I don't understand how predictions can trace the actual data so closely (see the code below)? Does that make sense? The model is $Y_t = \theta Y_{t-1} + Z_t$ where $Z_t$ is random noise. Hence the random noise term should have prevented such a close tracking. But it does not. Why? Data is here.

from statsmodels.tsa.ar_model import AR
import matplotlib.pyplot as plt  
import pandas as pd
dat = pd.read_csv("C:\\Users\\boris\\Documents\\lake.txt", sep='\t')
dat.drop(dat.index[:5], inplace = True)
model = AR(x)
model_fit = model.fit(maxlag=1, trend='nc')

predictions =model_fit.predict(start=1, end=len(x)-1)
plt.figure(num=None, figsize=(8, 5), dpi=70)
plt.plot(list(x), marker = '.', color = 'red')
plt.plot(list(predictions), marker = '.', color = 'blue')

Edit 1 and additional comment.

If I relabel the row indices:


and then run the prediction:

model = AR(x)
model_fit = model.fit(maxlag=1, trend='nc')

predictions =model_fit.predict(start=1,end=len(x)+30)

It returns predictions with row indices starting with 1 while the original x was starting with 0 index. However, the $predictions[1]$ is almost equal to $x[0]$. This leads to an unjustified shift when plotting both x and predictions. I tried putting $start=0$ but python complained bitterly. Is this some sort of a bug or it was supposed to be like this? If yes, why?

BTW, statsmodel is the package most commonly used for autoregression? Might there be better ones?

enter image description here

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    $\begingroup$ How large is the variance of the noise? if the noise is small it may produce this $\endgroup$ – Xiaomi Jun 15 '19 at 7:43
  • $\begingroup$ My reading of the manual is that the "predictions" for the data you give will equal the data themselves. The argument is supposed to be an array of future times--so when a time with a known value is included in the array, of course the model just spits back the actual value that was observed then. You can easily check this out by doing some testing. $\endgroup$ – whuber Jun 15 '19 at 12:17
  • $\begingroup$ @whuber But if one looks careful at the output, it is very close but not identical to the observed data. $\endgroup$ – Al Guy Jun 15 '19 at 16:08
  • $\begingroup$ Determining the predictions are not identical to the data requires a lot of faith in the accuracy of small details in a crude reproduction of the graph. It would be more helpful to describe the residuals by plotting them or summarizing them. $\endgroup$ – whuber Jun 16 '19 at 13:42
  • $\begingroup$ @whuber I did produce the list of numbers for the predicitions. $\endgroup$ – Al Guy Jun 16 '19 at 21:38

You are fitting an AR(1) model to this data, so you are postulating:

$$y_t = \phi y_{t-1} + \varepsilon_t$$

Your data looks close to non-stationary, which means that your parameter estimate for $\phi$ is probably close to 1. You can check with print(model_fit.params).

For the sake of argument, let's suppose that the estimate is exactly 1 (of course it won't be exactly, but I bet it's pretty close). Then you would have:

$$y_t = y_{t-1} + \varepsilon_t$$

But the error term $\varepsilon_t$ is assumed to be white noise, so your forecast of it will be zero. That means your forecast will be:

$$\hat y_t \approx y_{t-1}$$

As you noted, your original data starts at index 0, but your predictions start at index 1. That's not a mistake. The model is using the index-0 datapoint to forecast the index-1 datapoint, and because of the simplicity of the model and the parameter estimate close to 1, you will get a one-step-ahead forecast of $\hat y_1 \approx y_0$. In the same way, $\hat y_2 \approx y_1$, etc.

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My original answer below was applicable to general (i.e non-stationary) time series. In the OP case, if the data can be modeled as $Y_t = \theta Y_{t-1} + Z_t$, then it should be stationary, and therefore the variance is constant, so my statement about subsequent values $Z_{t+1}$, $Z_{t+s}$,..getting larger is not correct. The residual standard deviation $\sigma$ is a good estimate for all values of $Z$.

(To add to the confusion, the data in the plot looks almost, but not quite stationary - there seems to be a slight downward trend - I'm surprised statsmodels.tsa was able to fit an $AR(1)$ without throwing an error, or at least a warning)

Keep in mind that $Y_t = \theta Y_{t-1} + Z_t$ where $Z_t$ represents the "true" model underlying your component. Here by true, I mean the theoretical model that you have selected to represent your time series.

The estimated model, that is the one actually calculated by your software and plotted by the software, will be $\hat{Y}_t = \theta Y_{t-1}$.

