diff --git a/lib/node_modules/@stdlib/stats/incr/apcorr/README.md b/lib/node_modules/@stdlib/stats/incr/apcorr/README.md index e058faaeeb68..ad21f5263432 100644 --- a/lib/node_modules/@stdlib/stats/incr/apcorr/README.md +++ b/lib/node_modules/@stdlib/stats/incr/apcorr/README.md @@ -28,10 +28,14 @@ The [Pearson product-moment correlation coefficient][pearson-correlation] betwee -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `n`, the sample [Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/covariance/README.md b/lib/node_modules/@stdlib/stats/incr/covariance/README.md index bb7f88bc6062..e08d6db8d1ad 100644 --- a/lib/node_modules/@stdlib/stats/incr/covariance/README.md +++ b/lib/node_modules/@stdlib/stats/incr/covariance/README.md @@ -28,10 +28,14 @@ For unknown population means, the [unbiased sample covariance][covariance] is de -
+```math +\operatorname{cov_n} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n) +``` + + @@ -39,10 +43,14 @@ For known population means, the [unbiased sample covariance][covariance] is defi -
+```math +\operatorname{cov_n} = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu_x)(y_i - \mu_y) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/covmat/README.md b/lib/node_modules/@stdlib/stats/incr/covmat/README.md index 247cb447df1c..10c289142cb6 100644 --- a/lib/node_modules/@stdlib/stats/incr/covmat/README.md +++ b/lib/node_modules/@stdlib/stats/incr/covmat/README.md @@ -28,10 +28,14 @@ A [covariance matrix][covariance-matrix] is an M-by-M matrix whose elements spec -
+```math +\operatorname{cov_{jkn}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_{ij} - \bar{x}_{jn})(x_{ik} - \bar{x}_{kn}) +``` + + @@ -39,10 +43,14 @@ For known population means, the [unbiased sample covariance][covariance-matrix] -
+```math +\operatorname{cov_{jkn}} = \frac{1}{n} \sum_{i=0}^{n-1} (x_{ij} - \mu_{j})(x_{ik} - \mu_{k}) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/cv/README.md b/lib/node_modules/@stdlib/stats/incr/cv/README.md index 52ce0a7149fc..7580372fd6d7 100644 --- a/lib/node_modules/@stdlib/stats/incr/cv/README.md +++ b/lib/node_modules/@stdlib/stats/incr/cv/README.md @@ -28,10 +28,14 @@ The [corrected sample standard deviation][sample-stdev] is defined as -
+```math +s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2} +``` + + @@ -39,10 +43,14 @@ and the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i +``` + + @@ -50,10 +58,14 @@ The [coefficient of variation][coefficient-of-variation] (also known as **relati -
+```math +c_v = \frac{s}{\bar{x}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/ewmean/README.md b/lib/node_modules/@stdlib/stats/incr/ewmean/README.md index 759ee35af68c..945c0c06cb96 100644 --- a/lib/node_modules/@stdlib/stats/incr/ewmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/ewmean/README.md @@ -28,10 +28,14 @@ An [exponentially weighted mean][moving-average] can be defined recursively as -
+```math +\mu_t = \begin{cases} x_0 & \textrm{if}\ t = 0 \\ \alpha x_t + (1-\alpha) \mu_{t-1} & \textrm{if}\ t > 0 \end{cases} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/ewstdev/README.md b/lib/node_modules/@stdlib/stats/incr/ewstdev/README.md index 364435095396..e738bc6a5337 100644 --- a/lib/node_modules/@stdlib/stats/incr/ewstdev/README.md +++ b/lib/node_modules/@stdlib/stats/incr/ewstdev/README.md @@ -28,10 +28,14 @@ An [exponentially weighted variance][moving-average] can be defined recursively -
+```math +S_n = \begin{cases} 0 & \textrm{if}\ n = 0 \\ (1 - \alpha) (S_{n-1} + \alpha(x_n - \mu_{n-1})^2) & \textrm{if}\ n > 0 \end{cases} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/ewvariance/README.md b/lib/node_modules/@stdlib/stats/incr/ewvariance/README.md index 4ff38ef7f769..4361fc41a891 100644 --- a/lib/node_modules/@stdlib/stats/incr/ewvariance/README.md +++ b/lib/node_modules/@stdlib/stats/incr/ewvariance/README.md @@ -28,10 +28,14 @@ An [exponentially weighted variance][moving-average] can be defined recursively -
+```math +S_n = \begin{cases} 0 & \textrm{if}\ n = 0 \\ (1 - \alpha) (S_{n-1} + \alpha(x_n - \mu_{n-1})^2) & \textrm{if}\ n > 0 \end{cases} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/gmean/README.md b/lib/node_modules/@stdlib/stats/incr/gmean/README.md index a54f23a5d2ef..4865224a1065 100644 --- a/lib/node_modules/@stdlib/stats/incr/gmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/gmean/README.md @@ -28,10 +28,14 @@ The [geometric mean][geometric-mean] is defined as the nth root of a product of -
+```math +\biggl( \prod_{i=0}^{n-1} \biggr)^{\frac{1}{n}} = \sqrt[n]{x_0 x_1 \cdots x_{n-1}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/grubbs/README.md b/lib/node_modules/@stdlib/stats/incr/grubbs/README.md index 257c8d0f10ad..5e7dfcacd798 100644 --- a/lib/node_modules/@stdlib/stats/incr/grubbs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/grubbs/README.md @@ -33,10 +33,14 @@ The [Grubbs' test][grubbs-test] statistic for a two-sided alternative hypothesis -
+```math +G = \frac{\max_{i=0,\ldots,N-1} |Y_i - \bar{Y}|}{s} +``` + + @@ -46,10 +50,14 @@ The [Grubbs' test][grubbs-test] statistic for the alternative hypothesis that th -
+```math +G = \frac{\bar{Y} - Y_{\textrm{min}}}{s} +``` + + @@ -57,10 +65,14 @@ The [Grubbs' test][grubbs-test] statistic for the alternative hypothesis that th -
+```math +G = \frac{Y_{\textrm{max}} - \bar{Y}}{s} +``` + + @@ -68,10 +80,14 @@ For a two-sided test, the hypothesis that a dataset does **not** contain an outl -
+```math +G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2_{\alpha/(2N),N-2}}{N - 2 + t^2_{\alpha/(2N),N-2}}} +``` + + @@ -81,10 +97,14 @@ For a one-sided test, the hypothesis that a dataset does **not** contain an outl -
