FITTING REGRESSION EQUATION
13.1 FITTING REGRESSION EQUATION
The fitting of a straight line may be done objectively by one of the following statistical methods:
(a) The method of least squares (b) The method of moments (c) The method of maximum likelihood In this chapter, the method of least squares is used. Two variables y (dependent) and x
(independent) can be correlated by plotting them on x– and y–axis. If they are plotted on a straight line, there is a close linear relationship; on the other hand, if the points depart appre- ciably (without a definite trend), the graph is called a scatter diagram or plot.
that the best line for fitting a series of observations is the one for which the sum of the squares BLOG
If the trend is a straight line, the relationship is linear and has the equation
...(13.1) Number of lines can be obtained depending on the values of a and b. The method of least
y = a + bx
squares is used to select the line that fits the data best. The principle of least squares states
of the departures is minimum. A departure is the difference between the observed value and the line. Since x is the independent variable, the departures of y are used.
The least squares line Eq. (13.1) may be obtained by solving for a and b, the two normal equations
Σy = na + b Σx
Σxy = a Σx + b Σx 2 ...(13.2) where n = number of pairs of observed values of x and y. The most commonly used statistical parameter for measuring the degree of association
of two linearly dependent variables x and y, is the correlation coefficient
where ∆x = x – x , ∆y = y – y σ x ,σ y = standard deviations of x and y, respectively x, y = middle of each class interval, respectively
HYDROLOGY
ANNACIVIL curve, given by
If r = 1, the correlation is perfect giving a straight line plot (regression line). r = 0, no relation exists between x and y (scatter plot). r → 1, indicates a close linear relationship. If a linear regression can not be fitted, a quadratic parabola can be used as the fitting
y = a + bx + cx 2 ...(13.4) From the principles of least squares, a, b and c can be obtained by solving the three
normal equations
Σy = na + bΣx + c Σx 2 Σxy = aΣx + b Σx 2 + c Σx 3
2 y Σx = aΣx 2 + b Σx 3 + c Σx 4 ...(13.5) where n = number of pairs of observed values of x and y.
Regardless of the type of curve fitted, the correlation coefficient r is given by Eq. (13.3). The variables x and y, for instance, may be precipitation and the corresponding runoff, or gauge height and the corresponding stream flow, and like that.
...(13.6) it can be transformed to a straight line by using logarithms of the variables as
For the exponential function y = cx m
...(13.7) By putting log x = X, log y = Y, log c = a and m = b the function becomes similar to Eq.
log y = log c + m log x
(13.1), can be solved for a and b from Eq. (13.2) and the exponential function can be deter- mined.
BLOG
Whichever fitting gives r → 1 by Eq. (13.3), that curve fitting is adopted. Statistical method can be applied to many kinds of meteorological data, such as precipitation, tempera- ture, floods, droughts, and water quality.