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Topic: Logistic curve

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Logistic function

Generalised logistic curve

Exponential growth

Sigmoid

List of exponential topics

Growth curve

List of curves

Logistic regression

Logistic map

Hubbert curve

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	[No title] (Site not responding. Last check: 2007-11-06)
		The sigmoid curve is the curve whose formula is the sigmoid function
		The sigmoid curve shows early exponential growth which slows to linear growth then decelerates until it reaches a saturation level at y = 1.
		Members of the family of curves with obtained by linear scaling and translation of the sigmoid curve are called logistic curves, and are found in a range of fields, from biology to economics.
	www.informationgenius.com /encyclopedia/l/lo/logistic_curve.html (182 words)

	Logistic Growth Curve -- AIDS Infections
		A logistic growth curve is an S-shaped (sigmoidal) curve that can be used to model functions that increase gradually at first, more rapidly in the middle growth period, and slowly at the end, leveling off at a maximum value after some period of time.
		The initial part of the curve is exponential; the rate of growth accelerates as it approaches the midpoint of the curve.
		This type of curve is frequently used to model biological growth patterns where there is an initial exponential growth period followed by a leveling off as more of the population is infected or as the food supply or some other factor limits further growth.
	www.nlreg.com /aids.htm (393 words)

	Intro
		The logistic curves from Figures 1 and 2 can be used to calculate a curve and a forecast for the percentage of the world population that uses the Internet.
		The curve thus calculated resembles a logistic and reaches a ceiling of 14% for the world users of Internet as a percentage of the world population, a discouragingly low figure.
		Logistics that cascade harmoniously show periods of low and high growth of comparable duration.[3] Accordingly, and given that Internet has had a decade of rapid growth, a decade of low growth can reasonably be expected before a new rapid-growth phase begins.
	ourworld.compuserve.com /homepages/tmodis/End_Internet_Boom.htm (1487 words)

	Logistic Curve (Site not responding. Last check: 2007-11-06)
		The logistic curve is used to model a variety of physical situations in which a quantity's growth is "self-limited," that is, the growth rate of the quantity depends on the size of the quantity in such a way that if the quantity grows beyond a certain level, the growth rate decreases.
		In situations of "controlled growth" the logistic curve frequently provides an easily constructed graph that is readily understood.
		Curves generated while using this demo procedure will be increasing (but not necessarily strictly increasing).
	astro.temple.edu /~dhill001/logistic/logistic.html (1621 words)

	Why logistic ogive and not autocatalytic curve?, J Linacre (Site not responding. Last check: 2007-11-06)
		In later work, Pearl exhorts researchers to use logistic curve or function in preference to autocatalytic curve because the latter is tied to a physical process.
		Ogive (pronounced oh-jive) indicates the shape of the logistic curve.
		The combination, logistic ogive, to identify the logistic curve is recent.
	www.rasch.org /rmt/rmt64k.htm (661 words)

	Logistic Model (Site not responding. Last check: 2007-11-06)
		Logistic model was developed by Belgian mathematician Pierre Verhulst (1838) who suggested that the rate of population increase may be limited, i.e., it may depend on population density:
		Logistic model has two equilibria: N = 0 and N = K. The first equilibrium is unstable because any small deviation from this equilibrium will lead to population growth.
		The second equilibrium is stable because after small disturbance the population returns to this equilibrium state.
	www.ento.vt.edu /~sharov/PopEcol/lec5/logist.html (399 words)

	JPT (Sep 2003): Forecasting Development of Heavy-Oil Reserves in Ultradeep Waters (Site not responding. Last check: 2007-11-06)
		The second characteristic is a roughly modified exponential curve (or logistic) can be used to describe the overall technological progress in the selected capability, including the extraordinary breakthroughs that have occurred in deepwater technologies during the last decade.
		in which u(x) is the logistic utility function for technology, a and b are constants, c is the technological risk-aversion coefficient, x is the variable of interest (e.g., capital-expense reduction), and e is the exponential constant.
		It is important to emphasize that the adjustment of the initial points on the S-curve for each logistic curve was made on the basis of consultation with specialists operating in deep water, because historical records of oil gravity and viscosity were not readily available.
	www.spe.org /spe/jpt/jsp/jptpapersynopsis/0,2439,1104_11038_1434505_1434577,00.html (1577 words)

	Laherrere: Multi-Hubbert Modeling
		The logistic curve was introduced by the Belgian mathematician, Verhulst, in 1845, as a law of population growth, and it was used extensively by the biometrician, Raymond Pearl [1925].
		The normal curve is generally called the Gaussian curve, although it is also known as the Laplace-Gauss curve or the de Moivre-Laplace-Gauss law, as Laplace, who was influenced by de Moivre’s work of 1718, discovered the law in 1780 but did not publish it until 1812.
		A comparison between a Hubbert curve and a Gauss curve with the same peak shows that the difference is quite small, while a Hubbert curve is close to a parabola on the upper half part of the curve.
	www.hubbertpeak.com /laherrere/multihub.htm (2068 words)

