附录B 外文参考文献（原文）
GROWTH CURVE MODELLING
Growth is a fundamental property of biological systems, occuring at the level of populations, individual animals and plants, as well as within organisms. Much research has been devoted to modelling growth processes, and there are many ways of doing this, including: mechanistic models, time series, stochastic differential equations etc.
Sometimes we simply wish to summarise growth observations in terms of a few parameters, perhaps in order to compare individuals or groups. Many growth phenomena in nature show an 'S' shaped pattern, with initially slow growth speeding up before slowing down to approach a limit. These patterns can be modelled using several mathematical functions. Here we explore the generalized logistic and Gompertz curves.
Examples of applications of the technique include:
- animal weights over time;
- growth of human populations;
- biomedical data;
- development of organisms;
- Growth Curve in Modern Enterprise Product.
THE GOMPERTZ CURVE
1 The Gompertz curve is a limiting case of the generalized logistic as T becomes very small or very large.
where Y=weight, height, size etc; X = time.
A = the lower asymptote;
C = the upper asymptote;
M = the time of maximum growth;
B = the growth rate.
Investigate the Gompertz curve.
The Java applet may take a little time to load!
FITTING GOMPERTZ CURVES
We use data from a different cow to illustrate fitting this curve
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***** Nonlinear regression analysis ***** Response variate: weight Explanatory: age Fitted Curve: A C*EXP(-EXP(-B*(X-M))) Constraints: C gt; 0 *** Estimates of parameters *** estimate s.e. B 0.031022 0.000848 M 35.63 1.57 C 664.7 19.3 A 8.2 17.5 *** Summary of analysis *** d.f. s.s. m.s. v.r. Regression 3 2545361. 848453.69 12336.39 Residual 78 5365. 68.78 Total 81 2550726. 31490.44 Percentage variance accounted for 99.8 Standard error of observations is estimated to be 8.29 |
GOMPERTZ CURVE - FITTED COW DATA
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RESIDUALS FROM GOMPERTZ CURVE - --------- --------- --------- --------- ----- I * I 2.5 I * I I * * I I 3 I I *** 3 I I ** * *2* 2 * 24 I I * ***** * 2* * 26* I 0.0 I ** 2* * ** * * *** *3 2* I I ** * 2 *** ** * ****4 I I * 2 * 2* *22 * I I *3 2 I I * *** I I 3 * I -2.5 I I I * I I I I I I I I I -5.0 I I - --------- --------- --------- --------- ----- 0.0 200.0 400.0 600.0 800.0 res v. fit using symbol * |
EXAMPLE : ANIMAL WEIGHTS OVER TIME
2 The weights of cows have been recorded every two weeks for 232 weeks, i.e. 116 observations in total.
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The weights of cows |
We wish to fit a curve which best summarises the distribution of the points for each cow on the graph.
EXAMPLE : Growth Curve in Modern Enterprise Product
The typical product life cycle may generally divide into four stages: introducing stage, growing stage, mature stage and winter.
- The first stage: introducing stage
Once the new product is put into the market, it enters the introducing stage. In this stage, the customer hasnrsquo;t understood the product yet, except for minority pursuing novelty, nearly nobody would actually purchase the product. In this stage, the production lot is small, but the advertising cost is big and the product selling price is high. So the sales volume is extremely limited, the enterprise usually cannot make a profit.
2. The second stage: growing period
After the product entered the introducing stage and the sale obtained the success, then it enters the growing stage. In this stage, demand grows and the sale-volume rises rapidly. The production cost drops in large scale and profit is in explosive growth.
3. The third stage: Mature stage
After the growing stage, along with purchasing population increases, the market demand tends to be saturated. The product then enters the mature stage. In this stage, the sale rate rises slowly until drops because competition aggravates to cause the advertising cost being high again. The profit drops.
