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Table Of Contents

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Quick Start Tutorial

The GluonTS toolkit contains components and tools for building time series models using MXNet. The models that are currently included are forecasting models but the components also support other time series use cases, such as classification or anomaly detection.

The toolkit is not intended as a forecasting solution for businesses or end users but it rather targets scientists and engineers who want to tweak algorithms or build and experiment with their own models.

GluonTS contains:

  • Components for building new models (likelihoods, feature processing pipelines, calendar features etc.)
  • Data loading and processing
  • A number of pre-built models
  • Plotting and evaluation facilities
  • Artificial and real datasets (only external datasets with blessed license)
In [1]:
# Third-party imports
%matplotlib inline
import mxnet as mx
from mxnet import gluon
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import json

Datasets

GluonTS datasets

GluonTS comes with a number of publicly available datasets.

In [2]:
from gluonts.dataset.repository.datasets import get_dataset, dataset_recipes
from gluonts.dataset.util import to_pandas
In [3]:
print(f"Available datasets: {list(dataset_recipes.keys())}")
Available datasets: ['constant', 'exchange_rate', 'solar-energy', 'electricity', 'traffic', 'm4_hourly', 'm4_daily', 'm4_weekly', 'm4_monthly', 'm4_quarterly', 'm4_yearly']

To download one of the built-in datasets, simply call get_dataset with one of the above names. GluonTS can re-use the saved dataset so that it does not need to be downloaded again: simply set regenerate=False.

In [4]:
dataset = get_dataset("m4_hourly", regenerate=True)
INFO:root:downloading and processing m4_hourly
saving time-series into /var/lib/jenkins/.mxnet/gluon-ts/datasets/m4_hourly/train/data.json
saving time-series into /var/lib/jenkins/.mxnet/gluon-ts/datasets/m4_hourly/test/data.json

In general, the datasets provided by GluonTS are objects that consists of three main members:

  • dataset.train is an iterable collection of data entries used for training. Each entry corresponds to one time series
  • dataset.test is an iterable collection of data entries used for inference. The test dataset is an extended version of the train dataset that contains a window in the end of each time series that was not seen during training. This window has length equal to the recommended prediction length.
  • dataset.metadata contains metadata of the dataset such as the frequency of the time series, a recommended prediction horizon, associated features, etc.
In [5]:
entry = next(iter(dataset.train))
train_series = to_pandas(entry)
train_series.plot()
plt.grid(which="both")
plt.legend(["train series"], loc="upper left")
plt.show()
../../_images/examples_basic_forecasting_tutorial_tutorial_8_0.png
In [6]:
entry = next(iter(dataset.test))
test_series = to_pandas(entry)
test_series.plot()
plt.axvline(train_series.index[-1], color='r') # end of train dataset
plt.grid(which="both")
plt.legend(["test series", "end of train series"], loc="upper left")
plt.show()
../../_images/examples_basic_forecasting_tutorial_tutorial_9_0.png
In [7]:
print(f"Length of forecasting window in test dataset: {len(test_series) - len(train_series)}")
print(f"Recommended prediction horizon: {dataset.metadata.prediction_length}")
print(f"Frequency of the time series: {dataset.metadata.freq}")
Length of forecasting window in test dataset: 48
Recommended prediction horizon: 48
Frequency of the time series: H

Custom datasets

At this point, it is important to emphasize that GluonTS does not require this specific format for a custom dataset that a user may have. The only requirements for a custom dataset are to be iterable and have a “target” and a “start” field. To make this more clear, assume the common case where a dataset is in the form of a numpy.array and the index of the time series in a pandas.Timestamp (possibly different for each time series):

In [8]:
N = 10  # number of time series
T = 100  # number of timesteps
prediction_length = 24
freq = "1H"
custom_dataset = np.random.normal(size=(N, T))
start = pd.Timestamp("01-01-2019", freq=freq)  # can be different for each time series

Now, you can split your dataset and bring it in a GluonTS appropriate format with just two lines of code:

