Centered R² vs Uncentered R²
Bondi Iceberg pool
1. Introduction: R² and Its Relation to RSQ
The coefficient of determination, denoted as R² (R-squared), is one of the most widely used validation metrics in statistics and machine learning (ML) for assessing how well predicted values $\hat{y}$ agree with observed values $y_{true}$. In spreadsheet software and many engineering reports, R² is often referred to as RSQ (the Excel function name RSQ()), which computes the squared Pearson correlation coefficient between two variables. The default RSQ in Excel and the standard R² reported by most regression libraries refer to the same quantity: the centered R², which measures the proportion of variance in $y$ explained by the model relative to its mean baseline.
However, a less commonly discussed variant — the uncentered R² — replaces the mean-based baseline with a zero baseline. This article compares the two, derives the uncentered formula, provides a geometric interpretation, and discusses when each is appropriate. In particular, we examine whether uncentered R² is more suitable for evaluating 1:1-line agreement between $y_{true}$ and $\hat{y}$ in same-physical-quantity comparisons (Kvalseth 1985; Legates & McCabe 1999).
2. Comparison of the Two Formulas
- The difference lies entirely in the denominator. Centered R² asks “how much variance around the mean has been explained?”, whereas uncentered R² asks “how much of the total sum of squares around zero has been explained?”. Centered R² uses the mean prediction as its baseline, while uncentered R² uses zero prediction as its baseline (Kvalseth 1985).
- Centered R² asks: “How much better is the model compared to a baseline that simply predicts the mean?”
Uncentered R² asks: “How close is the model’s prediction to the 1:1 identity line (y=x)?”
3. Pros and Cons
Centered R²
- Pros: Scale- and unit-invariant, allowing comparison of models across different physical quantities. Equivalent to the squared Pearson correlation coefficient, providing rich statistical intuition.
- Cons: When the variance of $y$ is small (nearly constant), the denominator approaches zero and R² collapses or even turns negative. Measures only relative performance against a mean predictor and is insensitive to systematic bias.
Uncentered R²
- Pros: When comparing models, datasets, or experiments at the same physical-quantity level, uncentered R² provides a consistent absolute baseline (zero prediction), enabling coherent cross-comparison. Robust against low-variance data. Sensitive to deviations from the 1:1 line, properly penalizing systematic bias (Legates & McCabe 1999).
- Cons: Scale-dependent, so unsuitable for comparing models across different units or value ranges. When $\bar{y}$ is large, $R^2_u$ tends to be artificially close to 1. Uncentered $R^2$ values tend to be significantly higher (often reaching 0.99 or above) because the denominator, $\sum y_i^2$, is generally much larger than the centered denominator, $\sum (y_i – \bar{y})^2$. This can lead to an overly optimistic assessment if interpreted in isolation. However, for min-max normalized data, the uniformity of scale renders the uncentered $R^2$ a superior metric for quantifying the absolute agreement between observed and predicted values.
Key question: When comparing models on data of the same physical quantity, is uncentered R² actually more accurate? The answer is essentially “yes.” For evaluating absolute agreement between $y_{true}$ and $\hat{y}$ in same-unit data, uncentered R² (1) is not distorted by changes in dataset variance, (2) is sensitive to constant bias, and (3) shares an absolute baseline (zero prediction) across datasets, enabling consistent model-to-model comparison.
4. Applications
Centered R²
- Standard performance metric in general regression and ML modeling.
- Comparing model performance across heterogeneous units or domains in normalized form.
- Variance-explanation evaluation based on Pearson correlation (Draper & Smith 1998).
Uncentered R²
- Regression Through the Origin (RTO): Models physically constrained to pass through zero, such as Hooke’s law and radioactive decay. The
noconstantoption in Stata, R, and EViews automatically returns uncentered R² (Eisenhauer 2003). When the intercept is fixed to zero, standard (centered) $R^2$ can produce negative values or misleading interpretations. In such cases, Uncentered $R^2$ is used as a robust alternative to evaluate the model’s fit relative to the origin. - Diagnostic tests in econometrics: Auxiliary regressions in the Breusch-Pagan and White tests use $nR^2$ as a chi-squared statistic, computed in uncentered form (Wooldridge 2010).
- Evaluating deviation from the 1:1 line in $Y_{true}$ vs $Y_{pred}$ plots: For ML prediction tasks involving same-physical-quantity data, uncentered R² measures absolute agreement. Suitable for calibration, Computational Fluid Dynamics (CFD) and Finite Element Method (FEM) validation, sensor calibration, and reproducibility studies.
