{"id":308565,"date":"2020-08-16T21:00:56","date_gmt":"2020-08-16T21:00:56","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=308565"},"modified":"-0001-11-30T00:00:00","modified_gmt":"2020-08-16T21:00:56","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=308565","title":{"rendered":"\u041f\u0440\u043e\u0441\u0442\u044b\u0435 \u0438 \u0431\u044b\u0441\u0442\u0440\u044b\u0435 \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u044f \u043a \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u043c \u0444\u0443\u043d\u043a\u0446\u0438\u044f\u043c"},"content":{"rendered":"\n<div class=\"post__text post__text-html post__text_v1\" id=\"post-content-body\" data-io-article-url=\"https:\/\/habr.com\/ru\/post\/515350\/\">\n<p>\u0417\u0430\u0434\u0430\u0447\u0430. \u0415\u0441\u0442\u044c <a href=\"https:\/\/en.wikipedia.org\/wiki\/HP_35s\" rel=\"nofollow\">\u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440<\/a>, \u043d\u043e \u043d\u0435\u0442 \u043f\u043e\u0434 \u0440\u0443\u043a\u043e\u0439 <a href=\"https:\/\/www.medcalc.org\/manual\/statistical_tables.php\" rel=\"nofollow\">\u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0442\u0430\u0431\u043b\u0438\u0446<\/a>. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u043d\u0443\u0436\u043d\u044b \u0442\u0430\u0431\u043b\u0438\u0446\u044b \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0442\u043e\u0447\u0435\u043a <a href=\"https:\/\/ru.wikipedia.org\/wiki\/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%A1%D1%82%D1%8C%D1%8E%D0%B4%D0%B5%D0%BD%D1%82%D0%B0\" rel=\"nofollow\">\u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0421\u0442\u044c\u044e\u0434\u0435\u043d\u0442\u0430<\/a> \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f <a href=\"https:\/\/ru.wikipedia.org\/wiki\/%D0%94%D0%BE%D0%B2%D0%B5%D1%80%D0%B8%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B8%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB\" rel=\"nofollow\">\u0434\u043e\u0432\u0435\u0440\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0433\u043e \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u0430.<\/a> \u0412\u0437\u044f\u0442\u044c \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440 \u0441 Excel? \u041d\u0435 \u0441\u043f\u043e\u0440\u0442\u0438\u0432\u043d\u043e.<\/p>\n<p>  <\/p>\n<p>\u0411\u043e\u043b\u044c\u0448\u0430\u044f \u0442\u043e\u0447\u043d\u043e\u0441\u0442\u044c \u043d\u0435 \u043d\u0443\u0436\u043d\u0430, \u043c\u043e\u0436\u043d\u043e \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u0442\u044c\u0441\u044f \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u043d\u044b\u043c\u0438 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u043c\u0438. \u0418\u0434\u0435\u044f \u043f\u0440\u0438\u0432\u0435\u0434\u0451\u043d\u043d\u044b\u0445 \u043d\u0438\u0436\u0435 \u0444\u043e\u0440\u043c\u0443\u043b \u0441\u043e\u0441\u0442\u043e\u0438\u0442 \u0432 \u0442\u043e\u043c, \u0447\u0442\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0430 \u0432\u0441\u0435 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u043c\u043e\u0436\u043d\u043e \u0442\u0430\u043a \u0438\u043b\u0438 \u0438\u043d\u0430\u0447\u0435 \u0441\u0432\u0435\u0441\u0442\u0438 \u043a \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u043c\u0443. \u0410\u043f\u043f\u0440\u043e\u043a\u0441\u0438\u043c\u0430\u0446\u0438\u0438 \u0434\u043e\u043b\u0436\u043d\u044b \u043e\u0431\u0435\u0441\u043f\u0435\u0447\u0438\u0432\u0430\u0442\u044c \u043a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u043a\u0443\u043c\u0443\u043b\u044f\u0442\u0438\u0432\u043d\u043e\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f, \u0442\u0430\u043a \u0438 \u0440\u0430\u0441\u0447\u0435\u0442 \u043e\u0431\u0440\u0430\u0442\u043d\u043e\u0439 \u043a \u043d\u0435\u0439 \u0444\u0443\u043d\u043a\u0446\u0438\u0438.