知识见闻 - 数学: 均方根误差和标准差
Root Mean Square Error (RMSE) or RMSD (Root mean square deviation)
均方根误差（RMSE）是一种常用的测量方法，用来测量估计值或模式预测的数字（总体值和样本）之间的差异。RMSE 描述了预测值和观测值之间差异的样本标准偏差。在对用于估算的数据样本进行计算时，这些差异中的每一个都被称为残差，而在样本外进行计算时，则被称为预测误差。均方根误差将预测不同时间的误差大小汇总为一个单一的预测能力指标。
The Root Mean Square Error or RMSE is a frequently applied measure of the differences between numbers (population values and samples) which is predicted by an estimator or a mode. The RMSE describes the sample standard deviation of the differences between the predicted and observed values. Each of these differences is known as residuals when the calculations are done over the data sample that was used to estimate, and known as prediction errors when calculated out of sample. The RMSE aggregates the magnitudes of the errors in predicting different times into a single measure of predictive power.
Root Mean Square Error Formula
预测模型相对于估计变量 xmodel 的 RMSE 定义为均方误差的平方根。
The RMSE of a predicted model with respect to the estimated variable xmodel is defined as the square root of the mean squared error.
Where, xobs is observed values, xmodel is modelled values at time i.
参考：
1，BYJU'S
Root Mean Square (RMS) - Definition, Formula and RMS Error
Root mean square deviation
统计学
均方根偏差（RMSD）或均方根误差（RMSE）是用于衡量真实或预测值与观察值之间差异的常用指标。
定义 均方根偏差是观察值与预测值之间差异的平方均值的平方根。
用途 用于比较不同模型在特定数据集上的预测误差。
特性 RMSD总是非负的，值为0表示完美拟合。
敏感性 对异常值敏感，较大的误差对RMSD的影响较大。
归一化 归一化RMSD可以方便不同数据集或模型的比较。
RMSE（均方根误差）是衡量预测模型的预测值与实际观测值之间差异的一种常用方法。它通过计算预测误差的平方的平均值，然后取平方根得出，是一种评估模型性能的指标。
RMSE的定义
RMSE定义为预测值与实际值之间偏差的平方的平均值的平方根。它实际上是残差的标准偏差，即度量数据点如何分布在模型最佳拟合线周围。
计算方法
RMSE的计算包括以下步骤：
1. 计算每个观测值的残差：这是实际观测值与预测值之差。
2. 求残差的平方：将每个残差值平方。
3. 计算平方误差的平均值：将所有残差的平方求和，然后除以观测次数。
4. 开平方：对上述结果开平方，以得到RMSE值。
RMSE的用途
RMSE主要用于评估模型的预测精确度和模型的拟合优度分析。它被广泛应用于各种领域，如气象学、经济学和金融学等，以评估模型与观察数据的拟合程度。RMSE数值越小，表示预测模型与实际情况的偏差越小，模型性能越好。RMSD常用于模型比较，选择最优模型。1
局限性
RMSE对异常值较为敏感，因为平方运算会使得大的偏差对最终的RMSE有更大的影响。它也会随变量的数量按比例降低，可能导致过拟合模型看起来更好。尽管如此，RMSE仍然是理解和评估模型预测能力的重要工具。
资料来源:
[1] 均方根误差RMSE（Root Mean Square Error） 原创 - CSDN博客, 均方根误差RMSE（Root Mean Square Error）-CSDN博客
[2] Root Mean Square Error (RMSE) - Statistics By Jim, https://statisticsbyjim.com/regression/root-mean-square-error-rmse/
[3] Root Mean Squared Error (RMSE) - SAP Help Portal, loading... | SAP Help Portal
[4] 深入理解并实践：计算均方根误差（RMSE） - 百度智能云, 深入理解并实践：计算均方根误差（RMSE）
[5] RMSE（Root Mean Squared Error，均方根误差）, RMSE（Root Mean Squared Error，均方根误差）
[6] Root Mean Square Error - an overview | ScienceDirect Topics, https://www.sciencedirect.com/topics/engineering/root-mean-square-error
RMSE, or Root Mean Square Error, is a frequently used measure for evaluating the difference between predicted values and actual observed values in statistical models. It quantifies the average magnitude of the prediction error in a model's predictions, having the same units as the predicted variable.
Definition of RMSE
The RMSE is defined as the square root of the mean of the squared differences between observed and predicted values. It essentially represents the standard deviation of the residuals, which are the error terms in a model indicating the dispersion of observed data points around the line of best fit.
Calculation Method
To calculate RMSE, one follows these steps:
1. Compute the Residual for Each Observation: This involves finding the difference between the actual and predicted values.
2. Square Each Residual: This emphasizes larger errors.
3. Calculate the Mean of the Squared Residuals: This is known as the Mean Squared Error (MSE).
4. Take the Square Root: This provides the RMSE, translating the error into the same units as the observed data.
Usage of RMSE
RMSE is extensively used across various fields as a measure of predictive power and model accuracy. Some common applications include:
Model Evaluation in Regression Analysis: It helps assess how well a regression model predicts outcomes.
Comparing Predictive Models: RMSE provides a uniform metric for comparing the predictive accuracy of different models, making it easier to determine which model performs better on a given dataset.
Machine Learning Applications: In supervised learning, RMSE is often used as a loss function to optimize model performance during training and evaluation phases.
Climatology and Meteorology: In these fields, RMSE helps verify the accuracy of quantitative predictions, such as weather forecasts.
Limitations
While RMSE is highly useful, it is sensitive to outliers due to its reliance on squared differences, which can disproportionately affect the error metric. Additionally, RMSE is not scale-invariant, meaning it can be challenging to compare across models with different units or scales without normalization. Despite these limitations, RMSE remains a cornerstone for evaluating model precision and accuracy in various predictive modeling tasks.
资料来源:
[1] Root Mean Square Error (RMSE) - Statistics By Jim, https://statisticsbyjim.com/regression/root-mean-square-error-rmse/
[2] Root mean square deviation - Wikipedia, https://en.wikipedia.org/wiki/Root_mean_square_deviation
[3] RMSE: Root Mean Square Error - Statistics How To, https://www.statisticshowto.com/probability-and-statistics/regression-analysis/rmse-root-mean-square-error/
[4] Root Mean Square Error (RMSE) - C3 AI, https://c3.ai/glossary/data-science/root-mean-square-error-rmse/
[5] Root Mean Squared Error (RMSE) - SAP Help Portal, loading... | SAP Help Portal
[6] Root Mean Square Error (RMSE) In AI: What You Need To Know, https://arize.com/blog-course/root-mean-square-error-rmse-what-you-need-to-know/
[7] A Practical Guide to Root Mean Square Error (RMSE) | Aporia, https://www.aporia.com/learn/root-mean-square-error-rmse-the-cornerstone-for-evaluating-regression-models/
标准差的定义
标准差（Standard Deviation, SD）是用于量化数据集分布的离散程度的统计量。它表示数据点与均值之间的平均距离，是方差的平方根。标准差在统计学中被广泛应用于描述数据的离散性，其计算方式也反映了数据与均值的偏离程度。
标准差被广泛用于统计学中，用于描述数据的分布特性，并常用于风险评估和决策过程，如在金融市场中用于评估资产的风险，或者在制造过程中用于控制产品质量。
参考：
平方平均数 - 维基百科，自由的百科全书