The article explores the impact of the diffusion of new methods of production on output and employment growth and income distribution within a Classical one‐sector framework. Disequilibrium paths are studied analytically and in terms of simulations. Diffusion by differential growth affects aggregate dynamics through several channels. The analysis reveals the non‐steady nature of economic change and shows that the adaptation pattern depends both on the innovation's factor‐saving bias and on the extent of the bias, which determines the strength of the selection pressure on non‐innovators. The typology of different cases developed shows various aspects of Schumpeter's concept of creative destruction. In this article, elements of the modern classical approach to economics and elements of evolutionary economics are combined to study the process of diffusion of a new and economically superior production method and its impact on output and employment growth and income distribution. This task is accomplished by adapting the Classical one‐commodity model of Kurz and Salvadori , chap. 2The study is inspired by two papers which explore the interface between the two schools of thought: Kurz , who invinnovation bias and the second one the innovation intensity. Both the implications of the innovation intensity and the impact of innovations on income distribution have not received much attention in the literature.The article adds to the literature on how the economy adapts by differential firm growth. The focus is on the question of how the type of innovation affects aggregate output growth, employment growth and income distribution. It is argued that innovations differ along two dimensions: The first dimension is the innovation's factor‐saving bias. The second dimension is the degree or extent of its bias, which is important because it determines the strength or intensity of the selection pressure on non‐innovators. In the article, we call the first dimension the Adaptation paths are explored analytically and in terms of simulations. The study of disequilibrium paths is relevant for two reasons: First, understanding the dynamics outside equilibrium is crucial. Second, it adds to our understanding of how equilibria come about and it illuminates phenomena characterising the process of adaptation. The article shows that (1) the diffusion process drives aggregate performance via a number of effects which differ in direction and magnitude; (2) both the pattern of disequilibrium paths and the long‐term impact of the diffusion process depend on the bias and the intensity of the innovation; (3) the process of creative destruction depends on the characteristics of the innovation and under certain circumstances the destructive side of this process outweighs the creative one.The article proceeds in three steps. Section 2 presents the basic relations and concepts needed to deal with multiple methods of production within a classical one‐sector framework and identifies the various channels through which diffusion affects aggregate growth. Based on this, a simple model of diffusion‐driven growth is specified in section 3. Section 4 explores the dynamics of the model and develops a typology of different cases allowing one to identify various forms and dimensions of Schumpeter's concept of creative destruction. Section 5 concludes.The classical one‐sector growth model of Kurz and Salvadori , chap. 2t, I production methods contribute to total output. Method i∈I uses ia units of circulating capital and il units of labour per unit of output. Given the uniform real wage rate tw, method i yields an ‘individual’ profit rateVariety is introduced into this framework by the assumption that in period i is identified with a firm, which produces xi,t units of output by employing κi,t=aixi,t units of productive capacity and Li,t=lixi,t units of labour. Aggregate output is then given by xt=∑ixi,t, aggregate productive capacity by κt=∑iκi,t and aggregate employment by Lt=∑iLi,t. The market share of firm i in period t is given by qi,t=xi,t/xt. Given the structure of production reflected by market shares, the average amount of capital and the average amount of labour needed to produce one unit of output are computed asEach production method and reflects the average conditions of production at time t and thus is an abstract measure of the state of technical knowledge actually implemented. In addition, the general rate of profitgiven value of q.The average production method s of its profits into expansion of its own capacity such that firm i's growth rate is given byAssume, that every firm ploughs back a fraction s is assumed to be uniform across firms. The growth rate of aggregate output then equals the weighted average of firm growth rates, where the weights are the firm market shares:The investment propensity i evolves according toIf individual rates of profit and thus firm growth rates differ, the structure of production changes over time. A little calculation shows that the market share of firm ri,t is larger  than r¯t, the market share of firm i increases (decreases). We refer to the gradual rise of some iq over time as the diffusion of production method i by differential growth.This implies that if the individual profit rate A further relation helps dealing with the problem at hand. It is given by the following link between the evolution of total productive capacity and the general rate of profit:As one can see by comparing equations gκ,t, and capital biased technical change, indicated by rate of change αt=(a¯t+1−a¯t)/a¯t. As both result from the process of investment, the two sources cannot be separated. Instead, the quantitative aspect of capacity accumulation and the qualitative change of the production structure appear as intertwined.This reveals that—given perfectly flexible labour supply—output growth has two sources: capacity growth, indicated by technology effect of diffusion. It may either be positive, negative or zero.The above framework allows to identify different channels through which a diffusion process affects aggregate growth dynamics. A direct and an indirect channel can be distinguished. Both channels emanate from the mechanism of differential growth indicated by equation a¯ and l¯ will change such that at least one of the distributive variables increases. Any change of the general profit rate in turn affects aggregate capacity accumulation and hence output. Whether and to what extent this general accumulation effect is expansionary depends on the dynamics of the real wage rate. If the real wage adjusts, two complications emerge: First, if the real wage rate changes, all ‘individual’ profit rates change. If they change to different extents, the speed of structural change varies. Thus there is a feedback between the dynamics of the real wage rate and the process of diffusion. We call it the wage feedback effect. Second, if the real wage rate increases such that some of the ‘individual’ profit rates turn negative, some firms are forced to decline or even to exit. This differential accumulation effect provides a further indirect channel, which dampens capacity accumulation and hence output growth.The indirect channel emanates from the fact that the diffusion process changes the average production method. In general, a priori. As some effects might accelerate whereas others might decelerate growth, it is not even clear whether the diffusion process speeds up or slows down growth. Rather, the implications depend (1) on the bias and intensity of the innovation, (2) on the behavioural responses of firms, and (3) on the dynamics of the real wage rate.Because of the multiple effects and complications involved, the consequences of a change of the production structure for aggregate dynamics are not obvious To shed some light on the relation between a change in the production structure and aggregate dynamics, the remaining parts of the article presents and analyses a simple model of diffusion‐driven growth in which we consider only two methods of production and only some of the above effects.For analytical convenience the following analysis concentrates on the stylized case of two rival methods of production; further, the problem of how the innovation emerges is set aside as both methods are assumed to be already in place.qt=x2,t/xt and the market share of the incumbent method 1 by 1−qt=x1,t/xt. The two central assumptions the model is based on concern a simple wage setting rule (section 3.1) and a refined version of the investment function (5), which includes the case of negative individual profit rates (section 3.2).Let method 1 denote the established one and method 2 the innovation. The two ‘individual’ profit rates are given by equation given value of q. But since the superior method 2 allows for a higher surplus of production, there is the question of how this additional surplus is distributed among capitalists and workers when q increases.As already mentioned, the Classical inverse relationship holds between the general profit rate and the real wage rate for a w adjusts to a change in q by taking the general profit rate as constant and exogenously given. As tr = r, equation We impose a wage adjustment rule which determines the wage rate endogenously but keeps things as simple as possible. It is assumed that the real wage rate general accumulation effect. Attention therefore focuses on the remaining channels which are put in sharp relief.This wage setting rule has some ‘Classical’ flavour, since one of the two distributive parameters is assumed to be constant and exogenously given, whereas the other is determined by the conditions of production, which are changing in our case. It may be motivated by the empirical observation that the general profit rate is a trend‐less magnitude. However, this assumption implies that the diffusion process does not influence aggregate growth via the q = 0), the wage rate is determined by method of production 1 only, which is characterised by  and an ‘individual’ profit rate r1=r. If q = 1, production method 2 just obtains a profit rate r2=r and the corresponding wage rate is determined by method of production 2 only. If q gradually rises from 0 to 1, a¯ and l¯ change such that the real wage rate increases via equation From the wage setting rule (10) it follows that the real wage rate is bound to rise during diffusion: In the LPP before the