+```math +G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2_{\alpha/N,N-2}}{N - 2 + t^2_{\alpha/N,N-2}}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/hmean/README.md b/lib/node_modules/@stdlib/stats/incr/hmean/README.md index 563217019c91..d92363ce294f 100644 --- a/lib/node_modules/@stdlib/stats/incr/hmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/hmean/README.md @@ -28,10 +28,14 @@ The [harmonic mean][harmonic-mean] of positive real numbers `x_0, x_1, ..., x_{n -
+```math +\begin{align}H &= \frac{n}{\frac{1}{x_0} + \frac{1}{x_1} + \cdots + \frac{1}{x_{n-1}}} \\ &= \frac{n}{\sum_{i=0}^{n-1} \frac{1}{x_i}} \\ &= \biggl( \frac{\sum_{i=0}^{n-1} \frac{1}{x_i}}{n} \biggr)^{-1}\end{align} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/kurtosis/README.md b/lib/node_modules/@stdlib/stats/incr/kurtosis/README.md index cab1830e279a..8903e46d015a 100644 --- a/lib/node_modules/@stdlib/stats/incr/kurtosis/README.md +++ b/lib/node_modules/@stdlib/stats/incr/kurtosis/README.md @@ -28,10 +28,14 @@ The [kurtosis][sample-excess-kurtosis] for a random variable `X` is defined as -
+```math +\operatorname{Kurtosis}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr] +``` + + @@ -41,10 +45,14 @@ For a sample of `n` values, the [sample excess kurtosis][sample-excess-kurtosis] -
+```math +g_2 = \frac{m_4}{m_2^2} - 3 = \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} +``` + + @@ -54,10 +62,14 @@ The previous equation is, however, a biased estimator of the population excess k -
+```math +G_2 = \frac{(n+1)n}{(n-1)(n-2)(n-3)} \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})^4}{\biggl(\sum_{i=0}^{n-1} (x_i - \bar{x})^2\biggr)^2} - 3 \frac{(n-1)^2}{(n-2)(n-3)} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/maape/README.md b/lib/node_modules/@stdlib/stats/incr/maape/README.md index 23ce84f18667..9fbd0e8e4f17 100644 --- a/lib/node_modules/@stdlib/stats/incr/maape/README.md +++ b/lib/node_modules/@stdlib/stats/incr/maape/README.md @@ -28,10 +28,14 @@ The [mean arctangent absolute percentage error][@kim:2016a] is defined as -
+```math +\operatorname{MAAPE} = \frac{1}{n} \sum_{i=0}^{n-1} \operatorname{arctan}\biggl( \biggl| \frac{a_i - f_i}{a_i} \biggr| \biggr) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mae/README.md b/lib/node_modules/@stdlib/stats/incr/mae/README.md index 29f4ef902b6f..b9bab87bd6b7 100644 --- a/lib/node_modules/@stdlib/stats/incr/mae/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mae/README.md @@ -28,10 +28,14 @@ The [mean absolute error][mean-absolute-error] is defined as -
+```math +\operatorname{MAE} = \frac{\sum_{i=0}^{n-1} |y_i - x_i|}{n} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mapcorr/README.md b/lib/node_modules/@stdlib/stats/incr/mapcorr/README.md index d7a8134d6a42..1ddd35557cd9 100644 --- a/lib/node_modules/@stdlib/stats/incr/mapcorr/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mapcorr/README.md @@ -28,10 +28,14 @@ The [Pearson product-moment correlation coefficient][pearson-correlation] betwee -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `W`, the sample [Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mape/README.md b/lib/node_modules/@stdlib/stats/incr/mape/README.md index f1520f512068..b4ab2c98fa1c 100644 --- a/lib/node_modules/@stdlib/stats/incr/mape/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mape/README.md @@ -28,10 +28,14 @@ The [mean absolute percentage error][mean-absolute-percentage-error] is defined -
+```math +\operatorname{MAPE} = \frac{100}{n} \sum_{i=0}^{n-1} \biggl| \frac{a_i - f_i}{a_i} \biggr| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mcovariance/README.md b/lib/node_modules/@stdlib/stats/incr/mcovariance/README.md index 287cb4513e5c..16f37e7f3622 100644 --- a/lib/node_modules/@stdlib/stats/incr/mcovariance/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mcovariance/README.md @@ -28,10 +28,14 @@ For unknown population means, the [unbiased sample covariance][covariance] for a -
+```math +\operatorname{cov_n} = \frac{1}{n-1} \sum_{i=j}^{j+W-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n) +``` + + @@ -41,10 +45,14 @@ For known population means, the [unbiased sample covariance][covariance] for a w -
+```math +\operatorname{cov_n} = \frac{1}{n} \sum_{i=j}^{j+W-1} (x_i - \mu_x)(y_i - \mu_y) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mcv/README.md b/lib/node_modules/@stdlib/stats/incr/mcv/README.md index 1c1ca59615a4..154e5de82597 100644 --- a/lib/node_modules/@stdlib/stats/incr/mcv/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mcv/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [corrected sample standard deviation][standard-dev -
+```math +s = \sqrt{\frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2} +``` + + @@ -39,10 +43,14 @@ and the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mda/README.md b/lib/node_modules/@stdlib/stats/incr/mda/README.md index c4149894793d..cfe9ba216975 100644 --- a/lib/node_modules/@stdlib/stats/incr/mda/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mda/README.md @@ -28,10 +28,14 @@ The [mean directional accuracy][mean-directional-accuracy] is defined as -
+```math +\operatorname{MDA} = \begin{cases} 1 & \textrm{if}\ N = 1 \\\frac{1}{N} \sum_{i=1}^{N} \delta_{\operatorname{sgn}(\Delta f_{i,i-1}),\ \operatorname{sgn}(\Delta a_{i,i-1})} & \textrm{if}\ N > 1 \end{cases} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/me/README.md b/lib/node_modules/@stdlib/stats/incr/me/README.md index 0e9cf38a512e..014338439855 100644 --- a/lib/node_modules/@stdlib/stats/incr/me/README.md +++ b/lib/node_modules/@stdlib/stats/incr/me/README.md @@ -28,10 +28,14 @@ The [mean error][mean-absolute-error] is defined as -
+```math +\operatorname{ME} = \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mean/README.md b/lib/node_modules/@stdlib/stats/incr/mean/README.md index 5669704e6bc2..1022d5cdfcff 100644 --- a/lib/node_modules/@stdlib/stats/incr/mean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mean/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] is defined as -
