1. 均方误差 (Mean Squared Error, MSE): 其中， 是实际值， 是模型预测值， 是样本数量。MSE 衡量了模型预测误差的平方的平均值。

2. 均方根误差 (Root Mean Squared Error, RMSE): RMSE是MSE的平方根，它有助于量化预测误差的大小。

3. 平均绝对误差 (Mean Absolute Error, MAE): 它衡量了模型预测误差的绝对值的平均值。

4. R^2分数 (R-squared, 也称为决定系数):
其中
它表示模型解释的因变量变异性的比例。

5. 调整R^2分数 (Adjusted R-squared): 它考虑到了模型中自变量的数量。当我们在模型中增加更多的自变量，即使这些变量对响应变量没有实质贡献，普通的R^2也可能增加。调整的R^2会对这种情况进行惩罚。

6. F统计量 (F-statistic): 在线性回归中，F统计量用于测试模型中的所有自变量是否联合显著地与因变量相关。

7. AIC (Akaike Information Criterion) 和 BIC (Bayesian Information Criterion): 这两个指标都是模型选择准则，通常用于比较多个模型。它们考虑了模型的拟合度和模型的复杂性，目标是找到一个既能良好拟合数据又不过于复杂的模型。

1. 均方误差 (MSE):
• 优点:
• MSE易于计算。
• 考虑了所有预测误差，尤其是大的误差（因为误差是平方的）。
• 缺点:
• 由于是平方误差，对异常值非常敏感，可能会导致误导。
• 适用范围和情形:
• 当你希望模型对大的误差给予更重的惩罚时，可以使用MSE。

2. 均方根误差 (RMSE):
• 优点:
• RMSE同样考虑了所有预测误差。
• 相比MSE，更具有解释性，因为它的单位与原始数据相同。
• 缺点:
• 同样对异常值敏感。
• 适用范围和情形:
• 当你需要一个直观且单位与目标变量相同的误差指标时，可以使用RMSE。

3. 平均绝对误差 (MAE):
• 优点:
• 直接考虑了预测误差的绝对值，因此对异常值不太敏感。
• 易于解释。
• 缺点:
• 由于不对误差进行平方，它可能不会强调大的误差。
• 适用范围和情形:
• 当数据中存在许多异常值，或者你不希望模型对大的误差给予过重的惩罚时，可以使用MAE。

4. R^2分数 (R-squared, 决定系数):
• 优点:
• 提供了一个范围在0到1之间的统计量，直观表示模型解释数据变异性的比例。
• 当接近1时，表示模型的拟合效果很好；接近0表示拟合效果差。
• 缺点:
• 总是增加，当增加更多的特征，即使这些特征对模型无益。这可能导致过拟合。
• 不总是代表模型的预测能力或准确性，只是模型解释变异性的能力。
• 适用范围和情形:
• 当你需要衡量模型解释数据变异性的能力时，可以使用分数。但应当结合其他指标，如调整后的、AIC或BIC等，进行模型的选择和评估。

1. Mean Squared Error (MSE): Where is the actual value, is the predicted value by the model, and is the number of samples. MSE measures the average of the squares of the prediction errors.

2. Root Mean Squared Error (RMSE): RMSE is the square root of MSE, which helps to quantify the magnitude of the prediction errors.

3. Mean Absolute Error (MAE): It measures the average of the absolute values of the prediction errors.

4. R-squared (also known as the coefficient of determination): Where It represents the proportion of the variance in the dependent variable that is explained by the model.

5. Adjusted R-squared: This takes into account the number of independent variables in the model. The ordinary R^2 might increase even if the new variables added to the model are not significantly associated with the response variable. The adjusted R^2 penalizes for such additions.

6. F-statistic: In linear regression, the F-statistic is used to test whether all independent variables in the model jointly have a significant correlation with the dependent variable.

7. AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion): Both of these criteria are used for model selection, typically for comparing multiple models. They consider both the fit of the model and its complexity, with the goal of identifying a model that fits the data well without being overly complex.