<\/p>\n<p><a name=\"habracut\"><\/a>  <\/p>\n<p>\u041d\u0430\u0447\u043d\u0451\u043c \u0441 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f. <\/p>\n<p>  <\/p>\n<p><math>$$display$$\\Phi(z)=P=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{z}{\\sqrt{2}}\\right)\\right]$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$z=\\Phi^{-1}(P)=\\sqrt{2}\\cdot\\mathrm{erf}^{-1}(2P-1)$$display$$<\/math><\/p>\n<p>  <\/p>\n<p>\u0414\u043b\u044f \u043d\u0435\u0433\u043e \u0442\u0440\u0435\u0431\u0443\u0435\u0442\u0441\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u044e <math>$inline$\\mathrm{erf}(x)$inline$<\/math>\u0438 \u043e\u0431\u0440\u0430\u0442\u043d\u0443\u044e \u043a \u043d\u0435\u0439. \u042f \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0441\u044f \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u0438\u0435\u043c [1]:<\/p>\n<p>  <\/p>\n<p><math>$$display$$\\mathrm{erf}(x)=\\mathrm{sign}(x)\\cdot\\sqrt{1-\\exp\\left(-x^{2}\\cdot\\frac{\\frac{4}{\\pi}+ax^{2}}{1+ax^{2}}\\right)}$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$ \\mathrm{erf}^{-1}(x)=\\mathrm{sign}(x)\\cdot\\sqrt{-t_2 + \\sqrt{t_2^{2}-\\frac{1}{a}\\cdot \\ln t_1}} $$display$$<\/math><\/p>\n<p>  <\/p>\n<p>\u0433\u0434\u0435 <math>$inline$t_1$inline$<\/math> \u0438 <math>$inline$t_2$inline$<\/math> \u2014 \u0432\u0441\u043f\u043e\u043c\u043e\u0433\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0435 \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0435: <\/p>\n<p>  <\/p>\n<p><math>$$display$$t_1=1-x^{2},\\:t_2=\\frac{2}{\\pi a}+\\frac{\\ln t_1}{2}$$display$$<\/math><\/p>\n<p>  <\/p>\n<p>\u0430 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 <math>$inline$a=0.147$inline$<\/math>. \u041d\u0438\u0436\u0435 \u0434\u0430\u043d \u043a\u043e\u0434 \u043d\u0430 \u044f\u0437\u044b\u043a\u0435 Octave.<\/p>\n<p>  <\/p>\n<pre><code class=\"matlab\">function y = erfa(x)   a  = 0.147;   x2 = x**2; t = x2*(4\/pi + a*x2)\/(1 + a*x2);   y  = sign(x)*sqrt(1 - exp(-t)); endfunction  function y = erfinva(x)   a  = 0.147;    t1 = 1 - x**2; t2 = 2\/pi\/a + log(t1)\/2;   y  = sign(x)*sqrt(-t2 + sqrt(t2**2 - log(t1)\/a)); endfunction  function y = normcdfa(x)   y = 1\/2*(1 + erfa(x\/sqrt(2))); endfunction  function y = norminva(x)   y = sqrt(2)*erfinva(2*x - 1); endfunction<\/code><\/pre>\n<p>  <\/p>\n<p>\u0422\u0435\u043f\u0435\u0440\u044c, \u043a\u043e\u0433\u0434\u0430 \u0435\u0441\u0442\u044c \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u043d\u043e\u0440\u043c\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f, \u043f\u0440\u0438\u0432\u0435\u0434\u0451\u043c \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 \u0438 \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u043c t-\u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0421\u0442\u044c\u044e\u0434\u0435\u043d\u0442\u0430 [2]:<\/p>\n<p>  <\/p>\n<p><math>$$display$$F_t(x,n)=\\Phi\\left(\\sqrt{\\frac{1}{t_1}\\cdot\\ln(1+\\frac{x^{2}}{n})}\\right)$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$t=F_t^{-1}(P,n)=\\sqrt{n\\cdot\\exp\\left(\\Phi^{-1}(P)^{2}\\cdot