[BUG] - ARIMA predict method returns wrong predictions when n_samples>1

[TueLindhart]

Describe the bug
When I use the predict function with n_samples > 1, then it seems to return "delayed" predictions that not are offset from the latest series observation. The predictions seem to align with earlier observations in the training series.

To Reproduce

forecaster = ARIMA(p=0,d=1,q=0)
forecaster.fit(series=series,future_covariates=future_covariates_train)

y_pred1 = forecaster.predict(n=24,future_covariates=future_covariates_test,num_samples=1) # Correct offset
y_pred2 = forecaster.predict(n=24,future_covariates=future_covariates_test,num_samples=2) # Wrong offset 

The data used in array format:

series = array([12.9  , 12.9  , 12.9  , 12.9  , 12.9  , 12.9  , 12.9  , 12.9  ,
       12.9  , 12.9  , 12.9  , 12.9  , 12.925, 12.95 , 12.975, 13.   ,
       13.025, 13.05 , 13.075, 13.1  , 13.125, 13.15 , 13.175, 13.2  ,
       13.3  , 13.4  , 13.5  , 13.6  , 13.65 , 13.7  , 13.75 , 13.8  ,
       13.875, 13.95 , 14.025, 14.1  , 14.15 , 14.2  , 14.25 , 14.3  ,
       14.3  , 14.3  , 14.3  , 14.3  , 14.3  , 14.3  , 14.3  , 14.3  ])

future_covariates_train = array([[16.84177464, 77.        , 13.65      ],
       [16.85733716, 77.        , 13.4       ],
       [16.87269715, 77.        , 13.15      ],
       [16.88785423, 77.        , 12.9       ],
       [16.904009  , 77.25      , 12.625     ],
       [16.92115493, 77.5       , 12.35      ],
       [16.93928579, 77.75      , 12.075     ],
       [16.95839566, 78.        , 11.8       ],
       [16.97972589, 78.5       , 11.725     ],
       [17.00328543, 79.        , 11.65      ],
       [17.02908464, 79.5       , 11.575     ],
       [17.05713531, 80.        , 11.5       ],
       [17.08744904, 80.5       , 11.45      ],
       [17.12004015, 81.        , 11.4       ],
       [17.15492448, 81.5       , 11.35      ],
       [17.1921195 , 82.        , 11.3       ],
       [17.23163856, 82.5       , 11.3       ],
       [17.27350111, 83.        , 11.3       ],
       [17.31772833, 83.5       , 11.3       ],
       [17.3643432 , 84.        , 11.3       ],
       [17.41028977, 84.        , 11.3       ],
       [17.45557761, 84.        , 11.3       ],
       [17.50021616, 84.        , 11.3       ],
       [17.54421474, 84.        , 11.3       ],
       [17.58913722, 84.25      , 11.175     ],
       [17.63498688, 84.5       , 11.05      ],
       [17.68176728, 84.75      , 10.925     ],
       [17.72948236, 85.        , 10.8       ],
       [17.77811794, 85.25      , 10.775     ],
       [17.82767766, 85.5       , 10.75      ],
       [17.87816548, 85.75      , 10.725     ],
       [17.92958561, 86.        , 10.7       ],
       [17.98353051, 86.5       , 11.075     ],
       [18.04001078, 87.        , 11.45      ],
       [18.09903865, 87.5       , 11.825     ],
       [18.16062807, 88.        , 12.2       ],
       [18.22303155, 88.25      , 12.45      ],
       [18.28624752, 88.5       , 12.7       ],
       [18.35027463, 88.75      , 12.95      ],
       [18.41511178, 89.        , 13.2       ],
       [18.47355594, 88.25      , 13.15      ],
       [18.52587616, 87.5       , 13.1       ],
       [18.57232717, 86.75      , 13.05      ],
       [18.61315034, 86.        , 13.        ],
       [18.65023175, 85.5       , 12.75      ],
       [18.683676  , 85.        , 12.5       ],
       [18.71358437, 84.5       , 12.25      ],
       [18.74005494, 84.        , 12.        ]])

future_covariates_test = array([[18.76617668, 84.        , 11.85      ],
       [18.79195331, 84.        , 11.7       ],
       [18.8173885 , 84.        , 11.55      ],
       [18.84248586, 84.        , 11.4       ],
       [18.86723634, 84.        , 11.325     ],
       [18.8916446 , 84.        , 11.25      ],
       [18.91571521, 84.        , 11.175     ],
       [18.93945269, 84.        , 11.1       ],
       [18.9643633 , 84.25      , 11.225     ],
       [18.99044204, 84.5       , 11.35      ],
       [19.01768421, 84.75      , 11.475     ],
       [19.04608539, 85.        , 11.6       ],
       [19.07249807, 84.75      , 11.625     ],
       [19.09696678, 84.5       , 11.65      ],
       [19.11953511, 84.25      , 11.675     ],
       [19.14024571, 84.        , 11.7       ],
       [19.16210988, 84.23529412, 11.64117647],
       [19.18512517, 84.47058824, 11.58235294],
       [19.20928947, 84.70588235, 11.52352941],
       [19.23460093, 84.94117647, 11.46470588],
       [19.26105805, 85.17647059, 11.40588235],
       [19.28865963, 85.41176471, 11.34705882],
       [19.31740479, 85.64705882, 11.28823529],
       [19.34729295, 85.88235294, 11.22941176]])

Expected behavior
The offset of the first prediction should be close to the last entry in the training series.

System (please complete the following information):

• Python version: 3.10.6
• darts version 0.23.1

Additional context
Add any other context about the problem here.

[TueLindhart]

I believe it is fixed by:

forecast = self.model.simulate(
               nsimulations=n,
               repetitions=num_samples,
               initial_state=self.model.states.predicted[-1, :],
               anchor="end", <-- Add this
               exog=future_covariates.values(copy=False)
               if future_covariates
               else None,
)

in darts.models.forecasting.arima.py

[madtoinou]

Hi @TueLindhart,

Thank you for spotting and fixing this bug, would you like to open a PR with the fix? If you don't have the time, I would be happy to do it for you.

[TueLindhart]

Sorry, I currently do not have time to do so. :-)

[madtoinou]

Fixed by #1893.