Day 3 - Regression Evaluation Metrics: RMSE, MAE, MAPE, R², and Residual Analysis

📌 Key Takeaways

• Every regression metric differs only in how it compresses the residual y − ŷ. Do you square it, take the absolute value, or use a ratio?
• MAE is robust to outliers and RMSE is sensitive to large errors. Both are in the same unit as the target, and always RMSE ≥ MAE.
• MAPE is a scale-free relative error, but it's unusable when actual values are near 0 because the denominator blows up.
• R² is explanatory power relative to "predicting the target mean." If it's negative, it's a strong warning that the model is worse than the mean.
• Even when the metric looks good, if the residual plot shows a pattern (U-shape, funnel, skew), the model's assumptions are broken.

Starting From the Error

If classification asks "was it right or wrong," regression measures "how far off was it." For each sample i, the error is as follows.

residual_i = y_i (actual value) - ŷ_i (predicted value)

Every regression metric gathers these errors and compresses them into one score. The difference lies in how the error is handled — do you square it, take the absolute value, or look at it as a ratio?

MAE and RMSE

MAE  = (1/n) Σ |y_i - ŷ_i|
MSE  = (1/n) Σ (y_i - ŷ_i)²
RMSE = √[ (1/n) Σ (y_i - ŷ_i)² ]

Errors [1, 1, 1, 1, 6]
MAE  = (1+1+1+1+6)/5 = 2.0
RMSE = √((1+1+1+1+36)/5) = √8 = 2.83  ← the error of 6 stands out more

• Even for the same set of errors, RMSE ≥ MAE, and the larger the gap between them, the more it suggests large errors (outliers) exist.
• If all error magnitudes are equal, RMSE = MAE. In other words, the gap itself is a diagnostic signal.
• RMSE is often used as the default evaluation/tuning metric for SageMaker built-in regression (XGBoost, Linear Learner, etc.).

The choice between RMSE and MAE is a question of "how heavily do you penalize large errors." RMSE penalizes large errors strongly through squaring, so it fits problems where a single large mistake is fatal (stockouts from under-forecasting demand, etc.). MAE reflects large errors only proportionally, so it is stable when outliers are a data quality problem (sensor errors, etc.).

MAPE: Mean Absolute Percentage Error

MAPE = (100%/n) Σ |y_i - ŷ_i| / |y_i|

It converts the error into a ratio (%) of the actual value and averages it.

• Scale-free: lets you compare prediction performance across different units/scales (e.g., several stores with different sales volumes)
• Relative error perspective: predicting 1100 for 1000 gives a MAPE of 10%
• Trap: if the actual value y_i is 0 or close to 0, the denominator blows up or becomes undefined

[How the same absolute error of 1.0 looks depending on the actual value]
 
Actual y = 1000, predicted 1001  →  |1|/1000  =   0.1%
Actual y =   10, predicted 11    →  |1|/10    =  10.0%
Actual y =    1, predicted 2     →  |1|/1     = 100.0%
Actual y =  0.1, predicted 1.1   →  |1|/0.1   = 1000.0%   ← a single case dominates the average
Actual y =    0, predicted 1     →  |1|/0     = undefined (∞)

• It's useful when comparing items of different scale like demand forecasting, but it's unsuitable when there are many values near 0. Data where actual values of 0 are common, such as stockouts, off-seasons, and new products, is the typical case.
• MAPE is also asymmetric. Under-prediction is capped at 100% (when you predict 0), but over-prediction has no upper limit. So minimizing MAPE can bias the model toward predicting low.

⚠️ Trap: Don't pick MAPE just because you see "I want to compare stores with different scales." If the passage also says "some stores' sales are close to 0" or "there are days with no demand," MAPE is the wrong answer, and if scale correction is needed, consider RMSLE or per-store normalization.

R²: Coefficient of Determination

R² = 1 - (SS_res / SS_tot)
SS_res = Σ (y_i - ŷ_i)²        (sum of squared model errors)
SS_tot = Σ (y_i - ȳ)²          (variance relative to the mean line)

It measures "what % of the target's variance the model explains."

• It has no unit, which is good for comparing across different problems, but it doesn't tell you the absolute size of the error. Even with an R² of 0.9, the RMSE could be too large for the business to bear.
• So look at R² (relative explanatory power) and RMSE/MAE (absolute error) together.

R² vs. Adjusted R²

Adding features raises R² even when it means nothing. Even an unrelated column reduces SS_res slightly by fitting the noise.

Adjusted R² = 1 - (1 - R²) × (n - 1) / (n - p - 1)
   n = number of samples,  p = number of features (explanatory variables)

The key point is that R² measures "how much better than the baseline." The baseline is always the target mean (ȳ). So a negative R² means the model is worse than the simple mean, a strong warning that the model captured no signal at all. RMSE/MAE tell you the absolute size of the error, and R² tells you relative explanatory power, complementing each other.

