# Electricity Load Forecast

In this example we will show how to perform electricity load forecasting considering a model capable of handling multiple seasonalities (MSTL).

## Introduction

Some time series are generated from very low frequency data. These data generally exhibit multiple seasonalities. For example, hourly data may exhibit repeated patterns every hour (every 24 observations) or every day (every 24 * 7, hours per day, observations). This is the case for electricity load. Electricity load may vary hourly, e.g., during the evenings electricity consumption may be expected to increase. But also, the electricity load varies by week. Perhaps on weekends there is an increase in electrical activity.

In this example we will show how to model the two seasonalities of the time series to generate accurate forecasts in a short time. We will use hourly PJM electricity load data. The original data can be found here.

## Libraries

In this example we will use the following libraries:

`StatsForecast`

. Lightning ⚡️ fast forecasting with statistical and econometric models. Includes the MSTL model for multiple seasonalities.`DatasetsForecast`

. Used to evaluate the performance of the forecasts.`Prophet`

. Benchmark model developed by Facebook.`NeuralProphet`

. Deep Learning version of`Prophet`

. Used as benchark.

```
!pip install statsforecast
!pip install datasetsforecast
!pip install prophet
!pip install "neuralprophet[live]"
```

## Forecast using Multiple Seasonalities

### Electricity Load Data

According to the dataset’s page,

PJM Interconnection LLC (PJM) is a regional transmission organization (RTO) in the United States. It is part of the Eastern Interconnection grid operating an electric transmission system serving all or parts of Delaware, Illinois, Indiana, Kentucky, Maryland, Michigan, New Jersey, North Carolina, Ohio, Pennsylvania, Tennessee, Virginia, West Virginia, and the District of Columbia. The hourly power consumption data comes from PJM’s website and are in megawatts (MW).

Let’s take a look to the data.

```
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
pd.plotting.register_matplotlib_converters()
plt.rc("figure", figsize=(10, 8))
plt.rc("font", size=10)
```

```
df = pd.read_csv('https://raw.githubusercontent.com/panambY/Hourly_Energy_Consumption/master/data/PJM_Load_hourly.csv')
df.columns = ['ds', 'y']
df.insert(0, 'unique_id', 'PJM_Load_hourly')
df['ds'] = pd.to_datetime(df['ds'])
df = df.sort_values(['unique_id', 'ds']).reset_index(drop=True)
df.tail()
```

unique_id | ds | y | |
---|---|---|---|

32891 | PJM_Load_hourly | 2001-12-31 20:00:00 | 36392.0 |

32892 | PJM_Load_hourly | 2001-12-31 21:00:00 | 35082.0 |

32893 | PJM_Load_hourly | 2001-12-31 22:00:00 | 33890.0 |

32894 | PJM_Load_hourly | 2001-12-31 23:00:00 | 32590.0 |

32895 | PJM_Load_hourly | 2002-01-01 00:00:00 | 31569.0 |

```
df.plot(x='ds', y='y')
```

We clearly observe that the time series exhibits seasonal patterns.
Moreover, the time series contains `32,896`

observations, so it is
necessary to use very computationally efficient methods to display them
in production.

### MSTL model

The
`MSTL`

(Multiple Seasonal-Trend decomposition using LOESS) model, originally
developed by Kasun Bandara, Rob J Hyndman and Christoph
Bergmeir, decomposes the time series
in multiple seasonalities using a Local Polynomial Regression (LOESS).
Then it forecasts the trend using a custom non-seasonal model and each
seasonality using a
`SeasonalNaive`

model.

`StatsForecast`

contains a fast implementation of the
`MSTL`

model. Also, the decomposition of the time series can be calculated.

```
from statsforecast import StatsForecast
from statsforecast.models import MSTL, AutoARIMA, SeasonalNaive
from statsforecast.utils import AirPassengers as ap
```

First we must define the model parameters. As mentioned before, the
electricity load presents seasonalities every 24 hours (Hourly) and
every 24 * 7 (Daily) hours. Therefore, we will use `[24, 24 * 7]`

as
the seasonalities that the MSTL model receives. We must also specify the
manner in which the trend will be forecasted. In this case we will use
the
`AutoARIMA`

model.

```
mstl = MSTL(
season_length=[24, 24 * 7], # seasonalities of the time series
trend_forecaster=AutoARIMA() # model used to forecast trend
)
```

Once the model is instantiated, we have to instantiate the
`StatsForecast`

class to create forecasts.

```
sf = StatsForecast(
models=[mstl], # model used to fit each time series
freq='H', # frequency of the data
)
```

#### Fit the model

Afer that, we just have to use the `fit`

method to fit each model to
each time series.

```
sf = sf.fit(df=df)
```

#### Decompose the time series in multiple seasonalities

Once the model is fitted, we can access the decomposition using the
`fitted_`

attribute of
`StatsForecast`

.
This attribute stores all relevant information of the fitted models for
each of the time series.

