http://weblab.com.cityu.edu.hk/blog/chengjun/files/2012/04/Bass-Diffusion-Model.pdf

### Introduction

I had made slides to understand the bass diffusion model which is proposed by Frank M. Bass in 1969. The model proposes that diffusion is motivated both by inovativeness and imitation: first, the innovative early adopters adopted the products, after which the followers will imitate them and adopt the products.

**slides**will show you how to deprive the equation of Bass diffusion model by both discrete-time model and continuous model (using Hazard rate).

“The probability of adopting by those who have not yet adopted is a linear function of those who had previously adopted.”

# basss diffusion model # chengjun, 20120424@canberra # refer to http://en.wikipedia.org/wiki/Bass_diffusion_model # and http://book.douban.com/subject/4175572/discussion/45634092/ # BASS Diffusion Model # three parameters: # the total number of people who eventually buy the product, m; # the coefficient of innovation, p; # and the coefficient of imitation, q # example T79 <- 1:10 Tdelt <- (1:100) / 10 Sales <- c(840,1470,2110,4000, 7590, 10950, 10530, 9470, 7790, 5890) Cusales <- cumsum(Sales) Bass.nls <- nls(Sales ~ M * ( ((P+Q)^2 / P) * exp(-(P+Q) * T79) ) /(1+(Q/P)*exp(-(P+Q)*T79))^2, start = list(M=60630, P=0.03, Q=0.38)) summary(Bass.nls) # get coefficient Bcoef <- coef(Bass.nls) m <- Bcoef[1] p <- Bcoef[2] q <- Bcoef[3] # setting the starting value for M to the recorded total sales. ngete <- exp(-(p+q) * Tdelt) # plot pdf Bpdf <- m * ( (p+q)^2 / p ) * ngete / (1 + (q/p) * ngete)^2 plot(Tdelt, Bpdf, xlab = "Year from 1979",ylab = "Sales per year", type='l') points(T79, Sales) # plot cdf Bcdf <- m * (1 - ngete)/(1 + (q/p)*ngete) plot(Tdelt, Bcdf, xlab = "Year from 1979",ylab = "Cumulative sales", type='l') points(T79, Cusales) # when q=0, only Innovator without immitators. Ipdf<- m * ( (p+0)^2 / p ) * exp(-(p+0) * Tdelt) / (1 + (0/p) * exp(-(p+0) * Tdelt))^2 # plot(Tdelt, Ipdf, xlab = "Year from 1979",ylab = "Isales per year", type='l') Impdf<-Bpdf-Ipdf plot(Tdelt, Bpdf, xlab = "Year from 1979",ylab = "Sales per year", type='l', col="red") lines(Tdelt,Impdf,col="green") lines(Tdelt,Ipdf,col="blue") # when q=0, only Innovator without immitators. Icdf<-m * (1 - exp(-(p+0) * Tdelt))/(1 + (0/p)*exp(-(p+0) * Tdelt)) # plot(Tdelt, Icdf, xlab = "Year from 1979",ylab = "ICumulative sales", type='l') Imcdf<-m * (1 - ngete)/(1 + (q/p)*ngete)-Icdf plot(Tdelt, Imcdf, xlab = "Year from 1979",ylab = "Cumulative sales", type='l', col="red") lines(Tdelt,Bcdf,col="green") lines(Tdelt,Icdf,col="blue")