innovative method has entered the system  extra profits have a cost advantage (disadvantage) compared to the average conditions of production and are economically superior (inferior).By comparing total extra profits sum up to zero:An important implication of the proposed definitions of the general rate of profit and of the rate of extra profits, given by equations This relation is central for the analysis of the dynamics of growth and income distribution (see section 4).i invests, it ploughs back part of its profits into building capacity i. Although firm investment does not adapt to differential profitability, over time the fraction of total investment which flows into capacity expansion of the innovative process increases due to differential growth. (3) Firms operate at full capacity throughout. As we only consider circulating capital, productive capacity of i is determined by last period's investments into i.We implement a simple form of retained earning dynamics based on the following assumptions: (1) Investment behaviour is uniform across firms.i at time t, xi,t, is the sum of wage payments, capital investment to maintain the output‐level and profits:According to equation i, xi,t+1, the following variation of the classical investment hypothesis formulated at the level of firms is adopted: Let s∈aixi,t and capitalist consumption (1−s)aixi,t. As net investment is positive, the firm accumulates and its output increases. Hence, Ci,t>0 and xi,t+1>xi,t.Case 2: −1<r+ρi,t≤0. In this case net investment is negative because the amount of investible resources xi,t−wlixi,t is smaller than replacement investments aixi,t. As capitalist consumption is non‐negative, the firm devotes the whole amount of its investible resources aixi,t to capacity construction. This implies that Ci,t=0 and that xi,t+1≤xi,t.Case 3: r+ρi,t≤−1. Since now wlixi,t≥xi,t, the firm using production method i is just able to pay for the total wage bill or even fails to pay for it. As no resources remain with the firm, it is no longer able to continue its business but is forced to exit. In this case the firm pays its workers what it can and then leaves the market. Hence, xi,t+1=0.To determine next period's output produced by method of production i is given bySumming up, output growth of firm decide on how much to expand its capacity. This decision is reflected by its propensity to invest. Conversely, a negative rate of profit implies a loss and enforces a firm to reduce its capacity.The investment function is illustrated in figure r+ρi,t≤0), we assume that firms stay in business as long as they can. This assumption relies on arguments put forth by Kahn /a1=4. The incumbent method of production is located at the origin. The line BCD¯ is characterised by Θl=−1 (i.e. by l2=0) and the line BAE¯ is characterised by Θa=−1 (i.e. by a2=0). The downward sloping iso‐profit rate lineΔDEB are potential innovations, since they permeate the system if introduced. Methods of production lying above DE¯ are inferior compared to the incumbent method and do not succeed if introduced. They can be termed inventions that will not become innovations.ΔDEB can be partitioned according to the innovation bias. Table The two measures Θnew LPP: r+ρ1|q=1=0. And the linenew LPP: r+ρ1|q=1=−1. These two lines classify potential innovations into three types of innovation intensity differing with respect to intensity of selection pressure on the incumbent firm and its ability to survive. The ultimate fate of the incumbent firm is one central dimension of the process of creative destruction which takes one of three forms:The magnitude of the two biases determines the innovation intensity. Three types thereof can be distinguished according to what happens to the ‘individual’ rate of profit of the incumbent method after the diffusion process is accomplished. The lineΔDEF in figure If the invading method lies within the triangle ΔDFA, the ‘individual’ rate of profit of the incumbent firm lies between −1 and 0 in the new LPP. It is still able to continue business but it asymptotically declines in absolute terms due to losses. Thus only the innovative firm survives as time approaches infinity.If the invading method lies within the triangle ΔDAB implies an ‘individual’ rate of profit smaller than −1 for the incumbent method in the new LPP. Thus the incumbent firm is forced to exit in finite time; firm exit is the strongest evidence of creative destruction in this setting.Any invading method lying within the triangle ΔDEF is bigger for a higher r. However, this wedge is very narrow for reasonable values of r. Thus the cases of medium and high selection pressures or intensity are decisive. In the following analysis we focus on the case of medium intensity and abstract from the latter cases; in this way we keep the number of firms constant.Note that the above comparative analysis is independent of the proposed wage adjustment rule but solely relies on the assumption that the diffusion process has no lasting effect on the general rate of profit. Yet, the intensity of some innovation and thus the incumbent firm's ability to survive depends on the general rate of profit: A high general rate of profit tends to protect the incumbent firm against the more severe consequences of decline and exit as the wedge representing innovations of low intensity ρ¯t=(1−qt)ρ1,t+qtρ2,t denotes the market share weighted average rate of extra profits, and the growth rate of aggregate productive capacity byIn this section, we explore how innovation bias and innovation intensity shape the aggregate output growth rate. Because of equations r is constant and therefore the general accumulation effect does not affect aggregate growth. The two effects we focus on are the technology effect and the differential accumulation effect. They are analysed in the following.These two equations form the basis for the analysis of how the diffusion process affects output growth. Because of the proposed wage setting rule the general rate of profit s = 1, in which the kink in the investment function and thus the asymmetry between firm growth and decline vanishes. The aggregate growth rate is then given by gt=r+ρ¯t. From equation Consider the extreme case tq and ρ2,t are strictly positive implying that the sign of ρ¯t is the negative of the sign of Θa. Thus three cases can be distinguished:ρ¯t<0 in the case of labour saving and capital using innovations (Θa>0);ρ¯t=0 in the case of pure labour saving innovations (Θa=0);ρ¯t>0 in the case of capital saving innovations (Θa<0).It furthermore holds that Figure ΔDAF, the real wage rate lies above the maximum wage rate process 1 can pay without making losses, given by w^1=(1−a1)/l1, for all qt∈. This effect depends on the innovation intensity and dampens aggregate growth via decelerating aggregate capacity accumulation. If the invading method lies within the area s < 1 and qt∈ scenario the innovation is not introduced and output at time T is given by x^T=(1+sr^)Tx0 with r^=r for some initial output x0 and propensity to save s. The two output levels at time T > 0 are related byTo assess the long‐term impact of the diffusion process on growth, the non‐steady growth path is compared with the steady‐state output path defined by a2=0.2, one gets Δ0.3(38)=0.967 and Δ0.2(38)=0.768. Thus, for s = 0.3 (s = 0.2) the long‐term output is 3.3% (23.2%) smaller than BAU output. How these two output paths compare with their respective BAU paths is illustrated in figure This product series provides an assessment of the overall long‐term impact of diffusion‐driven growth. For the first two examples of figure The analysis and numerical examples show that although the diffusion process does not affect the LPP growth rate, short‐term fluctuations have long‐term implications on the output level. Moreover, the differential accumulation effect may outweigh the technology effect. If they are working in opposite direction, the economy may get on a path on which the output level is persistently below the BAU output.Summing up, non‐constant aggregate growth is a further aspect of creative destruction. The effects shaping the time profile of aggregate output depend on the bias and the intensity of the invading innovation. As the gradual replacement process involves both growth‐enhancing and growth‐depressing effects, the growth regime may flip during the process and the destructive impact may well predominate the creative one.qt=x2,t/xt of the innovation evolves according toBased on the above discussion, this section explores how the two dimensions characterising an innovation shape the pattern of diffusion. As the growth rate of the innovating firm is given by equation Generally, the mechanism of differential growth generates a sigmoid diffusion pattern and mimics the stylised fact of S‐shaped curves of diffusion processes  profit rates to different extents if the two methods of production differ with respect to their capital intensity wage feedback effect does not alter the speed of adaptation. Note that although the speed of diffusion might be affected, the direction at which market shares change does not vary in the case of two methods of production. In addition, also the intensity of the innovation bias determines the magnitude of the profit rate differential Δρt and thus the speed of diffusion.It follows that if the innovation is capital saving and labour using, the increase of the wage rate decelerates the diffusion speed. But if the innovation is capital using and labour saving, the increase of the wage rate accelerates the diffusion speed. Only in the case of a neutral innovation, this relative decline of incumbent firms in the case of a low innovation intensity; (2) an asymptotic diffusion path accelerated by the absolute decline of incumbent firms in the case of medium innovation intensity. This can be seen by looking at the negative term (1−s)(1−qt) in the denominator of Case 2 which indicates the impact of the incumbent firm's absolute decline; (3) in the case of high intensity the diffusion process is accomplished in finite time by the extinction of the incumbent method of production.The speed of the diffusion process and its overall pattern provides a further element involved in the process of creative destruction, which depends on the bias and intensity of the innovation and takes different forms: (1) an asymptotic diffusion path due to absolute growth but L=L1+L2. Given equations λt=(l¯t+1−l¯t)/l¯t is the rate at which the average labour coefficient changes. From equation This section explores how the innovation bias and its intensity affect the evolution of total employment Restricting our attention to the latter, letqt+1>qt and Θa,Θl≥−1, three cases can be distinguished:0<Γt<1 in cases for which Θa>Θl, e.g. in the case of a dominantly labour saving innovation.