+```math +\mu = \frac{1}{n} \sum_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/meanabs/README.md b/lib/node_modules/@stdlib/stats/incr/meanabs/README.md index 3d1c6f0da186..96c2ae5dab5f 100644 --- a/lib/node_modules/@stdlib/stats/incr/meanabs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/meanabs/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] of absolute values is defined as -
+```math +\mu = \frac{1}{n} \sum_{i=0}^{n-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/meanabs2/README.md b/lib/node_modules/@stdlib/stats/incr/meanabs2/README.md index 515d8f8d60f1..c5a93e3c13e4 100644 --- a/lib/node_modules/@stdlib/stats/incr/meanabs2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/meanabs2/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] of squared absolute values is defined as -
+```math +m = \frac{1}{n} \sum_{i=0}^{n-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/meanstdev/README.md b/lib/node_modules/@stdlib/stats/incr/meanstdev/README.md index 483cb6bd1267..4bf9c2ee1a67 100644 --- a/lib/node_modules/@stdlib/stats/incr/meanstdev/README.md +++ b/lib/node_modules/@stdlib/stats/incr/meanstdev/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i +``` + + @@ -39,10 +43,14 @@ and the [corrected sample standard deviation][sample-stdev] is defined as -
+```math +s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/meanvar/README.md b/lib/node_modules/@stdlib/stats/incr/meanvar/README.md index d70d6581c52a..ad09f19d50a3 100644 --- a/lib/node_modules/@stdlib/stats/incr/meanvar/README.md +++ b/lib/node_modules/@stdlib/stats/incr/meanvar/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i +``` + + @@ -39,10 +43,14 @@ and the [unbiased sample variance][sample-variance] is defined as -
+```math +s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mgmean/README.md b/lib/node_modules/@stdlib/stats/incr/mgmean/README.md index 30ad5893e2e2..531630d29e5a 100644 --- a/lib/node_modules/@stdlib/stats/incr/mgmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mgmean/README.md @@ -28,10 +28,14 @@ The [geometric mean][geometric-mean] is defined as the nth root of a product of -
+```math +\biggl( \prod_{i=0}^{n-1} \biggr)^{\frac{1}{n}} = \sqrt[n]{x_0 x_1 \cdots x_{n-1}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mgrubbs/README.md b/lib/node_modules/@stdlib/stats/incr/mgrubbs/README.md index 7fc0c90311de..e6039fe468db 100644 --- a/lib/node_modules/@stdlib/stats/incr/mgrubbs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mgrubbs/README.md @@ -33,10 +33,14 @@ For a window of size `W`, the [Grubbs' test][grubbs-test] statistic for a two-si -
+```math +G = \frac{\max_{i=0,\ldots,W-1} |Y_i - \bar{Y}|}{s} +``` + + @@ -46,10 +50,14 @@ The [Grubbs' test][grubbs-test] statistic for the alternative hypothesis that th -
+```math +G = \frac{\bar{Y} - Y_{\textrm{min}}}{s} +``` + + @@ -57,10 +65,14 @@ The [Grubbs' test][grubbs-test] statistic for the alternative hypothesis that th -
+```math +G = \frac{Y_{\textrm{max}} - \bar{Y}}{s} +``` + + @@ -68,10 +80,14 @@ For a two-sided test, the hypothesis that a dataset does **not** contain an outl -
+```math +G > \frac{W-1}{\sqrt{W}} \sqrt{\frac{t^2_{\alpha/(2W),W-2}}{W - 2 + t^2_{\alpha/(2W),W-2}}} +``` + + @@ -81,10 +97,14 @@ For a one-sided test, the hypothesis that a dataset does **not** contain an outl -
+```math +G > \frac{W-1}{\sqrt{W}} \sqrt{\frac{t^2_{\alpha/W,W-2}}{W - 2 + t^2_{\alpha/W,W-2}}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mhmean/README.md b/lib/node_modules/@stdlib/stats/incr/mhmean/README.md index b63ff18e1ffd..d86243fbc6d9 100644 --- a/lib/node_modules/@stdlib/stats/incr/mhmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mhmean/README.md @@ -28,10 +28,14 @@ The [harmonic mean][harmonic-mean] of positive real numbers `x_0, x_1, ..., x_{n -
+```math +\begin{align}H &= \frac{n}{\frac{1}{x_0} + \frac{1}{x_1} + \cdots + \frac{1}{x_{n-1}}} \\ &= \frac{n}{\sum_{i=0}^{n-1} \frac{1}{x_i}} \\ &= \biggl( \frac{\sum_{i=0}^{n-1} \frac{1}{x_i}}{n} \biggr)^{-1}\end{align} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmaape/README.md b/lib/node_modules/@stdlib/stats/incr/mmaape/README.md index 7911300db5d6..09494fb585ab 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmaape/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmaape/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean arctangent absolute percentage error][@kim:2 -
+```math +\operatorname{MAAPE} = \frac{1}{W} \sum_{i=0}^{W-1} \operatorname{arctan}\biggl( \biggl| \frac{a_i - f_i}{a_i} \biggr| \biggr) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmae/README.md b/lib/node_modules/@stdlib/stats/incr/mmae/README.md index f6af92f0c604..f38d4c724916 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmae/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmae/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean absolute error][mean-absolute-error] is defi -
+```math +\operatorname{MAE} = \frac{1}{W} \sum_{i=0}^{W-1} |y_i - x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmape/README.md b/lib/node_modules/@stdlib/stats/incr/mmape/README.md index 2afa76f7582c..1d11461ea27f 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmape/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmape/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean absolute percentage error][mean-absolute-per -
+```math +\operatorname{MAPE} = \frac{100}{W} \sum_{i=0}^{W-1} \biggl| \frac{a_i - f_i}{a_i} \biggr| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmda/README.md b/lib/node_modules/@stdlib/stats/incr/mmda/README.md index 6f04c6521a8d..8159119a01ff 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmda/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmda/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean directional accuracy][mean-directional-accur -
