4/1/2019 1 LECTURE 09: EMPIRICAL MODELS Total Growth Logaritmic phase Stationary phase Lag phase Growth rate Growth Time course Why the accumulation of plant biomass with time 1. increases slowly at the beginning 2. increases rapidly thereafter, and 3. is almost constant at the end of growing period http://smtom.lecture.ub.ac.id/ Password: https://syukur16tom.wordpress.com/ Password: LEARNING OUTCOMES After the completion of this lecture and mastering the lecture materials, students should be able 1. to explain ‘classical’ and ‘functional’ approach of plant growth analysis. 2. to apply several empirical models to analyze the data of plant growth. 3. to decide what empirical models to be used to obtain information from the data of plant growth
Transcript
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LECTURE 09:EMPIRICAL MODELS
Total Growth
Logaritmicphase
Stationaryphase
Lag phaseGrowthrate
Gro
wth
Time course
Why the accumulation of plant biomass with time1. increases slowly at the beginning2. increases rapidly thereafter, and3. is almost constant at the end of growing period
http://smtom.lecture.ub.ac.id/Password:
https://syukur16tom.wordpress.com/Password:
LEARNING OUTCOMES
After the completion of this lecture and masteringthe lecture materials, students should be able1. to explain ‘classical’ and ‘functional’ approach of
plant growth analysis.2. to apply several empirical models to analyze the
data of plant growth.3. to decide what empirical models to be used to
3. EMPIRICAL MODELS1. Linear Model2. Exponential Model3. Power (Allometric) Model4. Polynomial Model5. Logarithmic Model
1. INTRODUCTIONWhat is an empirical model?1. Many problems in engineering and science
involve exploring the relationships betweentwo or more variables.
2. For example, in a chemical process, supposethat the product of a process is related to theprocess-operating temperature.
3. An empirical model is based only on data andis used to predict, not to explain, a system
4. An empirical model consists of a function thatcaptures the trend of the data.
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5. An empirical model is a model where thestructure is determined by the observedrelationship among experimental data. These models can be used to develop relationships
for forecasting and describing trends. These relationships and trends are not necessarily
mechanistically relevant. Source: EPA website, glossary of frequently used modeling terms.
6. Models are only useful if they help you solveproblems.
7. The best model, whether it is theoretical orempirical, is the model that predicts best foryour situation.
8. An example of an Empirical Model: Investigating the relationship of inflowing nutrients in a
lake to algal biomass production (eutrophication). Most early (circa 1970) lake eutrophication models
based on statistical relationships between mass loadingof nutrients and average algal biomass (e.g.,Vollenweider models with numerous adaptations byothers).
A Vollenweider Model
where:0.368 = conversion factorP = total phosphorus concentration in lake (mg/l)L = areal phosphorus load (lbs/ac/yr)Z = mean depth of lake (feet), andp = the flushing rate in times (per year)
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Conceptual model
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LR: loading rateFR: unloding rateSR: sedimentation rateP: phosphorus[ P]: P concentration[PL]: [P] in inflow river[PF]: [P] in outflow river
[P0]: initial [P] in lakeV0: initial water volumeRIR: river inflow rateROR: river outflow rate
Conceptual model ofmechanistic model
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9. Where does human knowledge ultimately comefrom? EXPERIENCE. Empiricists have always claimed
that sense experience is the ultimate starting point forall our knowledge.- Empiricists stress induction which involves generalizing from
observables (SpecificGeneral) Science REASON. Rationalists have claimed that reason,
not the senses, is the ultimate starting point for allknowledge.- Rationalists stress deduction
which involves inferring from firstprinciples (GeneralSpecific) Mathematics
2. INITIAL PLANT GROWTHANALYSIS
A. ‘Classical' Approach1. The 'classical' approach is one of the oldest
methods in plant growth analysis.2. This classical approach known as RGR (relative
growth rate) was introduced in the beginning of 20th
century (Blackman 1919, West et al. 1920)3. This is calculated by dividing the difference in “ln-
transformed plant weight “at two harvests by the timedifference between those harvests.
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4. The RGR is derived from the exponential modelon the basis of an assumption that“an increase in plant biomass (W or W) over a shortperiod (t or t) is determined by the size of plant” asfollows
5. Therefore, the use of RGR applies to plantgrowth which follow exponential pattern
Wt = W0ert
6. This approach, although straightforward, hasbeen considered unsatisfactory based on thefollowing reasonsa. The accumulation of plant biomass (W) with time
may be not exponential as the base of RGR.
b. The time course of the growth rate may be severelyobscured by fluctuations in RGR between adjacentharvest intervals (Causton & Venus 1981).
c. In the calculation of the net assimilation rate (NAR),a fixed relation between leaf area and plantweight has to be assumed (Evans 1972).
d. It is difficult to statistically evaluate differencesin RGR (Poorter and Lewis 1986).
