Should I truncate some observation points of a time series before producing forecast?

Machine Learning and Modeling
#1

Hello Everyone, I am working on hourly forecasts for shops and restaurants. Sometimes, I am confused about the combination of a number of observations of a time series and its frequency. Example: I have a set of 10 different observation data in a time series such as observeData <- c(2,3,5,7,4,6,8,5,7,9) and frequency of time series is 3. As the frequency is 3, before putting observeData to a forecast algorithm auto.arima(), should I truncate the number of observeData from 10 (3x3 + 1) to 9 (3x3)?. I mean from observeData <- c(2,3,5,7,4,6,8,5,7,9) to observeData <- c(3,5,7,4,6,8,5,7,9). Intuition said, auto.arima() or other forecast algorithms need a rectangular matrix data. And 10 (3x3 + 1) observation data can not form a rectangular matrix for computation while an observed data of 9 (3x3) can form a rectangular matrix.

#2
  1. How can hourly data give you a frequency of 3?
  2. There is no need to truncate the data.
#3

Dear Rob J Hyndman, Thanks for your question. It's nice that I have found you in this community. I am following your book Forecasting Principles and Practice (2nd edition) since 2020 for most of my implementation. There is a little story behind finding this book. Anyway, Let me give the answer to your question.

Answer to your question: Imagine, a restaurant opens only 3 days per week (Monday, Wednesday and Friday) and each day it's opening hour is from 11.00h to 14.00h. So, the hourly frequency is 3 instead of 24 and daily frequency is also three instead of 7. If there is a mistake please, explain it to me. I am learning and open a new startup business to analysis minute sales data. In my business (http://www.sale4cast.com/), we collect minute historical sales data from customers. Once minute historical sales data of 2 months is collected, we aggregate minute sales data into hourly sales data and slice the hourly sales data according to different weekdays. Because, different weekdays has different opening hours of a restaurant. For example, current Wednesday data goes with previous Wednesday data to make a Wednesday hourly time series. And then produce hourly sales forecasts for different weekdays.

In the subsection 2.1 of Forecasting Principles and Practice (2nd edition), I believe, daily frequency for a week was considered as 7 and a weekly period was 52 instead of 52.18. So, a rectangular matrix 7x52 is arrived at for computation instead of 7x52.18.