TY - JOUR
AU - Davies, William J.
AB - Abstract This paper demonstrates the use of a novel suite of data‐based, recursive modelling techniques for the investigation of biological and other time‐series data, including high resolution leaf elongation. The Data‐Based Mechanistic (DBM) modelling methodology rejects the common practice of empirical curve fitting for a more objective approach where the model structure is not assumed a priori, but instead is identified directly from the data series in a stochastic form. Further, this novel approach takes advantage of the latest techniques in optimal recursive estimation of non‐stationary and non‐linear time‐series. Here, the utility and ease of use of these techniques is demonstrated in the examination of two time‐series of leaf elongation in an expanding leaf of tomato (Lycopersicon esculentum L. cv. Ailsa Craig) growing in a root pressure vessel (RPV). Using this analysis, the component signals of the elongation series are extracted and considered in relation to physiological processes. It is hoped that this paper will encourage the wider use of these new techniques, as well as the associated Data‐Based Mechanistic (DBM) modelling strategy, in analytical plant physiology. Time‐series, data‐based mechanistic modelling, unobserved component model, tomato, leaf expansion. Introduction Why use models? Computer‐based modelling is becoming increasingly popular for the investigation of contemporary issues in environmental plant physiology and topics such as stomatal functioning and plant water flux have been identified as ideal targets for the application of such modelling techniques (Jones, 1992; Jones and Tardieu, 1998). Normally, computer models are developed either as a research tool in order to develop an understanding of the systems and to test hypotheses, or as a management tool for decision making and planning. Perhaps the most prevalent and easily recognized approach to modelling in both of these areas is the computer simulation model developed within a ‘reductionist’ modelling philosophy (Young et al., 1996). However, such a reductionist approach can lead to complex, highly parameterized simulation models that require extensive data collection in order to provide sufficient information for model identification, estimation and validation. Indeed, it is questionable whether they can be validated at all in a rigorous statistical sense, since the information content in experimental data will rarely support the estimation of the many parameters that characterize models of this kind. Nevertheless, such complex simulation models exist for a variety of applications in plant physiology, including the prediction of plant nutritional status (Le Bot et al., 1998); biomass production (Marcelis et al., 1998) and yield, for example, TOMGRO (Jones et al., 1991) and AFRC WHEAT (Weir et al., 1984). A common alternative approach to simulation modelling are the various forms of curve fitting as, for example, when the solution of the logistic and other growth equations are used to describe the expansion or elongation of a leaf over time (Richards, 1959). Although useful, this curve fitting approach often provides little additional insight into any transient dynamics, which may represent an important part of the system behaviour. Also, it is often based on a prior assumption about the specific nature of the growth curve and it precludes the use of the model in deterministic or stochastic simulation terms. The fitting of empirical curves to experimental data is one aspect of data‐based modelling. Such a data‐based, or ‘black‐box’ model is developed entirely on the basis of observational data and the parameters of the model are usually obtained through some process of statistical estimation or deterministic optimization. Typical examples are transfer function (Young, 1984), neural network and neuro‐fuzzy models (Jang et al., 1997). Such models can be useful in yielding information on the system dynamics and they provide a natural basis for both forecasting and control system design. On the other hand, they do not normally provide much information on the mechanistic nature of the underlying physiological processes. Why use a DBM approach? This paper introduces a Data‐Based Mechanistic (DBM) approach to modelling (Young, 1999a, and references therein) that is a hybrid between black‐box modelling at one extreme and simulation modelling at the other. It appears to be widely applicable and, in contrast to the conventional hypothetico‐deductive approach to modelling where the model structure is assumed a priori by the model builder (as in growth modelling by curve fitting), the inductive DBM analysis allows the data to expose mechanistically meaningful model structures, thus ensuring that the modelling