar in distant relatives than close relatives (e.g. in convergent evolution), or are randomly distributed across a phylogeny, are said to show low phylogenetic signal. Much research has focused on quantifying these differences in phylogenetic signal among species and traits [2–5]. However, there is still disagreement about the ubiquitous nature of phylogenetic signal in biological traits, especially in behaviour and ecology. For instance, some authors argue that closely related species will always occupy similar, although not necessarily identical, environments [6]. Others suggest that strong phylogenetic signal should be the a priori expectation when examining ecological variation in a phylogenetic context [7–9]. Still others stress that strong phylogenetic signal in behavioural and ecological traits occurs in some clades for some traits, but not in others [10]. Consequently, it is still important to ask whether behavioural and ecological traits always exhibit phylogenetic signal, especially when compared with other biological characteristics. Unfortunately, owing to confusions in terminology and interpretation, phylogenetic signal still remains a misunderstood concept, especially in evolutionary anthropology. Therefore, in this study, we first briefly review the term phylogenetic signal, describe how it can be measured, mechanisms that may underlie the pattern and potential pitfalls in its estimation. We end by exploring phylogenetic signal in a range of primate morphological, behavioural, life-history, ecological and climatic niche variables.

2. What is phylogenetic signal?

Phylogenetic signal can be defined as the tendency for related species to resemble each other, more than they resemble species drawn at random from a phylogenetic tree [4,11]. More simply, phylogenetic signal is the pattern we observe when close relatives are more similar than distant relatives [1]. Species’ traits can show high or low phylogenetic signal where phylogenetic signal is high, closely related species exhibit similar trait values and trait similarity decreases as phylogenetic distance increases [10]. Conversely, a trait that exhibits weak phylogenetic signal may vary randomly across a phylogeny or show numerous cases where distantly related species converge on a similar trait value, while closely related species exhibit notably different trait values [12]. Note that previous authors have referred to phylogenetic signal using different terms, including phylogenetic effects, phylogenetic constraints and phylogenetic inertia. These terms are inconsistently defined, and thus mean different things to different people (see [11] for a review). We therefore follow other authors in recommending that these terms are avoided [10,11,13].

3. How is phylogenetic signal measured?

Several methods have been developed for measuring phylogenetic signal, including both autocorrelation methods and methods that use an explicit model of trait evolution. For recent detailed comparisons of the strengths and weaknesses of different measures of phylogenetic signal, see Münkemüller et al. [5] and Hardy & Pavoine [14]. Here, we focus on the two most commonly used metrics to date: Blomberg's K [4] and Pagel's λ [2,15,16]. These metrics are for continuous characters only (some methods exist for discrete characters [17] but these are beyond the scope of this study and not yet widely used). Both K and λ use an explicit model of trait evolution, namely the constant–variance or Brownian motion model [18,19]. It is important to note that phylogenies with arbitrary branch lengths (e.g. all branch lengths equal) are not appropriate for estimating phylogenetic signal using K or λ.

Under Brownian motion, the magnitude and direction of trait change through time is independent of the current state of the trait and has an expected mean change of zero. Therefore, a trait changes gradually through time. In this scenario, the expected covariance between species’ trait values at the tips of the phylogeny is exactly proportional to the shared history of the species involved, i.e. the sum of their shared branch lengths. In addition, the expected variance of a trait value for a given species is proportional to total length of the tree, i.e. the summed branch length from the root to the tip for that species [19]. All of this means that a phylogeny can be represented as an n × n phylogenetic variance–covariance matrix, where n is the number of species in the phylogeny. The off-diagonals of the matrix represent the covariances between species pairs, i.e. the sum of their shared branch lengths. The diagonals of the matrix represent the species variances, i.e. the total length of the tree (figure 1). We return to phylogenetic variance–covariance matrices below.

Figure 1. Example of a phylogenetic variance–covariance matrix.

The Brownian motion model may not be a very realistic representation of the evolutionary process. However, trait evolution by random walk is not as unlikely as it first appears it is the expectation under both genetic drift and natural selection where there are many, constantly changing selection pressures, or where traits randomly evolve from one adaptive peak to another. Both of these scenarios seem reasonable over long time periods given that environments (and hence selection pressures) are constantly fluctuating. In addition, using the Brownian motion model greatly simplifies the math needed to fit evolutionary models, allowing for a more tractable interpretation of results. Note that trait variation across a phylogeny may also be described by other models of evolution [13,20–23], though Brownian motion has been the best studied to date.

