Probabilistic Forecasting

import warnings
warnings.filterwarnings('ignore')

Load Data

from sktime.datasets import load_longley
_, y = load_longley() # 16*5
y.head()

	GNPDEFL	GNP	UNEMP	ARMED	POP
Period
1947	83.0	234289.0	2356.0	1590.0	107608.0
1948	88.5	259426.0	2325.0	1456.0	108632.0
1949	88.2	258054.0	3682.0	1616.0	109773.0
1950	89.5	284599.0	3351.0	1650.0	110929.0
1951	96.2	328975.0	2099.0	3099.0	112075.0

Train Forecaster

import numpy as np
from sktime.forecasting.var import VAR

fh = np.arange(1, 5)

forecaster = VAR()
forecaster.fit(y)
y_pred = forecaster.predict(fh)

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()

	GNPDEFL		GNP		UNEMP		ARMED		POP
	0.9		0.9		0.9		0.9		0.9
	lower	upper	lower	upper	lower	upper	lower	upper	lower	upper
Period
1963	117.492977	122.208376	560486.177419	587020.649184	3917.719752	5862.537044	1788.057405	3282.655684	132006.104828	132980.082567
1964	118.889313	124.047653	579429.082278	614063.738258	4436.940396	6829.680676	1225.810443	3074.629283	134217.732787	135744.466444
1965	120.921592	127.333840	610262.428743	649634.201865	4541.784488	7222.018501	800.607954	2911.299938	136653.038041	138573.129357
1966	124.033250	130.795854	641781.113945	682795.273342	4963.131313	7730.252362	455.670406	2718.751331	139342.410520	141613.299569

def get_plots_prob(y, y_pred, y_prob, name):
    fig, ax = plot_series(y[name], y_pred[name], labels=["y", "y_pred"])
    ax.fill_between(
        ax.get_lines()[-1].get_xdata(), # x axis
        y_prob[name].iloc[:, 0], # y_low axis
        y_prob[name].iloc[:, 1], # y_high 
        alpha=0.2,
        color=ax.get_lines()[-1].get_c(),
        label=f"{coverage} cov.pred.intervals",
    )
    ax.legend()
    
get_plots_prob(y, y_pred, y_pred_ints, 'GNPDEFL')

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()

	GNPDEFL		GNP		UNEMP		ARMED		POP
	0.275	0.975	0.275	0.975	0.275	0.975	0.275	0.975	0.275	0.975
1963	118.993859	578574.890704	4536.743084	2806.934283	132316.115927	122.660049	557944.526676	6048.824766	1644.894698	133073.376727
1964	120.531180	603039.741171	5198.534849	2486.161665	134703.682681	124.541754	576111.541236	7058.873484	1048.718100	135890.707295
1965	122.962571	637102.409649	5394.886421	2239.479716	137264.191248	127.948048	606491.134659	7478.749399	598.431635	138757.048659
1966	126.185745	669740.719719	5843.888853	1998.426650	140065.220479	131.443622	637852.501021	7995.305887	238.897245	141830.820645

# re-order the columns of prediction
y_pred_quantiles = y_pred_quantiles.iloc[:, [0, 5, 6, 7, 2, 1, 8, 3, 4, 9]]
y_pred_quantiles

	GNPDEFL	UNEMP	ARMED		GNP	GNPDEFL	POP	GNP	UNEMP	POP
	0.275	0.975	0.275	0.975	0.275	0.975	0.275	0.975	0.275	0.975
1963	118.993859	122.660049	557944.526676	6048.824766	4536.743084	578574.890704	1644.894698	2806.934283	132316.115927	133073.376727
1964	120.531180	124.541754	576111.541236	7058.873484	5198.534849	603039.741171	1048.718100	2486.161665	134703.682681	135890.707295
1965	122.962571	127.948048	606491.134659	7478.749399	5394.886421	637102.409649	598.431635	2239.479716	137264.191248	138757.048659
1966	126.185745	131.443622	637852.501021	7995.305887	5843.888853	669740.719719	238.897245	1998.426650	140065.220479	141830.820645

def get_plots_prob(y, y_pred, y_prob, name, index_low, index_high):
    fig, ax = plot_series(y[name], y_pred[name], labels=["y", "y_pred"])
    ax.fill_between(
        ax.get_lines()[-1].get_xdata(), # x axis
        y_prob.iloc[:, index_low], # y_low axis
        y_prob.iloc[:, index_high], # y_high 
        alpha=0.2,
        color=ax.get_lines()[-1].get_c(),
        label=f"{coverage} cov.pred.intervals",
    )
    ax.legend()
get_plots_prob(y, y_pred, y_pred_quantiles, 'GNPDEFL', 0, 1)

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()

	GNPDEFL	GNP	UNEMP	ARMED	POP
1963	2.054577	6.505885e+07	349496.724319	206411.766443	87656.385238
1964	2.458700	1.108427e+08	529025.512014	315845.148830	215383.313893
1965	3.799322	1.432371e+08	663790.333183	411656.726839	340666.369241
1966	4.225843	1.554365e+08	707525.034070	473244.595983	476515.823957

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(['GNPDEFL', 'GNP', 'UNEMP', 'ARMED', 'POP'], dtype='object'),
       index=PeriodIndex(['1963', '1964', '1965', '1966'], dtype='period[A-DEC]'),
       mu=         GNPDEFL            GNP        UNEMP        ARMED            POP
1963  119.850676  573753.413302  4890.128398  2535.356545  132493.093697
1964  121.468483  596746.410268  5633.310536  2150.219863  134981.099616
1965  124.127716  629948.315304  5881.901495  1855.953946  137613.083699
1966  127.414552  662288.193643  6346.691838  1587.210869  140477.855045,
       sigma=       GNPDEFL           GNP       UNEMP       ARMED         POP
1963  1.433380   8065.906695  591.182480  454.325617  296.068210
1964  1.568024  10528.187862  727.341400  562.001022  464.094079
1965  1.949185  11968.169227  814.733290  641.604806  583.666317
1966  2.055686  12467.419205  841.145073  687.927755  690.301256)

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# y_pred_proba.mean()
y_pred_proba.var()

	GNPDEFL	GNP	UNEMP	ARMED	POP
1963	2.054577	6.505885e+07	349496.724319	206411.766443	87656.385238
1964	2.458700	1.108427e+08	529025.512014	315845.148830	215383.313893
1965	3.799322	1.432371e+08	663790.333183	411656.726839	340666.369241
1966	4.225843	1.554365e+08	707525.034070	473244.595983	476515.823957

Reference

• Probabilistic Forecasting