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Fix Greek character
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  • lib/node_modules/@stdlib/stats/incr/mvariance/lib

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lib/node_modules/@stdlib/stats/incr/mvariance/lib/main.js

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@@ -37,13 +37,13 @@ var isnan = require( '@stdlib/math/base/assert/is-nan' );
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* - The difference between the unbiased sample variance in a window \\(W_i\\) and the unbiased sample variance in a window \\(W_{i+1})\\) is given by
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*
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* ```tex
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* \delta s^2 = s_i^2 - s_{i+1}^2
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* \Delta s^2 = s_i^2 - s_{i+1}^2
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* ```
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*
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* - If we multiply both sides by \\(N-1\\),
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*
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* ```tex
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* (N-1)(\delta s^2) = (N-1)s_i^2 - (N-1)s_{i+1}^2
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* (N-1)(\Delta s^2) = (N-1)s_i^2 - (N-1)s_{i+1}^2
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* ```
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*
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* - If we substitute the definition of the unbiased sample variance having the form
@@ -62,14 +62,14 @@ var isnan = require( '@stdlib/math/base/assert/is-nan' );
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* we return
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*
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* ```tex
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* (N-1)(\delta s^2) = \biggl(\sum_{k=1}^N (x_k^2 - N\bar{x}_{i+1}^2) \biggr) - \biggl(\sum_{k=0}^{N-1} (x_k^2 - N\bar{x}_{i}^2) \biggr)
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* (N-1)(\Delta s^2) = \biggl(\sum_{k=1}^N (x_k^2 - N\bar{x}_{i+1}^2) \biggr) - \biggl(\sum_{k=0}^{N-1} (x_k^2 - N\bar{x}_{i}^2) \biggr)
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* ```
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*
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* - This can be further simplified by recognizing that subtracting the sums reduces to \\(x_N^2 - x_0^2\\); in which case,
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*
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* ```tex
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* \begin{align}
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* (N-1)(\delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 - N\bar{x}_{i}^2 \\
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* (N-1)(\Delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 - N\bar{x}_{i}^2 \\
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* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1}^2 - \bar{x}_{i}^2) \\
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* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1} - \bar{x}_{i})(\bar{x}_{i+1} + \bar{x}_{i})
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* \end{align}
@@ -84,14 +84,14 @@ var isnan = require( '@stdlib/math/base/assert/is-nan' );
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* and substituting into the equation above
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*
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* ```tex
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* (N-1)(\delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
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* (N-1)(\Delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
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* ```
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*
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* - Rearranging terms gives us the update equation
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*
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* ```tex
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* \begin{align}
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* (N-1)(\delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
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* (N-1)(\Delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
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* &= (x_N - x_0)(x_N + x_0 - \bar{x}_{i+1} - \bar{x}_{i}) \\
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* &= (x_N - x_0)(x_N - \bar{x}_{i+1} + x_0 - \bar{x}_{i})
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* \end{align}

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