In [9]:
from gluonts.dataset.common import ListDataset
In [10]:
# train dataset: cut the last window of length "prediction_length", add "target" and "start" fields
train_ds = ListDataset([{'target': x, 'start': start}
                        for x in custom_dataset[:, :-prediction_length]],
                       freq=freq)
# test dataset: use the whole dataset, add "target" and "start" fields
test_ds = ListDataset([{'target': x, 'start': start}
                       for x in custom_dataset],
                      freq=freq)

Training an existing model (Estimator)

GluonTS comes with a number of pre-built models. All the user needs to do is configure some hyperparameters. The existing models focus on (but are not limited to) probabilistic forecasting. Probabilistic forecasts are predictions in the form of a probability distribution, rather than simply a single point estimate.

We will begin with GulonTS’s pre-built feedforward neural network estimator, a simple but powerful forecasting model. We will use this model to demonstrate the process of training a model, producing forecasts, and evaluating the results.

GluonTS’s built-in feedforward neural network (SimpleFeedForwardEstimator) accepts an input window of length context_length and predicts the distribution of the values of the subsequent prediction_length values. In GluonTS parlance, the feedforward neural network model is an example of Estimator. In GluonTS, Estimator objects represent a forecasting model as well as details such as its coefficients, weights, etc.

In general, each estimator (pre-built or custom) is configured by a number of hyperparameters that can be either common (but not binding) among all estimators (e.g., the prediction_length) or specific for the particular estimator (e.g., number of layers for a neural network or the stride in a CNN).

Finally, each estimator is configured by a Trainer, which defines how the model will be trained i.e., the number of epochs, the learning rate, etc.

In [11]:
from gluonts.model.simple_feedforward import SimpleFeedForwardEstimator
from gluonts.trainer import Trainer
INFO:root:Using CPU
In [12]:
estimator = SimpleFeedForwardEstimator(
    num_hidden_dimensions=[10],
    prediction_length=dataset.metadata.prediction_length,
    context_length=100,
    freq=dataset.metadata.freq,
    trainer=Trainer(ctx="cpu",
                    epochs=5,
                    learning_rate=1e-3,
                    num_batches_per_epoch=100
                   )
)

After specifying our estimator with all the necessary hyperparameters we can train it using our training dataset dataset.train by invoking the train method of the estimator. The training algorithm returns a fitted model (or a Predictor in GluonTS parlance) that can be used to construct forecasts.

In [13]:
predictor = estimator.train(dataset.train)
INFO:root:Start model training
INFO:root:Number of parameters in SimpleFeedForwardTrainingNetwork: 483
INFO:root:Epoch[0] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 192.21it/s, avg_epoch_loss=5.44]
INFO:root:Epoch[0] Elapsed time 0.523 seconds
INFO:root:Epoch[0] Evaluation metric 'epoch_loss'=5.439317
INFO:root:Epoch[1] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 214.37it/s, avg_epoch_loss=4.71]
INFO:root:Epoch[1] Elapsed time 0.467 seconds
INFO:root:Epoch[1] Evaluation metric 'epoch_loss'=4.710298
INFO:root:Epoch[2] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 217.59it/s, avg_epoch_loss=4.72]
INFO:root:Epoch[2] Elapsed time 0.461 seconds
INFO:root:Epoch[2] Evaluation metric 'epoch_loss'=4.724998
INFO:root:Epoch[3] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 220.99it/s, avg_epoch_loss=4.59]
INFO:root:Epoch[3] Elapsed time 0.454 seconds
INFO:root:Epoch[3] Evaluation metric 'epoch_loss'=4.585091
INFO:root:Epoch[4] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 218.43it/s, avg_epoch_loss=4.48]
INFO:root:Epoch[4] Elapsed time 0.459 seconds
INFO:root:Epoch[4] Evaluation metric 'epoch_loss'=4.481494
INFO:root:Loading parameters from best epoch (4)
INFO:root:Final loss: 4.481494476795197 (occurred at epoch 4)
INFO:root:End model training

With a predictor in hand, we can now predict the last window of the dataset.test and evaluate our model’s performance.

