最小化期望自定义分布在Mathematica

这与6月份的一个问题有关:

在Mathematica中计算自定义分布的期望

我有一个自定义混合分布，使用第二个自定义分布，沿着@Sasha在过去一年的许多回答中讨论的路线定义。

定义发行版的代码如下:

nDist /: CharacteristicFunction[nDist[a_, b_, m_, s_],
t_] := (a b E^(I m t - (s^2 t^2)/2))/((I a + t) (-I b + t));
nDist /: PDF[nDist[a_, b_, m_, s_], x_] := (1/(2*(a + b)))*a*
b*(E^(a*(m + (a*s^2)/2 - x))* Erfc[(m + a*s^2 - x)/(Sqrt[2]*s)] +
E^(b*(-m + (b*s^2)/2 + x))*
Erfc[(-m + b*s^2 + x)/(Sqrt[2]*s)]);
nDist /: CDF[nDist[a_, b_, m_, s_],
x_] := ((1/(2*(a + b)))*((a + b)*E^(a*x)*
Erfc[(m - x)/(Sqrt[2]*s)] -
b*E^(a*m + (a^2*s^2)/2)*Erfc[(m + a*s^2 - x)/(Sqrt[2]*s)] +
a*E^((-b)*m + (b^2*s^2)/2 + a*x + b*x)*
Erfc[(-m + b*s^2 + x)/(Sqrt[2]*s)]))/ E^(a*x);


nDist /: Quantile[nDist[a_, b_, m_, s_], p_] :=
Module[{x},
x /. FindRoot[CDF[nDist[a, b, m, s], x] == #, {x, m}] & /@ p] /;
VectorQ[p, 0 < # < 1 &]
nDist /: Quantile[nDist[a_, b_, m_, s_], p_] :=
Module[{x}, x /. FindRoot[CDF[nDist[a, b, m, s], x] == p, {x, m}]] /;
0 < p < 1
nDist /: Quantile[nDist[a_, b_, m_, s_], p_] := -Infinity /; p == 0
nDist /: Quantile[nDist[a_, b_, m_, s_], p_] := Infinity /; p == 1
nDist /: Mean[nDist[a_, b_, m_, s_]] := 1/a - 1/b + m;
nDist /: Variance[nDist[a_, b_, m_, s_]] := 1/a^2 + 1/b^2 + s^2;
nDist /: StandardDeviation[ nDist[a_, b_, m_, s_]] :=
Sqrt[ 1/a^2 + 1/b^2 + s^2];
nDist /: DistributionDomain[nDist[a_, b_, m_, s_]] :=
Interval[{0, Infinity}]
nDist /: DistributionParameterQ[nDist[a_, b_, m_, s_]] := !
TrueQ[Not[Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0]]
nDist /: DistributionParameterAssumptions[nDist[a_, b_, m_, s_]] :=
Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0
nDist /: Random`DistributionVector[nDist[a_, b_, m_, s_], n_, prec_] :=


RandomVariate[ExponentialDistribution[a], n,
WorkingPrecision -> prec] -
RandomVariate[ExponentialDistribution[b], n,
WorkingPrecision -> prec] +
RandomVariate[NormalDistribution[m, s], n,
WorkingPrecision -> prec];


(* Fitting: This uses Mean, central moments 2 and 3 and 4th cumulant \
but it often does not provide a solution *)


nDistParam[data_] := Module[{mn, vv, m3, k4, al, be, m, si},
mn = Mean[data];
vv = CentralMoment[data, 2];
m3 = CentralMoment[data, 3];
k4 = Cumulant[data, 4];
al =
ConditionalExpression[
Root[864 - 864 m3 #1^3 - 216 k4 #1^4 + 648 m3^2 #1^6 +
36 k4^2 #1^8 - 216 m3^3 #1^9 + (-2 k4^3 + 27 m3^4) #1^12 &,
2], k4 > Root[-27 m3^4 + 4 #1^3 &, 1]];
be = ConditionalExpression[


