[Download]

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', 'exchange_rate_nips', 'electricity_nips', 'traffic_nips', 'solar_nips', 'wiki-rolling_nips', 'taxi_30min', 'm4_hourly', 'm4_daily', 'm4_weekly', 'm4_monthly', 'm4_quarterly', 'm4_yearly', 'm5']

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)
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
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)
  0%|          | 0/100 [00:00<?, ?it/s]
learning rate from ``lr_scheduler`` has been overwritten by ``learning_rate`` in optimizer.
100%|██████████| 100/100 [00:00<00:00, 136.00it/s, epoch=1/5, avg_epoch_loss=5.47]
100%|██████████| 100/100 [00:00<00:00, 148.79it/s, epoch=2/5, avg_epoch_loss=5.03]
100%|██████████| 100/100 [00:00<00:00, 149.71it/s, epoch=3/5, avg_epoch_loss=4.73]
100%|██████████| 100/100 [00:00<00:00, 139.29it/s, epoch=4/5, avg_epoch_loss=4.67]
100%|██████████| 100/100 [00:00<00:00, 147.24it/s, epoch=5/5, avg_epoch_loss=4.69]

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_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:
 [678.2261  580.6828  557.8809  464.1978  514.54205 427.6854  468.7312
 524.2117  458.48663 557.7436  608.4296  686.28156 747.6431  769.39136
 863.8387  816.5097  784.5161  786.9378  830.2128  817.2316  841.11505
 774.81525 730.3486  672.6686  634.10144 584.23193 635.45276 515.27515
 505.85257 528.20746 483.71988 456.2665  523.257   539.4473  647.9726
 678.9236  806.6993  887.2035  828.574   813.1903  841.1396  877.74475
 826.8884  855.6368  894.4874  733.8635  702.4635  737.7713 ]
0.5-quantile (median) of the future window:
 [692.8289  566.53143 576.77386 472.06    488.82272 437.56717 481.01553
 535.8437  460.34384 534.44073 608.01184 681.93115 767.3685  773.1614
 861.2773  834.272   813.4933  796.3204  835.0291  814.61163 877.8365
 761.68146 736.03906 670.8835  630.60297 606.00415 639.9325  522.4326
 515.87866 536.9678  476.2396  446.25412 512.9928  535.16864 630.8398
 663.8413  785.49115 899.86707 826.87384 839.3145  829.1985  885.3145
 827.94977 837.6075  868.597   747.5629  717.9478  735.87335]

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 [00:00<00:00, 3689.97it/s]

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

In [28]:
print(json.dumps(agg_metrics, indent=4))
{
    "MSE": 9314655.693985011,
    "abs_error": 9661779.157546997,
    "abs_target_sum": 145558863.59960938,
    "abs_target_mean": 7324.822041043146,
    "seasonal_error": 336.9046924038305,
    "MASE": 3.981241892904796,
    "MAPE": 0.268757251908333,
    "sMAPE": 0.1931702274585429,
    "OWA": NaN,
    "MSIS": 38.419487705566205,
    "QuantileLoss[0.1]": 5599168.646871282,
    "Coverage[0.1]": 0.09395128824476649,
    "QuantileLoss[0.5]": 9661779.25660801,
    "Coverage[0.5]": 0.4928039452495974,
    "QuantileLoss[0.9]": 7226813.397608182,
    "Coverage[0.9]": 0.8873289049919484,
    "RMSE": 3051.9920861602855,
    "NRMSE": 0.41666433246556306,
    "ND": 0.06637712687922445,
    "wQuantileLoss[0.1]": 0.03846669662297575,
    "wQuantileLoss[0.5]": 0.06637712755978083,
    "wQuantileLoss[0.9]": 0.0496487346692062,
    "mean_wQuantileLoss": 0.05149751961732093,
    "MAE_Coverage": 0.008638620504562577
}

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 MAPE sMAPE OWA MSIS QuantileLoss[0.1] Coverage[0.1] QuantileLoss[0.5] Coverage[0.5] QuantileLoss[0.9] Coverage[0.9]
0 0.0 2972.818685 1972.183838 31644.0 659.250000 42.371302 0.969693 0.062990 0.061077 NaN 6.452755 1155.814429 0.0000 1972.183838 0.708333 1468.937256 1.000000
1 1.0 147499.937500 16444.986328 124149.0 2586.437500 165.107988 2.075029 0.143409 0.130307 NaN 12.027847 4360.103979 0.1875 16444.986694 0.979167 8701.296729 1.000000
2 2.0 39184.164062 6833.100586 65030.0 1354.791667 78.889053 1.804512 0.093007 0.099979 NaN 11.619019 3765.794153 0.0000 6833.100586 0.187500 2184.618018 0.812500
3 3.0 252178.020833 19680.253906 235783.0 4912.145833 258.982249 1.583141 0.081148 0.082231 NaN 7.438812 9895.550391 0.0000 19680.253418 0.416667 8119.014551 0.958333
4 4.0 77691.661458 9567.276367 131088.0 2731.000000 200.494083 0.994135 0.073843 0.069873 NaN 6.455980 4699.394312 0.0625 9567.276245 0.708333 7498.198047 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,
        )