GluonTS comes with the make_evaluation_predictions function that automates the process of prediction and model evaluation. Roughly, this function performs the following steps:

  • Removes the final window of length prediction_length of the dataset.test that we want to predict
  • The estimator uses the remaining data to predict (in the form of sample paths) the “future” window that was just removed
  • The module outputs the forecast sample paths and the dataset.test (as python generator objects)
In [14]:
from gluonts.evaluation.backtest import make_evaluation_predictions
In [15]:
forecast_it, ts_it = make_evaluation_predictions(
    dataset=dataset.test,  # test dataset
    predictor=predictor,  # predictor
    num_eval_samples=100,  # number of sample paths we want for evaluation
)

First, we can convert these generators to lists to ease the subsequent computations.

In [16]:
forecasts = list(forecast_it)
tss = list(ts_it)

We can examine the first element of these lists (that corresponds to the first time series of the dataset). Let’s start with the list containing the time series, i.e., tss. We expect the first entry of tss to contain the (target of the) first time series of dataset.test.

In [17]:
# first entry of the time series list
ts_entry = tss[0]
In [18]:
# first 5 values of the time series (convert from pandas to numpy)
np.array(ts_entry[:5]).reshape(-1,)
Out[18]:
array([605., 586., 586., 559., 511.], dtype=float32)
In [19]:
# first entry of dataset.test
dataset_test_entry = next(iter(dataset.test))
In [20]:
# first 5 values
dataset_test_entry['target'][:5]
Out[20]:
array([605., 586., 586., 559., 511.], dtype=float32)

The entries in the forecast list are a bit more complex. They are objects that contain all the sample paths in the form of numpy.ndarray with dimension (num_samples, prediction_length), the start date of the forecast, the frequency of the time series, etc. We can access all these information by simply invoking the corresponding attribute of the forecast object.

In [21]:
# first entry of the forecast list
forecast_entry = forecasts[0]
In [22]:
print(f"Number of sample paths: {forecast_entry.num_samples}")
print(f"Dimension of samples: {forecast_entry.samples.shape}")
print(f"Start date of the forecast window: {forecast_entry.start_date}")
print(f"Frequency of the time series: {forecast_entry.freq}")
Number of sample paths: 100
Dimension of samples: (100, 48)
Start date of the forecast window: 1750-01-30 04:00:00
Frequency of the time series: H

We can also do calculations to summarize the sample paths, such computing the mean or a quantile for each of the 48 time steps in the forecast window.

In [23]:
print(f"Mean of the future window:\n {forecast_entry.mean}")
print(f"0.5-quantile (median) of the future window:\n {forecast_entry.quantile(0.5)}")
Mean of the future window:
 [646.3635  564.07806 516.47705 470.01544 442.45718 482.9354  458.0251
 466.258   488.88632 587.02155 620.6604  623.71326 745.921   769.87134
 840.40796 901.7982  877.3948  866.97565 848.72174 869.6038  801.9432
 763.5759  737.21625 721.7439  669.8018  608.6251  594.5139  479.12234
 494.51913 482.62512 450.70828 466.81805 472.70602 576.6179  596.49384
 685.1739  757.09766 790.1473  850.8337  856.11743 859.3353  848.19806
 875.84094 884.9561  835.264   776.19617 782.8473  711.8572 ]
0.5-quantile (median) of the future window:
 [652.0278  563.43286 524.8412  471.33475 456.0021  489.3306  460.7348
 468.74442 498.16525 550.70667 625.85284 614.1444  757.3362  779.84485
 840.82855 912.9583  848.29596 863.57715 843.6569  868.46295 796.4493
 765.87024 765.19214 724.41736 666.53357 606.1739  592.04193 485.34973
 484.34372 474.08792 418.392   470.84436 470.52344 576.14746 608.3898
 679.33673 752.2393  811.48865 846.0428  868.93414 864.40063 863.4485
 875.76697 876.3205  830.6541  784.1353  786.2227  701.494  ]

Forecast objects have a plot method that can summarize the forecast paths as the mean, prediction intervals, etc. The prediction intervals are shaded in different colors as a “fan chart”.

