Multivariate Timeseries Forecast with Lead and Lag Timesteps Using LSTM

Why Multivariate and how can it help to make better predictions?

We can consider multivariate timeseries as regression problem with independent variables being the features of the previous lag (till t-1)along with the independent values of time t. With this approach there is lot more control on the forecast than just the previous timestamps.

From the above figure we can see that, along with the lag features, lead=2 (t+2) timesteps is also considered to make the forecast. This gives us more control on the factors effecting the forecast. In many cases we know that some of the future factors also effects our current time predictions. With these approaches, the decision making team can really simulate the forecast based on various input values of independent features.

Implementation of Forecast model using LSTM

Now let us see how to implement the multivariate timeseries with both lead and lag feature.

The major difference between using a LSTM for a regression task to timeseries is, that in timeseries lead and lag timestamp data needs to be considered. Lets define a function which can just do this based on the lead and lag as a parameter

	Unnamed: 0	Date_excel	Rainfall_Terni	Flow_Rate_Lupa	doy	Month	Year	ET01	Infilt_	Infiltsum	...	Infilt_M6	Infilt_M6_diff	Rainfall_Terni_scale_12_calculated_index	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP
0	2010-01-01	2010-01-01	40.8	82.24	1.0	1.0	2010.0	1.338352	1.934648	1.934648	...	20.984370	12.824615	1.074801	0.105768	4.548065	0.607917	1.000000	0.0	2.094607	0.88
1	2010-01-02	2010-01-02	6.8	88.90	2.0	1.0	2010.0	1.701540	1.571460	3.506108	...	5.949230	1.517793	1.074801	0.105766	4.546129	0.622538	0.999998	0.0	2.996092	0.84
2	2010-01-03	2010-01-03	0.0	93.56	3.0	1.0	2010.0	0.938761	2.334239	5.840347	...	0.000000	0.000000	1.074801	0.105764	4.544194	0.637159	0.999996	0.0	1.934498	0.84
3	2010-01-04	2010-01-04	4.2	96.63	4.0	1.0	2010.0	0.996871	2.276129	8.116476	...	3.701564	0.792433	1.074801	0.105761	4.542258	0.651780	0.999994	0.0	1.625804	0.84
4	2010-01-05	2010-01-05	26.0	98.65	5.0	1.0	2010.0	1.278242	1.994758	10.111234	...	13.467998	1.974067	1.074801	0.105759	4.540323	0.666401	0.999992	0.0	1.993541	0.89

5 rows × 41 columns

	Unnamed: 0	Date_excel	Rainfall_Terni	Flow_Rate_Lupa	doy	Month	Year	ET01	Infilt_	Infiltsum	...	Infilt_M6	Infilt_M6_diff	Rainfall_Terni_scale_12_calculated_index	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP
3828	2020-06-25	2020-06-25	0.0	74.29	177.0	6.0	2020.0	4.030210	-4.030210	-541.652567	...	0.0	0.0	0.122602	0.127096	4.345	1.160797	1.040964	15.897778	5.772770	0.52
3829	2020-06-26	2020-06-26	0.0	73.93	178.0	6.0	2020.0	4.171681	-4.171681	-545.824247	...	0.0	0.0	0.122602	0.127512	4.272	1.149976	1.036377	16.560185	6.107339	0.51
3830	2020-06-27	2020-06-27	0.0	73.60	179.0	6.0	2020.0	4.449783	-4.449783	-550.274031	...	0.0	0.0	0.122602	0.127928	4.199	1.139156	1.030895	17.222592	6.540321	0.50
3831	2020-06-28	2020-06-28	0.0	73.14	180.0	6.0	2020.0	4.513588	-4.513588	-554.787618	...	0.0	0.0	0.122602	0.128345	4.126	1.128336	1.024516	17.885000	6.593228	0.49
3832	2020-06-29	2020-06-29	0.0	72.88	181.0	6.0	2020.0	4.510906	-4.510906	-559.298525	...	0.0	0.0	0.122602	0.128761	4.053	1.117516	1.017240	18.547407	6.479413	0.48

5 rows × 41 columns

0   -0.077870
1   -0.077870
2   -0.051091
3   -0.032286
4   -0.020689
Name: α1, dtype: float64

