This requirement can be satisfied by adding sufficient noise to predictions $\hat{y}$ to decorrelate them from orthogonal values $v$. Ideally, if $\hat{y}$ is already decorrelated from $v$, no noise would be added to $\hat{y}$, thus $\hat{y}$ would be maximally correlated with $y$.

Mathematically, we want to create $\hat{y}'=\hat{y}+\epsilon$ from $\epsilon \sim \mathcal{N}(0, \sigma_{\epsilon})$, to satisfy $$r_{\hat{y}'v} = \frac{\sigma_{\hat{y}'v}}{\sigma_{\hat{y}'}\sigma_{v}} < \delta$$ for arbitrary threshold $\delta$. Now, lets expand this inequality to find a lower-bound for std of noise $\epsilon$, i.e. $\sigma_{\epsilon}$.
$$\begin{align*}
\sigma_{\hat{y}'}^2&=\sigma_{\hat{y}}^2 + \sigma_{\epsilon}^2,\\
\sigma_{\hat{y}'v}&={\Bbb E}\left[(\hat{y}+\epsilon - \mu_{\hat{y}} - \overbrace{\mu_{\epsilon}}^{=0})(v-\mu_{v})\right]\\
&={\Bbb E}\left[(\hat{y} - \mu_{\hat{y}})(v-\mu_{v})\right]+\overbrace{{\Bbb E}\left[\epsilon(v-\mu_{v})\right]}^{=0}&\\
&=\sigma_{\hat{y}v},\\
r_{\hat{y}'v} &= \frac{\sigma_{\hat{y}'v}}{\sigma_{\hat{y}'}\sigma_{v}} =\frac{\sigma_{\hat{y}v}}{\sigma_{v} \sqrt{\sigma_{\hat{y}}^2+\sigma_{\epsilon}^2}} < \delta\\
&\Rightarrow \sigma_{\hat{y}}\sqrt{\left(\frac{r_{\hat{y}v}}{\delta}\right)^2 - 1} < \sigma_{\epsilon}
\end{align*}$$

Since all the variables in the left side of inequality can be calculated, we can sample noises from $\mathcal{N}(0, \sigma_{\epsilon})$ and add them to $\hat{y}$ to satisfy the original inequality.

Here is a code that does the exact same thing:

```
import numpy as np
import pandas as pd
from sklearn.datasets import make_regression
from xgboost import XGBRegressor
ORTHO_VAR = 'ortho_var'
IND_VARNM = 'indep_var'
TARGET = 'target'
CORRECTED_VARNM = 'indep_var_fixed'
seed = 245
# Create regression dataset with two correlated targets
X, y = make_regression(n_samples=10000, n_features=20, random_state=seed, n_targets=2)
indep_vars = ['var{}'.format(i) for i in range(X.shape[1])]
# Pull into dataframe
df = pd.DataFrame(X, columns=indep_vars)
df[TARGET] = y[:, 0]
df[ORTHO_VAR] = y[:, 1]
# Fit a model to predict TARGET
xgb = XGBRegressor(n_estimators=10)
xgb.fit(df[indep_vars], df[TARGET])
df['yhat'] = xgb.predict(df[indep_vars])
delta = 0.01
# std of noise required to be added to y_hat to bring the correlation
# of y_hat with ORTHO_VAR below delta
std_y_hat = np.std(df['yhat'], ddof=1)
corr_y_hat_ortho_var = np.corrcoef(df['yhat'], df[ORTHO_VAR])[1, 0]
corr_y_hat_target = np.corrcoef(df['yhat'], df[TARGET])[1, 0]
std_noise_lower_bound = std_y_hat * np.sqrt((corr_y_hat_ortho_var / delta)**2 - 1.0)
std_noise = max(0, std_noise_lower_bound) + 1
print('delta: ', delta)
print('std_y_hat: ', std_y_hat)
print('corr_y_hat_target: ', corr_y_hat_target)
print('corr_y_hat_ortho_var: ', corr_y_hat_ortho_var)
print('std_noise_lower_bound: ', std_noise_lower_bound)
print('std_noise: ', std_noise)
# add noise
np.random.seed(seed)
noises = np.random.normal(0, std_noise, len(df['yhat']))
noises -= np.mean(noises) # remove slight deviations from zero mean
print('noise_samples: mean:', np.mean(noises), ', std: ', np.std(noises))
df['yhat'] = df['yhat'] + noises
# measure new correlation
corr_y_hat_ortho_var = np.corrcoef(df['yhat'], df[ORTHO_VAR])[1, 0]
corr_y_hat_target = np.corrcoef(df['yhat'], df[TARGET])[1, 0]
print('new corr_y_hat_target: ', corr_y_hat_target)
print('new corr_y_hat_ortho_var: ', corr_y_hat_ortho_var)
# Correlation should be low or preferably zero
assert corr_y_hat_ortho_var < delta, corr_y_hat_ortho_var
assert -delta < corr_y_hat_ortho_var, corr_y_hat_ortho_var
```

which outputs:

```
delta: 0.01
std_y_hat: 69.48568725585938
corr_y_hat_target: 0.8207672834857673
corr_y_hat_ortho_var: 0.7663936356880843
std_noise_lower_bound: 5324.885500165032
std_noise: 5325.885500165032
noise_samples: mean: 1.1059455573558807e-13 , std: 5373.914830034988
new corr_y_hat_target: -0.004125016071865934
new corr_y_hat_ortho_var: -0.000541131379457552
```

You can experiment with other `delta`

s. By comparing `std_y_hat`

with `std_noise_lower_bound`

, you can see that a huge noise must be added to $\hat{y}$ to decorrelate it from $v$ bellow $0.01$, which dramatically decolerates $\hat{y}$ from $y$ too.

Note: `Assertion`

might fail for too small thresholds $\delta$ due to insufficient sample count.

2What is the correlation of TARGET with ORTHO_VAR? – Esmailian – 2019-04-15T04:26:42.873

Good question. They are indeed correlated (let’s say 50%) the predictions will likely suffer in terms of accuracy by being made orthogonal. – Chris – 2019-04-15T10:39:55.087