In [24]:
def plot_prob_forecasts(ts_entry, forecast_entry):
    plot_length = 150
    prediction_intervals = (50.0, 90.0)
    legend = ["observations", "median prediction"] + [f"{k}% prediction interval" for k in prediction_intervals][::-1]

    fig, ax = plt.subplots(1, 1, figsize=(10, 7))
    ts_entry[-plot_length:].plot(ax=ax)  # plot the time series
    forecast_entry.plot(prediction_intervals=prediction_intervals, color='g')
    plt.grid(which="both")
    plt.legend(legend, loc="upper left")
    plt.show()
In [25]:
plot_prob_forecasts(ts_entry, forecast_entry)
../../_images/examples_basic_forecasting_tutorial_tutorial_38_0.png

We can also evaluate the quality of our forecasts numerically. In GluonTS, the Evaluator class can compute aggregate performance metrics, as well as metrics per time series (which can be useful for analyzing performance across heterogeneous time series).

In [26]:
from gluonts.evaluation import Evaluator
In [27]:
evaluator = Evaluator(quantiles=[0.1, 0.5, 0.9])
agg_metrics, item_metrics = evaluator(iter(tss), iter(forecasts), num_series=len(dataset.test))
Running evaluation: 100%|██████████| 414/414 [00:01<00:00, 231.01it/s]

Aggregate metrics aggregate both across time-steps and across time series.

In [28]:
print(json.dumps(agg_metrics, indent=4))
{
    "MSE": 5034469.591562721,
    "abs_error": 7953677.470357895,
    "abs_target_sum": 145558863.59960938,
    "abs_target_mean": 7324.822041043146,
    "seasonal_error": 336.9046924038305,
    "MASE": 2.536846195117526,
    "sMAPE": 0.1743546019103364,
    "MSIS": 30.967944734084643,
    "QuantileLoss[0.1]": 3787897.6270955084,
    "Coverage[0.1]": 0.12595611916264088,
    "QuantileLoss[0.5]": 7953677.5408091545,
    "Coverage[0.5]": 0.45752818035426734,
    "QuantileLoss[0.9]": 5193677.740578365,
    "Coverage[0.9]": 0.8553743961352657,
    "RMSE": 2243.762374130273,
    "NRMSE": 0.30632312451521804,
    "ND": 0.05464234381656192,
    "wQuantileLoss[0.1]": 0.02602313272735439,
    "wQuantileLoss[0.5]": 0.054642344300567205,
    "wQuantileLoss[0.9]": 0.035680944548074245,
    "mean_wQuantileLoss": 0.038782140525331944,
    "MAE_Coverage": 0.037684514224369296
}

Individual metrics are aggregated only across time-steps.

In [29]:
item_metrics.head()
Out[29]:
item_id MSE abs_error abs_target_sum abs_target_mean seasonal_error MASE sMAPE MSIS QuantileLoss[0.1] Coverage[0.1] QuantileLoss[0.5] Coverage[0.5] QuantileLoss[0.9] Coverage[0.9]
0 NaN 1483.695150 1554.802002 31644.0 659.250000 42.371302 0.764473 0.048000 4.290071 706.483167 0.020833 1554.802002 0.645833 1012.952271 1.000000
1 NaN 129432.614583 15294.315430 124149.0 2586.437500 165.107988 1.929837 0.117354 13.994565 4296.688916 0.333333 15294.315186 1.000000 6706.406543 1.000000
2 NaN 29361.510417 6647.462891 65030.0 1354.791667 78.889053 1.755488 0.099744 16.357854 2900.356738 0.000000 6647.462646 0.062500 2617.788062 0.666667
3 NaN 185483.354167 15404.260742 235783.0 4912.145833 258.982249 1.239166 0.064297 7.316111 8151.388867 0.020833 15404.260498 0.354167 5320.524707 0.916667
4 NaN 87159.604167 10558.778320 131088.0 2731.000000 200.494083 1.097162 0.073427 6.242598 3710.298560 0.125000 10558.778076 0.666667 5508.893164 1.000000
In [30]:
item_metrics.plot(x='MSIS', y='MASE', kind='scatter')
plt.grid(which="both")
plt.show()
../../_images/examples_basic_forecasting_tutorial_tutorial_46_0.png

