Polar coordinates - ReplaceAll issue
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From: Andrej on 14 Feb 2010 05:58 Hi, this is my first post to this group. Yes, I'm a newbie in Mathematica. I'm trying to compute the probability of circular disc (in R^2). Suppose that vertical and horizontal deviations from the center of the disc follows bivariate normal distribution. I hope that the code below is self explained. The main issue is the transformation from Cartesian to polar coordinates as follows: # First I set up the appropriate bivariate normal distribution: Needs["MultivariateStatistics`"] X = {x1, x2}; mu = {0, 0}; Sigma = sigma^2 ({{1,rho}, {rho,1}}); dist = MultinormalDistribution[mu, Sigma]; cond = {sigma > 0, -1 < rho < 1, r > 0, 0 < theta < 2 \[Pi]}; f = Simplify[PDF[dist, X], cond] domain[f] = {{x1, -\[Infinity], \[Infinity]}, {x2, -\[Infinity], \ [Infinity]}} && cond; # Transformation to polar coordinates: Omega = {x1 \[RightArrow] r Cos[theta], x2 \[RightArrow] r Sin[theta]}; g = Simplify[(f /. Omega) Jacob[X /. Omega, {r, theta}], cond] And the message I get: ReplaceAll::reps: {x1\[RightArrow]r Cos[theta],x2\[RightArrow]r Sin[theta]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >> Thanks in advance for any suggestions or pointers. Best, Andrej
From: David Park on 14 Feb 2010 08:16 You don't want to use \[RightArrow]! The arrow that is used in rules is entered as "->" (but without the quotes) and Mathematica automatically converts it to an arrow (but not a \[RightArrow]). omega = {x1 -> r Cos[theta], x2 -> r Sin[theta]} % // FullForm {x1 -> r Cos[theta], x2 -> r Sin[theta]} List[Rule[x1, Times[r, Cos[theta]]], Rule[x2, Times[r, Sin[theta]]]] David Park djmpark(a)comcast.net http://home.comcast.net/~djmpark/ From: Andrej [mailto:andrej.kastrin(a)gmail.com] Hi, this is my first post to this group. Yes, I'm a newbie in Mathematica. I'm trying to compute the probability of circular disc (in R^2). Suppose that vertical and horizontal deviations from the center of the disc follows bivariate normal distribution. I hope that the code below is self explained. The main issue is the transformation from Cartesian to polar coordinates as follows: # First I set up the appropriate bivariate normal distribution: Needs["MultivariateStatistics`"] X = {x1, x2}; mu = {0, 0}; Sigma = sigma^2 ({{1,rho}, {rho,1}}); dist = MultinormalDistribution[mu, Sigma]; cond = {sigma > 0, -1 < rho < 1, r > 0, 0 < theta < 2 \[Pi]}; f = Simplify[PDF[dist, X], cond] domain[f] = {{x1, -\[Infinity], \[Infinity]}, {x2, -\[Infinity], \ [Infinity]}} && cond; # Transformation to polar coordinates: Omega = {x1 \[RightArrow] r Cos[theta], x2 \[RightArrow] r Sin[theta]}; g = Simplify[(f /. Omega) Jacob[X /. Omega, {r, theta}], cond] And the message I get: ReplaceAll::reps: {x1\[RightArrow]r Cos[theta],x2\[RightArrow]r Sin[theta]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >> Thanks in advance for any suggestions or pointers. Best, Andrej
From: Patrick Scheibe on 14 Feb 2010 08:17 Hi, if you want to replace something, then you have to use a Rule or a RuleDelayed! Paste to following into a notebook Rule[a, b] a -> b RuleDelayed[a, b] a :> b and see, that this has *nothing* to do with an \[RightArrow]. Changing this and your sample is syntactically correct: Needs["MultivariateStatistics`"] X = {x1, x2}; mu = {0, 0}; Sigma = sigma^2 ({{1, rho}, {rho, 1}}); dist = MultinormalDistribution[mu, Sigma]; cond = {sigma > 0, -1 < rho < 1, r > 0, 0 < theta < 2 \[Pi]}; f = Simplify[PDF[dist, X], cond] domain[f] = {{x1, -\[Infinity], \[Infinity]}, {x2, -\[Infinity], \[Infinity]}} && cond; Omega = {x1 :> r Cos[theta], x2 :> r Sin[theta]}; g = Simplify[(f /. Omega) Jacob[X /. Omega, {r, theta}], cond] Cheers Patrick On Sun, 2010-02-14 at 05:58 -0500, Andrej wrote: > Hi, > > this is my first post to this group. Yes, I'm a newbie in Mathematica. > I'm trying to compute the probability of circular disc (in R^2). > Suppose that vertical and horizontal deviations from the center of the > disc follows bivariate normal distribution. I hope that the code below > is self explained. The main issue is the