is as objective as possible. Data‐Based Mechanistic (DBM) modelling The DBM modelling philosophy is captured in a suite of novel statistical time‐series modelling algorithms that are available in the CAPTAIN Toolbox (see http://www.es.lancs.ac.uk/cres/captain/), within the MATLAB™ software environment (The Mathworks: MATLAB™ is a well‐known and widely available numeric computation and visualization software package). These algorithms allow for non‐stationary time‐series analysis and forecasting (i.e. where it is assumed that the temporal and spectral characteristics of the time‐series may vary over time), as well as the modelling of multivariate discrete and continuous‐time transfer function models. Although many of these techniques are used extensively in environmental, economic and engineering fields, for time‐series modelling, forecasting, and signal processing, they have so far been under‐exploited for the study of biological data. A typical prior example where DBM modelling has proven useful in plant physiology research is the modelling of stomatal conductance to a reduction in humidity over leaves (Jarvis et al., 1999), where the transfer function identification and estimation tools exposed and quantified the nature of the stomatal feedback mechanisms. When used within MATLAB™, the algorithms are computationally fast and quite easy to use, as demonstrated in this paper. The DBM approach is, therefore, able to facilitate the kind of iterative approach to combined experimental planning and modelling which can make research more efficient and help avoid the possibility of prejudged scientific inference. This paper demonstrates the use of these algorithms in modelling time‐series of accumulating leaf length and provides the basis, in line with DBM modelling philosophy, to explore the regulatory components of leaf elongation which may not be revealed so easily by traditional analysis. Materials and methods Measurement and analysis of tomato leaf elongation In the experiments, high resolution measurements of leaf elongation were made using a linearly variable differential transformer (LVDT) system. This system allowed high spatial and temporal resolution measurement of the linear increase in size of a currently expanding leaf within the apex of a 4‐week‐old tomato plant. The leaf was fixed in such a way as to remove the influence of main stem and petiole elongation from the linear elongation of the leaf. Care was also taken to ensure that the analysis took place during the period when leaves expanded in an approximately linear manner with respect to time (between days 8–15 of a 20 d period of elongation, unpublished observation). As the leaf increases in length over time, it displaces the core of the LVDT, which is attached to the leaf via a cantilever system designed to amplify these relatively small increases in leaf length over time. All experiments were conducted in a growth cabinet, with a day‐night temperature of 25/12 °C, a relative humidity of 30/55% and a 12 h photoperiod with a photosynthetic photon flux density (PPFD) of 800 μm m−2 s−1 provided by 400 W tungsten halide lamps (Osram powerstar, HQI‐T). The time‐series data used for DBM analysis in this paper were obtained by frequent (5 min intervals) measurement of the currently expanding leaf of a plant which was either well‐watered or allowed to dry the soil over 7 d. Temperature was measured at 10 min intervals using a Tiny‐Talk data logger (Orion Ltd., Beds UK). The experimental equipment was assembled on a vibration‐proof plate to minimize any changes in LVDT output that were not related to any aspect of leaf expansion. Any effect not related to leaf expansion was quantified by coupling the LVDT to a mature, expanded leaf. No change in output was recorded under these circumstances. The Dynamic Harmonic Regression (DHR) model In order to develop a dynamic model of leaf elongation, the Dynamic Harmonic Regression (DHR) algorithm (Young et al., 1999), one of the tools for modelling non‐stationary time‐series in CAPTAIN, was first employed to analyse the nature of the data. The DHR algorithm is based on recursive estimation (Young, 1984), where the parameters in the model are updated sequentially whilst working through the data, one sample at a time, normally in temporal order. The DHR model is a particular example of the generic Unobserved Component (UC) model (Young, 1999b). In the present context, it is used to identify objectively any significant periodic components in the accumulating leaf length (ACLL) series and represent them by a time variable parameter (TVP or ‘dynamic’) harmonic regression model. In general, the periodic components in this DHR model will represent the fundamental and harmonic frequency components associated with the periodicity in the