(a) Blomberg's K

Blomberg's K measures phylogenetic signal by quantifying the amount of observed trait variance relative to the trait variance expected under Brownian motion [4]. More precisely, K is the ratio of two mean squared errors (MSEs). MSE0, the mean squared error of the tip data in relation to the phylogenetic mean of the data, is divided by MSE, the mean squared error extracted from a generalized least-squares (GLS) model that uses the phylogenetic variance–covariance matrix in its error structure. If trait similarities are effectively predicted by the phylogeny, MSE will be small so the ratio of MSE0 to MSE (and K) will be large. Conversely, if trait similarities are not predicted by the phylogeny, MSE will be large so the ratio of MSE0 to MSE (and K) will be small. Finally, to make values of K comparable among different phylogenies, the observed MSE0 to MSE ratio is standardized by the expected mean squared error ratio under Brownian motion.

K varies continuously from zero (the null expectation), indicating that there is no phylogenetic signal in the trait (i.e. that the trait has evolved independently of phylogeny and thus close relatives are not more similar on average than distant relatives), to infinity. Where K = 1 indicates that there is strong phylogenetic signal and the trait has evolved according to the Brownian motion model of evolution, while K > 1 indicates that close relatives are more similar than expected under a Brownian motion model of trait evolution. We can test whether K is significantly different from zero (i.e. no phylogenetic signal) by randomizing the trait data across the phylogeny and calculating the number of times the randomized trait data gives a higher value of K than our observed value. This number can then be divided by the total number of randomizations to get a p-value.

(b) Pagel's λ

Pagel's λ is a quantitative measure of phylogenetic dependence introduced by Pagel [15,16] and varies continuously from zero (the null expectation) to unity. λ = 0 indicates that there is no phylogenetic signal in the trait, i.e. that the trait has evolved independently of phylogeny and thus close relatives are not more similar on average than distant relatives. Where λ = 1 indicates that there is strong phylogenetic signal, and the trait has evolved according to the Brownian motion model of evolution. Intermediate values of λ indicate that although there is phylogenetic signal in the trait, it has evolved according to a process other than pure Brownian motion [2,15,16]. Note that although most implementations of the method constrain λ to be less than or equal to unity, λ can also be greater than unity, indicating that close relatives are more similar than expected under a Brownian motion model of trait evolution. The upper bound of lambda, however, is restricted because covariances cannot exceed variances in a phylogenetic variance–covariance matrix [2] (figure 1).

To estimate Pagel's λ, a maximum-likelihood approach is used to find the value of λ that best explains trait variation among species at the tips of the phylogeny. In practice, the λ parameter transforms the off-diagonal values, or the covariances between pairs of species, of the phylogenetic variance–covariance matrix. As described above (and in figure 1), under Brownian motion these off-diagonal/covariance values are equal to the sum of the shared branch lengths of the species. In terms of the phylogeny, these off-diagonals represent the internal branches of the tree. Thus, when λ = 1, the internal branch lengths stay the same, so the tree topology also stays the same. When λ < 1, these internal branches get shorter altering the tree topology, and when λ = 0, the internal branch lengths will also be equal to zero resulting in a star phylogeny (see the electronic supplementary material, S1). Because λ is estimated using maximum likelihood, we can test if λ is significantly different from zero (i.e. no phylogenetic signal) or unity (i.e. the Brownian expectation) using likelihood ratio tests comparing a model with the observed maximum-likelihood value of λ to a model with a fixed λ of zero or unity.

Note that λ is not just a measure of phylogenetic signal, it is also often used to transform the branch lengths of a phylogeny for use in other analyses, for example, to account for phylogenetic non-independence in a variety of statistical tests, including regression [2], principal components analysis [24], t-tests [25] and discriminant function analysis [26].

(c) Estimating Blomberg's K and Pagel's λ using R.