Create your own forecast model

For creating your own forecast model you need to:

  • Define the training and prediction network
  • Define a new estimator that specifies any data processing and uses the networks

The training and prediction networks can be arbitrarily complex but they should follow some basic rules:

  • Both should have a hybrid_forward method that defines what should happen when the network is called
  • The training network’s hybrid_forward should return a loss based on the prediction and the true values
  • The prediction network’s hybrid_forward should return the predictions

For example, we can create a simple training network that defines a neural network which takes as an input the past values of the time series and outputs a future predicted window of length prediction_length. It uses the L1 loss in the hybrid_forward method to evaluate the error among the predictions and the true values of the time series. The corresponding prediction network should be identical to the training network in terms of architecture (we achieve this by inheriting the training network class), and its hybrid_forward method outputs directly the predictions.

Note that this simple model does only point forecasts by construction, i.e., we train it to outputs directly the future values of the time series and not any probabilistic view of the future (to achieve this we should train a network to learn a probability distribution and then sample from it to create sample paths).

In [31]:
class MyTrainNetwork(gluon.HybridBlock):
    def __init__(self, prediction_length, **kwargs):
        super().__init__(**kwargs)
        self.prediction_length = prediction_length

        with self.name_scope():
            # Set up a 3 layer neural network that directly predicts the target values
            self.nn = mx.gluon.nn.HybridSequential()
            self.nn.add(mx.gluon.nn.Dense(units=40, activation='relu'))
            self.nn.add(mx.gluon.nn.Dense(units=40, activation='relu'))
            self.nn.add(mx.gluon.nn.Dense(units=self.prediction_length, activation='softrelu'))

    def hybrid_forward(self, F, past_target, future_target):
        prediction = self.nn(past_target)
        # calculate L1 loss with the future_target to learn the median
        return (prediction - future_target).abs().mean(axis=-1)


class MyPredNetwork(MyTrainNetwork):
    # The prediction network only receives past_target and returns predictions
    def hybrid_forward(self, F, past_target):
        prediction = self.nn(past_target)
        return prediction.expand_dims(axis=1)

Now, we need to construct the estimator which should also follow some rules:

  • It should include a create_transformation method that defines all the possible feature transformations and how the data is split during training
  • It should include a create_training_network method that returns the training network configured with any necessary hyperparameters
  • It should include a create_predictor method that creates the prediction network, and returns a Predictor object

A Predictor defines the predict method of a given predictor. Roughly, this method takes the test dataset, it passes it through the prediction network and yields the predictions. You can think of the Predictor object as a wrapper of the prediction network that defines its predict method.

Earlier, we used the make_evaluation_predictions to evaluate our predictor. Internally, the make_evaluation_predictions function invokes the predict method of the predictor to get the forecasts.

In [32]:
from gluonts.model.estimator import GluonEstimator
from gluonts.model.predictor import Predictor, RepresentableBlockPredictor
from gluonts.core.component import validated
from gluonts.support.util import copy_parameters
from gluonts.transform import ExpectedNumInstanceSampler, Transformation, InstanceSplitter
from gluonts.dataset.field_names import FieldName
from mxnet.gluon import HybridBlock
In [33]:
class MyEstimator(GluonEstimator):
    @validated()
    def __init__(
        self,
        freq: str,
        context_length: int,
        prediction_length: int,
        trainer: Trainer = Trainer()
    ) -> None:
        super().__init__(trainer=trainer)
        self.context_length = context_length
        self.prediction_length = prediction_length
        self.freq = freq