data; while any estimated temporal changes in the associated harmonic regression parameters allow for any temporal changes in the amplitude and phase characteristics of these components. This can be contrasted with conventional harmonic regression, where it is assumed that the amplitude and phase of the periodic components are time invariant. Since the DHR algorithm provides estimates of the periodic components, it can then be used in various ways (Young, 1999b): for example, forecasting; backcasting; data ‘repair’ (interpolation over gaps); identification and removal of ‘outliers’; and the removal of the periodic components from the series (often termed ‘seasonal adjustment’ when applied to economic and environmental time‐series). In the present context, the DHR model takes the following form (Young et al., 1999):   1 where St represents the total periodic component present in the data (e.g. a diurnal periodicity, as in the present example); Tt represents any long‐term, low‐frequency movements or ‘trends’ in the data; et is a residual series, assumed to be a serially uncorrelated, normally distributed, Gaussian sequence of purely random variables with zero mean value and variance σ2 (discrete‐time ‘white noise’). The subscript t denotes the sampled value of the variable at sample time t; and N is the total sample size. The most complex unobserved component in (1) is the total periodic component St, which is defined as the following linear sum of Rs sub‐components, each composed of a combination of sines and cosines at the appropriately defined periodic frequencies:   2 Here, each ait and bit is a stochastic Time Variable Parameter (TVP), as discussed later, and ωi, i=1, 2, … , Rs are the fundamental and harmonic frequencies associated with the periodicity in the series. In this form, the model is able to model a complex periodic component whose amplitude \(\left(A_{t}{=}\sqrt{{a_{it}^{2}}{+}{b_{it}^{2}}}\right)\) ; and phase \({(}\mathrm{tan}^{{-}1}\{b_{it}{/}a_{it}\}{)}\) is changing over the observation interval of N samples. The DHR algorithm estimates all of the components in (2) simultaneously using a combination of forward‐recursive ‘filtering’, followed by backwards‐recursive ‘fixed interval smoothing’ (Young et al., 1999). The smoothed estimates are particularly important in the present context because they are more accurate (i.e. they have a lower statistical error variance). Also, because the estimates of the parameters ait and bit, at any sampling instant t, are based on the whole of the N samples in the data set, they do not suffer from the phase lag inherent in conventional filtering. Results and discussion Accumulating leaf length (ACLL) series Figure 1 shows the ACLL profile of a currently expanding leaf within the apex of a tomato plant, which was either well‐watered or allowed to dry the soil for the 7 d experiment. Although on visual inspection both ACLL series in Fig. 1 appear approximately linear over time, there is some evidence of periodicity within both series. The plot illustrates that the plant subjected to soil drying exhibits a reduced rate of leaf elongation. However, it is difficult to ascertain visually how the series may be responding to any short‐term changes in environmental conditions. Fig. 1. View largeDownload slide Accumulating leaf length (mm) of an expanding leaf within the apex of a tomato plant which was either well‐watered (full line (A)), or allowed to dry in the soil from the beginning of the experiment (dotted line (B)). Fig. 1. View largeDownload slide Accumulating leaf length (mm) of an expanding leaf within the apex of a tomato plant which was either well‐watered (full line (A)), or allowed to dry in the soil from the beginning of the experiment (dotted line (B)). Modelling the components of the accumulating leaf length (ACLL) series Both measured ACLL series (Fig. 1) show some evidence of diurnal fluctuations. The DHR model estimates the frequencies associated with these periodic components by reference to the spectral properties of the series, as defined by the AutoRegressive (AR) spectrum. This is estimated by initial AR analysis of the series, in order to identify the number and values of the fundamental and harmonic frequencies associated with the periodicity. The well‐watered ACLL profile was considered first. In order to enhance the extraction of any diurnal signals and simplify the analysis, the data were subsampled every 60 min using the decimation combined optimal smoothing and subsampling tool in CAPTAIN. The AIC identified an AR(29) model and the associated AR(29) spectrum, presented in Fig. 2, confirms that the measured ACLL series does contain 24 h periodicity. This is indicated by the pronounced peaks at periods of 23.2, 12.2, 8, 6, 4.7, 3.3, and 2.7 (samples/cycle), which are close to the fundamental period (24 h) of the diurnal oscillation and some of its harmonics: 24/2, 24/3, 24/4, 24/5, 24/7, and 24/9 h. The pronounced power at periods greater than 30 h in the spectrum, arises as a result of the low frequency trend, in this case the underlying growth characteristics. As seen in Fig. 2, the above periodic components dominate the AR spectrum, indicating that the trend and fundamental 24 h period are the most important in shaping the observed series. As a result, the simplest DHR model (1) includes the trend and periodic components characterized by these fundamental and associated harmonic terms. The residual component et then represents the ‘noise’ in the series and is assumed to be completely random and so unpredictable. This assumption is tested statistically following model estimation. Also, the DHR model is often validated by ensuring that it is able to forecast the series well outside the observation interval used for estimation. Each of the TVPs (ait and bit, i=1, 2, …, 14, in this example) in the DHR model (1)–(2) are represented as a Random Walk (RW) process of the form:   where ηi,t is a zero mean, white noise input with variance \({{\sigma}_{{\eta}i}^{2}},\) uncorrelated with et in equation (1). In this manner, the estimation algorithm is instructed that the parameter in question is a stochastic variable that is likely to change by an unknown but small amount over each sampling interval, within the stochastic limits imposed by the variance \({{\sigma}_{{\eta}i}^{2}}.\) In turn, this allows for any changes in the amplitude and phase of the associated periodic components over the seven day period (Young, 1999b). The trend component Tt in equation (1) is modelled as an Integrated Random Walk (IRW) of the form:   This is simply an integrated version of the RW and is specified here in order to provide a smoother estimate of the changes in this low frequency component. It has the additional advantage that the second state variable dTt in the model is the rate of change of the trend component; in the present context, the elongation rate of the leaf (see next section). Clearly, the variances \({{\sigma}_{{\eta}i}^{2}}\) in the RW and IRW models are critical in estimating the variable parameters and need to be optimized against the data. In fact, within DHR analysis, it is the ratio of this variance to the variance σ2 of the residual et (the ‘Noise Variance Ratio’ or NVR) that is optimized in a maximum likelihood sense (Young et al., 1999b). The optimum NVRs, or ‘hyper‐parameters’ are then inserted into the recursive DHR algorithm to yield the estimates of the time variable parameters and, hence, the estimates of each unobserved component, which collectively make up the total DHR model output. The trend, the sum of the periodic components, and DHR model residuals obtained in the above manner are plotted in Fig. 3A, B, C, respectively. The total DHR model output ŷt, comprising all the estimated components except et, is plotted in Fig. 4, where it is compared with the observed well‐watered ACLL signal, sampled every hour. It is clear that, by breaking the ACLL series into its key components, the resultant DHR model provides an excellent explanation of the growth series, leaving only a small residual series that the algorithm has identified as noise on the data. The DHR modelling procedure outlined above was repeated for the ACLL series of the plant growing in drying soil, with an AIC identified AR(30) model spectrum in this case. The DHR analysis yields a trend and periodic components with a fundamental frequency of 24.2 h and harmonics of 12.1, 7.9, 6, and 4.8 h. Fig. 2. View largeDownload slide Autoregressive spectrum for the accumulating leaf length (mm) for the well‐watered plant, sampled every 60 min. Solid line, measured data; dashed line, modelled data. Fig. 2. View largeDownload slide Autoregressive spectrum for the accumulating leaf length (mm) for the well‐watered plant, sampled every 60 min. Solid line, measured data; dashed line, modelled data. Fig. 3. View largeDownload slide DHR model: (A) trend (mm); (B) sum of all the periodic components (mm); (C) DHR model residuals (mm). Fig. 3. View largeDownload slide DHR model: (A) trend (mm); (B) sum of all the periodic components (mm); (C) DHR model residuals (mm). Fig. 4. View largeDownload slide Plot showing the first 80 h section of the measured accumulating leaf length (mm) in the well‐watered plant (circles) plotted alongside the DHR model (full line) i.e. trend plus all periodic components. Fig. 4. View largeDownload slide Plot showing the first 80 h section of the measured accumulating leaf length (mm) in the well‐watered plant (circles) plotted alongside the DHR model (full line) i.e. trend plus all periodic