1. Mean Squared Error (MSE):
• Advantages:
• Easy to compute.
• Takes into account all prediction errors, especially large ones (due to the squaring of errors).
• Disadvantages:
• Because it squares the errors, it's sensitive to outliers which may be misleading.
• Applicability and Scenarios:
• When you want the model to penalize larger errors more heavily, use MSE.

2. Root Mean Squared Error (RMSE):
• Advantages:
• Considers all prediction errors.
• More interpretable than MSE since its unit is the same as the original data.
• Disadvantages:
• Also sensitive to outliers.
• Applicability and Scenarios:
• When you need an intuitive error metric with the same unit as the target variable, use RMSE.

3. Mean Absolute Error (MAE):
• Advantages:
• Directly considers the absolute values of prediction errors, so less sensitive to outliers.
• Easy to interpret.
• Disadvantages:
• Since it doesn't square the errors, it might not emphasize large errors as much.
• Applicability and Scenarios:
• Use MAE when there are many outliers in the data or when you don't want the model to penalize larger errors too heavily.

4. R-squared:
• Advantages:
• Provides a statistic between 0 and 1, intuitively indicating the proportion of variance in the data explained by the model.
• A value close to 1 means a good fit, and close to 0 means a poor fit.
• Disadvantages:
• R^2 always increases when more features are added to the model, even if they're not beneficial. This can lead to overfitting.
• Doesn't always represent the predictive power or accuracy of the model, only its ability to explain variance.
• Applicability and Scenarios:
• Use R^2 when you need to measure how well the model explains variance in the data. But it should be used in conjunction with other metrics like adjusted R^2, AIC, or BIC for model selection and evaluation.

1. 解释性：
• 是介于0和1之间的值，提供了一个直观的度量标准，表示模型解释的数据中的变异性百分比。一个值接近1表示模型解释了大部分的变异性，而一个值接近0表示模型没有很好地解释变异性。

2. 普遍性：
• 是多种回归模型评估的标准指标，因此在学术和业务报告中经常被使用，使得不同的模型和研究可以进行比较。

3. 简洁性：
• 与其他可能需要更多背景知识或复杂解释的指标相比，相对简单并且易于解释。

4. 考虑模型与基线比较：
• 实际上衡量了模型与仅使用目标变量的平均值进行预测的简单模型之间的比较。这为评估模型的相对效果提供了一个基准。

1. 不总是代表模型的预测准确性：
• 一个较高的值不一定意味着模型的预测准确性很好。而只是表示模型很好地解释了数据的变异性。

2. 可能的过拟合：
• 当在模型中加入更多的自变量时，即使这些变量对模型无益，也会增加。这可能会误导研究者认为模型更好了，而实际上可能导致过拟合。

3. 对异常值敏感：
• 如果数据中存在异常值，它可能会影响的值。

Why you would use instead of other metrics:

1. Interpretability:
• offers a value between 0 and 1, providing an intuitive measure that represents the percentage of variance in the data explained by the model. A value close to 1 suggests that the model explains a large portion of the variability, while a value close to 0 suggests it doesn’t explain the variability well.

2. Universality:
• is a standard metric for evaluating various regression models. Its frequent use in academic and business reports facilitates comparison across different models and studies.

3. Simplicity:
• Relative to other metrics that might require more background or complex explanations, is straightforward and easy to explain.

4. Comparison Against a Baseline:
• essentially measures the comparison between your model and a simple model that just predicts the mean of the target variable. This provides a benchmark for evaluating the relative effectiveness of the model.

1. Doesn’t Always Represent Predictive Accuracy:
• A higher doesn’t necessarily mean the model predicts very accurately. It just means it captures the variability in the data well.

2. Potential Overfitting:
• tends to increase when more variables, even irrelevant ones, are added to the model. This can mislead a researcher into thinking the model has gotten better when it might have just gotten more complex and overfit.

3. Sensitivity to Outliers:
• can be influenced by outliers in the data.