Both Blomberg's K and Pagel's λ are easy to estimate using R [27]. K can be estimated using the function phylosignal (or Kcalc ) in picante [28]. λ can be estimated using the function fitContinuous in GEIGER [29] or pgls (by fitting the model: trait∼1 ) in caper [30]. Both K and λ can also be estimated using the function phylosig in phytools [31]. Examples are available on the AnthroTree website (https://wiki.duke.edu/display/AnthroTree/The+AnthroTree+Website [13]).

4. How is phylogenetic signal interpreted?

Although people will often refer to ‘high’ or ‘low’ phylogenetic signal, the exact definition of high and low varies among studies. For example, low phylogenetic signal can refer to K and λ values from zero to any value less than unity high phylogenetic signal can refer to K and λ values from significantly greater than zero to infinity. Regardless of how high and low phylogenetic signal are defined, it is common to see the pattern of phylogenetic signal being used to provide information about evolutionary processes. Low phylogenetic signal is often interpreted as evolutionary lability [4] or high rates of trait evolution leading to large differences among close relatives. Adaptive radiations are expected to be characterized by low phylogenetic signal in ecological niche traits because in adaptive radiations close relatives rapidly diversify to fill new niches. Other kinds of divergent selection or convergent evolution may also result in a pattern where close relatives are, on average, less similar than distant relatives. High phylogenetic signal, i.e. K and λ = 1, is expected under genetic drift or neutral evolution, because these processes should approximate a Brownian motion model of evolution (i.e. gradual, random, non-directional trait change through time [19]). High phylogenetic signal is also often interpreted as evolutionary or phylogenetic conservatism [10]. Phylogenetic conservatism may be the result of stabilizing selection, pleiotropy, high levels of gene flow, limited genetic variation, low rates of evolution, physiological constraints or various biotic interactions (e.g. competition) restricting the evolution of new phenotypes [11,32–34]. Note, however, that the point at which phylogenetic signal is considered high enough to be phylogenetic conservatism varies among authors (e.g. K or λ > 1, [10] K or λ = 1, [35] 0 ≤ K or λ ≤ 1, [32]).

Unfortunately, interpreting phylogenetic signal is not as simple as it appears. Using simulations, Revell et al. [18] found that different evolutionary processes can produce similar K-values, especially when K is low. They also found no relationship between K and evolutionary rate under constant-rate genetic drift, their simplest evolutionary model. Although K or λ = 1 is usually interpreted as the result of neutral genetic drift, Hansen et al. [36] noted that this process is identical to one in which traits were evolving to an optimum which itself evolved according to a Brownian process (i.e. the Ornstein–Uhlenbeck model of evolution). This pattern could also arise during natural selection that randomly fluctuates, or where selective pressures themselves exhibit strong phylogenetic signal, thus producing strong phylogenetic signal in the trait(s) on which these selection pressures are acting. Finally, some evolutionary processes can also increase phylogenetic signal relative to the Brownian motion expectation for example, genetic drift occurring at different rates across the tree results in K > 1 [18].

Phylogenetic signal is also context (i.e. data and phylogeny) dependent and can be influenced by scale, convergent evolution, taxonomic inflation and cryptic species [10]. Because of this, using K or λ (or any other measure of phylogenetic signal) to infer evolutionary processes or rates must be performed with consideration of the traits involved, the hypotheses to be tested and any available external information [35]. Note that this only applies to studies whose aim is to quantify and report measures of phylogenetic signal. For analyses that use K or λ to account for phylogenetic non-independence in statistical tests, the evolutionary process underlying the phylogenetic signal is irrelevant.

There may be certain situations where interpreting the processes producing patterns of trait variation are less challenging. For instance, we would expect relatively weak phylogenetic signal in the ecological traits of species in adaptive radiations because closely related species have diversified into different niches so will exhibit distinct biological characteristics. In addition, examining the amount of phylogenetic signal in possible selective forces (e.g. climate/habitat characteristics) may help us to interpret phylogenetic signal in other traits. For instance, if the temperature conditions species experience drive variation in body size, strong phylogenetic signal in body mass may be the result of strong phylogenetic signal in temperature variables. However, if we estimate phylogenetic signal for the temperature niche space of species and find that it is low, then this suggests that correlation with temperature is not likely to explain the strong phylogenetic signal in body mass. A similar approach was used to examine the relative importance of phylogeny and environmental factors for explaining the behavioural diversity of Eulemur populations [37].