Regression Metric Selection Table

You need to choose a regression metric
├─ Does the data have extreme outliers, and are they errors?
│    ├─ Yes → MAE (robust because it doesn't square)
│    └─ No ↓
├─ Is a single large error fatal to the business? (stockouts, overcharging, etc.)
│    ├─ Yes → RMSE (penalizes large errors by squaring)
│    └─ No ↓
├─ Must items of widely varying scale be compared with one metric?
│    ├─ No actual values near 0 → MAPE
│    └─ Actual values of 0 are common → No MAPE, use RMSE/MAE or normalization
└─ Must you report "how much it explains" as a ratio?
     ├─ Single model → R²
     └─ Comparing models with different numbers of features → Adjusted R²

Residual Analysis: What a Single Number Hides

Even if one metric says the model is good, the model may have a systematic flaw. Drawing a residual plot (predicted value vs. residual) reveals it.

① Ideal — constant variance, no pattern     ② Funnel — heteroscedasticity
 Residual                                    Residual
  +│  ·  ·   ·  ·   ·                         +│        ·      ·
   │ ·   ·  ·   ·  ·                           │     ·     ·  ·   ·
  0├──·──·───·──·───·──→ Predicted            0├──·──·───·───·────·──→
   │  ·  ·  ·   ·  ·                           │     ·    ·   ·   ·
  -│   ·   ·  ·   ·                            -│        ·      ·
   → centered on residual 0, constant width     → spreads out as the prediction grows
 
③ U-shape/curve — nonlinearity not captured ④ Skew — systematic bias
 Residual                                    Residual
  +│ ·                    ·                    +│  ·  ·  ·  ·  ·  ·
   │  ·                 ·                       │ ·  ·  ·  ·  ·  ·
  0├────·─────────·────────→                  0├──────────────────────→
   │      ·   ·   ·                             │
  -│        · · ·                               │  (almost all above 0)
   → limit of a linear model, needs feature     → consistently under-predicting
     transformation

import matplotlib.pyplot as plt
 
residuals = y_true - y_pred
plt.scatter(y_pred, residuals, alpha=0.3)
plt.axhline(0, color="red", linestyle="--")
plt.xlabel("Predicted value")
plt.ylabel("Residual (actual - predicted)")
# Random scatter is good; if you see a pattern, the model's assumptions are violated

Look at the Distribution of Residuals Too

• If the residual histogram is symmetric around 0, it's normal. A long tail on one side means big mistakes in a particular direction keep recurring.
• If the mean deviates from 0, it signals that the whole model is consistently over- or under-predicting.
• If the residual distribution deviates greatly from a normal distribution or shows a pattern, you need to reexamine the model form or features even if the metric looks good.

Even with a low RMSE, if the residuals have a pattern, the model can't be trusted. That's because a single metric averages the errors into one number and erases "where and how" it's wrong. The residual plot restores the structure that the average hid — nonlinearity, heteroscedasticity, bias. On the exam, a passage like "the metric is good but it keeps being wrong in a particular range" points toward residual analysis / checking the model form.

Regression Metrics in SageMaker

Built-in regression algorithms emit RMSE as the default validation metric. An example of a regression objective metric in Automatic Model Tuning:

tuner = HyperparameterTuner(
    estimator=xgb_estimator,
    objective_metric_name="validation:rmse",  # regression default
    objective_type="Minimize",                 # it's an error, so minimize
    hyperparameter_ranges=ranges,
)

⚠️ Trap: Classification is usually Maximize (accuracy/F1/AUC), and regression is Minimize (RMSE/MAE). If you set objective_type the wrong way around, the tuner picks the worst model. If you use a regression metric where larger is better like R², then it's Maximize.

Passage Clue → Answer Mapping

Tomorrow we move on to tools that look into why a model is wrong — SageMaker Debugger, Clarify (bias detection, SHAP explainability), and error analysis.

📖 Glossary

• Residual : actual value − predicted value. The raw material of every regression metric, and because it has a sign it distinguishes over- from under-prediction.
• MAE (Mean Absolute Error) : the mean of the absolute values of the errors. Same unit as the target and robust to outliers.
• RMSE (Root Mean Squared Error) : square the errors, average them, then take the square root. Penalizes large errors heavily. Always at least MAE.
• MAPE (Mean Absolute Percentage Error) : the mean of the relative error obtained by dividing the error by the actual value (%). Breaks down when actual values are near 0.
• R² (Coefficient of Determination) : the ratio of how much better the model is than the baseline of predicting the target mean. Can be negative.
• Adjusted R² : R² penalized for the number of features. Used when comparing models with different numbers of explanatory variables.
• Heteroscedasticity : the phenomenon where the spread of errors differs across the prediction range. The funnel shape in a residual plot.
• Residual plot : a scatter plot with predicted values on the x-axis and residuals on the y-axis. Revives the structural flaws that a single metric number erased.
• RMSLE : compute RMSE after taking the log of actual and predicted values. Used when the target is heavily skewed and relative error matters.
• objective_type : the value in SageMaker tuning that specifies whether to raise or lower the objective metric. Error metrics are Minimize; accuracy, AUC, and R² are Maximize.