Γt=1 in the case of a neutral innovation (Θa=Θl).Γt>1 in the case for which Θa<Θl, e.g. in the case of a dominantly capital saving innovation.As This list indicates job creation and destruction as another aspect of the process of creative destruction. For example, the diffusion of a pure labour saving innovation of low intensity decelerates the job creation rate such that the resulting employment path lies below its BAU employment path. If a dominantly labour saving innovation of medium intensity diffuses, the incumbent firm's absolute decline results in the destruction of jobs, which outweighs the creation of new jobs due to the differential accumulation effect. Again, as some new jobs are created and some old jobs are destroyed at the same time, the overall long‐term impact of the diffusion process on the employment level may be negative.tω is defined as ωt=Wt/(Wt+Pt) with tW denoting total wage payments and tP total profits at time t. It evolves according toRt=(1−a¯t)/a¯t denotes the maximum rate of profits of the average production method.In this section, we explore the change of the income shares due to the diffusion of a process innovation. The wage share tR. This is due to the fact that the sum of total extra profits is zero throughout /ai denote the maximum rate of profits of production method i. The wage share in the LPP with method of production  being used then is given byTo see how it affects the wage share, let ω1) with the wage share which prevails after the diffusion is accomplished (ω2) shows the following:Θa>0), R2<R1 and the wage share falls: ω2<ω1.If the innovation is capital using (Θa=0), R2 = R1 and the wage share does not change: ω2=ω1.If the innovation is pure labour saving (Θa<0), R2>R1 and the wage share increases: ω2>ω1.If the innovation is capital saving (A comparison of the wage share before the innovative process enters the system (R2−R1=−Θa/a2, there is a symmetry between the technology effect on growth and the change in the wage share: Pure labour saving technical change neither affects aggregate growth nor does it affect the income shares, whereas capital using technical change dampens aggregate growth and reduces the wage share. All other forms of technical change increase both aggregate growth and the wage share. But, while the technology effect is related to the average rate of extra profits, the effect on income distribution arises from the change of the maximum rate of profit.Because the difference between the two maximum rates of profit is given by The article has discussed the consequences of a changing structure of production for firms and for the economy as a whole. The analysis uses a simple one‐commodity framework in which two methods of productions initially co‐exist. Adjustment of the structure of production is brought about by differential growth of firms. This mechanism implies that economically superior methods of production gradually supersede inferior ones.Our study shows that economic development proceeds in a non‐constant way and involves a number of effects that differ in direction and magnitude. It is demonstrated that the diffusion of new methods of production need not always result in higher output and employment growth. Rather, the overall impact on economic performance depends on the relative strength of counteracting tendencies, which can be traced back to the type of innovation. Both the innovation bias and intensity are major determinants of the process that unfolds after an innovation has upset the system. The typology of different cases developed reveals that the process of creative destruction involves different forms and dimensions. These findings suggest that a steady‐state analysis is not sufficient to fully grasp the mechanisms and consequences involved in the process of technical change.There are two issues that deserve further attention: First, firms are assumed not to respond to profit differentials: neither do they try to imitate nor to adjust their investment flows. How the above results change if firm behaviour is assumed to be less inertial is a topic for further investigation. Second, by assuming a given and constant general rate of profit one effect on aggregate dynamics of the diffusion process is excluded from the analysis. An exploration of the general accumulation effect is linked to the question of how labour market conditions interact with the process of diffusion. For a study along this line the wage adjustment mechanism proposed may serve as a benchmark.


Generation of ultrashort coherent radiation pulses in the extreme ultraviolet (EUV) and x-ray regime is of remarkable interest in the synchrotron radiation user community. In this work, a novel technique is proposed for directly imprinting strong coherent microbunching on the electron beam with very small laser-induced energy spread. Theoretical analysis and numerical simulations demonstrated that this technique can be used for the generation of megawatt-scale level, fully temporal coherent femtosecond EUV and soft x-ray radiation pulses at a storage ring light source. While synchrotron radiations as short as a few tens of picoseconds from the SR have already become a standard tool to study the structures of matters on the atomic scale, their pulse durations are still too long to measure the atomic motion and structural dynamics on the fundamental time scale of a vibrational period (~100 fs). Worth still, synchrotron radiation pulses are not suitable for some high-resolution spectroscopy and imaging experiments since they are incoherent in time.Evolutionary increases in storage ring (SR) source brightness over the past decades have supported a robust array of x-ray capabilities, impacting on many disciplines, such as physics, chemistry, biology and material science, etc.5. Although these high-gain FELs have met most criteria of future light sources, they usually have relatively low repetition rates and limited numbers of user beamlines. In comparison with the FEL, SR has preponderances in mature technology, high average flux, pulse-to-pulse stability and providing powerful support for a lot of users simultaneously. It is therefore worthwhile to develop new techniques to extend conventional synchrotron radiation sources toward short pulse duration.The urgent need for intense ultrashort radiation pulses in the extreme ultraviolet (EUV) and x-ray region has prompted new developments of advanced accelerator-based light sources such as high-gain free-electron lasers (FELs). To date, several EUV and x-ray FEL facilities were constructed worldwide and have been applied to diverse cutting edge researches18. Most of these methods employ external femtosecond lasers to manipulate the distributions of the relativistic electrons to precisely tailor the properties of the radiation pulses. One of the promising approaches is so-called “laser slicing” technique7. It has been applied in several SRs and immediately opened up many new experimental opportunities for capturing ultrafast dynamics. However, in the laser slicing method, only a fraction of the electrons participates in the femtosecond synchrotron radiation and the radiation pulse is incoherent, thus the average flux and brightness of the source will suffer a significant loss. A substantial gain in photon flux can been achieved by using coherent harmonic generation (CHG) technique10, which borrows the idea from seeded FELs to imprint coherent microbunching on the electron beam with external laser source to significantly enhance the output intensity and improve the temporal coherence. The CHG scheme consists of two short undulators separated by a small chicane and uses electron beams as the medium to convert the seed laser to high harmonic radiation. In the first undulator (modulator), an ultrashort laser pulse is used to interact with the electrons to generate energy modulation at the scale of optical laser wavelength. Then a magnetic chicane (dispersion section) converts the formed energy modulation into an associated density modulation (micro-bunching) giving rise to a short pulse of coherent radiation at harmonics of the seed in the second undulator (radiator). The density modulation of the electron beam that contains high harmonic components can be quantified by the bunching factor. Analytically, the bunching factor of CHG is given by9Jn is the nth order first class Bessel function, ks is the wave number of the seed laser, ξc is the momentum compaction of the dispersive chicane, γ is the relativistic parameter for the mean beam energy, Δγ is the energy modulation amplitude induced by the seed laser and σγ is the slice energy spread of the electron beam. The output power of CHG is proportional to the square of bn, which makes it orders of magnitude brighter than that of an equivalent incoherent source.Several methods have been developed in the last decade to improve the temporal properties of SRs16. Typically generating nth harmonic of the seed laser requires Δγ to be approximately n times larger than σγ. Because of the large inherent energy spread of the beam in SR, the harmonic number of a standard CHG is about one order of magnitude lower than what is required for reaching EUV and x-ray region. Further extension of the CHG output to shorter wavelength seems possible but very challenging. For example, it has been proposed that the frequency up-conversion efficiency of CHG can be enhanced by using a seed laser operating at TEM01 mode to introduce an angular modulation into the electron beam17. However, an intense seed laser with peak power up to terawatts level is needed for the realization of this technique and it is very challenging to operate a UV laser with the TEM01 mode. The phase-merging enhanced harmonic generation (PEHG) has also been proposed to improve the frequency up-conversion efficiency of the CHG18, however, a transverse-gradient undulator is typically needed in PEHG and it is found that the bunching factor of PEHG will be significantly