Box-cox Transform

Brief introductionEditThe general form of the Box-cox transformation is: The new variable obtained by the Box-cox transformation in the formula is the original continuous dependent variable, which is the transformation parameter. The above transformation requires the original variable value is positive, if the value is negative, you can first add a constant to all the original data to make it positive, and then do the above transformation. Different from the different transformations made. When the transform is a logarithmic transformation, it is the reciprocal transformation, and the square root transformation at the time. There are two methods for estimating parameters in the Box-cox transformation: (1) Maximum likelihood estimation and (2) Bayes method. By solving the values, you can determine which form of transformation to use. Transformation processEditThe Box-cox transformation is the following transformation of the regression dependent variable y: Here    is a pending transformation parameter. For different  &nbsp, the transformations made are not the same, so the Box-cox transformation is a family transformation, which includes the square root transformation (  ), the logarithmic transformation (  ) and the Reciprocal Transformation (  ) and other commonly used transformations, the n observations of the dependent variable  &nbsp, and applying the above transformation, we have to determine the transformation parameters  &nbsp, so that the    satisfies the requirement through the transformation of the dependent variable, so that the transformed vector     and regression independent variables have linear dependence, and the error is also normal distribution. The error components are equal variance and independent of each other, so the Box-cox transformation is an appropriate choice by parameter   . To achieve the "comprehensive treatment" of the original data, so that it satisfies a normal linear regression model of all the hypothetical conditions. The maximum likelihood method is used to determine  &nbsp, and because of  &nbsp, the likelihood function of fixed   ,   and    is,   Jacobi determinant for transformation when    fixed,   is not dependent on parameters    and    the remainder of the constant factor,   about     and    derivative numbers, so that they are equal to zero, can be obtained    and    maximum likelihood estimate residuals squared sum for the corresponding likelihood Max is the unary function of the type   , By seeking its maximum value to determine the  &NBSP, because    is the monotone function of x, the problem can be converted to the maximum value of   , the formula (3) logarithm, omit the    independent constant term, the formula (4) It is very convenient for the Box-cox transform to be implemented on the computer, because we only ask for the minimum value of the residual sum of squares and   , we can find the maximum value of   , although it is difficult to figure out the   to achieve the minimum value of    The analytic expression of  , but it is easy to calculate the corresponding  &nbsp for a series of    given values through the most common regression procedure of least squares estimation., Draw    about &NBSPThe curve of the;  can be approximated on the graph to find the    with the minimum value   . The specific steps of the Box-cox transform are as follows: (1) for a given    value, calculate  &nbsp, if   , use formula (6) to calculate, otherwise use formula (7); (2) Use formula (5) to calculate residuals squared and    ; (3) for a series of    values, repeat the above steps to get the corresponding residuals squared and a series of values   , to    the horizontal axis, to make the corresponding curve, using an intuitive method to find the    Point    with minimum value. (4) using formula (2) to find out   . SignificanceEditA significant advantage of the Box-cox transformation is that it is determined by the transformation parameters to determine the transformation form, and this process is completely based on the data itself without any prior information, which is undoubtedly more than empirical or through the selection of the logarithm, square root and other transformation methods to be objective and accurate. The purpose of the Box-cox transformation is to make the data satisfy the basic assumptions of the linear model, i.e. linearity, normality and variance homogeneity, but whether the data of Box-cox transform satisfies the above assumptions, still need to examine the verification[2].

Box-cox Transform