    def create_transformation(self):
        # Feature transformation that the model uses for input.
        # Here we use a transformation that randomly select training samples from all time series.
        return InstanceSplitter(
                    target_field=FieldName.TARGET,
                    is_pad_field=FieldName.IS_PAD,
                    start_field=FieldName.START,
                    forecast_start_field=FieldName.FORECAST_START,
                    train_sampler=ExpectedNumInstanceSampler(num_instances=1),
                    past_length=self.context_length,
                    future_length=self.prediction_length,
                )

    def create_training_network(self) -> MyTrainNetwork:
        return MyTrainNetwork(
            prediction_length=self.prediction_length
        )

    def create_predictor(
        self, transformation: Transformation, trained_network: HybridBlock
    ) -> Predictor:
        prediction_network = MyPredNetwork(
            prediction_length=self.prediction_length
        )

        copy_parameters(trained_network, prediction_network)

        return RepresentableBlockPredictor(
            input_transform=transformation,
            prediction_net=prediction_network,
            batch_size=self.trainer.batch_size,
            freq=self.freq,
            prediction_length=self.prediction_length,
            ctx=self.trainer.ctx,
        )
INFO:root:Using CPU

Now, we can repeat the same pipeline as in the case we had a pre-built model: train the predictor, create the forecasts and evaluate the results.

In [34]:
estimator = MyEstimator(
    prediction_length=dataset.metadata.prediction_length,
    context_length=100,
    freq=dataset.metadata.freq,
    trainer=Trainer(ctx="cpu",
                    epochs=5,
                    learning_rate=1e-3,
                    num_batches_per_epoch=100
                   )
)
In [35]:
predictor = estimator.train(dataset.train)
INFO:root:Start model training
INFO:root:Number of parameters in MyTrainNetwork: 128
INFO:root:Epoch[0] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 239.05it/s, avg_epoch_loss=3.04e+3]
INFO:root:Epoch[0] Elapsed time 0.420 seconds
INFO:root:Epoch[0] Evaluation metric 'epoch_loss'=3039.177498
INFO:root:Epoch[1] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 236.17it/s, avg_epoch_loss=1.17e+3]
INFO:root:Epoch[1] Elapsed time 0.424 seconds
INFO:root:Epoch[1] Evaluation metric 'epoch_loss'=1167.381439
INFO:root:Epoch[2] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 234.46it/s, avg_epoch_loss=950]
INFO:root:Epoch[2] Elapsed time 0.428 seconds
INFO:root:Epoch[2] Evaluation metric 'epoch_loss'=949.598687
INFO:root:Epoch[3] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 232.67it/s, avg_epoch_loss=745]
INFO:root:Epoch[3] Elapsed time 0.431 seconds
INFO:root:Epoch[3] Evaluation metric 'epoch_loss'=745.431857
INFO:root:Epoch[4] Learning rate is 0.001
100%|██████████| 100/100 [00:00<00:00, 238.09it/s, avg_epoch_loss=626]
INFO:root:Epoch[4] Elapsed time 0.421 seconds
INFO:root:Epoch[4] Evaluation metric 'epoch_loss'=626.136728
INFO:root:Loading parameters from best epoch (4)
INFO:root:Final loss: 626.1367277908325 (occurred at epoch 4)
INFO:root:End model training
In [36]:
forecast_it, ts_it = make_evaluation_predictions(
    dataset=dataset.test,
    predictor=predictor,
    num_eval_samples=100
)
In [37]:
forecasts = list(forecast_it)
tss = list(ts_it)
In [38]:
plot_prob_forecasts(tss[0], forecasts[0])
../../_images/examples_basic_forecasting_tutorial_tutorial_57_0.png

Observe that we cannot actually see any prediction intervals in the predictions. This is expected since the model that we defined does not do probabilistic forecasting but it just gives point estimates. By requiring 100 sample paths (defined in make_evaluation_predictions) in such a network, we get 100 times the same output.

