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These are the ACF and PACF plots of a time series (daily aggregates/counts). The first plot is un-differenced and the second plot is after a seasonal difference of 7 days.

ACF and PACF before seasonal differencing

ACF and PACF after seasonal differencing

My take looking at the seasonally differenced plots is to start with 8 seasonal auto-regression terms. However I also noticed that ACF peak before the decay, does that suggest anything to you?

Also, from the undifferenced plot, does seeing 7 significant peaks in the pacf suggest 7 immediate AR terms as well? Or perhaps 6, since the 7th one will already be included in the seasonal AR terms.

EDIT: I am not sure how to post data, I am pasting it directly on here.

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  • $\begingroup$ post your data and I will try and help $\endgroup$ – IrishStat Apr 26 at 10:47
  • $\begingroup$ The ACF and the PACF are summary statistics. Your data may auto-regressive or moving-average structure along with regular differencing , seasonal differencing, level/step shifts , local time trends, pulses , seasonal pulses AND/OR the need to partition due to changes in parameters over time or error variance changes over time or error variance-expected value linkages. Only your data knows for sure ... which is why I asked you to post your data. If the data is deemed confidential simply scale it by subtracting a constant and dividing it by another constant. $\endgroup$ – IrishStat Apr 26 at 12:06
  • $\begingroup$ @IrishStat Thanks, I've posted the data, it's daily counts of an observable, and they start from january until october. $\endgroup$ – Mike Apr 26 at 15:04
  • $\begingroup$ what is the starting date + describe the data $\endgroup$ – IrishStat Apr 26 at 16:12
  • $\begingroup$ @IrishStat Data is between January and October of 2003. It is daily counts (so sampling frequency is 1 hour but I have resampled at 1 day by summing) $\endgroup$ – Mike Apr 26 at 16:20
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I took 5609 daily crime values (I will send them to anybody who asks for them) for a US location and introduced them to AUTOBOX resulting in the following Actual-Forecast graph enter image description here . The Actual/Fit and Forecast is busier . enter image description here. Note that the OP posted data (5570 values) that had a number of omitted days which is a no-no when analyzing time series data.

As is expected crime is affected by not only day-of-the-week and month-of-the-year BUT major holidays also play a significant role. Additionally three level shifts were identified along with 3 fixed-days-of-the-month including the first day of each month . Note that day 5 has the sharpest drop-off in crime i.e Sunday suggesting that even criminals respect the sabbath with crime the highest on day 3 (Friday ).

Here is the acf of the original data illustrating the "fixed effect" of day-of-the-week enter image description here since the final model had no need for SARIMA with the acf of the model's residuals here enter image description here.

The equation is here in two parts enter image description here and here enter image description here . Note that no arima structure was needed to generate a gaussian error structure. In other words it is more important to predict based upon the actual type of day than weighting what occurred last week or any seasonal differences.

The forecasts are presented for the next 365 days ( starting at May 11, 2018 ) enter image description here

Weather and social habits (holidays ) play a major role in the occurrence of crime. Note that August has the most crime while December has the least . The monthly indicators could well be a proxy for temperature .

Box and Jenkins loudly ( but I guess not loudly enough ! ) that their suggested tools (acf/pacf) for model identification PREMISED no deterministic structure in the data among other things. Your data is replete with deterministic structure.

Following Abraham Wald ... " You need to know the assumptions underlying the test"

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  • $\begingroup$ Thank you very much for your detailed answer and time. 1) When you say the data is replete with deterministic structure, do you mean the weekly and monthly seasonality? I thought the ACF and PACF plots were ideal for identifying that deterministic structure, unless I'm misunderstanding the term. 2) By level shifts, do you mean a change in the expectation of the random variable for prolonged periods of time? 3) I'm trying to understand your output, is this saying that you have 56 variables in total? What does fixed_eff and month_eff stand for? Do you model holidays with dummy vars? $\endgroup$ – Mike Apr 29 at 17:08
  • $\begingroup$ Deterninistic structure that was extracted from the data : DAILY ;MONTHLY ; 3 SPECIFIC DAYS OF THE MONTH & LEVEL SHIFTS ... no weekly . The ACF deals with autoregressive sructure NOT fixed effects. You are misunderstanding . Level shifts are permanent changes in the expected value .fixed_eff n101017 means first type then day 01 of 07 days . Month effect zz is the effect of the the zzth month.Yes the holidays are modeled with one or more dummies depending upon whether or not it is a lead , contemporaneous or lag effect e.g. xmas is 1 day before (-1) , on and the following 2 days (1 and 2 ) . $\endgroup$ – IrishStat Apr 29 at 17:29
  • $\begingroup$ So the fact that the ACF shows weekly seasonality is not a 'deterministic' aspect of the data, however its response to a holiday, which is a fixed event, is. When you produce this model, I imagine you have a predictor matrix, in this case having 56 columns, are any of these columns auto-regressive or moving average terms? i.e. previous time steps and previous residuals? In other words, did you not identify any ARMA structure at all, and simply used other predictors that AUTOBOX identified to get normally distributed residuals? I am also curious how Lower and Upper limits were produced. $\endgroup$ – Mike Apr 29 at 21:10
  • $\begingroup$ I will take a look at the autobox website to see what is specified there, my models do not provide confidence intervals, I am merely running regressions with some autoregressive orders and whatever additional predictors improve my residual structure and other error metrics. To capture seasonality I try to use ACF/PACF or transfer to the frequency domain with Fourier to try and see something there, but I've seen your answers on many other posts and it is clear that I don't have a good enough grasp of what an ARMA structure is and how to identify it. $\endgroup$ – Mike Apr 29 at 21:13
  • $\begingroup$ if i can help in any way please contact me at my contact info , Fourier is not going to help you.. $\endgroup$ – IrishStat Apr 29 at 21:47

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