Orthogonal signals can still make the same bet

A quantitative researcher often rejects a new stock-selection signal because it looks too correlated with one already in the library. That sounds like diversification discipline. But the correlation being measured is usually across stocks on each date, while profit-and-loss correlation is measured across dates after every cross-section has collapsed to one payoff.

Nunes shows these are not noisy estimates of the same object. They are inner products over different axes. Without extra assumptions, neither one bounds or even orders the other.

The information coefficient (IC) is the cross-sectional Pearson correlation between a signal's scores and the returns realized next. After demeaning and scaling the signal to Euclidean norm one, its projection onto the realized return direction is exactly that IC. The remainder is transverse: it changes how signals resemble one another without changing that signal's same-date payoff.

Therefore signal similarity equals the product of the two realized ICs plus their transverse inner product. Normalizing the transverse term gives the ordinary partial correlation between the two signals after controlling for realized returns.

Return dispersion is the cross-sectional magnitude of that date's demeaned returns. For the normalized signal, realized PnL is simply dispersion times IC. PnL correlation then centers and normalizes those two time series. The transverse component has disappeared; variable dispersion and Pearson centering have entered.

def diagnose(ic1, ic2, dispersion, similarity):
    mean = lambda xs: sum(xs) / len(xs)
    products = [x * y for x, y in zip(ic1, ic2)]
    pnl1 = [a * x for a, x in zip(dispersion, ic1)]
    pnl2 = [a * y for a, y in zip(dispersion, ic2)]
    transverse = mean([s - p for s, p in zip(similarity, products)])
    pnl_corr = pearson(pnl1, pnl2)
    return mean(similarity), mean(products), transverse, pnl_corr

The site runs the TypeScript implementation above. The Python and C++ tabs are line-for-line translations of the same four diagnostics.

The non-identifiability theorem can be made concrete in four dates. Two mean-zero IC series are built with any requested temporal correlation. Their transverse angle is then chosen date by date so their signal similarity stays exactly 0.4. Constant dispersion makes PnL correlation equal the IC-series correlation.

The paper generates 12 signals over 1,000 dates and 500 assets with seed 20260906. Signals share one of three transverse feature families, while their IC paths load differently on a common factor. That separation is imposed deliberately to isolate the mechanism.

A second quiet trap appears when signals are combined. The best ex post squared correlation between realized returns and a rank-K signal span is an R². Even an uninformative isotropic span has expected R² equal to K/(d−1). With 12 signals and 500 assets, that mechanical baseline is 0.0240. The paper's raw 0.0334 becomes only 0.0096 after null adjustment.

The sharpest buried caveat is the ex post status. The same future-return direction defines IC, transverse geometry, and the optimal span fit. Those quantities are excellent forensic diagnostics, but using the raw R² or its optimizing weights as a library score leaks the answer into the feature-selection problem.

• Compute signal, IC-moment, transverse, and PnL matrices side by side.
• Repeat them at every forward-return horizon; the geometry is horizon-specific.
• Use block bootstrap or HAC inference for overlapping returns and persistent signals.
• Control multiplicity: 50 signals already create 1,225 pairwise comparisons.
• Estimate signed, regularized combination rules strictly out of sample.

The linear-regression page provides the projection geometry underneath IC and partial correlation. The lesson also echoes bagging: useful diversity concerns the dependence of errors—or here, realized payoffs—not merely the resemblance of inputs.