+```math +\operatorname{MDA} = \begin{cases} 1 & \textrm{if}\ W = 1 \\ \frac{1}{W} \sum_{i=1}^{W} \delta_{\operatorname{sgn}(\Delta f_{i,i-1}),\ \operatorname{sgn}(\Delta a_{i,i-1})} & \textrm{if}\ W > 1 \end{cases} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mme/README.md b/lib/node_modules/@stdlib/stats/incr/mme/README.md index 3345b06e35eb..bb7549b4e00d 100644 --- a/lib/node_modules/@stdlib/stats/incr/mme/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mme/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean error][mean-absolute-error] is defined as -
+```math +\operatorname{ME} = \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i) +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmean/README.md b/lib/node_modules/@stdlib/stats/incr/mmean/README.md index f185f856c526..96afb5cff895 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmean/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmeanabs/README.md b/lib/node_modules/@stdlib/stats/incr/mmeanabs/README.md index 21eaf0980d14..980173891d9a 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmeanabs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmeanabs/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] of absolute val -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmeanabs2/README.md b/lib/node_modules/@stdlib/stats/incr/mmeanabs2/README.md index d755192a27c7..ccb238bcc9c4 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmeanabs2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmeanabs2/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] of squared abso -
+```math +m = \frac{1}{W} \sum_{i=0}^{W-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmeanstdev/README.md b/lib/node_modules/@stdlib/stats/incr/mmeanstdev/README.md index 35af20b0e892..7c9222bc2f19 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmeanstdev/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmeanstdev/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i +``` + + @@ -39,10 +43,14 @@ and the [corrected sample standard deviation][standard-deviation] is defined as -
+```math +s = \sqrt{\frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmeanvar/README.md b/lib/node_modules/@stdlib/stats/incr/mmeanvar/README.md index ab0fbbcfcce1..6aa1876606d0 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmeanvar/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmeanvar/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i +``` + + @@ -39,10 +43,14 @@ and the [unbiased sample variance][sample-variance] is defined as -
+```math +s^2 = \frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmpe/README.md b/lib/node_modules/@stdlib/stats/incr/mmpe/README.md index d9bcb6f76983..23d25210808f 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmpe/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmpe/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean percentage error][mean-percentage-error] is -
+```math +\operatorname{MPE} = \frac{100}{W} \sum_{i=0}^{W-1} \frac{a_i - f_i}{a_i} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mmse/README.md b/lib/node_modules/@stdlib/stats/incr/mmse/README.md index 1990dbf40fb9..60c376770415 100644 --- a/lib/node_modules/@stdlib/stats/incr/mmse/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mmse/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [mean squared error][mean-squared-error] is define -
+```math +\operatorname{MSE} = \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i)^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mpcorr/README.md b/lib/node_modules/@stdlib/stats/incr/mpcorr/README.md index eb3118924206..6c16c9a28ab1 100644 --- a/lib/node_modules/@stdlib/stats/incr/mpcorr/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mpcorr/README.md @@ -28,10 +28,14 @@ The [Pearson product-moment correlation coefficient][pearson-correlation] betwee -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `W`, the [sample Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mpcorr2/README.md b/lib/node_modules/@stdlib/stats/incr/mpcorr2/README.md index b732d7597f5c..8af4b8ded3d6 100644 --- a/lib/node_modules/@stdlib/stats/incr/mpcorr2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mpcorr2/README.md @@ -28,10 +28,14 @@ The [Pearson product-moment correlation coefficient][pearson-correlation] betwee -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `W`, the sample [Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mpcorrdist/README.md b/lib/node_modules/@stdlib/stats/incr/mpcorrdist/README.md index 7908ad13f278..6c24906bc1c2 100644 --- a/lib/node_modules/@stdlib/stats/incr/mpcorrdist/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mpcorrdist/README.md @@ -28,10 +28,14 @@ The [sample Pearson product-moment correlation distance][pearson-correlation] is -
+```math +d_{x,y} = 1 - r_{x,y} = 1 - \frac{\operatorname{cov_n(x,y)}}{\sigma_x \sigma_y} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mpe/README.md b/lib/node_modules/@stdlib/stats/incr/mpe/README.md index 1d54fff023ff..6e4021f4bce1 100644 --- a/lib/node_modules/@stdlib/stats/incr/mpe/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mpe/README.md @@ -28,10 +28,14 @@ The [mean percentage error][mean-percentage-error] is defined as -
+```math +\operatorname{MPE} = \frac{100}{n} \sum_{i=0}^{n-1} \frac{a_i - f_i}{a_i} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mprod/README.md b/lib/node_modules/@stdlib/stats/incr/mprod/README.md index 7dcb8c86e021..8f341f1597f4 100644 --- a/lib/node_modules/@stdlib/stats/incr/mprod/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mprod/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving product is defined as -
+```math +\prod_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mrmse/README.md b/lib/node_modules/@stdlib/stats/incr/mrmse/README.md index b51cc29bfb95..f84d0e9356d3 100644 --- a/lib/node_modules/@stdlib/stats/incr/mrmse/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mrmse/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [**root mean squared error**][root-mean-squared-er -