	Unnamed: 0	Date_excel	Rainfall_Terni	Flow_Rate_Lupa	doy	Month	Year	ET01	Infilt_	Infiltsum	...	Rainfall_Terni_scale_12_calculated_index	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP	α1_OK	α4
0	2010-01-01	2010-01-01	40.8	82.24	1.0	1.0	2010.0	1.338352	1.934648	1.934648	...	1.074801	0.105768	4.548065	0.607917	1.000000	0.0	2.094607	0.88	-0.005	0.004800
1	2010-01-02	2010-01-02	6.8	88.90	2.0	1.0	2010.0	1.701540	1.571460	3.506108	...	1.074801	0.105766	4.546129	0.622538	0.999998	0.0	2.996092	0.84	-0.015	-0.010000
2	2010-01-03	2010-01-03	0.0	93.56	3.0	1.0	2010.0	0.938761	2.334239	5.840347	...	1.074801	0.105764	4.544194	0.637159	0.999996	0.0	1.934498	0.84	-0.015	-0.011667
3	2010-01-04	2010-01-04	4.2	96.63	4.0	1.0	2010.0	0.996871	2.276129	8.116476	...	1.074801	0.105761	4.542258	0.651780	0.999994	0.0	1.625804	0.84	-0.015	-0.012500
4	2010-01-05	2010-01-05	26.0	98.65	5.0	1.0	2010.0	1.278242	1.994758	10.111234	...	1.074801	0.105759	4.540323	0.666401	0.999992	0.0	1.993541	0.89	-0.015	-0.015000

5 rows × 43 columns

	Unnamed: 0	Date_excel	Rainfall_Terni	Flow_Rate_Lupa	doy	Month	Year	ET01	Infilt_	Infiltsum	...	Rainfall_Terni_scale_12_calculated_index	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP	α1_OK	α4
3822	2020-06-19	2020-06-19	0.2	76.39	171.0	6.0	2020.0	3.463793	-3.263793	-518.987061	...	0.122602	0.124598	4.783	1.206395	1.049658	11.923333	5.152927	0.58	0.004987	0.003689
3823	2020-06-20	2020-06-20	0.0	76.01	172.0	6.0	2020.0	3.333015	-3.333015	-522.320075	...	0.122602	0.125014	4.710	1.205237	1.050450	12.585741	5.203840	0.56	0.004880	0.004098
3824	2020-06-21	2020-06-21	0.0	75.64	173.0	6.0	2020.0	3.412032	-3.412032	-525.732107	...	0.122602	0.125430	4.637	1.204077	1.050345	13.248148	5.040534	0.56	0.004372	0.003658
3825	2020-06-22	2020-06-22	0.0	75.31	174.0	6.0	2020.0	3.742202	-3.742202	-529.474309	...	0.122602	0.125847	4.564	1.193257	1.049344	13.910555	5.448369	0.56	0.005726	0.004991
3826	2020-06-23	2020-06-23	0.0	74.88	175.0	6.0	2020.0	3.917863	-3.917863	-533.392172	...	0.122602	0.126263	4.491	1.182437	1.047447	14.572963	5.861305	0.54	0.004014	0.004748
3827	2020-06-24	2020-06-24	0.0	74.58	176.0	6.0	2020.0	4.230186	-4.230186	-537.622357	...	0.122602	0.126679	4.418	1.171617	1.044654	15.235370	6.209193	0.52	0.003896	0.004502
3828	2020-06-25	2020-06-25	0.0	74.29	177.0	6.0	2020.0	4.030210	-4.030210	-541.652567	...	0.122602	0.127096	4.345	1.160797	1.040964	15.897778	5.772770	0.52	0.004858	0.004624
3829	2020-06-26	2020-06-26	0.0	73.93	178.0	6.0	2020.0	4.171681	-4.171681	-545.824247	...	0.122602	0.127512	4.272	1.149976	1.036377	16.560185	6.107339	0.51	0.004474	0.004310
3830	2020-06-27	2020-06-27	0.0	73.60	179.0	6.0	2020.0	4.449783	-4.449783	-550.274031	...	0.122602	0.127928	4.199	1.139156	1.030895	17.222592	6.540321	0.50	0.006270	0.004874
3831	2020-06-28	2020-06-28	0.0	73.14	180.0	6.0	2020.0	4.513588	-4.513588	-554.787618	...	0.122602	0.128345	4.126	1.128336	1.024516	17.885000	6.593228	0.49	0.004800	0.005100