4. The fourth stage: winter
Along with the technical development, appearance of new product and substitutions as well as changing of consumption habits, the productrsquo;s sale-volume and the profit drop continually, thus the
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附录A 外文参考文献（译文）
成长曲线模型
成长是生物系统的一项根本属性，不管是人类、动物还是植物都会成长，包括内部器官。 Much research has been devoted to modelling growth processes, and there are many ways of doing this, including: mechanistic models, time series, stochastic differential equations etc.大量的研究一直致力于塑造成长过程的模型。有许多建模的方法，包括：机理模型，时间序列，随机微分方程等。
Sometimes we simply wish to summarise growth observations in terms of a few parameters, perhaps in order to compare individuals or groups.有时，为了比较个体和群体，我们只想根据几个参数来建立成长模型。Many growth phenomena in nature show an 'S' shaped pattern, with initially slow growth speeding up before slowing down to approach a limit.许多成长现象呈现“ S ”形模式，最初增长缓慢，加快后放缓，最后接近极限。 These patterns can be modelled using several mathematical functions.这些模式可以用几个数学函数来建模。 Here we explore the generalised logistic and Gompertz curves.在这里，我们来探讨一下罗吉斯缔曲线和龚柏兹曲线。
Examples of applications of the technique include:成长曲线模型的应用举例：
- animal weights over time ;动物的动物的体重变化；
- growth of human populations;人口增长；
- biomedical data;生物医学数据;
- development of organisms.有机体的发展；
- 现代企业产品生命周期曲线预测模型。
龚柏兹曲线
1 龚柏兹曲线是Logistic曲线T值极小或极大的极限情况。
where Y=weight, height, size etc; X = time.其中Y =体重，身高，尺寸等; x =时间。
A = the lower asymptote; A =下渐近线;
C = the upper asymptote; C =上渐近线;
M = the time of maximum growth; M =成长最高峰的时间;
B = the growth rate. B =增长率。
Gompertz曲线的拟合
2我们使用不同奶牛的体重数据来说明此曲线的拟合
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***** Nonlinear regression analysis ***** Response variate: weight Explanatory: age Fitted Curve: A C*EXP(-EXP(-B*(XM))) Constraints: C gt; 0 *** Estimates of parameters *** estimate se B 0.031022 0.000848 M 35.63 1.57 C 664.7 19.3 A 8.2 17.5 *** Summary of analysis *** dfssmsvr Regression 3 2545361. *****非线性回归分析***** 因变量：重量 自变量：年龄 拟合曲线：A C*EXP(-EXP(-B*(X-M))) 制约因素： C gt; 0 ***参数估计*** 估计值 实际值 B 0.031022 0.000848 M 35.63 1.57 C 664.7 19.3 A 8.2 17.5 ***分析概要*** d.f. s.s. m.s. v.r. 回归 3 2545361. 848453.69 12336.39 残值 78 5365. 68.78 总数 81 2550726. 31490.44 百分比差额占99.8%，标准误差估计为8.29%。 |
Gompertz曲线拟合
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Gompertz曲线的残值 - --------- --------- --------- --------- ----- I * I 2.5 I * I I * * I I 3 I I *** 3 I I ** * *2* 2 * 24 I I * ***** * 2* * 26* I 0.0 I ** 2* * ** * * *** *3 2* I I ** * 2 *** ** * ****4 I I * 2 * 2* *22 * I I *3 2 I I * *** I I 3 * I -2.5 I I I * I I I I I I I I I -5.0 I I - --------- --------- --------- --------- ----- 0.0 200.0 400.0 600.0 800.0 |
动物的体重变化
- 每隔两周纪录下奶牛的体重，持续232周。116次的观察记录如下图所示：
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奶牛的体重 |
We wish to fit a curve which best summarises the distribution of the points for each cow on the graph.我们希望构造一个能概括所有奶牛成长情况的曲线。
现代企业产品生命周期曲线预测模型
4典型的产品生命周期一般可以分成四个阶段：引入期、成长期、成熟期和衰退期。
1. 第一阶段：引入期
新产品投入市场，便进入了引入期。此时顾客对产品还不了解，除了少数追求新奇的顾客外，几乎没有人实际购买该产品。在此阶段产品生产批量小，制造成本高，广告费用大，产品销售价格偏高，销售量极为有限，企业通常不能获利。
2．第二阶段：成长期
当产品进入引入期，销售取得成功之后，便进入了成长期。这是需求增长阶段，需求量和销售额迅速上升，生产成本大幅度下降，利润迅速增长。
3．第三阶段：成熟期
经过成长期之后，随着购买产品的人数增多，市场需求趋于饱和，产品便进入了成熟期阶段。此时，销售增长速度缓慢直至转而下降，由于竞争的加剧，导致广告费用再度提高，利润下降。
4．第四阶段：衰退期
随着科技的发展、新产品和替代品的出现以及消费习惯的改变等原因，产品的销售量和利润持续下降，产品从而进人了衰退期。产品的需求量和销售量迅速下降，同时市场上出现替代品和新产品，使顾客的消费习惯发生改变。此时成本较高的企业就会由于无利可图而陆续停止生产，该类产品的生命周期也就陆续结束，以至最后完全撤出市场。
产品生命周期是一个很重要的概念，它和企业制定产品策略以及营销策略有着直接的联系。管理者要想使他的产品有一个较长的销售周期，以便赚到足够的利润来补偿在推出该产品时所做出的一切努力和经受的一切风险，就必须认真研究和运用产品的生命周期理论，此外，产品生命周期也是营销人员用来描述产品和市场运作方法的有力工具。但是，在开发市场营销战略的过程中，产品生命周期却显得有点力不从心，因为战略既是产品生命周期的原因又是其结果，产品现状可以使人想到最好的营销战略，此外，在预测产品性能时产品生命周期的运用也受到限制。
一个典型的生长曲线
下面的渐近线是成长的初始阶段。
The upper asymptote is the mature level.上面的渐近线是成长的成熟阶段。
The point of inflexion is the point of maximum growth.拐点是成长的最高峰。
The curve can be modelled in a number of ways.该曲线可以应用在许多不同的领域。
Logistic曲线
5 Logistic曲线是一种被广泛使用并且灵活多变的建模方程。
where Y=weight, height, size etc; X = time.其中Y =体重，身高，尺寸等; x =时间。
It has five parameters:它有5个参数：
A , the lower asymptote; A、下渐近
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