In [39]:
evaluator = Evaluator(quantiles=[0.1, 0.5, 0.9])
agg_metrics, item_metrics = evaluator(iter(tss), iter(forecasts), num_series=len(dataset.test))
Running evaluation: 100%|██████████| 414/414 [00:01<00:00, 253.26it/s]
In [40]:
print(json.dumps(agg_metrics, indent=4))
{
    "MSE": 7059718.912101227,
    "abs_error": 9248744.83713913,
    "abs_target_sum": 145558863.59960938,
    "abs_target_mean": 7324.822041043146,
    "seasonal_error": 336.9046924038305,
    "MASE": 4.798471894715852,
    "sMAPE": 0.23087650267089702,
    "MSIS": 191.93887487643667,
    "QuantileLoss[0.1]": 10660522.745114041,
    "Coverage[0.1]": 0.507799919484702,
    "QuantileLoss[0.5]": 9248744.730575085,
    "Coverage[0.5]": 0.507799919484702,
    "QuantileLoss[0.9]": 7836966.71603613,
    "Coverage[0.9]": 0.507799919484702,
    "RMSE": 2657.0131561776707,
    "NRMSE": 0.36274098418905465,
    "ND": 0.06353955099965448,
    "wQuantileLoss[0.1]": 0.07323856810560213,
    "wQuantileLoss[0.5]": 0.06353955026755172,
    "wQuantileLoss[0.9]": 0.053840532429501334,
    "mean_wQuantileLoss": 0.06353955026755172,
    "MAE_Coverage": 0.269266639828234
}
In [41]:
item_metrics.head(10)
Out[41]:
item_id MSE abs_error abs_target_sum abs_target_mean seasonal_error MASE sMAPE MSIS QuantileLoss[0.1] Coverage[0.1] QuantileLoss[0.5] Coverage[0.5] QuantileLoss[0.9] Coverage[0.9]
0 NaN 2.371210e+03 1970.007324 31644.0 659.250000 42.371302 0.968623 0.069210 38.744924 2032.522949 0.541667 1970.007324 0.541667 1907.491699 0.541667
1 NaN 1.708196e+05 16645.630859 124149.0 2586.437500 165.107988 2.100346 0.126000 84.013860 29239.730664 0.895833 16645.630859 0.895833 4051.531055 0.895833
2 NaN 4.289714e+04 8005.822754 65030.0 1354.791667 78.889053 2.114209 0.119511 84.568373 2194.095776 0.166667 8005.822632 0.166667 13817.549487 0.166667
3 NaN 2.013706e+05 16957.785156 235783.0 4912.145833 258.982249 1.364137 0.070868 54.565476 7099.391162 0.166667 16957.785889 0.166667 26816.180615 0.166667
4 NaN 1.413158e+05 14113.250000 131088.0 2731.000000 200.494083 1.466507 0.108526 58.660293 14054.674902 0.500000 14113.249512 0.500000 14171.824121 0.500000
5 NaN 2.449111e+05 19287.675781 303379.0 6320.395833 212.875740 1.887611 0.065188 75.504441 22415.683594 0.562500 19287.675781 0.562500 16159.667969 0.562500
6 NaN 9.521857e+06 115624.031250 1985325.0 41360.937500 1947.687870 1.236766 0.059011 49.470634 40486.117578 0.208333 115624.025391 0.208333 190761.933203 0.208333
7 NaN 5.320664e+06 92311.296875 1540706.0 32098.041667 1624.044379 1.184175 0.066309 47.366984 61802.628516 0.333333 92311.298828 0.333333 122819.969141 0.333333
8 NaN 9.081012e+06 119211.750000 1640860.0 34184.583333 1850.988166 1.341758 0.070482 53.670319 182406.514062 0.687500 119211.742188 0.687500 56016.970312 0.687500
9 NaN 7.976385e+02 1067.646240 21408.0 446.000000 10.526627 2.112987 0.050965 84.519497 503.818030 0.208333 1067.646301 0.208333 1631.474573 0.208333
In [42]:
item_metrics.plot(x='MSIS', y='MASE', kind='scatter')
plt.grid(which="both")
plt.show()
../../_images/examples_basic_forecasting_tutorial_tutorial_62_0.png