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--- interval_statistics [Mon May 28 16:25:08 2012]
Alexander
+++ interval_statistics [Thu Jan 14 17:19:58 2021] (current)
Alexander
@@ Line 1: / Line 1: @@
-====== Calculating Integrals and Statistics (Pro) on Intervals using Fit Plot ======
+====== Calculating Integrals and Statistics on Intervals using Fit Plot ======
-Setting of intervals in ''Fit interval'' tab of Fit Plot was initially intended for specifying the range of data which are used for fitting by sum of fit curves. However, this tab can also be used to calculate integrals and statistics on these intervals (Statistics is only available in Pro edition). Data-Baseline is used to calculate the results.
+Setting of intervals in ''Fit interval'' tab of Fit Plot was initially intended for specifying the range of data which are used for fitting by sum of fit curves. However, this tab can also be used to calculate integrals and statistics on these intervals. Data-Baseline is used to calculate the results.
+===== Peak Moments =====
 MagicPlot can integrate data on selected intervals and calculate peak moments (x mean, variance, skewness, kurtosis). Spectrum line is treated as probability distribution curve: //x// values are treated as 'independent variable' and //y// values are treated as 'probability'. Standard statistical formulas are used to calculate moments (see below).
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 All statistical data are summarized in the intervals table:
-{{:statistics-on-intervals.png|Statistics on intervals}}
+{{:statistics-on-intervals.png?nolink|Statistics on intervals}}
 ===== Managing Intervals =====
@@ Line 19: / Line 21: @@
 ===== Computational Formulas =====
+Central moments are calculated as follows (see table). All sums are calculated using [[wp>Compensated_summation|compensated summation]]. Central moments are calculated on second pass after Mean calculation.
+^  Property  ^  Formula  ^
+| //n// | The number of non-NaN (x,y) points  |
+| Y Sum (normalization) | <m>s = sum{j}{}{y_j}</m> |
+| X Mean (first moment) | <m>nu_1 = 1/s sum{j}{}{y_j x_j}</m>  |
+| 2, 3, 4<sup>th</sup> [[wp>Central moment|Central moments]] | <m>mu_k = 1/s sum{j}{}{y_j (x_j − nu_1)^k}, ~ k = 2...4</m> |
 MagicPlot uses the following formulas to calculate intervals statistics:
 ^  Property  ^  Formula  ^
 | Integral | Calculated using \\ [[integration|Trapezoidal rule]] |
 | X Mean (expected value) | <m>mu = nu_1</m> |
-| Variance | <m>sigma^2 = mu_2</m> |
+| Variance | <m>sigma^2 = {n / {n − 1}} mu_2</m> |
-| Standard deviation | <m>sigma = sqrt{mu_2}</m> |
+| Standard deviation | <m>sigma = sqrt{sigma^2}</m> |
-| Skewness | <m>gamma_1 = {mu_3} / {sigma^3}</m> |
+| Skewness | <m>gamma_1 = sqrt{n (n − 1)} / {n − 2} {mu_3} / {sigma^3}</m> |
-| Kurtosis | <m>gamma_2 = {mu_4} / {sigma^4} − 3</m> |
+| Kurtosis | <m>gamma_2 = {n − 1} / {(n − 2)(n − 3)} ({(n + 1) {mu_4} / {sigma^4} − 3 (n − 1)})</m> |
-| Y Sum | <m>s = sum{k}{}{y_k}</m> |
-Intermediate values are calculated as follows:
-^  Property  ^  Formula  ^
-| X Mean | <m>nu_1 = 1/s sum{k}{}{y_k x_k}</m>  |
-| [[wp>Central moment|Central moments]] | <m>mu_n = 1/s sum{k}{}{y_k (x_k - nu_1)^n}, ~ n = 2...4</m> |
 ===== See Also =====

MagicPlot Manual

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