transformation from Cartesian > to polar coordinates as follows: > > # First I set up the appropriate bivariate normal distribution: > Needs["MultivariateStatistics`"] > X = {x1, x2}; > mu = {0, 0}; > Sigma = sigma^2 ({{1,rho}, {rho,1}}); > dist = MultinormalDistribution[mu, Sigma]; > cond = {sigma > 0, -1 < rho < 1, r > 0, 0 < theta < 2 \[Pi]}; > f = Simplify[PDF[dist, X], cond] > domain[f] = {{x1, -\[Infinity], \[Infinity]}, {x2, -\[Infinity], \ > [Infinity]}} && cond; > > # Transformation to polar coordinates: > Omega = {x1 \[RightArrow] r Cos[theta], x2 \[RightArrow] r > Sin[theta]}; > g = Simplify[(f /. Omega) Jacob[X /. Omega, {r, theta}], cond] > > And the message I get: > > ReplaceAll::reps: {x1\[RightArrow]r Cos[theta],x2\[RightArrow]r > Sin[theta]} is neither a list of replacement rules nor a valid > dispatch table, and so cannot be used for replacing. >> > > Thanks in advance for any suggestions or pointers. > > Best, Andrej >
From: bsyehuda on 14 Feb 2010 08:16 \[Rule] and \[RightArrow] appear identically on screen. Replace the the \[RightArrow] with \[Rule] (shortcut is ESC -> ESC, but better always is -> only ) after this change it would work. yehuda On Sun, Feb 14, 2010 at 12:58 PM, Andrej <andrej.kastrin(a)gmail.com> wrote: > Hi, > > this is my first post to this group. Yes, I'm a newbie in Mathematica. > I'm trying to compute the probability of circular disc (in R^2). > Suppose that vertical and horizontal deviations from the center of the > disc follows bivariate normal distribution. I hope that the code below > is self explained. The main issue is the transformation from Cartesian > to polar coordinates as follows: > > # First I set up the appropriate bivariate normal distribution: > Needs["MultivariateStatistics`"] > X = {x1, x2}; > mu = {0, 0}; > Sigma = sigma^2 ({{1,rho}, {rho,1}}); > dist = MultinormalDistribution[mu, Sigma]; > cond = {sigma > 0, -1 < rho < 1, r > 0, 0 < theta < 2 \[Pi]}; > f = Simplify[PDF[dist, X], cond] > domain[f] = {{x1, -\[Infinity], \[Infinity]}, {x2, -\[Infinity], \ > [Infinity]}} && cond; > > # Transformation to polar coordinates: > Omega = {x1 \[RightArrow] r Cos[theta], x2 \[RightArrow] r > Sin[theta]}; > g = Simplify[(f /. Omega) Jacob[X /. Omega, {r, theta}], cond] > > And the message I get: > > ReplaceAll::reps: {x1\[RightArrow]r Cos[theta],x2\[RightArrow]r > Sin[theta]} is neither a list of replacement rules nor a valid > dispatch table, and so cannot be used for replacing. >> > > Thanks in advance for any suggestions or pointers. > > Best, Andrej > >
From: Bob Hanlon on 15 Feb 2010 05:45 The symbol for Rule that looks like a RightArrow is not entered as RightArrow. It is the infix form of Rule and is entered as -> (minus sign followed by greater than) and will display as a RightArrow in TraditionalForm. Look at the documentation for Rule. Bob Hanlon ---- Andrej <andrej.kastrin(a)gmail.com> wrote: ============= Hi, this is my first post to this group. Yes, I'm a newbie in Mathematica. I'm trying to compute the probability of circular disc (in R^2). Suppose that vertical and horizontal deviations from the center of the disc follows bivariate normal distribution. I hope that the code below is self explained. The main issue is the transformation from Cartesian to polar coordinates as follows: # First I set up the appropriate bivariate normal distribution: Needs["MultivariateStatistics`"] X = {x1, x2}; mu = {0, 0}; Sigma = sigma^2 ({{1,rho}, {rho,1}}); dist = MultinormalDistribution[mu, Sigma]; cond = {sigma > 0, -1 < rho < 1, r > 0, 0 < theta < 2 \[Pi]}; f = Simplify[PDF[dist, X], cond] domain[f] = {{x1, -\[Infinity], \[Infinity]}, {x2, -\[Infinity], \ [Infinity]}} && cond; # Transformation to polar coordinates: Omega = {x1 \[RightArrow] r Cos[theta], x2 \[RightArrow] r Sin[theta]}; g = Simplify[(f /. Omega) Jacob[X /. Omega, {r, theta}], cond] And the message I get: ReplaceAll::reps: {x1\[RightArrow]r Cos[theta],x2\[RightArrow]r Sin[theta]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >> Thanks in advance for any suggestions or pointers. Best, Andrej -- Bob Hanlon
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