r statistics forecasting facebook-prophet

Box Cox Transformation with zero Data Daily

I have a daily data of sales with zero values (by holidays and sundays) and I want to apply boxCox.lambda() function, but clearly with the zero values this is impossible. Mi options actually are:

1 - Change the zero values by values approaching zero, but I do not know how this can affect my forecast.

Any suggestions I will be grateful.

This is my data:

Solution

I'd recommend you just drop all the Sundays from your data. As we know they will alway s be zero there is no point in spending time and effort on forecasting them.

The periodicity is very strong even with them removed, and diagnosing the data by looking at acf plots etc. is much more straight forward.

# Removing every Sunday and creating a ts object of appropriate frequency
x6 <- x[seq_along(x) %% 7 != 0]
x6.ts <- ts(x6, frequency=6)

# Plenty of periodic structure left
par(mfcol=c(2, 1))
sp <- split(x6.ts, (seq_along(x6.ts)-1) %% 6 + 1)
stripchart(sp, vertical=TRUE, col=rainbow(6, alpha=0.2, start=0.97), pch=16,
  method="jitter", group.names=c("Mon", "Tue", "Wed", "Thu", "Fri", "Sat"))
plot.default(x6.ts, type="p", pch=16, col=rainbow(6, alpha=0.6, start=0.97))

The we could f.ex apply a SARIMA model

acf(x6.ts, adj=c(0.5))
title("x6.ts", cex.main=0.9)

acf(diff(x6.ts, lag=6))
title("diff(x6.ts, lag=6)", cex.main=0.9)

I see a seasonal random walk there, and once we take the seasonal difference we see that there's at least a couple of seasonal autoregressive components, and maybe a non-seasonal autoregression.

aa6.1 <- arima(x6.ts, order=c(0, 0, 0), seasonal=c(1, 1, 0))
aa6.2 <- arima(x6.ts, order=c(0, 0, 0), seasonal=c(2, 1, 0))
aa6.3 <- arima(x6.ts, order=c(1, 0, 0), seasonal=c(2, 1, 0))
aa6.4 <- arima(x6.ts, order=c(1, 0, 0), seasonal=c(3, 1, 0))

dummy11 <- model.matrix(~ as.factor(seq_along(x6.ts) %% 11))[,2]
aa6.5 <- arima(x6.ts, order=c(1, 0, 0), seasonal=c(3, 1, 0),
  xreg=dummy11)


AIC(aa6.1, aa6.2, aa6.3, aa6.4, aa6.5)
#       df      AIC
# aa6.1  2 5244.846
# aa6.2  3 5195.019
# aa6.3  4 5192.212
# aa6.4  5 5179.310
# aa6.5  6 5164.567

acfr <- function(x){
    a <- acf(residuals(x), plot=FALSE)
    a$acf[1, 1, 1] <- 0
    plot(a, main="", frame.plot=FALSE, ylim=c(-0.2, 0.2))
    mod <- paste(paste(names(x$call), 
      as.character(x$call), sep="=")[-1], collapse=", ")
    text(-0.1, 0.19, pos=4, xpd=NA,
      paste0("AIC: ", round(x$aic), "\n", "Mod: ", mod))
}

par(mfcol=c(5, 1))
k <- lapply(list(aa6.1, aa6.2, aa6.3, aa6.4, aa6.5), acfr)

Seems like (1 0 0) (3 1 0)[6] does a decent job, but there's a persistent autocorrelation at lag 11. This is an artefact of the removal of Sundays, but we can address it by including an external regressor of dummys.