decreased for a very small energy modulation due to the transverse-longitudinal phase space coupling during the modulation process. While decades have passed and tremendous progresses have been made for generating femtosecond radiation pulses from the SR, there is still no clear path on how to extend the wavelength of the ultrashort coherent synchrotron radiation to the EUV and x-ray region and improve the output intensity simultaneously.The CHG technique has been applied to the SR to generate coherent radiation in the ultraviolet range and belowWe report here a novel technique to introduce strong coherent microbunching into the electron beam from the SR. It is found that sufficient large bunching factor at ultra-high harmonics of the seed can be generated by a seed laser with relatively low power. With realistic beam parameters of the SR, it shows that coherent femtosecond EUV and x-ray radiation pulses with peak power of the order of about tens MW can be produced in a short undulator.b is added upstream of the modulator, and a dogleg (D) that consists of two dipole magnets of opposite polarity is adopted instead of the chicane as the dispersion section. The first dipole is used to introduce an angular dispersion into the electron beam. Then an ultrashort UV seed laser pulse is employed to interact with the electron beam in the modulator. After that, the energy modulation is converted into density modulation by the dogleg. The dispersive properties of the first dipole and the dogleg allow, if the parameters are chosen rightly, the full compensation of the initial beam energy spread to produce very sharp micro-bunches in the electron beam. This kind of electron beam would help to initiate intense coherent radiation at very high-harmonics of the seed in the following radiator. The proposed scheme should be inserted in a long drift section of the SR and the transverse dispersions should be zero upstream of the first dipole. The transverse dispersion generated by the dogleg can be fully compensated by another reversed dogleg (D*) after the radiator. The dispersion induced by the first dipole can be corrected by another two dipoles with bending angles of −2b and b elsewhere in the ring. Since the microbunching originates from the laser-beam interaction, it is easier to facilitate precision pump-probe experiments due to intrinsic synchronization of the CHG radiation with an optical pump laser, as shown in Fig. The layout of the proposed scheme is similar to a conventional CHG scheme , where x, y and z are the horizontal, vertical and longitudinal coordinates, x′ = dx/dz and y′ = dy/dz are the horizontal and vertical divergences and δ is the relative energy deviation with respect to the reference particle19. Here we leave  out for simplicity, i.e.,  is used in the following and assuming Gaussian distributions for y, y′ and δ. The electron beam is first sent through a thin-lens magnet dipole whose transport matrix isb is the bending angle of the dipole. After that, the electron interacts with the seed laser in the modulator and gets an energy change δ1 = δ + Δγsin(ksz)/γ. Considering electrons in one seed wavelength range, the electron beam around the zero crossing of the seed laser gets an almost linear energy chirp h = ksΔγ/γ, and the energy change of the electron can be written as δ1 ≈ δ + hz. Then the corresponding transport matrix for this part of electron beam can be derived asLM is the length of the modulator and ξM is the momentum compaction generated in the undulator. The momentum compaction of the modulator is usually much smaller than that of the following dispersion section, ξM is usually ignored in a linac-based FEL. However, the energy spread in the SR is about one order of magnitude larger than that of a linac, as a result, the longitudinal phase space is more sensitive to the momentum compaction factor and ξM should be considered here to get more accurate optimum parameters for the proposed scheme. After the modulator, the energy modulation is converted into density modulation by a dogleg with transport matrix ofLD is the length of the dispersion section, η and ξD are, respectively, the dispersion and the momentum compaction generated in the dogleg. It is easy to show the transport matrix for the whole CHG beam line:L = LM + LD and ξ = ξM + ξD are the total length and momentum compaction of the beam line before the radiator. According to the equation z + ξδ. In order to generate sufficient bunching factor at very high harmonics, the optimized condition is 1 + hξD = 0 and the bunching factor is determined by the product of ξ and σγ, as shown in equation  numerical simulations are necessary to show the possible performance of the proposed scheme. The electron beam is tracked through the beam line using the code ELEGANTy′ due to the correlation. The energy modulation amplitude induced by the seed laser is about 0.6 MeV, which is much smaller than the intrinsic beam energy spread (1.5 MeV). One can find in equation , the micro-bunches in each slice will be shifted apart due to the different energy  and get very sharp micro-bunches around the zero-crossing phase of the seed, as shown in Fig. th harmonic and is about 0.06 at the 30th harmonic. These bunching factors are sufficiently large to effectively suppress the electron beam shot noise and to generate coherent radiation in the following radiator. For small harmonic numbers, the bunching factor distribution from simulations fits quite well with the one-dimensional analytical results which are directly calculated by equation  and a peak power of about 200 MW. The laser waist is assumed to be 1 mm. The dipole magnet in the dispersion section has a length of 0.1 m and a bending angle of about 10 mrad. The distance between two dipoles in the dispersion section is 0.85 m. For comparison purpose, we carried out 3D simulations for both conventional CHG and the proposed scheme. The simulation results are summarized in Figs equation  that thergy Fig.  and resurgy Fig. . Howeverequation . Howeverrd generation and diffraction-limited SRs. The large bunching factor offered by the strong microbunching is responsible for the initially steep quadratic power growth in Fig. rd generation SR case is about 7.8 MW (~1010 photons/pulse). In the diffraction-limited SR case, the radiation peak power exceeds 27 MW at the exit of the radiator. Due to the harmonic pulse shortening effect22, the output pulse durations in these two cases are both around 16 fs (FWHM). From Fig. Here we adopt a 3 m long radiator with a period length of 3 cm to generate coherent radiation pulses at 13 nm with the proposed method. The simulation results are illustrated in Fig. rd generation SR case are shown in Fig. The performance of the proposed technique may be affected by some practical issues such as the fluctuation of the seed laser power and the instabilities of the magnetic fields of the dipoles. Here we choose 5% (rms) and 0.1% (rms) for the criteria of the laser power and the magnetic field stabilities, respectively, which are feasible with state-of-the-art commercial laser and accelerator technologies. The resulting fluctuations of the radiation pulse energy at the exit of the radiator for the 323 or improve the performance of the enhanced SASE24 with a relative small laser induced energy spread. Further investigations on these topics are ongoing.In conclusion, a novel technique is proposed to generate stable, coherent and ultra-short EUV and x-ray radiation pulses with high intensity by fully using the advantage of the electron beam from a SR. This kind of light source would allow one to perform many challenging experiments which require ultrafast and narrow-bandwidth radiation pulses. One unique advantage of the proposed scheme is that it doesn’t require high power seed lasers, which would allow the use of a laser with much higher repetition rate compared to the conventional CHG. Furthermore, the small modulation amplitude makes it possible to reuse the same bunch at a high rate without blowing up its energy spread. In addition to SR, the proposed scheme has other potential applications involving the use of electron beam with large intrinsic energy spread from laser-plasma accelerators. And it can also be used to enhance the frequency up-conversion efficiency of a seeded high gain FEL with the help of the flat-beam technique


Many ecological- and individual-level analyses of voting behaviour use multiple regressions with a considerable number of independent variables but few discussions of their results pay any attention to the potential impact of inter-relationships among those independent variables—do they confound the regression parameters and hence their interpretation? Three empirical examples are deployed to address that question, with results which suggest considerable problems. Inter-relationships between variables, even if not approaching high collinearity, can have a substantial impact on regression model results and how they are interpreted in the light of prior expectations. Confounded relationships could be the norm and interpretations open to doubt, unless considerable care is applied in the analyses and an extended principal components method for doing that is introduced and exemplified. Quantitative analyses of voting behaviour are heavily dependent on regression modelling of data at both individual and ecological scales. The goal in most cases is to identify the influences on decisions whether to vote or abstain and, if the former, which party to support. The independent variables are selected to represent the expected influences based on theory (often relatively weak), prior investigations, and the local (time and space) context. In the interpretation of those regressions—usually either ordinary least squares or binomial/multinomial logistic—emphasis is placed on the sign, magnitude and statistical significance of the coefficients for the independent variables.Such regression analyses can produce substantial insights but also have their disadvantages, some of which may be acknowledged in discussions of the output, but frequently their full import is not realised, with implications—often serious but unacknowledged—for the interpretation of the results. Many of these implications reflect the need for care not only in specifying models but also in assessing the results. This paper focuses on one issue only