	Logistic Curve Fitting
		Since L is the "limiting population" for the "S" shaped logistic curve, a value of L that is appropriate to the problem at hand can usually be obtained by guessing.
		Use the method of "data linearization" to find the logistic curve that fits the data for the population of the U.S. for the years 1900-1990.
		Various curves can be fit, but they all depend on the value of L. No one knows this value in advance and it must be estimated.
	math.fullerton.edu /mathews/n2003/LogisticEquationMod.html (202 words)

	Fitting a Logistic Curve to Data -- from Mathematica Information Center
		In particular, in Italy much attention is being paid to the problem of algal blooms in the Adriatic Sea and this has led also to an increase of interest in mathematical ecology at all levels, from high school teaching to advanced research.
		The logistic differential equation, dealt with in the next section, is a classical but still useful model for describing the dynamics of a one-species population in and environment with limited resources.
		We will explore the problem of fitting, in the least-squares sense, a logistic curve (i.e., the solution to the logistic equation) to such a set of measured data.
	library.wolfram.com /infocenter/Articles/3198 (143 words)

	Logistic Equation: The parable of the parabola
		The logistic map is the simplest model in population dynamics that incorporates the effects of both birth and death rates.
		For the case of graphical iteration plots below, the logistic curve (parabola) is plotted in yelow, the y=x (diagonal line) is plotted in red, and the movement of the iterates is followed by the blue line.
		As mentioned earlier, an m period limit cycle of the logistic equation is the fixed point of the m composition of the logistic equation, therefore to determine the stability of m period limit cycle, we evaluate the slope of f
	chaos.phy.ohiou.edu /~thomas/chaos/logistic.html (1617 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The ordinary logistic curve is sometimes known as the autocatalytic or inverse exponential curve.
		For the logistic and generalized logistic curves, you are not allowed to constrain the sense of the curve by the SENSE option.
		In fact, the logistic curve with parameters α, β, γ and μ is the same as the logistic curve with parameters (α+γ), -β, -γ and μ; GenStat will report only one of the two possible versions.
	www.genstat.co.uk /7thEdition/help/server/FITCURVE.htm (1742 words)

	Population Growth of Lemna and Azolla
		The logistic model is an exponential at the beginning and a log curve at the end, while the logistic estimate of Nt is a negative exponential curve
		Although the experiment is not perfectly in accordance with the initial hypotheses because of the curve of the mean total, the experiment in general, follows the rules that was introduced in the introduction.
		The logistic estimate of the population size is a negative exponential curve.
	www.sftext.com /ecology/lemnaazolla.html (1583 words)

	A Loglet Applet
		Note that the algorithm is iterative, so if the curve doesn't look right, you may need to do this a few times before the parameters converge to their best values.
		Logistic curves are used to model growth and diffusion, like the height of a sunflower, the aggregate length of railroads, or the cumulative works of an author.
		Logistic curves can be fit to time-series data using Levenberg-Marquardt least-squares regression.
	phe.rockefeller.edu /applets/loglet-demo-Jan98 (459 words)

	logistic_function (Site not responding. Last check: 2007-11-06)
		This function is also called the standard logistic function and is often encountered in many technical domains, especially in artificial neural networks as...
		An example of a Hubbert curve is: The Hubbert curve closely resembles, but is different from, the shape of the probability density function of the normal distribution.
		Fit A Five-Parameter Logistic Function In their paper andquot;A five-parameter logistic equation for investigating asymmetry of curvature in...
	logistic_function.networklive.org (364 words)

	Growth rates and the ``bell'' view
		Just as the differential equation (1) reveals the mechanism propelling its integral equation (2), the rates of change of the component logistics provide clues to the mechanisms propelling the composite logistic.
		Panel B of Figure 6 shows the derivative of the component logistics of our test function (Panel A is shown again for comparison purposes).
		Loglet Lab can decompose a logistic curve into its discrete components; each component can be transformed using the Fisher-Pry transform, or their rate of change can be plotted.
	phe.rockefeller.edu /LogletLab/whitepaper_OLD/node12.html (197 words)

	[No title]
		One parameter family of curves 2 2 The function X + c/X is sketched for c = 1,.5,.25, 0, -.25, -.5, -1.
		The US Population and logistic growth The population of the US from 1790 is frequently modeled using logistic growth.
		Using 1790, 1850, and 1910 as exact values the appropriate logistic curve is drawn.
	archives.math.utk.edu /software/msdos/diff.equations/ode_ss/ode_ss.readme (703 words)

	THE HUBBERT CURVE : ITS STRENGTHS AND WEAKNESSES
		He referred to a bell-shaped curve, of which the most commonly used are the Normal or Gauss curve, and also to the derivative of the logistic curve (Bartlett 1999), but he gave no equations.
		The classic logistic curve was discovered by Verhulst in 1845 in connection with population studies.
		A trend-line (Excel) and a Hubbert curve are plotted on the basis of a peak of 3.5 Gb/a in 1938 and a half life of 70 years.
	dieoff.org /page191.htm (4798 words)