5. Problems, misconceptions and other considerations with phylogenetic signal

Earlier we discussed some issues in using the pattern of phylogenetic signal to infer an evolutionary process. However, there are several other common problems and misconceptions associated with phylogenetic signal. We will briefly outline some of these below.

One issue involves the sample size of the trait under investigation. Irrespective of tree shape, K has good power (greater than 80%) to detect significant phylogenetic signal when sample sizes are greater than 20 [4]. For λ, power is good only for sample sizes greater than 30 [2]. This difference is partially due to significance in λ being determined by likelihood ratio tests, which can give notably imprecise p-values at low sample sizes. Like any statistic, when sample sizes are large, even small K and λ values will yield statistically significant p-values thus as the number of species increases, the ability to detect significant levels of phylogenetic signal increases. Consequently, it is important to think about statistical significance versus biological significance when sample sizes are extremely high or low. For large sample sizes, it may be more useful to focus on the actual measure of phylogenetic signal rather than placing too much emphasis on the significance or non-significance of p-values. Similarly for small sample sizes, finding K and λ of zero should not be taken to mean that there is no phylogenetic signal in the variable, or to justify the use of non-phylogenetic statistical analyses (see below). Instead, we would advise performing such analyses with several sets of branch lengths (i.e. in phylogenetic generalized least-squares models (PGLSs), we would advise comparing models with λ = 1 to those with the maximum-likelihood estimate of λ and/or λ = 0).

Another problem involves error in both the phylogeny and the measurement of the trait under investigation. Although errors in tree topology obviously affect measures of phylogenetic signal, simulations show that polytomies and missing branch length information, by far the most common problems in phylogenetic trees, have negligible effects on estimates of K or λ [5]. In terms of measurement error, Blomberg et al. [4] noted that K is sensitive to measurement error and suggested that it may obscure significant phylogenetic signal. Recently, Hardy & Pavoine [14] confirmed this using simulations to show that measurement error substantially decreases the power of K to detect significant phylogenetic signal and also biases values of K downwards. This bias was stronger in trees with many short branches near the tips [14]. Measurement error may be particularly problematic in comparative studies, where data are generally species averages and the raw data come from multiple sources.

Another issue to consider is the phylogenetic/taxonomic scale of the analysis. Measures of phylogenetic signal in any biological trait may vary at different phylogenetic/taxonomic scales. Therefore, a trait may exhibit high levels of phylogenetic signal at one level, e.g. at the level of genera, yet this pattern may break down at higher or lower levels of analysis. In addition, measuring phylogenetic signal does not account for variation within species. For instance, body mass is phylogenetically conserved across primates, yet we know that body mass varies within species geographically [38–42]. Within-species variation in behaviour and ecology is also well documented in many species, both in a geographical context, as well as between sexes [37,43–48].

One final common misconception about phylogenetic signal involves analyses that correct for the phylogenetic non-independence of species. In analyses using multiple species, species are not independent data points because they share characteristics with their close relatives owing to common ancestry [1]. This violates the assumptions of many statistical models thus phylogenetic comparative methods are used to account for this statistical problem. Some authors argue that unless the variables in the analysis show significant phylogenetic signal, standard non-phylogenetic methods should be employed [49]. However, it is important to note that this refers to phylogenetic signal in the residuals of an analysis, not of the raw variables themselves [50]. Thus, a K or λ of zero for variable X does not necessarily mean that the regression of variable Y on variable X should be performed non-phylogenetically, unless the residuals from the regression also have no significant phylogenetic signal. Also note that a PGLS with a λ = 0 is exactly the same as a non-phylogenetic GLS model.

