Detailed explanation of Box-Cox transformation

1 What is Box-Cox Transformation

The box-cox transformation is a method widely used in data transformation and normalization, which can bring the data closer to the normal distribution. It was invented by two statisticians box and cox, suitable for continuous, positive, skewed distribution data.

The mathematical formula of box-cox transformation is :

$y^{(\lambda)}= \begin {cases} \dfrac{y^\lambda - 1}{\lambda}, & \text{if } \lambda \neq 0 \\ \ln{(y)}, & \text{if } \ lambda = 0 \end{cases}$

Among them, $y$ is the original data, $\lambda$ is a parameter of the box-cox transformation. When $\lambda=0$ , use logarithmic transformation, otherwise use the above formula for transformation.

The main role of box-cox transformation :

The main function of the box-cox transformation is to normalize the data to make the data more in line with statistical assumptions. In practical applications, box-cox transformation is often used to solve the problem that the data in regression analysis and variance analysis do not satisfy the normal distribution, thereby improving the accuracy and reliability of the model.

It should be noted that the parameter $\lambda of the box-cox transformation$ needs to be determined by many trials and tests on the original data. Usually the maximum likelihood method or cross-validation method is used to select the best $\lambda$ value.

2 Cox-Box transformation with python

from scipy import stats

# 假设有一组数据x
x = [1, 2, 3, 4, 5]

# 进行Box-Cox变换 convert_res是输出结果
convert_res, _ = stats.boxcox(x)

print(convert_res)

The output is:

[ 0.          0.82603196  1.44077472  1.98810691  2.48402687]

Among them, xt is the transformed data, and _ is the transformed parameter. If you want to restore the data, you can use the inv_boxcox function:

# 还原数据
from scipy.special import inv_boxcox
x_inv = inv_boxcox(convert_res, _)

print(x_inv)

Note : The boxcox function can only handle positive data. If there are negative numbers or zeros in the data, you need to perform translation or add one operation first.

3 The difference between Box-Cox transformation and other normalization methods

The main difference between the box-cox transformation and other normalization methods is their goals and how they are applied .

3.1 box-cox transformation

The box-cox transformation is a method to adjust the data distribution shape by transforming the data with a power function . This transformation involves a value called the exponential parameter lambda (λ), which can be automatically optimized to maximize normality or symmetry of the data.
The box-cox transformation is suitable for data sets with skewed or non-normal distribution characteristics , and can make the data more consistent with the assumptions of the linear model . This is because the box-cox transformation can normalize the data by compressing or expanding it into a wider interval to better fit the assumptions of the linear model.

3.2 Other normalization methods

In data processing, other common normalization methods include:

Min-max scaling: scale the data to the [0,1] interval.
z-score normalization (z-score normalization): Scale the data to a normal distribution with mean 0 and variance 1.
Median absolute deviation normalization (mad): Scales the data to the range of the median ± constant times mad.

These methods usually do not change the shape of the data distribution, but only adjust their scale or position to better suit certain algorithms or processing steps. These methods are often used when feature scaling or preprocessing data.

Generally speaking, box-cox transformation and other normalization methods are methods to adjust the shape and proportion of data distribution, but their goals and application ranges are slightly different. The box-cox transformation can change the data distribution shape to conform to the assumption of the linear model , while other normalization methods are mainly used to adjust the data scale or position to suit various algorithms or statistical processes.

4 Advantages and disadvantages of Box-Cox transformation

The box-cox transformation is a data transformation method designed to make the data more conform to the normal distribution . Its advantages and disadvantages are as follows:

advantage:

Improve the prediction accuracy of the model : after performing box-cox transformation on the non-normally distributed data, the data can be more in line with the normal distribution, thereby improving the accuracy of the model prediction.
Statistical inference is more reliable : When performing statistical inference, if the data is assumed to conform to a normal distribution, but in fact it does not, it may lead to errors in the results. After transforming the data into a normal distribution by box-cox transformation, the results of statistical inference are more reliable.
Dealing with heteroscedasticity : For data with heteroscedasticity, box-cox transformation can make the data smoother, making it easier to deal with heteroscedasticity.

shortcoming:

Data must be positive : The box-cox transformation requires data to be positive, so it cannot handle datasets containing negative numbers.
Parameters need to be selected : The parameter λ in the box-cox transformation needs to be selected according to the data set, and different λ values may lead to different results. Therefore, multiple trials are required to find the most suitable value of λ.
The data range affects the transformation effect : the box-cox transformation is sensitive to the data range. If the data set range is small, the transformation effect may be poor, or numerical problems may occur.

After 5 box-cox transformation, the model performance may be improved as follows:

Enhance data stability: box-cox transformation can convert non-normally distributed data into approximately normally distributed data, which can reduce the influence of data noise and outliers on the model, thereby enhancing data stability.
Improve prediction accuracy: Since non-normally distributed data may not meet the assumptions in some specific scenarios (such as linear regression), the prediction accuracy of the model can be improved through box-cox transformation. For example, in a linear regression problem, if the variables do not follow a normal distribution, the residuals of the model will not follow a normal distribution, which can lead to erroneous confidence intervals and hypothesis testing results. Through the box-cox transformation, the data can be approximated to a normal distribution, thereby avoiding the occurrence of this problem.
Reduce the risk of overfitting: box-cox transformation can compress the data range, and can delete negative values, making the data more consistent with the assumptions of the model. This reduces the complexity of the model and reduces the risk of overfitting.