in that context—confounding associated with collinearity; without careful assessments of a regression outcome, misleading interpretations can result.Collinearity (sometimes termed multicollinearity) is usually defined as when two or more independent variables included in the model are highly correlated so that the values of one can be accurately predicted by that of another. This has clear implications for the size, perhaps the sign, and also the standard error of the regression coefficients associated with those collinear variables, and hence for their interpretation. The result is frequently termed confounding, the situation when the relationship between two variables is distorted because of the strength of the relationships between either one or both of them and a third variable included in the analysis  can occur when variables are either introduced or removed from a model, analysts need to be alert to these possibilities in their statistical practice. An important influence on what will happen is the extent of the interrelationships between included predictor variables. The degree of collinearity can readily be, though frequently is not, assessed by analysts using the Variance Inflation Factor (VIF) statistic (Allison 2]). A VIF of 1.8 tells us that the variance of that predictor variable (i.e. its standard error) is 80% greater than would be the case with no collinearity effect: VIFs of 2.5 or greater are generally considered indicative of considerable collinearity suggesting that there will be difficulty in separating out the independent contribution of variables with such large VIFs—although some authors  their coefficients are nevertheless often interpreted. Hence the need for care interpreting regression coefficients when collinearity may be present—and, as demonstrated here, even when that collinearity is relatively small there can be substantial impacts indicative of confounding although two independent variables are only relatively weakly correlated.Y is being regressed against X1 and X2, then the partial regression coefficient between Y and X1 involves, in effect, the regression of the residuals of the regression of Y on X2 on the residuals of the regression of X1 on X2. What is frequently not recognised when such regression results are reported is that the greater the correlation between X1 and X2 the greater the likelihood that the relationship between Y and X1 is, in effect, modelling little more than random noise . The results may be—and often are—expressed as regressing Y on X1, holding constant the effect of X2, but if X1 and X2 are closely inter-related there is little left to analyse separately.For clarity, the nature of the partial regression coefficients  in any multiple regression equation needs to be fully appreciated. They indicate the relationship between the relevant independent variable and the dependent—holding constant the impact of all other independents. Thus, for example, if But how closely? The conventional wisdom—when it is applied—regarding collinearity in voting analyses suggests that it should only be addressed when the VIF values are relatively large; in other circumstances it is assumed, without any detailed investigation, that any relationships among two or more of the independent variables do not substantially influence the outcome of statistical modelling and hence the interpretation of the regression coefficients. Even where collinearity is neither ‘perfect’ nor ‘almost perfect’—as Baguley’s  web tutoTo illustrate those arguments, three examples are presented of analyses in which one or more of the impacts of confounding—spurious inflation; masking or suppression; reverse interpretation—occurs in standard analyses of voting behaviour. The first two—an ecological, ordinary least squares, analysis of voting for a political party in England and Wales,The United Kingdom Independence Party’s (UKIP) success at recent British elections—notably for the European Parliament in 2014 and in the 2015 general election—is generally linked to the attractiveness of its right-wing populist appeal to those who have gained least from globalisation over recent decades in particular among: older people ; those with few, if any, formal educational qualifications; and those living in areas with high levels of economic and social deprivation , the three R2 values all show an increase, by as much as 0.11 when household deprivation is the additional variable (in Model 2b). But two features of that last regression equation raise immediate concerns regarding confounding. First, the regression coefficient for qualifications increases from 0.52 , but it is clearly sufficient to suggest that UKIP performed less well on average in the more deprived areas when qualifications are taken into account, whereas the regression with deprivation as the only independent variable indicated the opposite conclusion. Which is correct?At the second stage . Confounding with substantial changes in the estimated coefficient is thus common in these two-independent-variable regressions, despite the relatively weak collinearity.Of the other two-independent-variable regressions in that second block of Table At the third stage, two of the other three variables were added to that for qualifications in three three-independent-variable models . Further variables then look at the proximate influences on the vote—what led some who voted for the party last time to desert it at the subsequent election, for example, or, for some of those who voted otherwise at the first election of the pair, what stimulated them to switch their allegiance at the next contest. Thus, the first independent variable in this analysis—X2 and X3 in these analyses are coded, respectively, 1 if the respondent strongly liked the party  and 1 if the respondent strongly liked the party’s leader (Carwyn Jones)—and 0 otherwise. The results of the two regressions deploying those variables separately  again show the expected positive relationships—much stronger for liking the party than for liking its leader  of only 1.27: nevertheless, the partial regression coefficient for X3 in Model 2a is substantially smaller at 0.703 than the 1.149 recorded in the regression of X3 alone on Y—which would be interpreted as saying that the impact of the Labour party leader’s image on whether respondents voted Labour in 2011 was substantially reduced once the impact of his party’s image was taken into account.Other variables commonly included in such analyses ask respondents how well they like either or both of the party itself and its leader in the legislature—with the latter often presented as a short-cut heuristic deployed by voters Apart from general evaluations of parties and their leaders, many studies also ask respondents to assess the governing party’s (or parties’) performance on particular issues. The 2011 Welsh Election Study included six such assessments, of running Wales generally, and of handling the major issues of the previous 4 years—the NHS, schools, University tuition fees, the economy, and Welsh interests. These are introduced as binary variables X5–X9; all of their regression coefficients are substantially smaller than those in their single-variable regressions above and two of them, for the school and economy policy areas (X6 and X8), are statistically insignificant. When the general variable X4 is added (Model 2b), not only do the coefficients for X5–X9 reduce further, with four of them statistically insignificant, but in one case—variable X8, handling of the economy—the insignificant coefficient is also negative.The first six binary logistic regressions in Table 1, whether the respondent voted for a Labour candidate in 2007.Finally, a full regression including all nine independent variables Table , Model 31–X3 and X4–X9 respectively) each resulted in single-component solutions, accounting for 53 and 60% of the variation respectively. For the first, all three variables had a loading of 0.70 or greater on that component, and the scores related to it (FIa) were positively related to whether respondents voted Labour in 2011 (the first regression reported in Table b) were also positively related to Y. When both sets of scores were included in a regression, each was positively and significantly related to the probability of a Labour vote in 2011, with variation on FIa having more than twice the impact than variation on FIb.There are clearly sufficiently strong interrelationships in this data set, despite the low VIF values, to influence the regression outcomes and thus the interpretation of how the various factors influenced voter choice at the 2011 National Assembly of Wales election; there is substantial confounding. This is further illustrated by again deploying principal components factor analyses. Three were undertaken Table : the fir4–X9, and the second for X1–X3 although there were some substantial cross-loadings on both components—notably for X4. Regressing the two sets of component scores on Y  than it was to variations in evaluations of government performance (FIab).When all nine variables were included in a single principal components analysis, two components, together accounting for 59% of the variation, were extracted and simple structure was obtained using a direct oblimin rotation. The first component has its heaviest loadings for variables XAs was concluded from the ecological regression example, therefore, substantial confounding effects—some involving spurious inflation, some masking or suppression, and some reverse interpretation—appear in this set of multinomial regression analyses, despite the low levels of collinearity among the variables. Care is needed when running such analyses, therefore: confounding can mask the true relationships unless it is taken into account in the model structure.To exemplify further confounding and its impact on the nature of regression outcomes—some difficult to interpret, others nonsensical—we use a simulated data set comprising 1500 observations.Y—is voting for Labour (coded 1 if voted Labour and 0 otherwise). There are two independent variables: X1—whether the respondent is a member of the Working Class (coded 1 if Working Class and 0 otherwise); and X2—whether the respondent considers Labour is the best party to tackle the problems of the economy (coded 1 if Labour is best and 0 otherwise). The latter of those independent variables appears in four different scenarios—X21, …, X24—each of which has a closer correlation with X1 than the previous version.  