In this case we are fitting a single model for a single time series, so
by accessing the fitted_ location [0, 0] we will find the relevant
information of our model. The
`MSTL`

class generates a `model_`

attribute that contains the way the series
was decomposed.

```
sf.fitted_[0, 0].model_
```

data | trend | seasonal24 | seasonal168 | remainder | |
---|---|---|---|---|---|

0 | 22259.0 | 26183.898892 | -5215.124554 | 609.000432 | 681.225229 |

1 | 21244.0 | 26181.599305 | -6255.673234 | 603.823918 | 714.250011 |

2 | 20651.0 | 26179.294886 | -6905.329895 | 636.820423 | 740.214587 |

3 | 20421.0 | 26176.985472 | -7073.420118 | 615.825999 | 701.608647 |

4 | 20713.0 | 26174.670877 | -7062.395760 | 991.521912 | 609.202971 |

… | … | … | … | … | … |

32891 | 36392.0 | 33123.552727 | 4387.149171 | -488.177882 | -630.524015 |

32892 | 35082.0 | 33148.242575 | 3479.852929 | -682.928737 | -863.166767 |

32893 | 33890.0 | 33172.926165 | 2307.808829 | -650.566775 | -940.168219 |

32894 | 32590.0 | 33197.603322 | 748.587723 | -555.177849 | -801.013195 |

32895 | 31569.0 | 33222.273902 | -967.124123 | -265.895357 | -420.254422 |

Let’s look graphically at the different components of the time series.

```
sf.fitted_[0, 0].model_.tail(24 * 28).plot(subplots=True, grid=True)
plt.tight_layout()
plt.show()
```

We observe that there is a clear trend towards the high (orange line).
This component would be predicted with the
`AutoARIMA`

model. We can also observe that every 24 hours and every `24 * 7`

hours
there is a very well defined pattern. These two components will be
forecast separately using a
`SeasonalNaive`

model.

#### Produce forecasts

To generate forecasts we only have to use the `predict`

method
specifying the forecast horizon (`h`

). In addition, to calculate
prediction intervals associated to the forecasts, we can include the
parameter `level`

that receives a list of levels of the prediction
intervals we want to build. In this case we will only calculate the 90%
forecast interval (`level=[90]`

).

```
forecasts = sf.predict(h=24, level=[90])
forecasts.head()
```

ds | MSTL | MSTL-lo-90 | MSTL-hi-90 | |
---|---|---|---|---|

unique_id | ||||

PJM_Load_hourly | 2002-01-01 01:00:00 | 29956.744141 | 29585.187500 | 30328.298828 |

PJM_Load_hourly | 2002-01-01 02:00:00 | 29057.691406 | 28407.498047 | 29707.884766 |

PJM_Load_hourly | 2002-01-01 03:00:00 | 28654.699219 | 27767.101562 | 29542.298828 |

PJM_Load_hourly | 2002-01-01 04:00:00 | 28499.009766 | 27407.640625 | 29590.378906 |

PJM_Load_hourly | 2002-01-01 05:00:00 | 28821.716797 | 27552.236328 | 30091.197266 |

Let’s look at our forecasts graphically.

```
_, ax = plt.subplots(1, 1, figsize = (20, 7))
df_plot = pd.concat([df, forecasts]).set_index('ds').tail(24 * 7)
df_plot[['y', 'MSTL']].plot(ax=ax, linewidth=2)
ax.fill_between(df_plot.index,
df_plot['MSTL-lo-90'],
df_plot['MSTL-hi-90'],
alpha=.35,
color='orange',
label='MSTL-level-90')
ax.set_title('PJM Load Hourly', fontsize=22)
ax.set_ylabel('Electricity Load', fontsize=20)
ax.set_xlabel('Timestamp [t]', fontsize=20)
ax.legend(prop={'size': 15})
ax.grid()
```

