@@ -37,13 +37,13 @@ var isnan = require( '@stdlib/math/base/assert/is-nan' );
37
37
* - 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
38
38
*
39
39
* ```tex
40
- * \delta s^2 = s_i^2 - s_{i+1}^2
40
+ * \Delta s^2 = s_i^2 - s_{i+1}^2
41
41
* ```
42
42
*
43
43
* - If we multiply both sides by \\(N-1\\),
44
44
*
45
45
* ```tex
46
- * (N-1)(\delta s^2) = (N-1)s_i^2 - (N-1)s_{i+1}^2
46
+ * (N-1)(\Delta s^2) = (N-1)s_i^2 - (N-1)s_{i+1}^2
47
47
* ```
48
48
*
49
49
* - 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' );
62
62
* we return
63
63
*
64
64
* ```tex
65
- * (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)
65
+ * (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)
66
66
* ```
67
67
*
68
68
* - This can be further simplified by recognizing that subtracting the sums reduces to \\(x_N^2 - x_0^2\\); in which case,
69
69
*
70
70
* ```tex
71
71
* \begin{align}
72
- * (N-1)(\delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 - N\bar{x}_{i}^2 \\
72
+ * (N-1)(\Delta s^2) &= x_N^2 - x_0^2 - N\bar{x}_{i+1}^2 - N\bar{x}_{i}^2 \\
73
73
* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1}^2 - \bar{x}_{i}^2) \\
74
74
* &= x_N^2 - x_0^2 - N(\bar{x}_{i+1} - \bar{x}_{i})(\bar{x}_{i+1} + \bar{x}_{i})
75
75
* \end{align}
@@ -84,14 +84,14 @@ var isnan = require( '@stdlib/math/base/assert/is-nan' );
84
84
* and substituting into the equation above
85
85
*
86
86
* ```tex
87
- * (N-1)(\delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
87
+ * (N-1)(\Delta s^2) = x_N^2 - x_0^2 - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
88
88
* ```
89
89
*
90
90
* - Rearranging terms gives us the update equation
91
91
*
92
92
* ```tex
93
93
* \begin{align}
94
- * (N-1)(\delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
94
+ * (N-1)(\Delta s^2) &= (x_N - x_0)(x_N + x_0) - (x_N - x_0)(\bar{x}_{i+1} + \bar{x}_{i})
95
95
* &= (x_N - x_0)(x_N + x_0 - \bar{x}_{i+1} - \bar{x}_{i}) \\
96
96
* &= (x_N - x_0)(x_N - \bar{x}_{i+1} + x_0 - \bar{x}_{i})
97
97
* \end{align}
0 commit comments