The correlations (Nagelkerke r2) between X1 and each of those four, derived from binary logistic regressions, are:The dependent variable in this data set—Y is regressed against X1 and X24.There is virtually no correlation between the two variables in the first example, therefore, and only a slight one in the second; correlation is more substantial in the third example, and even more so in the fourth. The VIF values are thus:Economic Competence and Vote: Models 2a–d) each shows a similar relationship (though with varying intensity) between opinions on Labour’s ability to manage the economy and voting Labour; all of those relationships are positive and statistically significant (i.e. the regression coefficient is at least twice the size of its standard error).The first binomial regression in Table X2, which are increasingly correlated with X1, as shown above. In the first case—X21, with virtual nil correlation between the two (0.020)—the two independent variables clearly make additive contributions to a statistical explanation of variation in the values of Y; the coefficient, standard error and exponent for X1 are virtually unchanged from those in the first regression in Table 2 value is (slightly) increased.The final block of four regressions in Table X21 by X22, which has a higher correlation with X1—though not large (0.149: VIF 1.18). Nevertheless, the coefficient for X1 is reduced by about 40% compared to the regression with X21 and the associated exponent is almost halved; there is an extremely large coefficient, and associated exponent, for X22.The next regression—Model 3b—replaces X23 and X24, produce results that can only be considered nonsensical, although the VIFs suggest that problems should only appear for that with X24. There are very large  coefficients for X1 and comparable large  coefficients (with meaninglessly large exponents) for X23 and X24. The correlations between X1 and each of the other two variables mean that only residual noise is being regressed against X1 once the common variance shared by the two collinear variables is held constant—and the result is very substantial spurious inflation, producing nonsense results.The final two regressions (Models 3c–d), involving Many ecological- and individual-level analyses of voting behaviour use multiple regressions with a considerable number of independent variables but few discussions of their results pay any attention to the impact of collinearity among those independent variables, let alone report VIF values. Very few indeed explore various combinations of independent variables in their data to reveal the impact of collinearity and identify the likely impact of any one independent variable on the dependent; whether the relationships between the individual independent variables and the dependent are affected by confounding, and therefore difficult to interpret, is very rarely addressed. More importantly, as the examples in this paper have illustrated, even where collinearity is low substantial confounding can nevertheless occur as a result of interrelationships among the variables included in a model. Because most analysts only report the final model (and may have done no prior explorations of those interrelationships of the type reported here) it is rarely clear whether the results incorporate any substantial confounding that substantially impacts upon interpretations of the size and sign of partial regression coefficients and their statistical significance—and hence on the substantive and theoretical appreciation of the empirical tests.Three examples have been used here to indicate the potential pitfalls of such practice. With little or only mild collinearity the impact on the interpretation should be slight, according to ‘standard practice’; regression coefficients may change in their size reflecting the results of partialling out the effect of other variables, but the standard errors are not inflated and considerable confidence can be expressed in the interpretations; there is little or no confounding. But as the links between independent variables strengthen  unexpected results appear: change in the direction of the regression coefficients, for example; increase in the standard errors; and, in logistic regressions, inflation in the values of the exponents associated with the regression coefficients, in some cases to nonsense levels.All of this suggests care is needed in conducting such analyses. One regression model incorporating all of the selected independent variables should not be run and then reported without careful exploration, involving not only calculation of the VIF values but also running regressions with only some of the variables included. This could lead to decisions to eliminate some of the independent variables from the final version (one of Kennedy’s—2008—‘What to do’ suggestions) but if it is considered necessary to include them all to assess their joint impact an approach such as that deployed here using principal components analysis might be used. This approach, rarely used in the behavioural social sciences, offers a clear way forward in the analysis of voting patterns that avoids any confounding impacts of inter-relationships among the independent variables and provides a much clearer test of the strength of the impact of each independent variable on the dependent  using quantitative methods, notably regression. Confounding can have a substantial impact on the nature of model results and how they are interpreted in the light of prior expectations; indeed, confounded relationships could be the norm and interpretations open to doubt.conceptual status of the introduced variable in terms of confounders, mediators and modifiers. Changes involving one or more of reduction, increase, and reversal of sign in the original relation do not necessarily mean that the introduced variable is a confounder. Confounders are a nuisance and need to be nullified to prevent distortion of results; their impact occurs when the third variable is associated with both the exposure—the main predictor of interest—and the outcome but conceptually does not lie on the ‘causal’ pathway from the exposure to the outcome. Mediator variables, like confounders, show associations with both the exposure and the outcome, but are seen as lying on the causal pathway between exposure and outcome—a mediator variable is one that explains the relationship between the two other variables. Mediators are seen as intervening variables that produce the outcome; changes in the exposure lead to changes in the mediator which in turn result in changes in the outcome. The general statistical procedure for evaluating mediation is that the relation between exposure and outcome should be reduced after including the mediator variable. This reduction of original association when the third variable is included is not spurious, rather we have the development of an explanation. Finally, with effect modification, the third variable is interacting with exposure to modify the effect so that an exposure has a different effect among different subgroups. Effect modification is associated with the outcome but not the exposure. The results are not spurious but of real interest as the nature of the effect differs according to the presence of a third factor. In practice, in statistical analysis moderators are simply interaction terms that change the nature of the effect of the exposure on outcome.While we have concentrated on the scale of the changes that come about as variables are either introduced to or removed from a model, it is also important to see this in a wider context. There is a large literature  the apparent influence of one of more of the variables by conducting separate regressions using subsets of the independent variables only; and if those explorations indicate considerable confounding effectsAdopt an alternative analytical procedure, such as that introduced here using principal components factor analysis, to circumvent those confounding effects and thereby identify the ‘true’ relationships.The message from this cautionary tale, therefore, is that in multiple regression analyses—as illustrated here with studies of voting behaviour—exploratory procedures should be deployed when empirically testing models in which the outcome is believed to be influenced by a number of contributory factors (independent variables) that are not structured in a causal path. Those procedures should:Following these steps is in line with the general strategy set out by Franzosi , p. 21 o


This work presents the design of a real-time system to model visual objects with the use of self-organising networks. The architecture of the system addresses multiple computer vision tasks such as image segmentation, optimal parameter estimation and object representation. We first develop a framework for building non-rigid shapes using the growth mechanism of the self-organising maps, and then we define an optimal number of nodes without overfitting or underfitting the network based on the knowledge obtained from information-theoretic considerations. We present experimental results for hands and faces, and we quantitatively evaluate the matching capabilities of the proposed method with the topographic product. The proposed method is easily extensible to 3D objects, as it offers similar features for efficient mesh reconstruction. The images captured of hand gestures, which are effectively a 2D projection of a 3D object, can become very complex for any recognition system. Systems that follow a model-based method , 32 requHowever, for the accurate analysis of the hand’s properties, a suitable segmentation that separates the object of interest from the background is needed. Segmentation is a pre-processing step in many computer vision applications. These applications include visual surveillance , 18, 20,We decided to use NNs to represent the geometric properties of objects, and more specifically the self-organising maps (SOMs), due to their incremental nature. One of these SOM-based methods is the growing cell structures (GCS) algorithm , which iAlthough the use of the SOM-based techniques of NG, GCS or GNG for various data inputs has already been studied and successful results have been reported , 31, 32,In this work, we extend the