Root[2 Root[
864 - 864 m3 #1^3 - 216 k4 #1^4 + 648 m3^2 #1^6 +
36 k4^2 #1^8 -
216 m3^3 #1^9 + (-2 k4^3 + 27 m3^4) #1^12 &,
2]^3 + (-2 +
m3 Root[
864 - 864 m3 #1^3 - 216 k4 #1^4 + 648 m3^2 #1^6 +
36 k4^2 #1^8 -
216 m3^3 #1^9 + (-2 k4^3 + 27 m3^4) #1^12 &,
2]^3) #1^3 &, 1], k4 > Root[-27 m3^4 + 4 #1^3 &, 1]];
m = mn - 1/al + 1/be;
si =
Sqrt[Abs[-al^-2 - be^-2 + vv ]];(*Ensure positive*)
{al,
be, m, si}];


nDistLL =
Compile[{a, b, m, s, {x, _Real, 1}},
Total[Log[
1/(2 (a +
b)) a b (E^(a (m + (a s^2)/2 - x)) Erfc[(m + a s^2 -
x)/(Sqrt[2] s)] +
E^(b (-m + (b s^2)/2 + x)) Erfc[(-m + b s^2 +
x)/(Sqrt[2] s)])]](*, CompilationTarget->"C",
RuntimeAttributes->{Listable}, Parallelization->True*)];


nlloglike[data_, a_?NumericQ, b_?NumericQ, m_?NumericQ, s_?NumericQ] :=
nDistLL[a, b, m, s, data];


nFit[data_] := Module[{a, b, m, s, a0, b0, m0, s0, res},


(* So far have not found a good way to quickly estimate a and \
b.  Starting assumption is that they both = 2,then m0 ~=
Mean and s0 ~=
StandardDeviation it seems to work better if a and b are not the \
same at start. *)


{a0, b0, m0, s0} = nDistParam[data];(*may give Undefined values*)


If[! (VectorQ[{a0, b0, m0, s0}, NumericQ] &&
VectorQ[{a0, b0, s0}, # > 0 &]),
m0 = Mean[data];
s0 = StandardDeviation[data];
a0 = 1;
b0 = 2;];
res = {a, b, m, s} /.
FindMaximum[
nlloglike[data, Abs[a], Abs[b], m,
Abs[s]], {{a, a0}, {b, b0}, {m, m0}, {s, s0}},
Method -> "PrincipalAxis"][[2]];
{Abs[res[[1]]], Abs[res[[2]]], res[[3]], Abs[res[[4]]]}];


nFit[data_, {a0_, b0_, m0_, s0_}] := Module[{a, b, m, s, res},
res = {a, b, m, s} /.
FindMaximum[
nlloglike[data, Abs[a], Abs[b], m,
Abs[s]], {{a, a0}, {b, b0}, {m, m0}, {s, s0}},
Method -> "PrincipalAxis"][[2]];
{Abs[res[[1]]], Abs[res[[2]]], res[[3]], Abs[res[[4]]]}];


dDist /: PDF[dDist[a_, b_, m_, s_], x_] :=
PDF[nDist[a, b, m, s], Log[x]]/x;
dDist /: CDF[dDist[a_, b_, m_, s_], x_] :=
CDF[nDist[a, b, m, s], Log[x]];
dDist /: EstimatedDistribution[data_, dDist[a_, b_, m_, s_]] :=
dDist[Sequence @@ nFit[Log[data]]];
dDist /: EstimatedDistribution[data_,
dDist[a_, b_, m_,
s_], {{a_, a0_}, {b_, b0_}, {m_, m0_}, {s_, s0_}}] :=
dDist[Sequence @@ nFit[Log[data], {a0, b0, m0, s0}]];
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] :=
Module[{x}, x /. FindRoot[CDF[dDist[a, b, m, s], x] == p, {x, s}]] /;
0 < p < 1
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] :=
Module[{x},
x /. FindRoot[ CDF[dDist[a, b, m, s], x] == #, {x, s}] & /@ p] /;
VectorQ[p, 0 < # < 1 &]
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] := -Infinity /; p == 0
dDist /: Quantile[dDist[a_, b_, m_, s_], p_] := Infinity /; p == 1
dDist /: DistributionDomain[dDist[a_, b_, m_, s_]] :=
Interval[{0, Infinity}]
dDist /: DistributionParameterQ[dDist[a_, b_, m_, s_]] := !
TrueQ[Not[Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0]]
dDist /: DistributionParameterAssumptions[dDist[a_, b_, m_, s_]] :=
Element[{a, b, s, m}, Reals] && a > 0 && b > 0 && s > 0
dDist /: Random`DistributionVector[dDist[a_, b_, m_, s_], n_, prec_] :=
Exp[RandomVariate[ExponentialDistribution[a], n,
WorkingPrecision -> prec] -
RandomVariate[ExponentialDistribution[b], n,
WorkingPrecision -> prec] +
RandomVariate[NormalDistribution[m, s], n,
WorkingPrecision -> prec]];

