normalization（统计）

In statistics and applications of statistics, normalization can have a range of meanings.^[1] In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging. In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. In the case of normalization of scores in educational assessment, there may be an intention to align distributions to a normal distribution. A different approach to normalization of probability distributions is quantile normalization, where the quantiles of the different measures are brought into alignment.

In another usage in statistics, normalization refers to the creation of shifted and scaled versions of statistics, where the intention is that these normalized values allow the comparison of corresponding normalized values for different datasets in a way that eliminates the effects of certain gross influences, as in an anomaly time series. Some types of normalization involve only a rescaling, to arrive at values relative to some size variable. In terms of levels of measurement, such ratios only make sense for ratio measurements (where ratios of measurements are meaningful), not interval measurements (where only distances are meaningful, but not ratios).

In theoretical statistics, parametric normalization can often lead to pivotal quantities – functions whose sampling distribution does not depend on the parameters – and to ancillary statistics – pivotal quantities that can be computed from observations, without knowing parameters.

在统计学和应用统计学中，normalization有着宽泛的意义。最简单的理解，比如评级的标准化，意味着不同尺度上测量的数据，调整为理论上的共同尺度，这通常要先于平均运算。在复杂的案例中，normalization通常也意味着复杂的调整，目的就是要使得调整后的数据的概率分布，保证某种尺度上的一致。举个例子，在教育评估中，不同科目难易不同，不同的学生选择了不同的科目，得了不同的分数，如何评价他们的好坏？要想使不同科目的分数具有科比性，就需要以‘标准分布（normal distribution）’作为比较的基准。与概率分布标准化不同的一种方法，就是‘分位点标准化（ quantile normalization）’，也就是使得不同测量方法的分位点保持一致（我估计是不是类似于举重、拳击的轻量级、重量级的分位）。

在统计学的另一个术语中，标准化normalization特指经过平移和缩放后的统计版本，目的是这些标准化的数据使得来源于不同数据集合中的经归一化后，能够互相比较。以这样的方式消除总体影响效果，比如“异常事件序列（ anomaly time series）”。某些类型的标准化只包括一个缩放因子，相对于尺度变量，使其达到某个某个量值。根据测量等级，这样的比率只对比率测量有意义（其中，测量的比率才是有意义的），而不是间隔测量（其中，只有距离是有意义的，而不是比率）

在理论统计学中，参数标准化常常可以导致基准量—采样分布函数不依赖于参数；并且产生一些辅助统计—基准量，这些基准量可以从观察数据计算得到，不需要知道具体参数。

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Examples[edit]

There are various normalizations in statistics – nondimensional ratios of errors, residuals, means and standard deviations, which are hence scale invariant – some of which may be summarized as follows. Note that in terms of levels of measurement, these ratios only make sense for ratio measurements (where ratios of measurements are meaningful), not interval measurements (where only distances are meaningful, but not ratios). See also Category:Statistical ratios...

在统计学上，有多种不同的标准化：比如无量纲的误差、残差、均值和标准差等的比率。因为是无量纲比率，所以是尺度不变的。某些比率可以概括如下。注意，根据测量等级，这些比率只是对“比率测量（ratio measurement）”有意义，其中的测量比率是有意义的。See also Category:Statistical ratios...

Name	Formula	Use
Standard score		Normalizing errors when population parameters are known. Works well for populations that are normally distributed
Student's t-statistic		Normalizing residuals when population parameters are unknown (estimated).
Studentized residual		Normalizing residuals when parameters are estimated, particularly across different data points in regression analysis.
Standardized moment		Normalizing moments, using the standard deviation {\displaystyle \sigma } as a measure of scale.
Coefficient of variation		Normalizing dispersion, using the mean {\displaystyle \mu } as a measure of scale, particularly for positive distribution such as the exponential distribution and Poisson distribution.
Feature scaling		Feature scaling is used to bring all values into the range [0,1]. This can be generalized to restrict the range of values in the dataset between any arbitrary points a and b usings .

Note that some other ratios, such as the variance-to-mean ratio {\displaystyle \left({\frac {\sigma ^{2}}{\mu }}\right)}, are also done for normalization, but are not nondimensional: the units do not cancel, and thus the ratio has units, and are not scale invariant.

Other types[edit]

Other non-dimensional normalizations that can be used with no assumptions on the distribution include:

Assignment of percentiles. This is common on standardized tests. See also quantile normalization.
Normalization by adding and/or multiplying by constants so values fall between 0 and 1. This used for probability density functions, with applications in fields such as physical chemistry in assigning probabilities to |ψ|².

References[edit]

Jump up^ Dodge, Y (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entry for normalization of scores)

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