Angle Between Vectors

The angle between two nonzero vectors \(a\), \(b\) is defined as \[\theta = \text{arccos} \bigg( \frac{a^Tb}{||a|| \ ||b||} \bigg)\] where arccos denotes the inverse cosine, normalized to lie in the interval \([0, \pi]\). In other words, we define \(\theta\) as the unique number between 0 and \(\pi\) that satisfies \[a^Tb = ||a|| \ ||b|| \ \text{cos} \ \theta\] The angle between \(a\) and \(b\) is written as \(\angle(a, b)\), and is sometimes expressed in degrees (the default unit is radians). …

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Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states for all vectors \(u\) and \(v\), \[|\langle \mathbf{u}, \mathbf{v} \rangle| \leq ||\mathbf{u}|| \ ||\mathbf{v}||\] This means that the dot product of the two vectors is less than the product of the norms of the vectors. import numpy as np u = np.array([3, 5, 5]) v = np.array([5, 2, 6]) np.dot(u, v) <= np.linalg.norm(u) * np.linalg.norm(v) ## True

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Standardization

For any vector \(x\), we refer to \(\tilde{x} = x - \mathbf{avg}(x)\mathbf{1}\) as the de-meaned version of \(x\), since it has average or mean value zero. If we then divide by the RMS value of \(\tilde{x}\) (which is the standard deviation of \(x\)), we obtain the vector \[z = \frac{1}{\mathbf{std}(x)} (x - \mathbf{avg}(x)\mathbf{1})\] This vector is called the standardized version of \(x\). It has mean zero, and standard deviation one. …

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Chebyshev Inequality For Standard Deviation

The Chebyshev inequality goes like this. Suppose that \(x\) is an \(n\)-vector, and that \(k\) of its entries satisfy \(|x_i| \geq a\), where \(a > 0\). Then \(k\) of its entries satisfy \(x_i^2 \geq a^2\). It follows that \[||x||^2 = x_1^2 + \cdots + x_n^2 \geq ka^2\] since \(k\) of the numbers in the sum are at least \(a^2\), and the other \(n - k\) numbers are nonnegative. We conclude that \(k \leq ||x||^2 / a^2\), which is the Chebyshev inequality. …

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Average, RMS, And Standard Deviation

The average, RMS value, and standard deviation of a vector are related by the formula \[\mathbf{rms}(x)^2 = \mathbf{avg}(x)^2 + \mathbf{std}(x)^2\] \(\mathbf{rms}(x)^2\) is the mean square value of the entries of \(x\), which can be expressed as the square of the mean value, plus the mean square fluctuation of the entries of \(x\) around their mean value. Examples Mean return and risk. Suppose that an \(n\)-vector represents a time seris of retun on an investment, expressed as the percentage, in \(n\) time periods over some interval of time. …

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