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I'm dealing with sales forecast for products in same category (f.i. all products belonging to $wine$ category). Data are time series with a daily frequency, $Y_{product_i} \in {0,1,2,..}$ . After some inspection I found out that time series models like Arima or Arimax do not fit particularly well my data and so I would like to try a new way.

I was considering, having count data, a Poisson Regression Model, which for a single product would look like this:

$Y_t|F_{t-1} \sim Poisson(\lambda_{t})$

$\lambda_t = \beta_0 + \beta_1*Y_{t-1} + \alpha_1*\lambda_{t-1} + \eta*X_t$

where $Y_t $ = sales at day t, $X_t$ = Exogenous regressors at day t.

My first question is:

Is there any package in $R$ implementing this? Is $glm$ sufficient?

Furthermore some products have very sparse time series while others sell always at least a couple of decades of items: would it be proper to consider log linear models in the second case and Hurdle models in the first one?

I was then considering the fact that I probably will need a more complex model which models a covariance structure between the sales of different products. Unfortunately I don't have much theoretical and Rpackages background in this case. I'm thinking about mixed effects models but I'm just wondering.. Basically I would like a Poisson Regression model with an autoregressive term for each of my products, but I don't know how to insert a covariance structure over the products. Should I go Bayesian?

$Y_{i_t} \sim Poisson(\lambda_{i_t})$

$\lambda_{i_t} = \beta_{0_i} + \beta_{i_1}*Y_{i_{t-1}} +\alpha_{1_i}*\lambda_{i_{t-1}} + \eta_i*X_{i_t}$

$i=1,..,numberofproducts$

So my second question is about finding theoretical perspective about multivariate time series count data together with packages in R.

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  • $\begingroup$ Have you looked into Gaussian Cox processes? They may work nicely here, and can be fitted with the R package INLA (pdf) $\endgroup$ – LE Rogerson Nov 7 '16 at 13:53
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    $\begingroup$ arxiv.org/pdf/1405.3738.pdf This is the answer $\endgroup$ – Tommaso Guerrini Nov 9 '16 at 21:00
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I found a reference which answered my question: https://arxiv.org/pdf/1405.3738.pdf.

The model is quite complicated, here is the state space representation:

enter image description here

So, let's say I have L different products I'm studying across 1,..,T time periods.

$Y_{l,t} \sim z*\delta_0 + (1-z)NB(exp(\widetilde{\eta}_{l,t}),alpha_l)$ is the distribution for product l at time t

$\widetilde{\eta}_{l,t} = \eta_{l,t} + X_{l,t}\theta_l$ this is the Log of the mean of product l sales at time t, guaranteeing that it is positive.

$\eta_{l,t} = \mu_l + \phi_l(\eta_{l,t-1}-\mu) + \epsilon_{l,t}$

$\epsilon_{l,t} \sim N(0,\frac{1}{\tau_l})$

The other priors and hyperpriors are in the next images:

enter image description here enter image description here

P.S. Now I'm trying to write the JAGS code and any help would be much appreciated! ( https://stackoverflow.com/questions/40528715/runtime-error-in-jags )

Edit:

Here is the JAGS code:

model{


#hyperpriors 4
alpha_star ~ dunif(0.001,0.1)
tau_mu_star ~ dunif(1,10)
mu_star ~ dnorm(0,0.5)
beta_tau ~ dunif(2,25)
beta_0_tau ~ dunif(1,10)
beta_theta ~ dunif(2,25)
phiminus ~ dunif(1,50)
k_tau ~ dunif(5,10)
k_0_tau ~ dunif(1,5)
pointmass_0 ~ dnorm(0,10000)
k_theta ~ dunif(5,10)
phiplus ~ dunif(1,600)
theta_star ~ dmnorm(b0,B0)
#17
for (l in 1:L){
z[l] ~ dbeta(0.5,0.5)
phi[l] ~ dbeta(phiplus + phiminus, phiminus)
tau[l] ~ dgamma(k_tau,beta_tau)
tau_theta[l] ~ dgamma(k_tau,beta_tau)
mu[l] ~ dnorm(mu_star, tau_mu_star)
alpha[l] ~ dexp(alpha_star)
eps[1,l] ~ dnorm(0,tau[l])
eta[1,l] = mu_star + eps[1,l]
theta[l,1:8] ~ dmnorm(theta_star,thetavar*tau_theta[l])
#y[1,l] ~ inprod(1-z[l],dnegbin(exp(eta[1,l]),alpha[l]))
y[1,l] ~ dnegbin(exp(eta[1,l]),alpha[l])

#y[1,l] ~ dnegbin(exp(eta[1,l]),alpha[l])
ystar[1,l] ~ dnorm(z[l]*pointmass_0 + inprod((1-z[l]),y[1,l]),100000)
}

for (i in 2:N){

for (l in 1:L){
eps[i,l] ~ dnorm(0,tau[l])
}

    for(l in 1:L){
    eta[i,l] = mu[l]+ phi[l]*(eta[i-1,l]-mu[l]) + eps[i,l] 
    eta_star[i,l]= eta[i,l] + inprod(c(x[i,l],xshared[i,]),t(theta[l,]))
    #observations
#kobe[i,l] ~ dnegbin(dexp(eta_star[i,l]),alpha[l])
#   #y[i,l] = inprod(1-z[l],kobe[i,l])
    #y[i,l] ~ inprod(1-z[l],dnegbin(exp(eta_star[i,l]),alpha[l]))
    #y[i,l] ~ dnegbin(exp(eta_star[i,l]),alpha[l])
    y[i,l] ~ dnegbin(exp(eta_star[i,l]),alpha[l])
    ystar[i,l] ~ dnorm(z[l]*pointmass_0 + inprod((1-z[l]),y[i,l]),100000)
}
}

}

Which I call from R using runjags:

 parsamples <- run.jags('jags_model.txt', data=forJags, monitor=c('y','theta'), sample=100, method='rjparallel')
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  • $\begingroup$ in this model there is a share of information which happens through the hyperparameters. I'm thinking about a vector $\mathbf{\epsilon} \sim N(0,\Sigma)$, so to model the covariance between the innovations. $\endgroup$ – Tommaso Guerrini Nov 11 '16 at 14:00

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