+```math +\operatorname{RMSE} = \sqrt{ \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i)^2 } +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mrss/README.md b/lib/node_modules/@stdlib/stats/incr/mrss/README.md index 1ea3ade8efb6..53a3aa498613 100644 --- a/lib/node_modules/@stdlib/stats/incr/mrss/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mrss/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [residual sum of squares][residual-sum-of-squares] -
+```math +\operatorname{RSS} = \sum_{i=0}^{W-1} (y_i - x_i)^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mse/README.md b/lib/node_modules/@stdlib/stats/incr/mse/README.md index c40f953c4774..0f111f2411eb 100644 --- a/lib/node_modules/@stdlib/stats/incr/mse/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mse/README.md @@ -28,10 +28,14 @@ The [mean squared error][mean-squared-error] is defined as -
+```math +\operatorname{MSE} = \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i)^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mstdev/README.md b/lib/node_modules/@stdlib/stats/incr/mstdev/README.md index 5d196d50cd32..3644e0403caa 100644 --- a/lib/node_modules/@stdlib/stats/incr/mstdev/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mstdev/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [corrected sample standard deviation][standard-dev -
+```math +s = \sqrt{\frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/msum/README.md b/lib/node_modules/@stdlib/stats/incr/msum/README.md index 417ed272e399..49b8f6b2c0a4 100644 --- a/lib/node_modules/@stdlib/stats/incr/msum/README.md +++ b/lib/node_modules/@stdlib/stats/incr/msum/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum is defined as -
+```math +s = \sum_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/msumabs/README.md b/lib/node_modules/@stdlib/stats/incr/msumabs/README.md index 011a92b18a86..b91ad8268b07 100644 --- a/lib/node_modules/@stdlib/stats/incr/msumabs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/msumabs/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum of absolute values is defined as -
+```math +s = \sum_{i=0}^{W-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/msumabs2/README.md b/lib/node_modules/@stdlib/stats/incr/msumabs2/README.md index c771c1829f2d..7a35954aaa40 100644 --- a/lib/node_modules/@stdlib/stats/incr/msumabs2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/msumabs2/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum of squared absolute values is defined a -
+```math +s = \sum_{i=0}^{W-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/msumprod/README.md b/lib/node_modules/@stdlib/stats/incr/msumprod/README.md index 0a0d60ab915f..c46b928903b0 100644 --- a/lib/node_modules/@stdlib/stats/incr/msumprod/README.md +++ b/lib/node_modules/@stdlib/stats/incr/msumprod/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum of products is defined as -
+```math +s = \sum_{i=0}^{W-1} x_i y_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mvariance/README.md b/lib/node_modules/@stdlib/stats/incr/mvariance/README.md index fb0278b899ef..294749e6f75d 100644 --- a/lib/node_modules/@stdlib/stats/incr/mvariance/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mvariance/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [unbiased sample variance][sample-variance] is def -
+```math +s^2 = \frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/mvmr/README.md b/lib/node_modules/@stdlib/stats/incr/mvmr/README.md index 9964c3a1bce8..e136e75bb12b 100644 --- a/lib/node_modules/@stdlib/stats/incr/mvmr/README.md +++ b/lib/node_modules/@stdlib/stats/incr/mvmr/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [unbiased sample variance][sample-variance] is def -
+```math +s^2 = \frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2 +``` + + @@ -39,10 +43,14 @@ and the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i +``` + + @@ -50,10 +58,14 @@ The [variance-to-mean ratio][variance-to-mean-ratio] (VMR) is thus defined as -
+```math +F = \frac{s^2}{\bar{x}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/nansum/README.md b/lib/node_modules/@stdlib/stats/incr/nansum/README.md index 453f7058f1ae..bce0e49c37d3 100644 --- a/lib/node_modules/@stdlib/stats/incr/nansum/README.md +++ b/lib/node_modules/@stdlib/stats/incr/nansum/README.md @@ -28,10 +28,14 @@ The sum is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/nansumabs/README.md b/lib/node_modules/@stdlib/stats/incr/nansumabs/README.md index 74ae959439ef..de1752951b66 100644 --- a/lib/node_modules/@stdlib/stats/incr/nansumabs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/nansumabs/README.md @@ -28,10 +28,14 @@ The sum of absolute values is defined as -
+```math +s = \sum_{i=0}^{n-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/nansumabs2/README.md b/lib/node_modules/@stdlib/stats/incr/nansumabs2/README.md index 06d6c9884a17..587697def212 100644 --- a/lib/node_modules/@stdlib/stats/incr/nansumabs2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/nansumabs2/README.md @@ -28,10 +28,14 @@ The sum of squared absolute values is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/pcorr/README.md b/lib/node_modules/@stdlib/stats/incr/pcorr/README.md index d42e54d8c61e..1c8150b8be3f 100644 --- a/lib/node_modules/@stdlib/stats/incr/pcorr/README.md +++ b/lib/node_modules/@stdlib/stats/incr/pcorr/README.md @@ -28,10 +28,14 @@ The [Pearson product-moment correlation coefficient][pearson-correlation] betwee -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `n`, the [sample Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/pcorr2/README.md b/lib/node_modules/@stdlib/stats/incr/pcorr2/README.md index d0c8df57be45..32c6b3e6d527 100644 --- a/lib/node_modules/@stdlib/stats/incr/pcorr2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/pcorr2/README.md @@ -28,10 +28,14 @@ The [Pearson product-moment correlation coefficient][pearson-correlation] betwee -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `n`, the sample [Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/pcorrdist/README.md b/lib/node_modules/@stdlib/stats/incr/pcorrdist/README.md index 3314fe248e11..caaea710881a 100644 --- a/lib/node_modules/@stdlib/stats/incr/pcorrdist/README.md +++ b/lib/node_modules/@stdlib/stats/incr/pcorrdist/README.md @@ -28,10 +28,14 @@ The [sample Pearson product-moment correlation distance][pearson-correlation] is -