components. Interpreting the DHR models in relation to physically meaningful processes The DHR model identifies two key components of the ACLL series in both well‐watered and soil drying plants: (1) a low‐frequency trend representing an objective estimate of the underlying growth pattern; (2) a cyclical component with a 24 h periodicity representing diurnal fluctuations about this growth path. The next step within the DBM framework, is to consider whether the model is physically acceptable and provides any additional insight into the system. The trend component: The observed variation in growth between the well‐watered plant and the plant where the soil was allowed to dry, is reflected in the trends for each DHR model. The optimal estimate of the elongation rates for each plant is automatically estimated in the DHR analysis as the second, rate of change, state variable in the IRW model of the trend component (see previous description of the IRW model). The estimated elongation rates obtained in this manner are compared in Fig. 5. The smooth nature of these curves contrasts with the crude estimate of the elongation rates obtained by differencing the data directly, which are very noisy. Based on the DHR estimates, the elongation rate of the well‐watered plant has a mean value of 0.39 mm h−1 and fluctuates dramatically over the 7 d period. This can be compared with the soil drying series, which had a mean rate of 0.31 mm min−1 and remains relatively steady. During the first 24 h of analysis, the growth rates of the two plants increase in unison. During this stage of leaf development (0–8 d after emergence), the rate of cell division predominates over the rate of cell elongation, explaining the relatively slow growth rates in both plants (Lecoeur et al., 1995). The growth rate of the plant subjected to soil drying then remains fairly static, while the growth rate of the well‐watered leaf continues to increase rapidly. A restriction in leaf elongation in plants growing in drying soil is a well‐reported phenomenon (Passioura, 1988; Saab and Sharp, 1989; Van Volkenburgh and Boyer, 1985). However, the underlying mechanism that brings about this restriction is less well understood. For several decades, the idea of a totally hydraulic limitation to the elongation of leaves of plants growing in drying soil held sway in the literature (see Kramer, 1988, for review). More recent work, over the past 15–20 years, has cast doubt upon this purely hydraulic explanation: several investigations have now demonstrated that even if the water relations of the plant can be held constant in soil drying conditions, a restriction in leaf elongation can still be seen (Passioura, 1988). Work such as this and others (Gowing et al., 1990) has led to the suggestion that the plant can in some way ‘sense’ the water available within the soil and communicate this information from the roots to the shoots via a xylem‐borne signal. Periodic behaviour: The DHR analysis has identified periodic behaviour in the ACLL series of both plants, characterized by a fundamental period of 24 h. Diurnal patterns in leaf elongation rates have been reported in the literature (Thomas and Stoddart, 1994) but no sound understanding of the phenomenon is reported. One possible reason for lower elongation rates during the dark is the lower air temperature. However, other studies have found the opposite to be the case, with higher leaf elongation rates recorded at lower night‐time temperatures (Parish and Wolf, 1983). In the few studies in which leaf temperature has been held constant, leaf elongation has still been observed to have a diurnal signal, implicating some internal circadian mechanism, that is cued, but not driven by the light environment (Thomas and Stoddart, 1984; Schnyder and Nelson, 1988). Regulation of leaf expansion by light or temperature?: In the current investigation, visual inspection would suggest that the diurnal signal in ACLL mirrors that of ‘near‐leaf ’ temperature and photoperiod. Figure 6A presents the diurnal temperature fluctuation and Fig. 6B and C the diurnal signals of the ACLL series and of the plant grown in drying and well‐watered soil, respectively. This close correlation suggests that temperature and/or light were the key environmental variables driving the rate of ACLL in both well‐watered plants and plants growing in drying soil. Temperature variation is one possible explanation for the varying occurrence of growth peaks during either the dark or light photoperiod. In order to investigate these results further, therefore, an additional short experiment was conducted in which the ACLL of a well‐watered plant was recorded continuously for a 2 d period, as before, at a higher day‐time temperature of 34 °C. The DHR analysis identifies the same diurnal