t_1\\right)-n}$$display$$<\/math><\/p>\n<p>  <\/p>\n<p>\u0433\u0434\u0435 \u0432\u0441\u043f\u043e\u043c\u043e\u0433\u0430\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u043f\u0435\u0440\u0435\u043c\u0435\u043d\u043d\u0430\u044f <math>$inline$t_1$inline$<\/math> \u0435\u0441\u0442\u044c <\/p>\n<p>  <\/p>\n<p><math>$$display$$t_1=\\frac{n-1.5}{(n-1)^{2}}$$display$$<\/math><\/p>\n<p>  <\/p>\n<pre><code class=\"matlab\">function y = tcdfa(x,n)   t1 = (n - 1.5)\/(n - 1)**2;  y = normcdfa(sqrt(1\/t1*log(1 + x**2\/n))); endfunction  function y = tinva(x,n)   t1 = (n - 1.5)\/(n - 1)**2;   y  = sqrt(n*exp(t1*norminva(x)**2) - n); endfunction<\/code><\/pre>\n<p>  <\/p>\n<p>\u0418\u0434\u0435\u044f \u043f\u0440\u0438\u0431\u043b\u0438\u0436\u0435\u043d\u043d\u043e\u0433\u043e \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f <math>$inline$\\chi^{2}$inline$<\/math> \u043d\u0430\u0433\u043b\u044f\u0434\u043d\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0430\u043c\u0438 [3]:<\/p>\n<p>  <\/p>\n<p><math>$$display$$\\sigma^{2}=\\frac{2}{9n},\\:\\mu=1-\\sigma^{2}$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$F_{\\chi^{2}}(x,n)=\\Phi\\left(\\frac{\\left(\\frac{x}{n}\\right)^{1\/3}-\\mu}{\\sigma}\\right)$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$\\chi^2=F_{\\chi^2}^{-1}(P,n)=n\\cdot\\left(\\Phi^{-1}(P)\\cdot\\sigma + \\mu\\right)^3$$display$$<\/math><\/p>\n<p>  <\/p>\n<pre><code class=\"matlab\">function y = chi2cdfa(x,n)   s2 = 2\/9\/n; mu = 1 - s2;   y  = normcdfa(((x\/n)**(1\/3) - mu)\/sqrt(s2)); endfunction  function y = chi2inva(x,n)  s2 = 2\/9\/n; mu = 1 - s2;   y = n*(norminva(x)*sqrt(s2) + mu)**3; endfunction<\/code><\/pre>\n<p>  <\/p>\n<p>\u0420\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0424\u0438\u0448\u0435\u0440\u0430 (\u0434\u043b\u044f <math>$inline$n\/k\\geq3$inline$<\/math> \u0438 <math>$inline$n\\geq3$inline$<\/math>) \u043d\u0430\u0445\u043e\u0434\u0438\u0442\u0441\u044f \u0432 \u0434\u0432\u0430 \u0448\u0430\u0433\u0430. \u0421\u043d\u0430\u0447\u0430\u043b\u0430 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442 \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u0443\u0435\u0442\u0441\u044f \u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044e \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0424\u0438\u0448\u0435\u0440\u0430 \u0447\u0435\u0440\u0435\u0437 \u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 <math>$inline$\\chi^2$inline$<\/math> [4], \u0430 \u0435\u0433\u043e \u043c\u044b \u0443\u0436\u0435 \u0437\u043d\u0430\u0435\u043c, \u043a\u0430\u043a \u0432\u044b\u0447\u0438\u0441\u043b\u0438\u0442\u044c.<\/p>\n<p>  <\/p>\n<p><math>$$display$$\\sigma^2=\\frac{2}{9n},\\:\\mu=1-\\sigma^2$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$\\lambda=\\frac{2n+k\\cdot x\/3+(k-2)}{2n+4k\\cdot x\/3}$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$F_f(x;k,n)=\\Phi\\left(\\frac{\\left(\\lambda\\cdot x\\right)^{1\/3}-\\mu}{\\sigma}\\right)$$display$$<\/math><\/p>\n<p>  <\/p>\n<p>\u041d\u0430\u0439\u0434\u0451\u043c \u043e\u0431\u0440\u0430\u0442\u043d\u0443\u044e \u0444\u0443\u043d\u043a\u0446\u0438\u044e, \u0440\u0435\u0448\u0438\u0432 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u043e\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435.