In a white noise series \(a_{t}\) of variance \(\sigma^{2}\), an outlier of size \(\omega\) is identified by the ratio \(\omega / \sigma\). In a vector white noise \(\boldsymbol{a}_{t}\) of \(k\) time series with the same variance \(\sigma^{2}\) and a multivariate outlier \(\boldsymbol{\omega}\), the univariate time series obtained by projecting \(\boldsymbol{a}_{t}\) in the direction \(\boldsymbol{\omega}, y_{t}=\boldsymbol{\omega}^{\prime} \boldsymbol{a}_{t}\) has an outlier of size \(\boldsymbol{\omega}^{\prime} \boldsymbol{\omega}\) and variance \(\boldsymbol{\omega}^{\prime} \boldsymbol{\omega} \sigma^{2}\), and the ratio to identify the outlier is \(\sqrt{\omega^{\prime} \omega} / \sigma=\sqrt{k} \bar{\omega} / \sigma\), where \(\bar{\omega}=\sqrt{\omega^{\prime} \omega / k}\). Explain why these results prove that the multivariate outlier detection by projections can be more powerful than univariate detection if the direction of the outlier is well identified.