5. Derivation of Uncentered R²
Sum of Squares Decomposition
For observations $y_i$, predictions $\hat{y}_i$, and residuals $e_i = y_i – \hat{y}_i$ ($i = 1, \ldots, n$), define the uncentered sums of squares — Total Sum of Squares (TSS), Explained Sum of Squares (ESS), and Residual Sum of Squares (RSS) — as follows:
Ordinary Least Squares (OLS) Orthogonality Condition
From the OLS normal equations $X^T(y – X\hat{\beta}) = 0$, every column of the design matrix $X$ is orthogonal to the residual vector $e$:
The fitted-value vector and the residual vector are orthogonal. This holds regardless of whether the model includes an intercept.
Uncentered Decomposition
Squaring $y_i = \hat{y}_i + e_i$ and summing over $i$:
The cross term vanishes by orthogonality $\sum \hat{y}_i e_i = 0$, leaving:
Definition of Uncentered R²
Dividing both sides by $TSS_u$ gives $1 = ESS_u/TSS_u + RSS/TSS_u$, and uncentered R² is defined as the explained ratio:
Relation to Centered R²
Since $\sum y_i^2 = \sum (y_i – \bar{y})^2 + n\bar{y}^2$, we have $TSS_u = TSS_c + n\bar{y}^2$. For an intercept-included model:
When $\bar{y}$ is large or $n$ is large, $R^2_u$ moves closer to 1 than $R^2_c$. When $\bar{y} = 0$, the two coincide.
6. Geometric Interpretation
Vector Space Setup
Treating the $n$ observations as vectors $\mathbf{y}, \hat{\mathbf{y}}, \mathbf{e} = \mathbf{y} – \hat{\mathbf{y}} \in \mathbb{R}^n$, every sum of squares becomes a squared norm: $\sum y_i^2 = \|\mathbf{y}\|^2$, $\sum \hat{y}_i^2 = \|\hat{\mathbf{y}}\|^2$, $\sum e_i^2 = \|\mathbf{e}\|^2$.
OLS as Orthogonal Projection
Let $\mathcal{C}(X)$ denote the column space of the design matrix $X$. The OLS estimate is $\hat{\mathbf{y}} = P_X \mathbf{y}$ with $P_X = X(X^TX)^{-1}X^T$, the orthogonal projection of $\mathbf{y}$ onto $\mathcal{C}(X)$. The residual $\mathbf{e}$ lies in the orthogonal complement, automatically satisfying $\mathbf{e} \perp \hat{\mathbf{y}}$ (Strang 2009).
Uncentered Decomposition as the Pythagorean Theorem
Since $\mathbf{y} = \hat{\mathbf{y}} + \mathbf{e}$ and $\hat{\mathbf{y}} \perp \mathbf{e}$, a right triangle is formed:
y
/|
/ |
/ | e (perpendicular residual)
/ |
/____|
origin ŷ (on the C(X) plane)
This is the geometric identity behind $TSS_u = ESS_u + RSS$ — not merely an algebraic identity, but literally the Pythagorean theorem. The geometric meaning of uncentered R² becomes:
where $\theta_u$ is the angle between $\mathbf{y}$ and $\hat{\mathbf{y}}$ measured from the origin. $R^2_u \to 1$ means the two vectors point in the same direction; $R^2_u \to 0$ means they are orthogonal.
Centered R²: Subtracting the Mean Vector
Subtracting the mean vector $\bar{\mathbf{y}} = \bar{y} \cdot \mathbf{1}$ is geometrically equivalent to projecting onto the hyperplane $\mathbf{1}^\perp$, which removes the component along the constant vector $\mathbf{1}$. When the model contains an intercept, $\mathbf{1} \in \mathcal{C}(X)$, so a right triangle still forms within $\mathbf{1}^\perp$:
where $\theta_c$ is the angle between the two vectors after the mean has been removed — equivalent to the squared Pearson correlation coefficient.
Geometric Distinction Between the Two R²
y
/|
/ |
/ |
/ | e
/ |
/ |
*------ŷ-----→ (C(X) plane)
origin
/
*────────── 1 direction (constant vector)
ȳ·1
- Uncentered R²: angle between $\mathbf{y}$ and $\hat{\mathbf{y}}$ measured from the origin.
- Centered R²: angle measured after shifting the reference point to $\bar{y}\cdot\mathbf{1}$.
The fundamental distinction is whether the reference point is the origin or the mean vector. The same data, viewed from different reference points, yields different angles between the two vectors.