<\/p>\n<p>  <\/p>\n<p><math>$$display$$q=\\left(\\Phi^{-1}(P)\\cdot\\sigma+\\mu\\right)^3$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$b=2n+k-2-4\/3\\cdot kq$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$D=b^2+8\/3\\cdot knq$$display$$<\/math><\/p>\n<p>  <\/p>\n<p><math>$$display$$x=F_f^{-1}(P;k,n)=\\frac{-b+\\sqrt{D}}{2k\/3}$$display$$<\/math><\/p>\n<p>  <\/p>\n<pre><code class=\"matlab\">function y = fcdfa(x,k,n)   mu = 1-2\/9\/k; s = sqrt(2\/9\/k);   lambda = (2*n + k*x\/3 + k-2)\/(2*n + 4*k*x\/3);   normcdfa(((lambda*x)**(1\/3)-mu)\/s) endfunction  function y = finva(x,k,n)   mu = 1-2\/9\/k; s = sqrt(2\/9\/k);   q = (norminva(x)*s + mu)**3;   b = 2*n + k-2 -4\/3*k*q;   d = b**2 + 8\/3*k*n*q;   y = (sqrt(d) - b)\/(2*k\/3); endfunction<\/code><\/pre>\n<p>  <\/p>\n<p><strong>\u0421\u043f\u0438\u0441\u043e\u043a \u043b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u044b<\/strong> <\/p>\n<p>  <\/p>\n<ol>\n<li>Sergei Winitzki. A handy approximation for the error function and its inverse. February 6, 2008.<\/li>\n<li>Gleason J.R. A note on a proposed Student t approximation \/\/ Computational statistics &amp; data analysis. \u2013 2000. \u2013 Vol. 34. \u2013 \u2116. 1. \u2013 Pp. 63-66.<\/li>\n<li>Wilson E.B., Hilferty M.M. The distribution of chi-square \/\/ Proceedings of the National Academy of Sciences. \u2013 1931. \u2013 Vol. 17. \u2013 \u2116. 12. \u2013 Pp. 684-688.<\/li>\n<li>Li B. and Martin E.B. An approximation to the F-distribution using the chi-square distribution. Computational statistics &amp; data analysis. \u2013 2002. Vol. 40. \u2013 \u2116. 1. pp. 21-26.<\/li>\n<\/ol>\n<\/div>\n<p> \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438 <a href=\"https:\/\/habr.com\/ru\/post\/515350\/\"> https:\/\/habr.com\/ru\/post\/515350\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"\n<div class=\"post__text post__text-html post__text_v1\" id=\"post-content-body\" data-io-article-url=\"https:\/\/habr.com\/ru\/post\/515350\/\">\n<p>\u0417\u0430\u0434\u0430\u0447\u0430. \u0415\u0441\u0442\u044c <a href=\"https:\/\/en.wikipedia.org\/wiki\/HP_35s\" rel=\"nofollow\">\u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440<\/a>, \u043d\u043e \u043d\u0435\u0442 \u043f\u043e\u0434 \u0440\u0443\u043a\u043e\u0439 <a href=\"https:\/\/www.medcalc.org\/manual\/statistical_tables.php\" rel=\"nofollow\">\u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0442\u0430\u0431\u043b\u0438\u0446<\/a>. \u041d\u0430\u043f\u0440\u0438\u043c\u0435\u0440, \u043d\u0443\u0436\u043d\u044b \u0442\u0430\u0431\u043b\u0438\u0446\u044b \u043a\u0440\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0442\u043e\u0447\u0435\u043a <a href=\"https:\/\/ru.wikipedia.org\/wiki\/%D0%A0%D0%B0%D1%81%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%A1%D1%82%D1%8C%D1%8E%D0%B4%D0%B5%D0%BD%D1%82%D0%B0\" rel=\"nofollow\">\u0440\u0430\u0441\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0421\u0442\u044c\u044e\u0434\u0435\u043d\u0442\u0430<\/a> \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f <a href=\"https:\/\/ru.wikipedia.org\/wiki\/%D0%94%D0%BE%D0%B2%D0%B5%D1%80%D0%B8%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%B8%D0%BD%D1%82%D0%B5%D1%80%D0%B2%D0%B0%D0%BB\" rel=\"nofollow\">\u0434\u043e\u0432\u0435\u0440\u0438\u0442\u0435\u043b\u044c\u043d\u043e\u0433\u043e \u0438\u043d\u0442\u0435\u0440\u0432\u0430\u043b\u0430.<\/a> \u0412\u0437\u044f\u0442\u044c \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440 \u0441 Excel? 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