[英]Non-linear regression time interval estimation in R - diffusion models
你如何估计 R 中的非线性回归时间间隔?
我们正在寻求运行广义 Norton Bass 扩散模型,其中我们有三个未知参数:m、p 和 q(潜在市场规模、创新参数和模仿参数)。 我们想运行扩展的 Bass 扩散模型回归(见图 1 和图 2)。
函数由 sales = m1*F1(t)-m1*F1(t)*F2(t-t2) 给出。
F(t) = ((1-e^-(p+g)*t)/((q/p)*e^-((p+g)*t)+1))
我们目前运行了以下代码,但不确定如何在回归中定义 F2(t-t2)? 你会如何建议这样做? 我们需要估计参数 m、q 和 p
GNB.model.s1 <- nls(s1 ~
M * (1 - (exp(-(P+Q) * t1)))/(1 + (Q/P) * (exp(-(P+Q) * t1)))
- M * (1 - (exp(-(P+Q) * t1)))/(1 + (Q/P) * (exp(-(P+Q) * t1)))
* ( (1 - (exp(-(P+Q) * t1)))/(1 + (Q/P) * (exp(-(P+Q) * t1)))
- (1 - (exp(-(P+Q) * t2)))/(1 + (Q/P) * (exp(-(P+Q) * t2)))),
start = list(M=20000, P=0.03, Q=0.38), trace = T)
其中 F(t) 由下式给出:
我考虑了来自https://pubsonline.informs.org/doi/abs/10.1287/mnsc.1120.1529 的参数这里是使用 DE 的解决方案:
fgt1 = function(params,t){
P=params[1];Q=params[2]
pf=(1-exp(-(P+Q)*t))/((Q/P)*exp(-(P+Q)*t)+1)
pf[t<0]=0
pf
}
fgt2 = function(params,t){
P=params[1];Q=params[3]
pf=(1-exp(-(P+Q)*t))/((Q/P)*exp(-(P+Q)*t)+1)
pf[t<0]=0
pf
}
st = 1:25 # time
pt2=11 # 2006-1984 1995-1984
Params0=c(0.009,0.33,0.5,5*10^7,21*10^7)
S1=function(params,t,t2){
m1=params[4]
m1*fgt1(params,t)-m1*fgt1(params,t)*fgt2(params,t-t2)
}
S2=function(params,t,t2){
m1=params[4];m2=params[5]
(m2 + m1*fgt1(params,t))*fgt2(params,t-t2)
}
#Simulate some data, use set.seed(324)
Sales = S1(Params0,st,pt2) + rnorm(length(st),sd=35)
Sales2 = S2(Params0,st,pt2) + ifelse(st<pt2,0,rnorm(length(st),sd=35))
sd=data.frame(Sales,Sales2)
library(NMOF)
algo1 <- list(printBar = FALSE,
nP = 200L,
nG = 1000L,
F = 0.50,
CR = 0.99,
min = c(.01,.3,.4,10000,10000),
max = c(.5,.5,.45,6*10^7,22*10^7)) # this appears critical
OF1 <- function(Param, data) {
t <- data$x
sg <- data$y
t2 <- data$t2
s1e <- data$model1(Param,t,t2);
s2e <- data$model2(Param,t,t2);
aux <- c(sg[,1],sg[,2]) - c(s1e,s2e); auxs <- sum(aux^2)
if (is.na(auxs)) auxs <- 1e10 # for bad solutions!
auxs
}
repair <- function(b,data) { # may be improved
b[1:5]=abs(b[1:5])
if(b[1]==0)b[1]=.01 # for Q/P
b
}
algo1$repair=repair
data1 <- list(x = st, y = sd, t2=11,model1 = S1,model2 = S2, ww = 1)
system.time(sol1 <- DEopt(OF = OF1, algo = algo1, data = data1))
sol1$xbest
OF1(sol1$xbest,data1)
plot(Sales,ylim=range(c(Sales,Sales2)),type="b",col=2)
points(data1$x[data1$x>=11],Sales2[data1$x>=11],col=3,pch=2)
lines(data1$x,data1$model1(sol1$xbest, data1$x,11),col=6,lwd=2)
lines(data1$x[data1$x>=11],data1$model2(sol1$xbest, data1$x[data1$x>=11],11),col=7,lwd=2)
#> sol1$xbest
#[1] 9.000012e-03 3.299996e-01 4.999998e-01 5.000003e+07 2.100000e+08
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