ARMA to ARIMA
- When there is a trend in data we take differences
- ARIMA – Auto regressive Integrated Moving average
- Integrated term includes order of difference, In the example below it is d=2
Below is the sample github gist and output pdf is avaialble at ARIMA model.pdf
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| — | |
| title: "ARIMA model" | |
| author: "Archit Vora" | |
| date: "April 3, 2018" | |
| output: html_document | |
| — | |
| “`{r setup, include=FALSE} | |
| knitr::opts_chunk$set(echo = TRUE) | |
| “` | |
| We will try to fit ARIMA model to BJsales dataset from 'dataset' package in R. | |
| Here is the plot of it. | |
| “`{r} | |
| library(datasets) | |
| plot(BJsales) | |
| “` | |
| Mean in changing over time and seems time series is not statioary. Let's take the difference. | |
| “`{r} | |
| plot(diff(BJsales)) | |
| “` | |
| Seems It is still not staionary, let's take one more diff. | |
| “`{r} | |
| plot(diff(diff(BJsales))) | |
| “` | |
| Now it seems stationary. Let's plot acf and pacf of this doubly differenced series. | |
| “`{r} | |
| focus<- diff(diff(BJsales)) | |
| acf(focus) | |
| pacf(focus) | |
| “` | |
| From ACF lag 1, 8 and 11 seems significant, while from PACF seems lag 1, 2, 3, 10 and 19 seems significant. | |
| Keeping parsimonious principle in mind we shall consider order of 0 and 1 for MA terms and oreder of (0, 1, 2, 3) for AR terms. | |
| Now let's try differenct models and check their AIC. | |
| “`{r} | |
| d <- 2 | |
| for (p in 0:3){ | |
| for (q in 0:1){ | |
| if(p+d+q<6){ # Ensures simple model | |
| mm<- arima(x = BJsales, order = c(p, d, q)) | |
| pval <- Box.test(mm$residuals, lag = log(length(mm$residuals))) | |
| sse <- sum(mm$residuals^2) | |
| aic <- mm$aic | |
| cat(p, d, q, "AIC = ", aic, "SSE = ", sse, "P-value = ", pval$p.value, "\n") | |
| } | |
| } | |
| } | |
| “` | |
| Seems like (0, 2, 1) has smallest AIC but does not have significant p-value. | |
| Let's leave it for this post and plots residual. | |
| “`{r} | |
| model <- arima(x = BJsales, order = c(0, 2, 1)) | |
| par(mfrow = c(2, 2)) | |
| plot(model$residuals) | |
| acf(model$residuals) | |
| pacf(model$residuals) | |
| qqnorm(model$residuals) | |
| “` | |
| There seems no significant correlation as well as QQ-plot also looks okay. | |
| Now let's plot the forecast. | |
| “`{r} | |
| library(forecast) | |
| fc <- forecast(model, h=20) | |
| plot(fc) | |
| “` | |
| All this can also be done by auto.arima routine of forecast package. | |
| “`{r} | |
| model <- auto.arima(BJsales, seasonal = FALSE) | |
| model | |
| fc <- forecast(model, h = 20) | |
| plot(fc) | |
| “` | |
| auto.arima has come up with different model, but it is okay. |
Data Stories 