# foo
#=
Supported dependent arithmetic between uncertain numbers:
| | independent | dependency known | dependency unknown | perfect/opposite | partial information |
|:-------------------------:|:------------------:|:------------------:|:------------------:|:--------------------:|:--------------------:|
| intervals | not known to exist | solutions exist | yes | solutions exist | solutions exist |
| probability distributions | yes | yes | yes | yes | yes |
| probability boxes | yes | yes | yes | yes | yes |
||independent|dependency known|dependency unknown|perfect/opposite|partial information|
|---------------------------|:------------------:|--------------------|--------------------|----------------------|----------------------|
|intervals|not known to exist|solutions exist|yes|solutions exist|solutions exist|
|probability distributions|yes|yes|yes|yes|yes|
|probability boxes|yes|yes|yes|yes|yes|
=#
Most of the fundamental binary operations can be performed between uncertain numbers of all types:
julia> a = normal(10,1);
julia> b = uniform(1,2);
julia> a + b
Pbox: ~ ( range=[7.9098, 15.09], mean=11.5, var=[0.38392, 2.1086])
julia> a - b
Pbox: ~ ( range=[4.9098, 12.09], mean=8.5, var=[0.38392, 2.1086])
julia> a * b
Pbox: ~ ( range=[6.9098, 26.18], mean=[12.194, 18.069], var=[0.6576, 32.544])
julia> a / b
Pbox: ~ ( range=[3.4549, 13.09], mean=[5.6205, 8.3861], var=[0.13574, 7.6644])<img src="./plots/sumFrechet.png" width="49.5%"/>
<img src="./plots/subFrechet.png" width="49.5%"/>
<img src="./plots/prodFrechet.png"width="49.5%"/>
<img src="./plots/divFrechet.png" width="49.5%"/>
By default all arithmetic operations in ProbabilityBoundsAnalysis.jl are Frechet operations, operations which consider all possible dependencies (copulas). Therefore we naturally get p-boxes even if we began with precise distributions. For independence:
julia> convIndep(a, b, op = +)
Pbox: ~ ( range=[7.9098, 15.09], mean=11.5, var=1.0833)
julia> convIndep(a, b, op = -)
Pbox: ~ ( range=[4.9098, 12.09], mean=8.5, var=1.0833)
julia> convIndep(a, b, op = *)
Pbox: ~ ( range=[6.9098, 26.18], mean=15.0, var=[10.052, 11.538])
julia> convIndep(a, b, op = /)
Pbox: ~ ( range=[3.4549, 13.09], mean=[6.919, 6.944], var=[2.3032, 2.6815])The resulting p-boxes are precise.
<img src="./plots/sumIndep.png" width="49.5%"/>
<img src="./plots/subIndep.png" width="49.5%"/>
<img src="./plots/prodIndep.png"width="49.5%"/>
<img src="./plots/divIndep.png" width="49.5%"/>
Perfect and opposite convolutions can also be performed:
julia> convPerfect(a, b, op = +)
Pbox: ~ ( range=[7.9098, 15.09], mean=[11.482, 11.518], var=[1.5807, 1.7096])
julia> convPerfect(a, b, op = *)
Pbox: ~ ( range=[6.9098, 26.18], mean=[15.234, 15.33], var=[18.661, 19.605])
julia> convOpposite(a, b, op = +)
Pbox: ~ ( range=[8.9048, 14.095], mean=[11.482, 11.518], var=[0.47843, 0.55648])
julia> convOpposite(a, b, op = *)
Pbox: ~ ( range=[12.162, 16.638], mean=[14.67, 14.766], var=[2.0885, 2.3469])<img src="./plots/sumPerfect.png" width="49.5%"/>
<img src="./plots/prodPerfect.png" width="49.5%"/>
<img src="./plots/sumOpposite.png"width="49.5%"/>
<img src="./plots/prodOpposite.png" width="49.5%"/>
Binary operations with a specified correlation coefficient may also be performed:
julia> conv(a, b, op = *, corr = -0.5)
Pbox: ~ ( range=[6.9098, 26.18], mean=15.0, var=[5.5032, 7.9116])
julia> conv(a, b, op = *, corr = 0.5)
Pbox: ~ ( range=[6.9098, 26.18], mean=15.0, var=[12.985, 16.804])<img src="./plots/prodCor1.png" width="49.5%"/>
<img src="./plots/prodCor2.png" width="49.5%"/>
This assumes that a and b follow a Gaussian Copula. You may however perform the operation with any copula by using the function
julia> C1 = clayton(5)
julia> convCorr(a, b, op = *, C = C1)
Pbox: ~ ( range=[6.9098, 26.18], mean=[15.024, 15.473], var=[15.738, 19.714])
julia> C2 = Frank(-2)
julia> convCorr(a, b, op = *, C = C2)
Pbox: ~ ( range=[6.9098, 26.18], mean=15.0, var=[6.78, 9.5713])where C is a copula (see section on dependence modelling).
<img src="./plots/prodCop1.png" width="49.5%"/>
<img src="./plots/prodCop2.png" width="49.5%"/>
Note that:
conv(a, b, op = op, corr = 0) == convIndep(a, b, op = op)
conv(a, b, op = op, corr = 1) == convPerfect(a, b, op = op)
conv(a, b, op = op, corr = -1) == convOpposite(a, b, op = op)
conv(a, b, op = op, corr = interval(-1,1)) == convFrechet(a, b, op = op)