signal in this series, with a fundamental frequency of 23.4 h and low frequency harmonics of 11.9, 7.9 and 5 h. In contrast to the experiment at the lower temperature, however, the peak in the 24 h ACLL periodicity occurs during the dark period when temperatures are lower; namely a 12 h ‘phase‐shift’ in relation to the ACLL series of the well‐watered plant growing at a lower ambient temperature. Figure 6D presents the photoperiod and Fig. 6E the periodic component for the higher temperature, well‐watered series in order to illustrate this ‘phase shift’ in visual terms. Here, it is clear that the peak in growth now coincides with the ‘lights off ’ period of the corresponding photoperiod. These observations suggest that the photoperiod is the cause of the diurnal variation but that temperature is modulating the phase of the diurnal signal. The existence of entrained diurnal oscillations in plants is well reported (Sweeney, 1987). However, it is clear that the DBM approach has allowed this insight to be gathered in a manner which is difficult using more conventional comparative analysis. Increasing the ambient temperature has shifted the maximum period of growth from day to night (a ‘day–night phase shift’), but the diurnal nature of the ACLL series itself would appear to be cued by the start of the photoperiod. This would suggest that the photoperiod, together with temperature is entraining some form of ‘phase modulator’ within the plant, which regulates leaf elongation. Numerous candidates for such modulators exist, including diurnal changes in cellular water relations and/or cell wall mechanical properties (Van Volkenburgh, 1999). The broader basis for this study is to identify potential chemical regulators of leaf elongation within the xylem sap. Several reports already confirm that xylem sap pH (Wilkinson, 1999), concentrations of nitrate ions (Gollan et al., 1992) and abscisic acid (Zhang and Davies, 1990) in the xylem sap show a distinct diurnal behaviour and are all capable of regulating leaf elongation via changes in cell wall properties or water relations (Bacon, 1999; Van Volkenburgh, 1999). Consequently, it may not be unreasonable to suggest that such plant growth regulators are responsible for the temperature‐independent diurnal pattern of leaf elongation revealed by the DHR analysis. The harmonics of the 24 h signal: The DHR analysis has identified the fundamental period of 24 h and its principal harmonics. Of these, those periodic components at 12 h and 24 h are clearly the most significant and dominate the AR spectrum. Figure 7 presents the photoperiod (A), the 12 h component (B), and the sum of the periodic components (C), for the lower temperature well‐watered ACLL series. The 12 h signal is seen to trough in the middle of the photoperiod and is qualitatively similar to the frequently observed phenomenon of midday partial stomatal closure. Prior to midday closure, water deficits may begin to develop and restrict leaf elongation. It may be speculated that the 12 h harmonic estimated in this study may also be the result of a similar turgor‐related phenomenon. Fig. 5. View largeDownload slide Estimated leaf elongation rates in well‐watered plants (solid line) and plants growing in drying soil (dashed line) determined via the DHR analysis (see text for details). Fig. 5. View largeDownload slide Estimated leaf elongation rates in well‐watered plants (solid line) and plants growing in drying soil (dashed line) determined via the DHR analysis (see text for details). Fig. 6. View largeDownload slide (A) Temperature variation with changing photoperiod; (B) DHR model; total periodic components (mm) for the tomato grown in drying conditions; (C) DHR model; total periodic components (mm) for the measured accumulating leaf length in the well‐watered tomato series; (D) photoperiod; (E) DHR model: total periodic components (mm) for the accumulating leaf length series of the well‐watered plant at a higher ambient temperature. Fig. 6. View largeDownload slide (A) Temperature variation with changing photoperiod; (B) DHR model; total periodic components (mm) for the tomato grown in drying conditions; (C) DHR model; total periodic components (mm) for the measured accumulating leaf length in the well‐watered tomato series; (D) photoperiod; (E) DHR model: total periodic components (mm) for the accumulating leaf length series of the well‐watered plant at a higher ambient temperature. Fig. 7. View largeDownload slide (A) Diurnal temperature variation; (B) DHR model; first harmonic (12 h period) (mm) for the well‐watered tomato; (C) DHR model; total periodic components (mm) for the accumulating leaf length (mm) series in the well‐watered plant. Dashed lines are added to highlight the influence of the 12 h component on the ‘shape’ of