In the next section we will plot different models so it is convenient to reuse the previous code with the following function.

```
def plot_forecasts(y_hist, y_true, y_pred, models):
_, ax = plt.subplots(1, 1, figsize = (20, 7))
y_true = y_true.merge(y_pred, how='left', on=['unique_id', 'ds'])
df_plot = pd.concat([y_hist, y_true]).set_index('ds').tail(24 * 7)
df_plot[['y'] + models].plot(ax=ax, linewidth=2)
colors = ['orange', 'green', 'red']
for model, color in zip(models, colors):
ax.fill_between(df_plot.index,
df_plot[f'{model}-lo-90'],
df_plot[f'{model}-hi-90'],
alpha=.35,
color=color,
label=f'{model}-level-90')
ax.set_title('PJM Load Hourly', fontsize=22)
ax.set_ylabel('Electricity Load', fontsize=20)
ax.set_xlabel('Timestamp [t]', fontsize=20)
ax.legend(prop={'size': 15})
ax.grid()
```

### Performance of the MSTL model

#### Split Train/Test sets

To validate the accuracy of the
`MSTL`

model, we will show its performance on unseen data. We will use a
classical time series technique that consists of dividing the data into
a training set and a test set. We will leave the last 24 observations
(the last day) as the test set. So the model will train on `32,872`

observations.

```
df_test = df.tail(24)
df_train = df.drop(df_test.index)
```

#### MSTL model

In addition to the
`MSTL`

model, we will include the
`SeasonalNaive`

model as a benchmark to validate the added value of the
`MSTL`

model. Including
`StatsForecast`

models is as simple as adding them to the list of models to be fitted.

```
sf = StatsForecast(
models=[mstl, SeasonalNaive(season_length=24)], # add SeasonalNaive model to the list
freq='H'
)
```

To measure the fitting time we will use the `time`

module.

```
from time import time
```

To retrieve the forecasts of the test set we only have to do fit and predict as before.

```
init = time()
sf = sf.fit(df=df_train)
forecasts_test = sf.predict(h=len(df_test), level=[90])
end = time()
forecasts_test.head()
```

ds | MSTL | MSTL-lo-90 | MSTL-hi-90 | SeasonalNaive | SeasonalNaive-lo-90 | SeasonalNaive-hi-90 | |
---|---|---|---|---|---|---|---|

unique_id | |||||||

PJM_Load_hourly | 2001-12-31 01:00:00 | 28345.212891 | 27973.572266 | 28716.853516 | 28326.0 | 23468.693359 | 33183.304688 |

PJM_Load_hourly | 2001-12-31 02:00:00 | 27567.455078 | 26917.085938 | 28217.824219 | 27362.0 | 22504.693359 | 32219.306641 |

PJM_Load_hourly | 2001-12-31 03:00:00 | 27260.001953 | 26372.138672 | 28147.865234 | 27108.0 | 22250.693359 | 31965.306641 |

PJM_Load_hourly | 2001-12-31 04:00:00 | 27328.125000 | 26236.410156 | 28419.839844 | 26865.0 | 22007.693359 | 31722.306641 |

PJM_Load_hourly | 2001-12-31 05:00:00 | 27640.673828 | 26370.773438 | 28910.572266 | 26808.0 | 21950.693359 | 31665.306641 |

```
time_mstl = (end - init) / 60
print(f'MSTL Time: {time_mstl:.2f} minutes')
```

```
MSTL Time: 0.22 minutes
```

Then we were able to generate forecasts for the next 24 hours. Now let’s look at the graphical comparison of the forecasts with the actual values.

```
plot_forecasts(df_train, df_test, forecasts_test, models=['MSTL', 'SeasonalNaive'])
```

Let’s look at those produced only by
`MSTL`

.

```
plot_forecasts(df_train, df_test, forecasts_test, models=['MSTL'])
```

We note that
`MSTL`

produces very accurate forecasts that follow the behavior of the time
series. Now let us calculate numerically the accuracy of the model. We
will use the following metrics: `MAE`

, `MAPE`

, `MASE`

, `RMSE`

, `SMAPE`

.

```
from datasetsforecast.losses import (
mae, mape, mase, rmse, smape
)
```

```
def evaluate_performace(y_hist, y_true, y_pred, models):
y_true = y_true.merge(y_pred, how='left', on=['unique_id', 'ds'])
evaluation = {}
for model in models:
evaluation[model] = {}
for metric in [mase, mae, mape, rmse, smape]:
metric_name = metric.__name__
if metric_name == 'mase':
evaluation[model][metric_name] = metric(y_true['y'].values,
y_true[model].values,
y_hist['y'].values, seasonality=24)
else:
evaluation[model][metric_name] = metric(y_true['y'].values, y_true[model].values)
return pd.DataFrame(evaluation).T
```

```
evaluate_performace(df_train, df_test, forecasts_test, models=['MSTL', 'SeasonalNaive'])
```

mase | mae | mape | rmse | smape | |
---|---|---|---|---|---|

MSTL | 0.341926 | 709.932048 | 2.182804 | 892.888012 | 2.162832 |

SeasonalNaive | 0.894653 | 1857.541667 | 5.648190 | 2201.384101 | 5.868604 |

We observe that
`MSTL`

has an improvement of about 60% over the
`SeasonalNaive`

method in the test set measured in `MASE`

.