$Z_t$ is a stochastic process (i.e. random variable) and hence cannot be estimated by a deterministic calculation, which is what your point forecasting model is.

For a model $\hat{Y}_t$ fitted on the data $[Y_0,...Y_{t-1}]$, a good estimate $\hat{Z}_t$ is the standard deviation $\sigma$ of the residuals $\hat{Y}-Y$. But $\sigma$ is a good estimate only for the first value $Z_t$. Subsequent values $Z_{t+1}$, $Z_{t+s}$,..will become larger and larger. Intuitively, this corresponds to the idea that the farther out into the future your forecast, the more uncertain it will be.

Sometimes, depending on the forecasting method used, you can estimate the value of $Z_t$ analytically. So you have a formula $Z_{t+k+1} = f(Z_{t+k})$ which you apply iteratively to get the value $Z_{t+h}$ at your forecast horizon $h$.

Other times, you can't calculate $Z_t$ analytically, so you have to simulate it instead. For this you need to use something like an MCMC model, or sample path simulation (here is a good example on how to do sample path simulation, it is regarding neural networks, not AR processes, and the code is in R not Python, but it is well written enough that it is still relevant to your question).

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  • $\begingroup$ Could you elaborate on the first sentence of paragraph 4? Are some words perhaps missing from it? $\endgroup$ – Richard Hardy Jun 15 '19 at 8:16
  • $\begingroup$ @RichardHardy I just realized I might be wrong (serves me right for answering cross-validated question at 1:00AM). If the process can be modeled by an $AR(1)$ model, then it is stationary. So it has constant variance, and what I said about widening forecast intervals doesn't hold, no? $\endgroup$ – Skander H. Jun 15 '19 at 8:28
  • $\begingroup$ Whoever upvoted this answer should downvote it after editing, it is wrong. What I said here is not entirely correct. See edit please. $\endgroup$ – Skander H. Jun 15 '19 at 8:37
  • $\begingroup$ @SkanderH. But I still don't know what the program does when outputting predictions which closely trace the observed values but are not idential to them. $\endgroup$ – Al Guy Jun 15 '19 at 16:10
  • $\begingroup$ @AlGuy you predictions are shifted one steps ahead, correct? (I can tell from the graph, it is too small). $\endgroup$ – Skander H. Jun 15 '19 at 17:36

When things are too good to be true , they often aren't true ! Here is the ar(1) model enter image description here and the residual plot ( always a good idea ! ) enter image description here and the Actual,Fit and Forecast graph enter image description here where the 1 period out forecast is heavily based upon the prior value .

It appears that the 1 period out forecast is upwards whereas I get the following downwards mean-reverting forecast enter image description here which is to be expected ( play on words here ! )


The 97 estimable equations lead to an intercept of 1.467 and regression coefficient of .836411 using the last previous value.

Having used data for the 98 periods . the approach is to predict the second from the first .. the third from the second all the way out to predict the 98th value from the 97th actual.

these 97 predictions are then the fitted values and in general the one-period out forecasts. . Now the optimal parameters that were found are the "best" possible combination of intercept and lag1 regression coefficient.

To illustrate how this works consider predicting the 99th value ( 1 period out)

enter image description here

To continue if we wish to predict the 100th vale we pretend that the first period out prediction (9.79768) is the actual value for period 99.

Thus we then plug in the value 9.79768 and obtain a prediction for period 100 ... getting 9.662

If this clears up your issues .. upvote my answer and accept it ..

if not call me on my landline (US) as I am the office as I cannot think of anything else to write about this.

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  • $\begingroup$ Who is "they" and to which "proof readers" are you referring?? $\endgroup$ – whuber Jun 15 '19 at 12:18
  • $\begingroup$ I believe the OP in his original question/comment pointed the egregious plot being presented in a Brockwell & Davis text. I guess this has been now taken down . The "proof readers" would have been the pre-publication reviewers of the text. $\endgroup$ – IrishStat Jun 15 '19 at 12:22
  • $\begingroup$ Thank you--but it looks like you had a different thread in mind. The record shows no reference to Brockwell & Davis was made in any version of the question. $\endgroup$ – whuber Jun 15 '19 at 12:25
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    $\begingroup$ ok ... so the "They" would be the those responsible for presenting a fit vs actual that is a tad to close for the OP & I .Perhaps the author of statsmodel can help here .. $\endgroup$ – IrishStat Jun 15 '19 at 12:28
  • $\begingroup$ @IrishStat But I still don't know what the program does when outputting predictions which closely trace the observed values but still not quite idential to them. $\endgroup$ – Al Guy Jun 15 '19 at 16:11

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