6. Phylogenetic signal in primate traits

Several researchers have examined primate behaviour and ecology in a phylogenetic perspective, although formally testing for phylogenetic signal using current methods has been rare. One of the first quantitative studies of primate behaviour in an explicitly phylogenetic context was conducted by DiFiore & Rendall [51] using social system traits of most primate genera. They found that Old World monkey behaviour was phylogenetically conserved, even though these species occupy a wide variety of habitats and a strong link between habitat variability and behaviour was expected. Additional studies focused on ecological niche space. For example, Fleagle & Reed [52] found a significant correlation between the ecological similarity and phylogenetic relatedness of primates within each continent. A more taxonomically narrow study found that closely related populations of Eulemur exhibited similar social organization characteristics independent of local habitat conditions [37], perhaps because social organization characteristics are associated with behaviours related to mating, which could be difficult to modify [37]. In contrast, no phylogenetic effect was found for diet, activity budget or ranging in Eulemur populations. The importance of phylogeny for explaining primate behavioural and social organization variation was also recently emphasized by Thierry [53]. He argued that there are several cases where the absence of measureable behavioural variation across species (especially across macaque species) can be best explained by the phylogenetic relationships of the species and not ecological factors. Finally, a recently published paper by Kamilar & Muldoon [12] explicitly tested if the climatic niche space of Malagasy primates exhibited significant phylogenetic signal. They found relatively low levels of phylogenetic signal for each climatic niche axis closely related species often occupied quite different climatic niches while distantly related species often converged on similar climatic niches. This may reflect the fact that Malagasy primates comprise an adaptive radiation and have rapidly diversified to fill a wide variety of niches.

Considering the mixed evidence for the idea that behavioural and ecological traits exhibit significant phylogenetic signal, a more detailed examination of this issue is warranted. To date, there have been only two broad-scale studies of phylogenetic signal in biological traits [2,4]. These studies found that, in a variety of taxa, the traits that exhibited the highest levels of phylogenetic signal were body size and morphology, followed by life-history, physiology and finally behavioural traits. Although the findings of these studies are important, they did have some significant limitations. For mammals in particular, both analyses used comparative datasets that were phylogenetically broad (e.g. across Mammalia, within Carnivora and Primates, etc.), but were fairly limited in terms of the number of species in each dataset (mean = 52). Therefore, in many cases, these datasets may have not sufficiently captured important trait variation. Second, in many ways, the traits examined were quite narrow in scope. For instance, 17 of the 60 traits examined were some measure of body mass/weight. Only four datasets quantified social organization (group size in 75 antelope species, 28 macropod species, 26 hystricognath rodent species and 15 mole rat species). Similarly, only five of the 60 datasets consisted of life-history traits, and these were only examined in carnivores (52 species), mole rats (15 species) and ‘mammals’ (26 species). Finally, Freckleton et al. [2] did not include any primate-only dataset, and Blomberg et al. [4] only examined four primate datasets, three of which were body mass or mass dimorphism and one being testis size.

Therefore, although previous studies provided us with a general picture of phylogenetic signal for many traits, we still have little knowledge about variation in phylogenetic signal across numerous biological traits, especially within a single mammalian order. Here, we address this issue by investigating phylogenetic signal for numerous traits in a taxonomically well-sampled primate dataset. In particular, we address the following questions: (i) What is the strength of phylogenetic signal in primate morphological, behavioural, ecological, life-history and climate niche traits? (ii) Is there variation in phylogenetic signal among and within these trait categories? (iii) What does phylogenetic signal tell us about primate trait evolution?

If the findings of previous studies that formally tested for phylogenetic signal hold true [2,4], then we would expect to find that phylogenetic signal is strongest in morphological traits, followed by life-history variables, and then behaviour. In addition, we predict that the climatic niche space of species should have low phylogenetic signal. This prediction is based on what we know about the macroecology of primate communities, where distantly related primate species are found sympatrically in the same study site (and consequently, climatic environment), especially in Africa and Asia [54,55]. Also, we expect to find variation in phylogenetic signal among traits within biological categories. For instance, the amount of leaves in a species’ diet may have relatively high phylogenetic signal compared with other food items because leaves comprise a very limited portion of the diet for small-bodied taxa (i.e. relatively invariable across closely related small species, such as callitrichines, cheirogaleids) and is often a significant component of the diet in primate clades with specialized anatomical traits that are related to leaf processing (e.g. colobines, Alouatta). We predict that this dietary specialization will also influence activity budgets. Specialized folivores typically spend a significant amount of time dedicated to resting. Therefore, we expect that resting time will have a stronger phylogenetic signal than other activity budget variables. In contrast to folivory, fruit-eating is not dependent on body mass and/or anatomical specializations, and therefore, the fruit component of a species’ diet can be more variable. This should also have ramifications for other aspects of behaviour and ecology.