	Logistic Growth Versus Exponential Growth
		The principal assumption of the logistic curve, that the population growth declines in a straight line as the population increases, is flatly contradicted by the increasing population growth rate throughout most of the history of world population up to 1965....
		In plant or animal populations that are claimed to show exponential growth, closer examination invariably shows that the supposedly exponential curve is actually the lower limb of a logistic curve, or a section of a Lotka-Volterra cycle.
		In fact, Logistic Growth uses variable rates of growth which are inexorably tied to the limit to growth.
	members.optusnet.com.au /exponentialist/Logistic_Vs_Exponential.htm (3900 words)

	PC AI 17.2 Paid Version Page 36
		In fact, if we think of neural nets as a smooth curve modeling tool, it is quite easy to see what the network is learning, for one or two input dimensions.
		The next three figures show the original data points in green, the generating logistic curve in blue, and what a neural network learned from the data in red.
		The red lines are the curves produced by giving the network every x-coordinate value from 1 to 200.
	www.pcai.com /Paid/Issues/PCAI-Online-Issues/17.2_OL/New_Folder/Po(HY79/17.2_PA/PCAI-17.2-Paid-pg.36-Art3.htm (708 words)

	Lec Notes: Peterson - Pop Growth Limited
		The earliest attempt to simulate this curve was the logistic equation of Verhulst (1839).
		While growth curves similar to those predicted by the logistic growth model are occasionally found, most metazoans do not show very good fits to the equation.
		At one time, it was suggested that the logistic curve per se described a fundamental law of population growth.
	www.cnr.uidaho.edu /wlf448/Peterson3.htm (948 words)

	Logistic Regression
		With some models, like the logistic curve, there is no mathematical solution that will produce least squares estimates of the parameters.
		We can pick the parameters of the model (a and b of the logistic curve) at random or by trial-and-error and then compute the likelihood of the data given those parameters (actually, we do better than trail-and-error, but not perfectly).
		Now if we go back up to the last column of the printout where is says odds ratio in the treatment column, you will see that the odds ratio is 3.50, which is what we got by finding the odds ratio for the odds from the two treatment conditions.
	luna.cas.usf.edu /~mbrannic/files/regression/Logistic.html (2682 words)

	Logistic Equation
		Mathematical details of the logistic equation are explained, and applied to Australia's population projections....
		For early years, the death rate is negligible (set d to zero in equation), and the curve is indistinguishable from an exponential growth curve.
		If the underlying data is truly exponential, fitting a logistic curve to it will give a d of 0.
	condellpark.com /bear/logistic_m.htm (812 words)

	Lab 9: PopGrow - Limited
		Deceleration in population growth is smooth as it approaches K; thus, when the curve is cut in half, the upper and lower halves are mirror images.
		The point where you could cut the curve in half is called the inflection point, and it occurs at K/2.
		The logistic equation provides a relatively good fit to many case histories of population growth observed in the lab and field; however, it is just a model and contains several oversimplifications.
	www.cnr.uidaho.edu /wlf448/poplim1.htm (749 words)

	The logistic growth model
		This is then the solution of the logistic growth model (18) and (19), the size of the population at time t.
		We have seen how Mathematica can be utilized to find the curve of least squares fit for a set of data points, where the curve is represented by a function which can be written as a linear sum of a specific set of known functions.
		Unfortunately, the function that represents the general logistic growth curve contains unknown parameters within its structure, and these parameters cannot be expressed as the coefficients of a sum of known functions.
	www.mathcs.emory.edu /ccs/ccs215/model/node7.html (1250 words)

	GROWTH CURVE MODELLING - Smart Module
		The Gompertz curve is a limiting case of the generalised logistic as T becomes very small or very large.
		A further example is available which illustrates the fitting of separate growth curves for each experimental unit.
		The data and the Genstat code, which fits growth curves for each experimental unit and forms a table of the medians for each of the five parameters of the Gompertz curve, is provided here.
	www.bioss.ac.uk /smart/unix/mgrow/slides/frames.htm (770 words)

	CSC.USAPop.html
		The results have led us to the conclusion that a logistic curve provides the best fit of the data for the period 1950-1990.
		This is an important conclusion because it suggests that of all the possible functions to choose from, we should try to fit a logistic function directly to the data.
		If instead of the logistic function the function were linear, we could use Maple's built-in least squares line-fitting routine.
	www.dartmouth.edu /~math3f98/csc98/chap5/CSC.USAPop5.html (439 words)

	Encyclopedia: Generalised logistic curve (Site not responding. Last check: 2007-11-06)
		People who viewed "Generalised logistic curve" also viewed:
		The generalised logistic (or Richards') curve is a widely used and flexible function for growth modelling.
		The logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops.
	www.nationmaster.com /encyclopedia/Generalised-logistic-curve (152 words)

	MATH 18
		The logistic function f increases if B is positive and decreases if B is negative.
		The graph of the logistic function is bounded by the horizontal lines y = D and y = A + D.
		The “S” shape is typical of what is called a logistic curve.
	www.emba.uvm.edu /~puterbau/math18fall2003/Logistic_Models.htm (798 words)

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