+```math +d_{x,y} = 1 - r_{x,y} = 1 - \frac{\operatorname{cov_n(x,y)}}{\sigma_x \sigma_y} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/pcorrdistmat/README.md b/lib/node_modules/@stdlib/stats/incr/pcorrdistmat/README.md index def3f9dc79c0..dd776c569007 100644 --- a/lib/node_modules/@stdlib/stats/incr/pcorrdistmat/README.md +++ b/lib/node_modules/@stdlib/stats/incr/pcorrdistmat/README.md @@ -28,10 +28,14 @@ A [sample Pearson product-moment correlation distance matrix][pearson-correlatio -
+```math +d_{x,y} = 1 - r_{x,y} = 1 - \frac{\operatorname{cov_n(x,y)}}{\sigma_x \sigma_y} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/pcorrmat/README.md b/lib/node_modules/@stdlib/stats/incr/pcorrmat/README.md index 2f3c96e86eb0..f50f140ead46 100644 --- a/lib/node_modules/@stdlib/stats/incr/pcorrmat/README.md +++ b/lib/node_modules/@stdlib/stats/incr/pcorrmat/README.md @@ -28,10 +28,14 @@ A [Pearson product-moment correlation matrix][pearson-correlation] is an M-by-M -
+```math +\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +``` + + @@ -41,10 +45,14 @@ For a sample of size `n`, the [sample Pearson product-moment correlation coeffic -
+```math +r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/prod/README.md b/lib/node_modules/@stdlib/stats/incr/prod/README.md index 08ef98d984d9..9659d8440c51 100644 --- a/lib/node_modules/@stdlib/stats/incr/prod/README.md +++ b/lib/node_modules/@stdlib/stats/incr/prod/README.md @@ -28,10 +28,14 @@ The product is defined as -
+```math +p = \prod_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/rmse/README.md b/lib/node_modules/@stdlib/stats/incr/rmse/README.md index 9817ad6c7334..9848470e3b56 100644 --- a/lib/node_modules/@stdlib/stats/incr/rmse/README.md +++ b/lib/node_modules/@stdlib/stats/incr/rmse/README.md @@ -28,10 +28,14 @@ The [**root mean squared error**][root-mean-squared-error] (also known as the ** -
+```math +\operatorname{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i)^2 } +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/rss/README.md b/lib/node_modules/@stdlib/stats/incr/rss/README.md index 140ae854a22f..48210ca22bc2 100644 --- a/lib/node_modules/@stdlib/stats/incr/rss/README.md +++ b/lib/node_modules/@stdlib/stats/incr/rss/README.md @@ -28,10 +28,14 @@ The [**residual sum of squares**][residual-sum-of-squares] (also referred to as -
+```math +\operatorname{RSS} = \sum_{i=0}^{n-1} (y_i - x_i)^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/skewness/README.md b/lib/node_modules/@stdlib/stats/incr/skewness/README.md index 6f9ac3fb9622..052c0454d72d 100644 --- a/lib/node_modules/@stdlib/stats/incr/skewness/README.md +++ b/lib/node_modules/@stdlib/stats/incr/skewness/README.md @@ -28,10 +28,14 @@ The [skewness][sample-skewness] for a random variable `X` is defined as -
+```math +\operatorname{Skewness}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^3 \biggr] +``` + + @@ -39,10 +43,14 @@ For a sample of `n` values, the [sample skewness][sample-skewness] is -
+```math +b_1 = \frac{m_3}{s^3} = \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^3}{\biggl( \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2 \biggr)^{3/2}} +``` + + @@ -52,10 +60,14 @@ An alternative definition for the [sample skewness][sample-skewness] which inclu -
+```math +G_1 = \frac{n^2}{(n-1)(n-2)} \frac{m_3}{s^3} = \frac{\sqrt{n(n-1)}}{n-2} \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^3}{\biggl( \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^2 \biggr)^{3/2}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/stdev/README.md b/lib/node_modules/@stdlib/stats/incr/stdev/README.md index 66875c61cd99..f2e210c05a09 100644 --- a/lib/node_modules/@stdlib/stats/incr/stdev/README.md +++ b/lib/node_modules/@stdlib/stats/incr/stdev/README.md @@ -28,10 +28,14 @@ The [corrected sample standard deviation][sample-stdev] is defined as -
+```math +s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/sum/README.md b/lib/node_modules/@stdlib/stats/incr/sum/README.md index eabac2662c4e..c07252897f16 100644 --- a/lib/node_modules/@stdlib/stats/incr/sum/README.md +++ b/lib/node_modules/@stdlib/stats/incr/sum/README.md @@ -28,10 +28,14 @@ The sum is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/sumabs/README.md b/lib/node_modules/@stdlib/stats/incr/sumabs/README.md index 09df427c1c09..13c73708df1b 100644 --- a/lib/node_modules/@stdlib/stats/incr/sumabs/README.md +++ b/lib/node_modules/@stdlib/stats/incr/sumabs/README.md @@ -28,10 +28,14 @@ The sum of absolute values is defined as -
+```math +s = \sum_{i=0}^{n-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/sumabs2/README.md b/lib/node_modules/@stdlib/stats/incr/sumabs2/README.md index 683d62a89c76..492407ae1b2c 100644 --- a/lib/node_modules/@stdlib/stats/incr/sumabs2/README.md +++ b/lib/node_modules/@stdlib/stats/incr/sumabs2/README.md @@ -28,10 +28,14 @@ The sum of squared absolute values is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/sumprod/README.md b/lib/node_modules/@stdlib/stats/incr/sumprod/README.md index 74883c26f371..4d79fda884f5 100644 --- a/lib/node_modules/@stdlib/stats/incr/sumprod/README.md +++ b/lib/node_modules/@stdlib/stats/incr/sumprod/README.md @@ -28,10 +28,14 @@ The sum of products is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i y_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/variance/README.md b/lib/node_modules/@stdlib/stats/incr/variance/README.md index afc80eddd62e..9a54f6efc6d3 100644 --- a/lib/node_modules/@stdlib/stats/incr/variance/README.md +++ b/lib/node_modules/@stdlib/stats/incr/variance/README.md @@ -28,10 +28,14 @@ The [unbiased sample variance][sample-variance] is defined as -