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B. ‘Functional' Approach1. The 'functional' approach, developed in the
1960's (Vernon and Allison 1963, Hughes andFreeman 1967), has been presented as asolution to the previous RGR problems.
2. The approach proposed a polynomial of theform;
Y = b0 + b1X +b2X2 + ... + bnXn
to analyze the growth data where Y = the ln-transformed weight of the plant, and X = the time
3. By differentiating this equation, an equationfor RGR is obtained.
4. Hunt (1982) mentions 12 advantages of thismethod namely, among others,(1) The functional approach provides a clearer
perception of ontogenetic drift;(2) Assumptions involved in the calculation of mean
values of NAR are avoided;(3) Statistical analyses may be integrated into the same
analytical procedure as the calculation of thederived quantities.
5. However, the functional approach does notnecessarily result in correct values for RGR,NAR and the confidence limits of theseparameters.
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6. Poorter and Lewis (1986) showed that thetesting of differences in RGR had only limitedbiological meaning.
7. Wickens and Cheeseman (1988) argued that thefunctional approach is of limited value if plantsare subjected to short-term environmentalchanges.
8. But even when plants are grown in a constantenvironment, the functional approach has somepitfalls.
9. The main problem of the functional approach isthe choice of the appropriate degree of thepolynomial to fit the data
Selecting a first degree polynomial will inevitably resultin a constant RGR, whereas
a quadratic function will invariably lead to a linearlyincreasing or decreasing value of RGR with lime.
10. This implies that complex growth patterns are'underfitted' if a too low order polynomial is used.
11. Hughes and Freeman (1967) proposed fitting all datawith a third degree polynomial to be able todescribe both simple and more complicated growthcurves.
12. However, Nicholls and Calder (1973) showed that ahigh degree of the polynomial may lead to'overfitting', resulting in an RGR with spuriousupward or downward trends, especially at the endsof the curve.
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1. The analysis of plant growth with the empiricalmodels (correlative or statistical models) is the useof available models to describe relationshipsamong variables without referring to the correlatedprocesses.
2. The empirical models are not derived fromassumptions concerning the actual relationshipbetween variables, and not based on physicalprinciples.
3. The first step in deriving an empirical model is toget the scatter plot of the data.
3. EMPIRICAL MODELS
3. If the data does not seem to be linear, try toplot one or both variables (X & Y) as logarithmsso that you can check if an exponential orpower (allometric) models are good fits.
4. The idea is to get a graph that looks reasonablylinear and then to get a linear model.
5. Models available in excel program that can beused directly area. Linear Modelb. Exponential Modelc. Power (Allometric) Modeld. Polynomial Modele. Logarithmic Model
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Keep in mind that: In Linear model, y depends linearly on x
y = ax + b In Power model, ln y depends linearly on ln x
y = axbln y = ln a+b lnx In Exponential model, ln y depends linearly on x
y = aebxln y = ln a+bx In Logarithmic model, y depends linearly on ln x
y = a ln(x) + bx =e[(y-b)/a]
A rapidly growing bacteria has been discovered, and its growth rate isshown in the chart.
The following procedure could be used for theapplication of empirical modeling to plant growth(1) Collect data(2) Open excel program, and type the data (columns and
rows)I. First Approach of Analysis(1) Block the data (highlight), click insert menu, then click
scatter, and choose (click) the graph you like(2) Set the performance of the graph to look it better
II. Second Approach of Analysis(1) Apply analysis with empirical models by clicking the data
in the figure, then click layout, trendline, and choose (click)a model you like.
(2) if model is not satisfactory, click the trendline and delete it,then go to the initial step
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A simple example of empirical model is the height oftomato plants in relation to temperature.- An experiment carried out on young tomato plants after 40
days of growth at Wageningen Univ. in a climate chamber. Inthe following table, the respective columns stand for averagedaily temperature, average nightly temperature and plantheight.
A statistical model can be defined from these data:Height = 36.3 + 4.48 * (Tday-21) + 1.37 * (Tnight-21)
The application of bars (standard deviation of datafrom replicates)
Linear model Exponetial model
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This analysis is applied to understand therelationship between the variables of plant growth
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QUESTIONS1. What is an empirical model?2. What is the best model?3. What is eutrophication?4. What are factor determining the accumulation of
phosphorous in lake based on Vollenweider model?5. What is the starting point for all knowledge?6. What is the condition required for RGR application?7. What is an advantage of functional approach?8. What is the main problem of the functional approach?9. What is the first step in deriving an empirical model?10.What is the formula of circumference?