7. Material and methods

J.M.K. collated data for a total of 213 primate species from various databases, published datasets, articles and books (see the electronic supplementary material, S2). The dataset contains 31 variables representing nine trait categories: (i) body mass (ii) brain size (iii) life-history (iv) sexual selection (v) social organization (vi) diet (vii) activity budget (viii) ranging patterns and (ix) climatic niche variables (table 1).

Table 1. Variables and trait categories examined in the study. Numbers in parentheses are the number of species with data for that variable.


Applications of statistical mechanics to non-brownian random motion

We analysed discrete and continuous Weierstrass–Mandelbrot representations of the Lévy flights occasionally interrupted by spatial localizations. We chose the discrete representation to easily detect by Monte Carlo simulation which stochastic quantity could be a candidate for describing the real processes. We found that the particle propagator is able to reveal surprisingly close, stable long-range algebraic tail. Unfortunately, long flights present in the system make, in practice, the particle mean-square displacement an irregular step-like function such a behavior was expected since it is an experimental reminiscence of divergence of the mean-square displacement, predicted by the theory. We developed the continuous representation in the context of random motion of a particle in an amorphous environment we established a correspondence between the stochastic quantities of both representations in which the latter quantities contain some material constants. The material constants appear due to the thermal average of the space-dependent stretch exponent which defines the probability of the particle passing a given distance. This averaging was performed for intermediate or even high temperatures, as well as for low or even intermediate internal friction regimes where long but not extremely long flights are readily able to construct a significant part of the Lévy distribution. This supplies a kind of self-cut-off of the length of flights. By way of example, we considered a possibility of observing the Lévy flights of hydrogen in amorphous low-concentration, high-temperature Pd85Si15H7.5 phase this conclusion is based on the results of a real experiment (Driesen et al., in: Janot et al. (Eds.), Atomic Transport and Defects in Metals by Neutron Scattering, Proceedings in Physics, Vol. 10, Springer, Berlin, 1986, p. 126 Richter et al., Phys. Rev. Lett. 57 (1986) 731 Driesen, Doctoral Thesis, Antwerpen University, 1987), performed by detecting the incoherent quasielastic scattering of thermal neutrons. We emphasize that the observed HWHM ∼k β , where exponent β is distinctly smaller than 2, could be caused by these long flights of hydrogen.


Firstly, SDE is just an informal way of writing a definition of process $X$, while more formal way is to use so called stochastic integration: $ X(t)=int_0^t F(s, X)ds + int_0^tG(s, X)dW, $ where the latter term is a limit of $ sum_i G(t_i, X_i)(W(t_)-W(t_i)). $ Let’s state for the sake of simplicity that $G$ depends solely on $t$. Then the terms of the sum are normally distributed and therefore, due to the property of normal distribution the limit is Gaussian too. Moreover, normal distribution is an infinitely divisible distribution, which means that if we have two independent normally distributed random variables, then the sum of them will be also normally distributed. There are different distributions that obey this law (e.g. Stable or Poisson). They form the class of Lévy processes. However, Weibull is not one of them.

Now let us define another non-Gaussian process, say $Y(t)$ and replace $W(t)$ with it. Will the sum preserve the distribution now? The answer lies in the definition of $Y(t)$ and how you assume the sum $ sum_i Y(t_)-Y(t_i) $ is distributed in the limit. Once you get it, you could write the SDE.


Appendix

(A1) (A2) (A3) (A4) (A5) (A6)
  1. If α is small then evolution is approximately Brownian: if α is small then % 1 - e -2αT ≈ 2αT, i.e. traits accrue variance as if evolving according to a Brownian process.
  2. If species i and j diverged recently, evolution is approximately Brownian: if two species diverged recently, then T - tij ≈ 0 and hence . Thus, recently diverged species provide little information relevant to estimating non-Brownian evolution according to an OU process.
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