+```math +s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/vmr/README.md b/lib/node_modules/@stdlib/stats/incr/vmr/README.md index 147edbb3d73f..5a9c41938761 100644 --- a/lib/node_modules/@stdlib/stats/incr/vmr/README.md +++ b/lib/node_modules/@stdlib/stats/incr/vmr/README.md @@ -28,10 +28,14 @@ The [unbiased sample variance][sample-variance] is defined as -
+```math +s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2 +``` + + @@ -39,10 +43,14 @@ and the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i +``` + + @@ -50,10 +58,14 @@ The [variance-to-mean ratio][variance-to-mean-ratio] (VMR) is thus defined as -
+```math +D = \frac{s^2}{\bar{x}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/incr/wmean/README.md b/lib/node_modules/@stdlib/stats/incr/wmean/README.md index 8e73ea0c4d95..b0c23a597420 100644 --- a/lib/node_modules/@stdlib/stats/incr/wmean/README.md +++ b/lib/node_modules/@stdlib/stats/incr/wmean/README.md @@ -28,10 +28,14 @@ The [weighted arithmetic mean][weighted-arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{\sum_{i=0}^{n-1} w_{i} x_{i}}{\sum_{i=0}^{n-1} w_{i}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cugmean/README.md b/lib/node_modules/@stdlib/stats/iter/cugmean/README.md index 5224fdef65b6..98b3e806ae69 100644 --- a/lib/node_modules/@stdlib/stats/iter/cugmean/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cugmean/README.md @@ -28,10 +28,14 @@ The [geometric mean][geometric-mean] is defined as the nth root of a product of -
+```math +\biggl( \prod_{i=0}^{n-1} \biggr)^{\frac{1}{n}} = \sqrt[n]{x_0 x_1 \cdots x_{n-1}} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cuhmean/README.md b/lib/node_modules/@stdlib/stats/iter/cuhmean/README.md index 86fb54eecf03..5f59eaf888df 100644 --- a/lib/node_modules/@stdlib/stats/iter/cuhmean/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cuhmean/README.md @@ -28,10 +28,14 @@ The [harmonic mean][harmonic-mean] of positive real numbers `x_0, x_1, ..., x_{n -
+```math +\begin{align}H &= \frac{n}{\frac{1}{x_0} + \frac{1}{x_1} + \cdots + \frac{1}{x_{n-1}}} \\ &= \frac{n}{\sum_{i=0}^{n-1} \frac{1}{x_i}} \\ &= \biggl( \frac{\sum_{i=0}^{n-1} \frac{1}{x_i}}{n} \biggr)^{-1}\end{align} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cumean/README.md b/lib/node_modules/@stdlib/stats/iter/cumean/README.md index d079b41a87b7..c1f5493fd3f3 100644 --- a/lib/node_modules/@stdlib/stats/iter/cumean/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cumean/README.md @@ -28,10 +28,14 @@ The cumulative [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x}_n = \frac{1}{n} \sum_{i=0}^n x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cumeanabs/README.md b/lib/node_modules/@stdlib/stats/iter/cumeanabs/README.md index 378a1510424f..fd75799cd19e 100644 --- a/lib/node_modules/@stdlib/stats/iter/cumeanabs/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cumeanabs/README.md @@ -28,10 +28,14 @@ The cumulative [arithmetic mean][arithmetic-mean] of absolute values is defined -
+```math +\bar{x}_n = \frac{1}{n} \sum_{i=0}^n |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cumeanabs2/README.md b/lib/node_modules/@stdlib/stats/iter/cumeanabs2/README.md index d1ebf756aa65..d332c9e082ca 100644 --- a/lib/node_modules/@stdlib/stats/iter/cumeanabs2/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cumeanabs2/README.md @@ -28,10 +28,14 @@ The cumulative [arithmetic mean][arithmetic-mean] of squared absolute values is -
+```math +\bar{x}_n = \frac{1}{n} \sum_{i=0}^n x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cuprod/README.md b/lib/node_modules/@stdlib/stats/iter/cuprod/README.md index 8adbfc9ff63d..3c96ac67d23d 100644 --- a/lib/node_modules/@stdlib/stats/iter/cuprod/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cuprod/README.md @@ -28,10 +28,14 @@ The cumulative product is defined as -
+```math +\begin{align*} p_0 &= x_0 \\ p_1 &= x_1 \cdot p_0 \\ p_2 &= x_2 \cdot p_1 \\ p_n &= x_n \cdot p_{n-1} = x_n \cdot \prod_{i=0}^{n-1} x_i \end{align*} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cusum/README.md b/lib/node_modules/@stdlib/stats/iter/cusum/README.md index a41e2964ea06..458c557c4863 100644 --- a/lib/node_modules/@stdlib/stats/iter/cusum/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cusum/README.md @@ -28,10 +28,14 @@ The cumulative sum is defined as -
+```math +\begin{align*} s_0 &= x_0 \\ s_1 &= x_1 + s_0 \\ s_2 &= x_2 + s_1 \\ s_n &= x_n + s_{n-1} = x_n + \sum_{i=0}^{n-1} x_i \end{align*} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cusumabs/README.md b/lib/node_modules/@stdlib/stats/iter/cusumabs/README.md index 4b6f0b3a7551..2053c847d33a 100644 --- a/lib/node_modules/@stdlib/stats/iter/cusumabs/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cusumabs/README.md @@ -28,10 +28,14 @@ The cumulative sum of absolute values is defined as -
+```math +\begin{align*} s_0 &= |x_0| \\ s_1 &= |x_1| + s_0 \\ s_2 &= |x_2| + s_1 \\ s_n &= |x_n| + s_{n-1} = |x_n| + \sum_{i=0}^{n-1} |x_i| \end{align*} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/cusumabs2/README.md b/lib/node_modules/@stdlib/stats/iter/cusumabs2/README.md index 0004591fe2f0..b6dce5e131de 100644 --- a/lib/node_modules/@stdlib/stats/iter/cusumabs2/README.md +++ b/lib/node_modules/@stdlib/stats/iter/cusumabs2/README.md @@ -28,10 +28,14 @@ The cumulative sum of squared absolute values is defined as -
+```math +\begin{align*} s_0 &= x_0^2 \\ s_1 &= x_1^2 + s_0 \\ s_2 &= x_2^2 + s_1 \\ s_n &= x_n^2 + s_{n-1} = x_n^2 + \sum_{i=0}^{n-1} x_i^2 \end{align*} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/mean/README.md b/lib/node_modules/@stdlib/stats/iter/mean/README.md index 28a6eb53714f..36a65d830f99 100644 --- a/lib/node_modules/@stdlib/stats/iter/mean/README.md +++ b/lib/node_modules/@stdlib/stats/iter/mean/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] is defined as -