method presented in  for objeThe remainder of the paper is organised as follows. Section GNG  is an unIdentifying the points of the image that belong to objects allows the GNG network to obtain an induced Delaunay triangulation of the objects. In other words, to obtain an approximation of the geometric appearance of the object. Let an object T of the object Given a domain The input distribution as the set of points in the image: The probability density function according to the standard potential field obtained for each point of the image: Learning takes place with our modified GNG algorithm where wrong edges in the network are eliminated and the final graph is normalised. Algorithms 1 and 2 describe our extended GNG. During this process, the neural network is obtained which preserves the topology of the object T. Therefore, from the visual appearance A and a set of connections (edges) C. To speed up the learning, we used the faster Manhattan distance [Considering:distance  comparedFigure N pixels, or reference vectors receptive field centre in the input distribution. We first analyse the quantisation error for each node with the Euclidean distance to its Best Matching Unit (BMU)We test the performance of the modified GNG by quantitative measures as shown in Table As reflected in Table Figure T composed by a translation s. The normalisation is given by Algorithm 2.To normalise the graph that represents the contour we must define a starting point, for example the node on the left-bottom corner. Taking that node as the first we must follow the neighbours until all the nodes have been added to the new list. If necessary we must apply a scale and a rotation to the list with respect to the centre of gravity of the list of nodes. We achieved the required alignment by applying a transformation x  and y is:The determination of accurate topology preservation requires the determination of best similarity threshold and best network map without overfitting. Let Figure In our experiments, the numerical value of K, and the network parameters as the mixture coefficients d-dimensional means and covariances p(Y) as is the case for the conditional entropy heuristic criterion [We can describe the optimum number of similarity thresholds, required for the accuracy of the map for different objects, as the unknown clusters th (MDL) , which driterion . The MDLd-dimensional means and covariances M describes the number of free parameters and is given for a Gaussian mixture by d-dimensional means and diagonal covariance matrices.The information-criterion MDL of Rissanen , is defiK, For all (a) Maximize the likelihood (b) Calculate the value of MDL(K) according to Eqs. Select the model parameters Figure EM cluster centres as part of the MDL(K) initialisation of the mixture coefficients, the measure is averaged over 10 runs and the minimal value for each configuration is selected. Algorithm 4 summarises the steps.The optimal number of similarity thresholds can be determined by applying the following iterative procedure:st and any subsequent frames are added to the TPG map and can be used for the learning in a sequence of k frames. The segmented frame and the stored shape and colour information in each node is given by:We can now use this optimal network to track objects locally wherever common regions are found. To do that, shape information and colour information from the 1In this section, different experiments are shown validating the capabilities of our extended GNG method to represent 2D and 3D hand models. The proposed method by considering elimination of noisy connections during the learning process is able to define an optimal number of nodes using the MDL criterion. The method has also been used in 3D datasets. First, a quantitative study is performed adding different levels of noise to the ground truth model (datasets). Using the ground truth models and the generated ones adding noise, we are able to measure the error produced by our method. In addition, our method is compared against the state-of-the-art algorithms Active Shape Models and Poisson surface reconstruction.All methods have been developed and tested on a desktop machine of 2.26 GHz Pentium IV processor. These methods have been implemented in MATLAB and C++. The Poisson surface reconstruction method has been implemented using the PCL libraryWe tested our modified GNG network on a dataset of hand images recorded from 5 participants each performing different gestures Fig.  that freFurthermore, we have performed the experiments having in mind specific applications, thus limiting its applicability. The quality and stability of the results at close range makes it worthwhile for webcam or green screen sign language applications which share a close range viewing distance and a relatively uncluttered background.We have also tested the system in a more generic background where shadows, changes in lighting and extremely cluttered backgrounds are common. Figure Tolerance. The values for the Tolerance were found by experimentation, and range from 0.5 to 0.7 based on the hand posture.To classify a region as a hand or face, we take into account domain knowledge information that always respects some proportions found in hands and human faces . To do tK are applied in the gesture (d) from Fig. Table Figure Table Finally, we added different levels of Gaussian noise to three different gestures to test the validity of the modified GNG in comparison with Kohonen map and the growing cell structures (GCS). The results of applying different levels of noise to the gestures are shown in Fig. active step is performed globally even if parts of the snake have already converged. Figure Our modified GNG network has been compared to the methodology of the active snake model. The snake converges when all the forces achieve an equilibrium state. The drawbacks with this method are that the snake has no a priori knowledge of the domain, which means it can deform to match any contour; this attribute is not desirable if we want to keep the specificity of the model or preserve the physical attributes such as geometry, topological relations, etc., and that the Figure The parameters for the snake are summarised in Table This section shows the result of applying an existing approach proposed by Orts-Escolano et al.  for perfIn , the oriThe edge creation, the neurons insertion and the neuron removal stages were extended considering the creation of triangular faces during this process. Algorithm 5 describes the extended CHL to produce triangular faces during the adaption process.Figure Figure Moreover, it can also be appreciated that the generated representation is accurate and implicitly it performs some typical computer vision preprocessing steps such as filtering, downsampling and 3D reconstruction.Figure Figure Finally, we performed some experiments using 3D human faces instead of hands to demonstrate that the method can also deal with different shapes. Figure In all our experiments, the parameters of the network are as follows: While 3D downsampling and reconstruction methods like Poisson or Voxelgrid are not able to deal with noisy data, GNG method is able to avoid outliers and obtain an accurate representation in presence of noise. This ability is due to the Hebbian learning rule used and its random nature that update vertex location based on the average influence of a large number of input patterns.Based on the capabilities of GNG to readjust to new input patterns without restarting the learning process, we developed an approach to minimise the user intervention by utilising an automatic criterion for maximum node growth. This automatic criterion for GNG is based on the object’s distribution and the similarity threshold (


Covariate misclassification is well known to yield biased estimates in single level regression models. The impact on hierarchical count models has been less studied. A fully Bayesian approach to modeling both the misclassified covariate and the hierarchical response is proposed. Models with a single diagnostic test and with multiple diagnostic tests are considered. Simulation studies show the ability of the proposed model to appropriately account for the misclassification by reducing bias and improving performance of interval estimators. A real data example further demonstrated the consequences of ignoring the misclassification. Ignoring misclassification yielded a model that indicated there was a significant, positive impact on the number of children of females who observed spousal abuse between their parents. When the misclassification was accounted for, the relationship switched to negative, but not significant. Ignoring misclassification in standard linear and generalized linear models is well known to lead to biased results. We provide an approach to extend misclassification modeling to the important area of hierarchical generalized linear models.  