#### Comparison with Prophet

One of the most widely used models for time series forecasting is
`Prophet`

. This model is known for its ability to model different
seasonalities (weekly, daily yearly). We will use this model as a
benchmark to see if the
`MSTL`

adds value for this time series.

```
from prophet import Prophet
# create prophet model
prophet = Prophet(interval_width=0.9)
init = time()
prophet.fit(df_train)
# produce forecasts
future = prophet.make_future_dataframe(periods=len(df_test), freq='H', include_history=False)
forecast_prophet = prophet.predict(future)
end = time()
# data wrangling
forecast_prophet = forecast_prophet[['ds', 'yhat', 'yhat_lower', 'yhat_upper']]
forecast_prophet.columns = ['ds', 'Prophet', 'Prophet-lo-90', 'Prophet-hi-90']
forecast_prophet.insert(0, 'unique_id', 'PJM_Load_hourly')
forecast_prophet.head()
```

```
23:41:40 - cmdstanpy - INFO - Chain [1] start processing
23:41:56 - cmdstanpy - INFO - Chain [1] done processing
```

unique_id | ds | Prophet | Prophet-lo-90 | Prophet-hi-90 | |
---|---|---|---|---|---|

0 | PJM_Load_hourly | 2001-12-31 01:00:00 | 25317.658386 | 20757.919539 | 30313.561582 |

1 | PJM_Load_hourly | 2001-12-31 02:00:00 | 24024.188077 | 19304.093939 | 28667.495805 |

2 | PJM_Load_hourly | 2001-12-31 03:00:00 | 23348.306824 | 18608.982825 | 28497.334752 |

3 | PJM_Load_hourly | 2001-12-31 04:00:00 | 23356.150113 | 18721.142270 | 28136.888630 |

4 | PJM_Load_hourly | 2001-12-31 05:00:00 | 24130.861217 | 19896.188455 | 28970.202276 |

```
time_prophet = (end - init) / 60
print(f'Prophet Time: {time_prophet:.2f} minutes')
```

```
Prophet Time: 0.30 minutes
```

```
times = pd.DataFrame({'model': ['MSTL', 'Prophet'], 'time (mins)': [time_mstl, time_prophet]})
times
```

model | time (mins) | |
---|---|---|

0 | MSTL | 0.217266 |

1 | Prophet | 0.301172 |

We observe that the time required for `Prophet`

to perform the fit and
predict pipeline is greater than
`MSTL`

.
Let’s look at the forecasts produced by `Prophet`

.

```
forecasts_test = forecasts_test.merge(forecast_prophet, how='left', on=['unique_id', 'ds'])
```

```
plot_forecasts(df_train, df_test, forecasts_test, models=['MSTL', 'SeasonalNaive', 'Prophet'])
```

We note that `Prophet`

is able to capture the overall behavior of the
time series. However, in some cases it produces forecasts well below the
actual value. It also does not correctly adjust the valleys.

```
evaluate_performace(df_train, df_test, forecasts_test, models=['MSTL', 'Prophet', 'SeasonalNaive'])
```

mase | mae | mape | rmse | smape | |
---|---|---|---|---|---|

MSTL | 0.341926 | 709.932048 | 2.182804 | 892.888012 | 2.162832 |

Prophet | 1.094768 | 2273.036373 | 7.343292 | 2709.400341 | 7.688665 |

SeasonalNaive | 0.894653 | 1857.541667 | 5.648190 | 2201.384101 | 5.868604 |

In terms of accuracy, `Prophet`

is not able to produce better forecasts
than the
`SeasonalNaive`

model, however, the
`MSTL`

model improves `Prophet`

’s forecasts by 69% (`MASE`

).