+```math +\mu = \frac{1}{n} \sum_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/meanabs/README.md b/lib/node_modules/@stdlib/stats/iter/meanabs/README.md index 1766ea752eca..3fd2635cec34 100644 --- a/lib/node_modules/@stdlib/stats/iter/meanabs/README.md +++ b/lib/node_modules/@stdlib/stats/iter/meanabs/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] of absolute values is defined as -
+```math +\mu = \frac{1}{n} \sum_{i=0}^{n-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/meanabs2/README.md b/lib/node_modules/@stdlib/stats/iter/meanabs2/README.md index 258740535e9f..2514b589e4e5 100644 --- a/lib/node_modules/@stdlib/stats/iter/meanabs2/README.md +++ b/lib/node_modules/@stdlib/stats/iter/meanabs2/README.md @@ -28,10 +28,14 @@ The [arithmetic mean][arithmetic-mean] of squared absolute values is defined as -
+```math +m = \frac{1}{n} \sum_{i=0}^{n-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/mmean/README.md b/lib/node_modules/@stdlib/stats/iter/mmean/README.md index 2ed37b78c73f..5463e04c06f9 100644 --- a/lib/node_modules/@stdlib/stats/iter/mmean/README.md +++ b/lib/node_modules/@stdlib/stats/iter/mmean/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] is defined as -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/mmeanabs/README.md b/lib/node_modules/@stdlib/stats/iter/mmeanabs/README.md index a76b60d52e21..82fda2e9926b 100644 --- a/lib/node_modules/@stdlib/stats/iter/mmeanabs/README.md +++ b/lib/node_modules/@stdlib/stats/iter/mmeanabs/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] of absolute val -
+```math +\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/mmeanabs2/README.md b/lib/node_modules/@stdlib/stats/iter/mmeanabs2/README.md index 2975ff7af3a8..f5f1e546fb6b 100644 --- a/lib/node_modules/@stdlib/stats/iter/mmeanabs2/README.md +++ b/lib/node_modules/@stdlib/stats/iter/mmeanabs2/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the [arithmetic mean][arithmetic-mean] of squared abso -
+```math +m = \frac{1}{W} \sum_{i=0}^{W-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/mprod/README.md b/lib/node_modules/@stdlib/stats/iter/mprod/README.md index 680fca56f63e..c0cfbf5d391d 100644 --- a/lib/node_modules/@stdlib/stats/iter/mprod/README.md +++ b/lib/node_modules/@stdlib/stats/iter/mprod/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving product is defined as -
+```math +\prod_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/msum/README.md b/lib/node_modules/@stdlib/stats/iter/msum/README.md index 49648ec1eaac..40a4e89683ec 100644 --- a/lib/node_modules/@stdlib/stats/iter/msum/README.md +++ b/lib/node_modules/@stdlib/stats/iter/msum/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum is defined as -
+```math +s = \sum_{i=0}^{W-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/msumabs/README.md b/lib/node_modules/@stdlib/stats/iter/msumabs/README.md index 61a91bd9472a..52f8ce68f357 100644 --- a/lib/node_modules/@stdlib/stats/iter/msumabs/README.md +++ b/lib/node_modules/@stdlib/stats/iter/msumabs/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum of absolute values is defined as -
+```math +s = \sum_{i=0}^{W-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/msumabs2/README.md b/lib/node_modules/@stdlib/stats/iter/msumabs2/README.md index 5b412c8a9bb8..5f3d1fcafae2 100644 --- a/lib/node_modules/@stdlib/stats/iter/msumabs2/README.md +++ b/lib/node_modules/@stdlib/stats/iter/msumabs2/README.md @@ -28,10 +28,14 @@ For a window of size `W`, the moving sum of squared absolute values is defined a -
+```math +s = \sum_{i=0}^{W-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/prod/README.md b/lib/node_modules/@stdlib/stats/iter/prod/README.md index 60e9a49a9347..c09d9c3a6980 100644 --- a/lib/node_modules/@stdlib/stats/iter/prod/README.md +++ b/lib/node_modules/@stdlib/stats/iter/prod/README.md @@ -28,10 +28,14 @@ The product is defined as -
+```math +p = \prod_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/stdev/README.md b/lib/node_modules/@stdlib/stats/iter/stdev/README.md index b3ab40b823aa..a3922aa8ba19 100644 --- a/lib/node_modules/@stdlib/stats/iter/stdev/README.md +++ b/lib/node_modules/@stdlib/stats/iter/stdev/README.md @@ -28,10 +28,14 @@ The [corrected sample standard deviation][sample-stdev] is defined as -
+```math +s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2} +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/sum/README.md b/lib/node_modules/@stdlib/stats/iter/sum/README.md index b87c75ec8ac5..0d19abc8668f 100644 --- a/lib/node_modules/@stdlib/stats/iter/sum/README.md +++ b/lib/node_modules/@stdlib/stats/iter/sum/README.md @@ -28,10 +28,14 @@ The sum is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/sumabs/README.md b/lib/node_modules/@stdlib/stats/iter/sumabs/README.md index c60c1362c70a..bfc2a8d2365b 100644 --- a/lib/node_modules/@stdlib/stats/iter/sumabs/README.md +++ b/lib/node_modules/@stdlib/stats/iter/sumabs/README.md @@ -28,10 +28,14 @@ The sum of absolute values is defined as -
+```math +s = \sum_{i=0}^{n-1} |x_i| +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/sumabs2/README.md b/lib/node_modules/@stdlib/stats/iter/sumabs2/README.md index 591de1937d6e..5fbaac529e33 100644 --- a/lib/node_modules/@stdlib/stats/iter/sumabs2/README.md +++ b/lib/node_modules/@stdlib/stats/iter/sumabs2/README.md @@ -28,10 +28,14 @@ The sum of squared absolute values is defined as -
+```math +s = \sum_{i=0}^{n-1} x_i^2 +``` + + diff --git a/lib/node_modules/@stdlib/stats/iter/variance/README.md b/lib/node_modules/@stdlib/stats/iter/variance/README.md index a37209404e9f..bc5911c98fbc 100644 --- a/lib/node_modules/@stdlib/stats/iter/variance/README.md +++ b/lib/node_modules/@stdlib/stats/iter/variance/README.md @@ -28,10 +28,14 @@ The [unbiased sample variance][sample-variance] is defined as -
+```math +s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2 +``` + +