Application of Curve Fitting methods in Ecology Based on Least Squares

                   Yan Ren-xiang 

( College of Life Science, FuJian Agriculture And Forestry University, FuZhou 350002, China)http://www.16sheji8.cn/

Abstract

     This thesis is an introduction to curve fitting which is the study of ways of constructing functions whose graphs are curves that "best" approximates a given collection of points and its application in ecology.
     As far as I know, data analysis is of great importance in the ecology. However,  field data is often accompanied by noise. Even though all control parameters (independent variables) remain constant, the resultant outcomes (dependent variables) vary. A process of quantitatively estimating the trend of the outcomes, also known as curve fitting, therefore becomes necessary. With clearly know the data type of ecology, we can analysis the data more correctly. And the application of method of least squares in the ecology is discuss.With the method, some programs which sove some calculation problems in ecology, such as Exponential Growth ,Power Law, mth degree Polynomial Grow Model  are developed.

1.Introduction

1.1The deviation of data of ecology[1][2][3]http://www.16sheji8.cn/
Ecological experiments, whether instructional exercises or original research, routinely involve the measurement of multiple quantities. Typically, some of those quantities are controlled by the experimenter (the "causes" or "independent variables") while others are simply observed (the "effects" or "dependent variables"). In many situations, the connections among the directly measured quantities are most readily understood by comparing values that can be calculated from them.
The heart of any measurement process is deciding whether the quantity being measured is greater than, equal to, or less than some reference value. The simplest type of measurement involves the direct physical comparison of a standard to the unknown quantity (e.g., measuring the length of an object with a ruler, or the temperature with a conventional mercury thermometer). Such "analog" measurements provide the experimenter with an immediate, intuitive feel for their uncertainty, because the experimenter must directly judge the comparison between the object and the standard. The markings on the scale will have some non-zero width, the object may not have well-defined boundaries (where is the edge of a cotton ball, anyway?), and, depending on the spacing between the finest markings on the scale and the visual aquity of the experimenter, it may well be possible to honestly estimate more precisely than the scale is marked (although rarely more precisely than to the tenth of the finest markings).
As what we discussed above, we can safely draw the conclusion that raw data we obtained in the ecological experiment usually have deviation between the accurate figure.
1.2 The method of least squares[4][5][6][7]

    The curve fitting process fits equations of approximating curves to the raw field data. Nevertheless, for a given set of data, the fitting curves of a given type are generally NOT unique. Thus, a curve with a minimal deviation from all data points is desired. This best-fitting curve can be obtained by the method of least squares.
    The method of least squares assumes that the best-fit curve of a given type is the curve that has the minimal sum of the deviations squared (least square error) from a given set of data.
Suppose that the data points are  ,  , ...,  where  is the independent variable and  is the dependent variable. The fitting curve  has the deviation (error)  from each data point, i.e.,      ,  , ...,  .  According to the method of least squares, the best fitting curve has the property that: 
 

1.3   The Least-Squares Line[4][5][6][7][8]

The least-squares line uses a straight line
 
to approximate the given set of data,  ,  , ...,  , where  . The best fitting curve  has the least square error, i.e., 
Please note that  and  are unknown coefficients while all  and  are given. To obtain the least square error, the unknown coefficients  and  must yield zero first derivatives. 
              

Expanding the above equations, we have:
     
where  stands for  .

1.4 The Least-Squares Parabola[4][5][6][7][8]

    
To approximate the given set of data,  ,  , ...,  , where  . The best fitting curve  has the least square error, i.e.,
 
Please note that  ,  , and  are unknown coefficients while all  and  are given. To obtain the least square error, the unknown coefficients  ,  , and  must yield zero first derivatives.

Expanding the above equations, we have