#### Comparison with NeuralProphet

`NeuralProphet`

is the version of `Prophet`

using deep learning. This
model is also capable of handling different seasonalities so we will
also use it as a benchmark.

```
from neuralprophet import NeuralProphet
neuralprophet = NeuralProphet(quantiles=[0.05, 0.95])
init = time()
neuralprophet.fit(df_train.drop(columns='unique_id'))
future = neuralprophet.make_future_dataframe(df=df_train.drop(columns='unique_id'), periods=len(df_test))
forecast_np = neuralprophet.predict(future)
end = time()
forecast_np = forecast_np[['ds', 'yhat1', 'yhat1 5.0%', 'yhat1 95.0%']]
forecast_np.columns = ['ds', 'NeuralProphet', 'NeuralProphet-lo-90', 'NeuralProphet-hi-90']
forecast_np.insert(0, 'unique_id', 'PJM_Load_hourly')
forecast_np.head()
```

```
WARNING - (NP.forecaster.fit) - When Global modeling with local normalization, metrics are displayed in normalized scale.
INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 99.973% of the data.
INFO - (NP.df_utils._infer_frequency) - Dataframe freq automatically defined as H
INFO - (NP.config.init_data_params) - Setting normalization to global as only one dataframe provided for training.
INFO - (NP.config.set_auto_batch_epoch) - Auto-set batch_size to 64
INFO - (NP.config.set_auto_batch_epoch) - Auto-set epochs to 76
```

```
Finding best initial lr: 0%| | 0/262 [00:00<?, ?it/s]
```

```
Training: 0it [00:00, ?it/s]
```

```
INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 99.973% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - H
INFO - (NP.df_utils.return_df_in_original_format) - Returning df with no ID column
INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 95.833% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - H
INFO - (NP.df_utils._infer_frequency) - Major frequency H corresponds to 95.833% of the data.
INFO - (NP.df_utils._infer_frequency) - Defined frequency is equal to major frequency - H
```

```
Predicting: 514it [00:00, ?it/s]
```

```
INFO - (NP.df_utils.return_df_in_original_format) - Returning df with no ID column
```

unique_id | ds | NeuralProphet | NeuralProphet-lo-90 | NeuralProphet-hi-90 | |
---|---|---|---|---|---|

0 | PJM_Load_hourly | 2001-12-31 01:00:00 | 25019.892578 | 22296.675781 | 27408.724609 |

1 | PJM_Load_hourly | 2001-12-31 02:00:00 | 24128.816406 | 21439.851562 | 26551.615234 |

2 | PJM_Load_hourly | 2001-12-31 03:00:00 | 23736.679688 | 20961.978516 | 26289.349609 |

3 | PJM_Load_hourly | 2001-12-31 04:00:00 | 23476.744141 | 20731.619141 | 26050.443359 |

4 | PJM_Load_hourly | 2001-12-31 05:00:00 | 23899.162109 | 21217.503906 | 26449.603516 |

```
time_np = (end - init) / 60
print(f'Prophet Time: {time_np:.2f} minutes')
```

```
Prophet Time: 2.95 minutes
```

```
times = times.append({'model': 'NeuralProphet', 'time (mins)': time_np}, ignore_index=True)
times
```

model | time (mins) | |
---|---|---|

0 | MSTL | 0.217266 |

1 | Prophet | 0.301172 |

2 | NeuralProphet | 2.946358 |

We observe that `NeuralProphet`

requires a longer processing time than
`Prophet`

and
`MSTL`

.

```
forecasts_test = forecasts_test.merge(forecast_np, how='left', on=['unique_id', 'ds'])
```

```
plot_forecasts(df_train, df_test, forecasts_test, models=['MSTL', 'NeuralProphet', 'Prophet'])
```

The forecasts graph shows that `NeuralProphet`

generates very similar
results to `Prophet`

, as expected.

```
evaluate_performace(df_train, df_test, forecasts_test, models=['MSTL', 'NeuralProphet', 'Prophet', 'SeasonalNaive'])
```

mase | mae | mape | rmse | smape | |
---|---|---|---|---|---|

MSTL | 0.341926 | 709.932048 | 2.182804 | 892.888012 | 2.162832 |

NeuralProphet | 1.084915 | 2252.578613 | 7.280202 | 2671.145730 | 7.615492 |

Prophet | 1.094768 | 2273.036373 | 7.343292 | 2709.400341 | 7.688665 |

SeasonalNaive | 0.894653 | 1857.541667 | 5.648190 | 2201.384101 | 5.868604 |

With respect to numerical evaluation, `NeuralProphet`

improves the
results of `Prophet`

, as expected, however,
`MSTL`

improves over `NeuralProphet`

’s foreacasts by 68% (`MASE`

).

ImportantThe performance of

`NeuralProphet`

can be improved using hyperparameter optimization, which can increase the fitting time significantly. In this example we show its performance with the default version.

## Conclusion

In this post we introduced
`MSTL`

,
a model originally developed by Kasun Bandara, Rob Hyndman and
Christoph Bergmeir capable of
handling time series with multiple seasonalities. We also showed that
for the PJM electricity load time series offers better performance in
time and accuracy than the `Prophet`

and `NeuralProphet`

models.