Comparison Table
| Aspect | Uncentered | Centered |
|---|---|---|
| Reference point | Origin 0 | Mean vector ȳ·1 |
| Right triangle | y, ŷ, e | ỹ, ŷ̃, e |
| Angle interpretation | cos²(∠yŷ) | cos²(∠ỹŷ̃) = r² |
| Required condition | e ⊥ ŷ | e ⊥ ŷ and 1 ∈ C(X) |
| Sensitivity to bias | Sensitive (origin-fixed) | Insensitive (mean-shift invariant) |
Geometric meaning of 1:1-line validation: Under a constant bias $\hat{\mathbf{y}} = \mathbf{y} + c\mathbf{1}$, the uncentered angle widens but the centered angle remains unchanged. Therefore, for evaluating absolute agreement, uncentered R² responds more sensitively.
7. Python Code
Centered R²
def centered_r2(y_true, y_pred):
"""
Centered R² (standard R²)
R²_c = 1 - SSE / SST_centered
"""
y_true = np.asarray(y_true, dtype=float)
y_pred = np.asarray(y_pred, dtype=float)
sse = np.sum((y_true - y_pred) ** 2)
sst_c = np.sum((y_true - np.mean(y_true)) ** 2)
if sst_c == 0:
return np.nan # undefined when y has zero variance
return 1.0 - sse / sst_c
# Example usage
y_true = np.array([10.1, 10.2, 10.3, 10.4, 10.5])
y_pred = np.array([10.0, 10.3, 10.2, 10.5, 10.4])
print(f"Centered R²: {centered_r2(y_true, y_pred):.6f}")
# Confirm equivalence with scikit-learn
from sklearn.metrics import r2_score
print(f"sklearn r2: {r2_score(y_true, y_pred):.6f}")</pre>Uncentered R²
def uncentered_r2(y_true, y_pred):
"""
Uncentered R² (origin-based R²)
R²_u = 1 - SSE / SST_uncentered
= ⟨ŷ, ŷ⟩ / ⟨y, y⟩ (when e ⊥ ŷ)
"""
y_true = np.asarray(y_true, dtype=float)
y_pred = np.asarray(y_pred, dtype=float)
sse = np.sum((y_true - y_pred) ** 2)
sst_u = np.sum(y_true ** 2)
if sst_u == 0:
return np.nan
return 1.0 - sse / sst_u
# Example usage — comparing predictions on a same-physical-quantity scale (e.g., voltage)
y_true = np.array([10.1, 10.2, 10.3, 10.4, 10.5])
y_pred = np.array([10.0, 10.3, 10.2, 10.5, 10.4])
print(f"Uncentered R²: {uncentered_r2(y_true, y_pred):.6f}")
# Constant-bias scenario: contrast between the two metrics
y_pred_biased = y_true + 0.5 # all predictions offset by +0.5
print(f"\nConstant bias +0.5 scenario:")
print(f" Centered R²: {centered_r2(y_true, y_pred_biased):.6f} "
f"(insensitive to bias)")
print(f" Uncentered R²: {uncentered_r2(y_true, y_pred_biased):.6f} "
f"(sensitive to bias)")</pre>Integrated Evaluation for 1:1-Line Agreement
def evaluate_1_to_1_line_agreement(y_true, y_pred):
"""Evaluate 1:1-line agreement on same-physical-quantity data."""
y_true = np.asarray(y_true, dtype=float)
y_pred = np.asarray(y_pred, dtype=float)
err = y_pred - y_true
rmse = np.sqrt(np.mean(err ** 2))
mae = np.mean(np.abs(err)) # Mean Absolute Error
bias = np.mean(err) # Mean Error (systematic bias)
# MAPE: Mean Absolute Percentage Error (%) — undefined when y_true contains 0
mask = y_true != 0
if mask.any():
mape = np.mean(np.abs(err[mask] / y_true[mask])) * 100.0
else:
mape = np.nan
return {
"CenteredR2": centered_r2(y_true, y_pred),
"UncenteredR2": uncentered_r2(y_true, y_pred),
"RMSE": rmse,
"MAE": mae,
"MAPE": mape,
"Bias": bias,
}
# Example usage
y_true = np.array([10.1, 10.2, 10.3, 10.4, 10.5])
y_pred = np.array([10.0, 10.3, 10.2, 10.5, 10.4])
results = evaluate_1_to_1_line_agreement(y_true, y_pred)
for k, v in results.items():
print(f"{k:>15s}: {v:.6f}")References
- Draper, N. R., & Smith, H. (1998). Applied Regression Analysis (3rd ed.). Wiley.
- Eisenhauer, J. G. (2003). Regression through the origin. Teaching Statistics, 25(3), 76–80.
- Kvalseth, T. O. (1985). Cautionary note about R². The American Statistician, 39(4), 279–285.
- Legates, D. R., & McCabe, G. J. (1999). Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resources Research, 35(1), 233–241.
- Strang, G. (2009). Introduction to Linear Algebra (4th ed.). Wellesley-Cambridge Press.
- Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.
