Probabilistic Forecasting

• produce low/high scenarios of forecasts
• quantify uncertainty around forecasts
• produce expected range of variation of forecasts

Train Forecaster

import numpy as np

from sktime.datasets import load_airline
from sktime.forecasting.theta import ThetaForecaster

# until fit, identical with the simple workflow
y = load_airline()

fh = np.arange(1, 13)

forecaster = ThetaForecaster(sp=12)
forecaster.fit(y, fh=fh)
y_pred = forecaster.predict()

Predict_interval

• predict_interval(fh=None, X=None, coverage=0.90)
• produces symmetric forecasting intervals
• 0.5 - coverage/2 = 0.05, 0.5 + coverage/2 = 0.95

coverage = 0.9
y_pred_ints = forecaster.predict_interval(coverage=coverage)
y_pred_ints.head()

	Coverage
	0.9
	lower	upper
1961-01	418.280121	464.281951
1961-02	402.215881	456.888055
1961-03	459.966113	522.110500
1961-04	442.589309	511.399214
1961-05	443.525027	518.409480

from sktime.utils.plotting import plot_series

plot_series(y, y_pred, labels=["y", "y_pred"], pred_interval=y_pred_ints)

(<Figure size 1600x400 with 1 Axes>,
 <Axes: ylabel='Number of airline passengers'>)

fig, ax = plot_series(y, y_pred, labels=["y", "y_pred"])
ax.fill_between(
    ax.get_lines()[-1].get_xdata(), # x axis
    y_pred_ints["Coverage"][coverage]["lower"], # y_low axis
    y_pred_ints["Coverage"][coverage]["upper"], # y_high
    alpha=0.2,
    color=ax.get_lines()[-1].get_c(),
    label=f"{coverage} cov.pred.intervals",
)
ax.legend()

<matplotlib.legend.Legend at 0x133d629d0>

# Multiple Coverages
coverages = [0.5, 0.8, 0.95]
y_pred_ints = forecaster.predict_interval(coverage=coverages)
y_pred_ints

	Coverage
	0.50		0.80		0.95
	lower	upper	lower	upper	lower	upper
1961-01	431.849266	450.712806	423.360378	459.201694	413.873755	468.688317
1961-02	418.342514	440.761421	408.253656	450.850279	396.979011	462.124925
1961-03	478.296822	503.779790	466.829089	515.247523	454.013504	528.063109
1961-04	462.886144	491.102379	450.188398	503.800124	435.998232	517.990291
1961-05	465.613670	496.320837	451.794965	510.139542	436.352089	525.582418
1961-06	530.331440	563.342111	515.476124	578.197428	498.874797	594.798754
1961-07	586.791063	621.954661	570.966896	637.778829	553.282845	655.462879
1961-08	584.116789	621.308897	567.379760	638.045925	548.675556	656.750129
1961-09	505.795123	544.910684	488.192511	562.513297	468.520987	582.184821
1961-10	437.370840	478.319605	418.943257	496.747188	398.349800	517.340645
1961-11	377.660798	420.364142	358.443627	439.581313	336.967779	461.057161
1961-12	426.638370	471.026993	406.662797	491.002565	384.339409	513.325954

fig, ax = plot_series(y, y_pred, labels=["y", "y_pred"])

colors = ['red', 'blue', 'yellow']

for i, coverage in enumerate(coverages):
    ax.fill_between(
        ax.get_lines()[-1].get_xdata(), # x axis
        y_pred_ints["Coverage"][coverage]["lower"], # y_low axis
        y_pred_ints["Coverage"][coverage]["upper"], # y_high
        alpha=0.4,
        color=colors[i],
        label=f"{coverage} cov.pred.intervals",
    )
    
ax.legend()

<matplotlib.legend.Legend at 0x133e81ad0>

Predict_quantiles

• predict_quantiles(fh=None, X=None, alpha=[0.05, 0.95])
• return quantile values of forecasting

y_pred_quantiles = forecaster.predict_quantiles(alpha=[0.275, 0.975])
y_pred_quantiles.head()

	Quantiles
	0.275	0.975
1961-01	432.922219	468.688317
1961-02	419.617696	462.124925
1961-03	479.746287	528.063109
1961-04	464.491077	517.990291
1961-05	467.360286	525.582418

fig, ax = plot_series(y, y_pred, labels=["y", "y_pred"])

colors = ['red', 'blue', 'yellow']

ax.fill_between(
    ax.get_lines()[-1].get_xdata(), # x axis
    y_pred_quantiles["Quantiles"][0.275], # y_low axis
    y_pred_quantiles["Quantiles"][0.975], # y_high
    alpha=0.2,
    color=ax.get_lines()[-1].get_c(),
    label=f"0.275-0.975",
)
ax.legend()

<matplotlib.legend.Legend at 0x13432e0d0>

Predict_var

• predict_var(fh=None, X=None, cov=False)
• produces variance forecasts
• not all estimators support cov

y_pred_var = forecaster.predict_var(cov=False) #
y_pred_var.head()

	0
1961-01	195.540049
1961-02	276.196509
1961-03	356.852968
1961-04	437.509428
1961-05	518.165887

predict_proba

• predict_proba(fh=None, X=None, marginal=True)
• forecasting mu values and their standard deviation

y_pred_proba = forecaster.predict_proba()
y_pred_proba # mu, sigma

Normal(columns=Index(['Number of airline passengers'], dtype='object'),
       index=PeriodIndex(['1961-01', '1961-02', '1961-03', '1961-04', '1961-05', '1961-06',
             '1961-07', '1961-08', '1961-09', '1961-10', '1961-11', '1961-12'],
            dtype='period[M]'),
       mu=         Number of airline passengers
1961-01                    441.281036
1961-02                    429.551968
1961-03                    491.038306
1961-04                    476.994261
1961-05                    480.967253
1961-06                    546.836776
1961-07                    604.372862
1961-08                    602.712843
1961-09                    525.352904
1961-10                    457.845222
1961-11                    399.012470
1961-12                    448.832681,
       sigma=                 0
1961-01  13.983564
1961-02  16.619161
1961-03  18.890552
1961-04  20.916726
1961-05  22.763257
1961-06  24.470847
1961-07  26.066814
1961-08  27.570551
1961-09  28.996409
1961-10  30.355365
1961-11  31.656036
1961-12  32.905336)

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
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# y_pred_proba.mean()
y_pred_proba.var()

	Number of airline passengers
1961-01	195.540049
1961-02	276.196509
1961-03	356.852968
1961-04	437.509428
1961-05	518.165887
1961-06	598.822347
1961-07	679.478807
1961-08	760.135266
1961-09	840.791726
1961-10	921.448185
1961-11	1002.104645
1961-12	1082.761105

Reference

• Forecasting
• Probabilistic Forecasting