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)
  0%|          | 0/100 [00:00<?, ?it/s]
learning rate from ``lr_scheduler`` has been overwritten by ``learning_rate`` in optimizer.
100%|██████████| 100/100 [00:00<00:00, 156.32it/s, epoch=1/5, avg_epoch_loss=2.56e+3]
100%|██████████| 100/100 [00:00<00:00, 155.86it/s, epoch=2/5, avg_epoch_loss=1.29e+3]
100%|██████████| 100/100 [00:00<00:00, 152.89it/s, epoch=3/5, avg_epoch_loss=1.13e+3]
100%|██████████| 100/100 [00:00<00:00, 152.75it/s, epoch=4/5, avg_epoch_loss=1.11e+3]
100%|██████████| 100/100 [00:00<00:00, 153.34it/s, epoch=5/5, avg_epoch_loss=1.07e+3]
In [36]:
forecast_it, ts_it = make_evaluation_predictions(
    dataset=dataset.test,
    predictor=predictor,
    num_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 [00:00<00:00, 5108.61it/s]
In [40]:
print(json.dumps(agg_metrics, indent=4))
{
    "MSE": 36599322.47120415,
    "abs_error": 19195483.781196594,
    "abs_target_sum": 145558863.59960938,
    "abs_target_mean": 7324.822041043146,
    "seasonal_error": 336.9046924038305,
    "MASE": 9.778880760580046,
    "MAPE": 0.5501193984298947,
    "sMAPE": 0.2713944442717952,
    "OWA": NaN,
    "MSIS": 391.1552287209777,
    "QuantileLoss[0.1]": 31880462.642330453,
    "Coverage[0.1]": 0.7308776167471819,
    "QuantileLoss[0.5]": 19195483.453031063,
    "Coverage[0.5]": 0.7308776167471819,
    "QuantileLoss[0.9]": 6510504.26373167,
    "Coverage[0.9]": 0.7308776167471819,
    "RMSE": 6049.737388614827,
    "NRMSE": 0.8259227807469394,
    "ND": 0.13187437237760977,
    "wQuantileLoss[0.1]": 0.21902110152511528,
    "wQuantileLoss[0.5]": 0.13187437012308864,
    "wQuantileLoss[0.9]": 0.044727638721061995,
    "mean_wQuantileLoss": 0.13187437012308864,
    "MAE_Coverage": 0.34362587224906066
}
In [41]:
item_metrics.head(10)
Out[41]:
item_id MSE abs_error abs_target_sum abs_target_mean seasonal_error MASE MAPE sMAPE OWA MSIS QuantileLoss[0.1] Coverage[0.1] QuantileLoss[0.5] Coverage[0.5] QuantileLoss[0.9] Coverage[0.9]
0 0.0 1.100802e+04 4351.830078 31644.0 659.250000 42.371302 2.139730 0.145146 0.132504 NaN 85.589174 7519.076111 0.895833 4351.829773 0.895833 1184.583435 0.895833
1 1.0 5.095260e+05 31574.125000 124149.0 2586.437500 165.107988 3.984025 0.277157 0.236661 NaN 159.360980 56540.375195 0.958333 31574.124023 0.958333 6607.872852 0.958333
2 2.0 6.791075e+04 10876.902344 65030.0 1354.791667 78.889053 2.872415 0.170119 0.171419 NaN 114.896625 7983.323694 0.437500 10876.902649 0.437500 13770.481604 0.437500
3 3.0 4.333768e+05 26023.630859 235783.0 4912.145833 258.982249 2.093421 0.120302 0.111649 NaN 83.736856 40228.492236 0.770833 26023.631104 0.770833 11818.769971 0.770833
4 4.0 1.821795e+05 17444.771484 131088.0 2731.000000 200.494083 1.812686 0.151376 0.139510 NaN 72.507426 23275.974097 0.604167 17444.770874 0.604167 11613.567651 0.604167
5 5.0 8.434385e+05 36431.312500 303379.0 6320.395833 212.875740 3.565393 0.122798 0.113709 NaN 142.615733 59667.046582 0.812500 36431.314941 0.812500 13195.583301 0.812500
6 6.0 2.973278e+07 216019.625000 1985325.0 41360.937500 1947.687870 2.310642 0.120222 0.112733 NaN 92.425658 278649.030859 0.541667 216019.623047 0.541667 153390.215234 0.541667
7 7.0 1.148527e+07 133652.953125 1540706.0 32098.041667 1624.044379 1.714508 0.093124 0.088091 NaN 68.580305 193292.737109 0.666667 133652.951172 0.666667 74013.165234 0.666667
8 8.0 3.820493e+07 263305.437500 1640860.0 34184.583333 1850.988166 2.963568 0.168200 0.151561 NaN 118.542740 473695.111328 0.979167 263305.447266 0.979167 52915.783203 0.979167
9 9.0 3.521698e+03 2278.766357 21408.0 446.000000 10.526627 4.509925 0.111392 0.104047 NaN 180.396988 3384.442816 0.687500 2278.766327 0.687500 1173.089838 0.687500
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