这使我能够拟合分布参数并生成PDF格式的和提供的。举个例子:

Plot[PDF[dDist[3.77, 1.34, -2.65, 0.40], x], {x, 0, .3},
PlotRange -> All]
Plot[CDF[dDist[3.77, 1.34, -2.65, 0.40], x], {x, 0, .3},
PlotRange -> All]

现在我定义了一个function来计算平均剩余寿命(参见这个问题的解释)。

MeanResidualLife[start_, dist_] :=
NExpectation[X \[Conditioned] X > start, X \[Distributed] dist] -
start
MeanResidualLife[start_, limit_, dist_] :=
NExpectation[X \[Conditioned] start <= X <= limit,
X \[Distributed] dist] - start

第一种方法不像第二种方法那样设置限制，需要很长时间来计算，但它们都有效。

现在我需要为相同的分布(或它的一些变化)找到MeanResidualLife函数的最小值或使其最小化。

我尝试了很多不同的方法:

FindMinimum[MeanResidualLife[x, dDist[3.77, 1.34, -2.65, 0.40]], x]
FindMinimum[MeanResidualLife[x, 1, dDist[3.77, 1.34, -2.65, 0.40]], x]


NMinimize[{MeanResidualLife[x, dDist[3.77, 1.34, -2.65, 0.40]],
0 <= x <= 1}, x]
NMinimize[{MeanResidualLife[x, 1, dDist[3.77, 1.34, -2.65, 0.40]], 0 <= x <= 1}, x]

这些要么似乎永远持续下去，要么会遇到:

幂::infy:无限表达式1/ 0。遇到。>>

MeanResidualLife函数应用于一个更简单但形状相似的分布，表明它有一个最小值:

Plot[PDF[LogNormalDistribution[1.75, 0.65], x], {x, 0, 30},
PlotRange -> All]
Plot[MeanResidualLife[x, LogNormalDistribution[1.75, 0.65]], {x, 0,
30},
PlotRange -> {{0, 30}, {4.5, 8}}]

还两个:

FindMinimum[MeanResidualLife[x, LogNormalDistribution[1.75, 0.65]], x]
FindMinimum[MeanResidualLife[x, 30, LogNormalDistribution[1.75, 0.65]], x]

当与LogNormalDistribution一起使用时，给我答案(如果先有一堆消息)。

对于如何让它为上面描述的自定义发行版工作，您有什么想法吗?

我是否需要添加约束或选项?

我需要在自定义分布的定义中定义其他东西吗?

也许FindMinimum或NMinimize只需要运行更长的时间(我已经运行了近一个小时，但没有用)。如果是这样，我是否只需要一些方法来加速找到函数的最小值?有什么建议吗?

Mathematica是否有另一种方法来做到这一点?

美国东部时间2月9日下午5:50增加:

任何人都可以从Wolfram技术会议2011研讨会“创建自己的发行版”在这里中下载关于在Mathematica中创建发行版的Oleksandr Pavlyk的演示。下载包括笔记本'ExampleOfParametricDistribution.nb'，它似乎列出了创建一个可以像Mathematica附带的发行版一样使用的发行版所需的所有部分。

它可能会提供一些答案。