the total periodic signal. Fig. 7. View largeDownload slide (A) Diurnal temperature variation; (B) DHR model; first harmonic (12 h period) (mm) for the well‐watered tomato; (C) DHR model; total periodic components (mm) for the accumulating leaf length (mm) series in the well‐watered plant. Dashed lines are added to highlight the influence of the 12 h component on the ‘shape’ of the total periodic signal. Interpolating gaps in a measured ACLL series One of the desirable features of the DHR algorithm is that it can function well even if there are large gaps in the data and this can help the analysis in this situation. In order to demonstrate this facility, a gap in the cumulative ACLL data was introduced between 120 and 140 h. Figure 8A presents the resultant interpolated series, together with the original measured ACLL series into which the gap in the data was introduced. Figure 8B compares the 24 h periodic component of the original series which was influenced by the LVDT adjustment, and the interpolated 24 h component. The accuracy of the reconstruction is clear from the figure. Fig. 8. View largeDownload slide (A) Enlarged view to compare the original well‐watered accumulating leaf length series (circles) with artificial missing samples in the data introduced and the interpolated signal (solid line) with standard error bands (dashed lines). (B) The original 24 h fundamental period component for the well‐watered plant (dashed lines) compared with the interpolated 24 h component signal from the series which had a gap in the data introduced. Fig. 8. View largeDownload slide (A) Enlarged view to compare the original well‐watered accumulating leaf length series (circles) with artificial missing samples in the data introduced and the interpolated signal (solid line) with standard error bands (dashed lines). (B) The original 24 h fundamental period component for the well‐watered plant (dashed lines) compared with the interpolated 24 h component signal from the series which had a gap in the data introduced. Conclusions So why should the plant physiologist use analytical techniques such as DHR over other, more conventional, curve‐fitting practices? Unlike curve fitting, with its associated a priori assumptions, the more objective nature of this DBM modelling technique assists in obtaining greater insight into the data set and prevents any mis‐interpretation associated with inappropriate inference about the behaviour of a system. Techniques such as DHR actually use the properties of the data to optimize the data processing and so ensure optimal attenuation of noise and extraction of the underlying signals. The unobserved component analysis ensures that the periodic components of interest are clearly defined and are, therefore, much easier to compare with potential regulators of a system. In this study, it has been demonstrated that, by increasing the ambient daytime temperature, the period of higher elongation rates is moved from the light period (at lower temperatures) to the dark period (at higher temperatures). The paper has introduced the concept of Data‐Based Mechanistic (DBM) modelling but this approach is not exploited to any great degree in the present paper. Typical examples which show the value of the DBM approach within a more complex setting are given elsewhere (Jarvis et al., 1999; AJ Jarvis, PC Young, WJ Davies, unpublished results). Also, further analysis of ACLL data, as described elsewhere (PC Young, J Butler, AJ Jarvis, unpublished results), shows how other tools available in CAPTAIN allow for the DBM modelling of the growth rate characteristics. This analysis makes no prior assumptions about the nature of the growth model, which is inferred objectively from the data, initially in non‐parametric form, and then provides a meaningful biological interpretation of these growth characteristics. It must be emphasized that this paper is focused primarily on demonstrating the potential application of the novel time‐series techniques available in the CAPTAIN Toolbox. Future experimental research on leaf growth, including further analysis of the resulting experimental data using other options in the Toolbox, will be aimed at testing the physiological interpretation of the analysis that has been presented in this paper. 1 To whom correspondence should be addressed. Fax: +44 1524 843854. 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TI - High‐resolution analysis of tomato leaf elongation: the application of novel time‐series analysis techniques
JF - Journal of Experimental Botany
DO - 10.1093/jexbot/52.362.1925
DA - 2001-09-01
UR - https://www.deepdyve.com/lp/oxford-university-press/high-resolution-analysis-of-tomato-leaf-elongation-the-application-of-FkcAXahdOU
SP - 1925
EP - 1932
VL - 52
IS - 362
DP - DeepDyve
ER -