在 Julia 中高效实现马尔可夫链

Question

我想尽可能高效地模拟随机游走者在网络中的运动。下面我展示了一个玩具模型，其中包含我迄今为止尝试过的三种方法。我应该注意，在我最初的问题中，网络的边缘是固定的，但是边缘的权重可能会更新（即邻居列表是相同的，但权重可能会改变）。

using QuantEcon
using LightGraphs
using Distributions
using StatsBase

n = 700 #number of nodes
#setting an arbitrary network and its transition matrix
G_erdos = erdos_renyi(n, 15/n)
A_erdos = adjacency_matrix(G_erdos) + eye(n, n);
A_transition = A_erdos ./ sum(A_erdos, 2);

##Method 1
#using QuantEcon library
function QE_markov_draw(i::Int, A::Array{Float64,2})
    d = DiscreteRV(A[i, :]);
    return rand(d, 1)   
end

##Method 2
#using a simple random draw
function matrix_draw(i::Int, A::Array{Float64,2}, choices::Array{Int64,1})
    return sample(choices, Weights(A[i, :]))
end

##Method 3
# The matrix may be sparse. Therefore I obtain first list of neighbors and weights
#for each node. Then run sample using the list of neighbors and weights.
function neighbor_weight_list(A::Array{Float64,2}, i::Int)
    n = size(A)[1]
    neighbor_list = Int[]
    weight_list = Float64[]
    for i = 1:n
        for j = 1:n
            if A[i, j] > 0
                push!(neighbor_list, j)
                push!(weight_list, A[i, j])
            end
        end
    end
    return neighbor_list, weight_list
end
#Using sample on the reduced list.
function neigh_weights_draw(i::Int, neighs::Array{Int,1}, weigh::Array{Float64,1})
    return sample(neighs, Weights(weigh))
end

neighbor_list, weight_list = neighbor_weight_list(A_transition, 1)
states = [i for i = 1:n];

println("Method 1")
@time for t = 1:100000
    QE_markov_draw(3, A_transition)
end

println("Method 2")
@time for t = 1:100000
    matrix_draw(3, A_transition, states)
end

println("Method 3")
@time for t = 1:100000
    neigh_weights_draw(3, neighbor_list, weight_list)
end

一般结果表明（第一次迭代后）方法 2 是最快的。方法 3 使用最少的内存，其次是方法 2，但这可能是因为它们“喂”neighbor_list 和 states。

Method 1
  0.327805 seconds (500.00 k allocations: 1.086 GiB, 14.70% gc time)
Method 2
  0.227060 seconds (329.47 k allocations: 554.344 MiB, 11.24% gc time)
Method 3
  1.224682 seconds (128.19 k allocations: 3.482 MiB)

我想知道哪种实现最有效，是否有办法改进它。

Answer 1

以下是我可以给出的一些建议：

在选项 2 中，改用视图并处理矩阵的转置（因此您处理的是列而不是行）：

# here A should be a transpose of your original A
function matrix_draw(i::Int, A::Array{Float64,2}, choices::Array{Int64,1})
    return sample(choices, Weights(view(A, i, :)))
end

这使我的测试速度提高了近 7 倍。

但总的来说，方法 3 应该是最快的，但似乎执行不正确。这是一个固定的方法

function neighbor_weight_list(A::Array{Float64,2})
    n = size(A)[1]
    neighbor_list = Vector{Int}[]
    weight_list = Vector{Float64}[]
    for i = 1:n
        push!(neighbor_list, Int[])
        push!(weight_list, Float64[])
        for j = 1:n
            if A[i, j] > 0
                push!(neighbor_list[end], j)
                push!(weight_list[end], A[i, j])
            end
        end
    end
    return neighbor_list, weight_list
end

function neigh_weights_draw(i::Int, neighs::Vector{Vector{Int}}, weigh::Vector{Vector{Float64}})
    return sample(neighs[i], Weights(weigh[i]))
end

neighbor_list, weight_list = neighbor_weight_list(A_transition)

当我运行此代码时，它比固定方法 2 快 4 倍。另请注意，您可以使用方法 3 而无需创建邻接矩阵，而是直接使用来自 LightGraphs 的 neighbors 函数。