Misclassification and measurement error is well known to cause bias in estimation. The exposure variable in epidemiologic studies is often subject to misclassification –3. WhilePartner violence can have impacts on a wide range of outcomes on the abused partner, for instance, depression and even suicide attempts , 7. The Our analysis uses India National Health Survey (NFHS-3) 2005-2006. It is a nationally representative household survey that provides a set of key variables for the study, including the number of surviving children a woman has and the occurrence of partner violence in the previous generation which is subject to misclassification. Thus our interest is in determining if previous generation spousal abuse impacts number of children correcting for potential misclassification in reported spousal abuse.Our paper is organized as follows. We first describe the National Family and Health Survey. Next, we discuss the proposed Bayesian hierarchical model that accounts for covariate misclassification. We discuss the results of a simulation experiment in which we compare the proposed model to the naïve model that ignores misclassification. We apply the proposed model to the data from National Family and Health Survey and conclude with a discussion.According to the National Family and Health Survey in 2005, total lifetime prevalence of domestic violence was 33.5% and 8.5% for sexual violence among women aged 15–49. More notably, lifetime prevalence of domestic abuse ranged from 18% to 45% in states across India. In , it is ei = 1, 2,…, n denoting subjects and j = 1, 2, …, N denoting clusters. The count data at level 1 are assumed to be Poisson:rij is an offset, often time or space, and λij is the rate of occurrence which is related to covariates through a log-linear model:xij, is assumed to be potentially misclassified. Other covariates (zijk) are assumed to be measured without error. βj0 represents a random intercept for each cluster:uj ~ N. β1 is the exposure effect of primary interest while βk, k > 1, represents the impact of other covariates. The exposure model for binary covariate xij is assumed to be a logistic model:vj ~ N and γj0. Finally, the measurement model depends on several assumptions and the amount of data available. Here, we assume nondifferential misclassification and either one or two tests available used as surrogates for the true exposure, xij. In the one test case, we assume the fallible method has sensitivity S1 and specificity C1 and conditional on the true value we have fallible assessment xij1 which takes values of 0 and 1 with probability xij1 and xij2 with sensitivities S1 and S2 and specificities C1 and C2 and probability function:We follow the general approach of  for measγ0, γ1,…, γq have independent normal prior distributions with mean 0 and variance 10. Similarly, the Poisson regression coefficients, β0, β1,…, βp, are also assumed to be independent and normally distributed but we assume a variance of 100 for these. We assume uniform priors for the random effects standard deviations σ and ν [S1 and C1.Because we are using the Bayesian framework we require prior distributions for the model parameters. In the absence of prior information or expert opinion, diffuse normal prior distributions are often used for logistic and Poisson regression coefficients; see, for example, , 18. Thu σ and ν . FinallyThe joint posterior is proportional to the product of the likelihoods from the outcome, exposure, and measurement models along with the prior distributions. The resulting marginal posteriors for all parameters of interest are not available in closed form. Packages such as OpenBUGS and JAGS can be used to fit the model. Our OpenBUGS code is available from the first author upon request.νj ~ N and uj ~ N. These random effect standard deviations of 0.1 are relatively small. We also considered the cases of 0.2 and 0.3 and found the results to be similar. For the measurement model we consider two cases. The first is with a single diagnostic test with sensitivity S1 and specificity C1. As a second case, we assume two conditionally independent tests, the first with sensitivity S1 and specificity C1 and the second with sensitivity S2 and specificity C2. For the first test we fix S1 = 0.55 and C1 = 0.95. For the second test we have S2 = 0.7 and C2 = 0.8.To determine the size of the impact of the covariate misclassification on the hierarchical count model we performed a simulation experiment. We generate the simulated data in order to mimic the real data. Specifically, we assume a single exposure variable that is subject to misclassification along with four other covariates that are assumed to be measured correctly. The true exposure model is a binomial model with a logit link: N, the number of clusters, to be 25 and ni = 200 for all clusters. We simulated 50 data sets and fit three models. The first is the naïve model where the misclassified covariate is assumed to be measured without error. The second model is the case of a single fallible diagnostic test. Finally, we consider the case of two conditionally independent diagnostic tests. For the case of a single diagnostic test we require informative priors for the sensitivity and specificity. We use a beta  for S1 and a beta  for C1. Note the prior mean for S1 is 0.55 and the prior mean for C1 is 0.94. So the priors are centered near the truth, but the 95% prior interval for S1 is , which allows for a wide range of possible values. These priors are consistent with [S1, C1, S2, and C2.For the simulation we setent with  which acβ1, are displayed β1 = 0.4 The naïve model performs particularly poorly illustrating the problem of ignoring the misclassification. The average posterior mean across the 50 data sets is approximately 0 and none of the 95% intervals contained the true value. The two corrected models performed significantly better with both having empirical coverage over 90% and significantly less bias. The main cost of the misclassification model over the naïve is larger variability. For instance, the posterior standard deviation for β1 averages only 0.025 for the naïve model but is about 0.11 for both misclassification models. Accounting for misclassification and measurement error commonly inflates variability in estimates, but this is preferred to the large bias and low coverage of the naïve model. Though β1 has the largest degree of bias for the naïve model, most of the other parameters are also somewhat biased and coverages of the intervals below nominal. The performance for all the parameters are provided in Appendix.The results across the 50 simulations for the main parameter of interest,  S ~ Beta  and C ~ Beta . The priors indicate a substantial degree of underreporting with a prior mean of only 40% for the sensitivity. Conversely, the specificity is expected to be quite high, with a prior mean over 94%. We also consider the naïve model and report the posterior estimates of the response model; however, it is unnecessary to consider the exposure model.We apply the proposed Poisson regression model with a misclassified covariate to the India National Health Survey (NFHS-3) 2005-2006 data accounting for the 27 states as a cluster random effect. We consider five covariates: a binary indicator for whether a parent reported being abused, the age of the female, the education level of the female, a binary indicator as to whether the family considers themselves religious, and a continuous variable indicating their wealth. The number of children each woman surveyed has is the outcome variable and we assume the counts are distributed Poisson. All of the variables are considered to be reported correctly other than the binary variable indicating whether a female was abused or not. Because the one test Poisson model is nonidentifiable we must supply the prior on the sensitivity and specificity relatively informative beta distributions. Information from , 15, 19 β1. For the naïve model, the posterior of β1 is positive and the 95% interval is completely above 0 indicating statistical significance. Thus the inference would be that parental partner violence would lead to an increase in number of children. For the misclassification model, the coefficient is not significant as the interval does contain 0. This is not unusual because misclassification is well known to bias estimates towards the null. What is particularly interesting here is that though the interval contains 0, the posterior mean is negative, thus by accounting for the misclassification the majority of the posterior shifted from positive to negative. This further illustrates the importance of accounting for covariate misclassification in hierarchical count models. The other covariates in the response model are all significant and are very similar in both models. Age has a positive relationship with number of children while education, religiosity, and wealth all have negative relationships. The naïve model has no estimates for the γ parameters or the sensitivity and specificity since these are all concerned with the misclassification model. For the misclassification model, we see that age, education, and wealth all reduce the probability of abuse among the parents and are statistically significant. Religiosity appears to have a positive relationship with abuse, but the 95% interval contains 0, so the relationship is not strong. The standard deviation of the random slopes for the outcome model is relatively modest with a posterior mean of 0.13. This indicates only a small degree of heterogeneity of the baseline average number of children across the 27 states. For the misclassification model, the random effect standard deviation has a posterior mean of 1.12 indicating considerably more heterogeneity of the baseline probability of abuse across the 27 states. Finally, the sensitivity and specificity both have posteriors that are centered around larger values than the priors. This indicates there is less misclassification in the data than was expected, but still considerable underreporting since the sensitivity is still centered around a value less than 0.5.We fit the data with both the proposed model and a naïve model that does not account for the misclassification. The models were fit in OpenBUGS and inferences were based on 20,000 iterations after 10,000 burn-ins. The results for both models are provided in In this paper, we have addressed the problem of covariate misclassification in a hierarchical count model. Through simulation, we illustrated that ignoring the misclassification can lead to biased estimates and undercoverage of interval estimators. Our real data example demonstrated an extreme possibility in that the naïve model yielded results that were statistically significant  while the bias corrected model had a point estimate that was negative, though not statistically significant. There are several extensions to the model we have proposed. In some cases, the count response may also be subject to misclassification. Accounting for under- or overreporting of the response in a hierarchical model such as this would be an interesting follow-up work. An important limitation to note is that while the information for our priors for the example is from developing countries including studies from India specifically, we have not matched subjects with our current data; thus there is no guarantee that the populations match perfectly. This is one reason why the two diagnostic case approaches are preferred because the information on the sensitivity and specificity come from the current data. However, the one diagnostic test case essentially works as a Monte Carlo sensitivity analysis where the priors dictate a range of likely values for the sensitivity and specificity. In that sense, our results are robust with respect to a large number of possible true values of the sensitivity and specificity.