Evaluate upper bound prediction results using classic error calculation instead of PI metrics

Question

I have the following data over time:

that means data collected for a single variable like CPU usage in lowest, highest, and average mode over time every 5 mins (data granularity = 5mins) like the following data frame:

|    | timestamp           |   min cpu |     max cpu |     avg cpu |
|---:|:--------------------|----------:|------------:|------------:|
|  0 | 2017-01-01 00:00:00 |    715147 | 2.2233e+06  | 1.22957e+06 |
|  1 | 2017-01-01 00:05:00 |    700474 | 2.21239e+06 | 1.21132e+06 |
|  2 | 2017-01-01 00:10:00 |    705954 | 2.21306e+06 | 1.20663e+06 |
|  3 | 2017-01-01 00:15:00 |    688383 | 2.18757e+06 | 1.19037e+06 |
|  4 | 2017-01-01 00:20:00 |    688277 | 2.18368e+06 | 1.18099e+06 |

I sliced the dataframe and worked on a univariate time-series data problem as follows:

|    | timestamp           |     avg cpu |
|---:|:--------------------|------------:|
|  0 | 2017-01-01 00:00:00 | 1.22957e+06 |
|  1 | 2017-01-01 00:05:00 | 1.21132e+06 |
|  2 | 2017-01-01 00:10:00 | 1.20663e+06 |
|  3 | 2017-01-01 00:15:00 | 1.19037e+06 |
|  4 | 2017-01-01 00:20:00 | 1.18099e+06 |

I split data and applied PI (Prediction Interval) using a regression:

|                     |        pred |   lower_bound |   upper_bound |
|:--------------------|------------:|--------------:|--------------:|
| 2017-01-25 00:00:00 | 1.15232e+06 |   1.12482e+06 |   1.1874e+06  |
| 2017-01-25 00:05:00 | 1.14453e+06 |   1.10052e+06 |   1.18994e+06 |
| 2017-01-25 00:10:00 | 1.14033e+06 |   1.08739e+06 |   1.20795e+06 |
| 2017-01-25 00:15:00 | 1.13669e+06 |   1.0843e+06  |   1.20252e+06 |
| 2017-01-25 00:20:00 | 1.1271e+06  |   1.06837e+06 |   1.19865e+06 |

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question:

Since I'm interested in upper_bound upper prediction limit only, (if you see this image from Jason for Relationship between prediction, actual value and prediction interval) then the frame of my problem changed from PI to the simple problem of target prediction. then:

1. Does it mean I can easily use normal target prediction metrics ($MSE$, $MAE$, $MAPE$, $R^2 score$ ) instead of PI metrics ($PICP$, $PINC$, $ACE$, $MPIW$, $PINAW$, $PINRW$, $score$ ) to evaluate the used regression/predictive model as they treat in skforecast package? ref.

# Prediction error
# ==============================================================================
from sklearn.metrics import mean_squared_error
error_mse = mean_squared_error(
                y_true = data_test['avg cpu'],
                y_pred = predictions.iloc[:, 0]
            )
print(f"Test error (MSE): {error_mse}")

In other words (regardless of the way is treated in skforecast package):

2. Is it fine in academics and papers that one treats the evaluation classic use of normal target prediction metrics ($MSE$, etc ) instead of PI metrics ($PICP$, etc. ) for evaluation if you are interested in only one target amongst upper\lower\targetin PI tasks? (It would be great if you cite an example paper)

Thanks in advance

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Related materials found:

• Time Series Forecasting: Prediction Intervals Estimate the range of a future observation with confidence
• Upper Bound for Size of Prediction Interval
• imbalanced regression problem + lower bound prediction + custom error weighting
• Why is the lower bound of the confidence interval of a model's error relatively constant compared to the upper bound?
• Upper & lower bound of confidence interval of mean
• Upper Bound and Lower Bound on Means when Distributions are bounded?
• Confidence Interval Upper and Lower Bound
• Confidence Intervals vs Prediction Intervals: Confusing the two can be costly. See how they differ and when to use each!

Answer 1

Since I'm interested in upper_bound upper prediction limit only ... Does it mean I can easily use normal target prediction metrics ($MSE$, $MAE$, $MAPE$, $R^2 score$ ) instead of PI metrics ($PICP$, $PINC$, $ACE$, $MPIW$, $PINAW$, $PINRW$, $score$ ) to evaluate the used regression/predictive model

You can use those metrics (and especially for training as explained here: Could a mismatch between loss functions used for fitting vs. tuning parameter selection be justified?).

However to evaluate the models you might want to use a cost function that is closest to your target goal. If this is accuracy of the upper bound of a prediction interval, then use that as metric.

Is it fine in academics and papers that one treats the evaluation classic use of normal target prediction metrics (MSE , etc ) instead of PI metrics (PICP , etc. ) for evaluation if you are interested in only one target amongst upper\lower\target in PI tasks? (It would be great if you cite an example paper)

If the paper wants to treat a study about PI then it should be about PI in it's text and not about MSE.