10 rows × 43 columns

82.24

4.409641801706855

4.409641801706855

9.068627239003707

-0.0031319072300002304

0.0031319072300002304

Lupa['α1']= Lupa.log_Flow.shift(-1) -Lupa.log_Flow

	Unnamed: 0	Date_excel	Rainfall_Terni	Flow_Rate_Lupa	doy	Month	Year	ET01	Infilt_	Infiltsum	...	Rainfall_Terni_scale_12_calculated_index	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP	α1_OK	α4
2378	2016-07-06	2016-07-06	0.2	145.36	188.0	7.0	2016.0	4.27521	-4.07521	-422.631172	...	0.068649	0.129659	4.941290	-0.614314	0.919147	39.429314	5.688181	0.46	0.015	0.006657
2908	2017-12-18	2017-12-18	0.0	55.51	352.0	12.0	2017.0	0.85813	-0.85813	-790.968845	...	0.645122	0.104350	4.419677	-0.177246	0.991808	0.000000	2.144621	0.64	0.015	0.003307

2 rows × 43 columns

Lupa['α4']= Lupa['α1'].rolling(4, center=True).mean().fillna( Lupa['α1'].median() )

Lupa['α10']= Lupa['α1'].rolling(10, center=True).mean().fillna( Lupa['α1'].median() )

Index(['Unnamed: 0', 'Date_excel', 'Rainfall_Terni', 'Flow_Rate_Lupa', 'doy',
       'Month', 'Year', 'ET01', 'Infilt_', 'Infiltsum', 'Rainfall_Ter', 'P5',
       'Flow_Rate_Lup', 'Infilt_m3', 'Week', 'log_Flow', 'Lupa_Mean99_2011',
       'Rainfall_Terni_minET', 'Infiltrate', 'log_Flow_10d', 'log_Flow_20d',
       'α10', 'α20', 'log_Flow_10d_dif', 'log_Flow_20d_dif', 'α10_30',
       'Infilt_7YR', 'Infilt_2YR', 'α1', 'α1_negatives', 'ro', 'Infilt_M6',
       'Infilt_M6_diff', 'Rainfall_Terni_scale_12_calculated_index', 'SMroot',
       'Neradebit', 'smian', 'DroughtIndex', 'Deficit', 'PET_hg', 'GWETTOP',
       'α1_OK', 'α4'],
      dtype='object')

	Rainfall_Terni_minET	log_Flow	Lupa_Mean99_2011	Infilt_M6	α1_OK	α10	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP
0	39.461648	4.409642	117.814892	20.984370	-0.005	0.004800	0.105768	4.548065	0.607917	1.000000	0.0	2.094607	0.88
1	5.098460	4.487512	120.382310	5.949230	-0.015	0.004800	0.105766	4.546129	0.622538	0.999998	0.0	2.996092	0.84
2	0.000000	4.538603	118.858733	0.000000	-0.015	-0.029459	0.105764	4.544194	0.637159	0.999996	0.0	1.934498	0.84
3	3.203129	4.570889	121.065519	3.701564	-0.015	-0.027267	0.105761	4.542258	0.651780	0.999994	0.0	1.625804	0.84
4	24.721758	4.591578	119.763396	13.467998	-0.015	-0.028786	0.105759	4.540323	0.666401	0.999992	0.0	1.993541	0.89

	Rainfall_Terni_minET	log_Flow	Lupa_Mean99_2011	Infilt_M6	α1_OK	α10	SMroot	Neradebit	smian	DroughtIndex	Deficit	PET_hg	GWETTOP
3814	4.009934	4.370966	163.535143	5.104967	0.006212	0.003989	0.121267	5.367	1.215667	1.024150	6.624074	3.286200	0.67
3815	0.000000	4.364753	163.327754	0.000000	0.004333	0.004190	0.121683	5.294	1.214508	1.027398	7.286481	5.497035	0.58
3816	0.000000	4.360420	162.317328	0.000000	0.004994	0.004186	0.122100	5.221	1.213350	1.030798	7.948889	5.540033	0.57
3817	0.000000	4.355426	161.169102	0.219185	0.006052	0.005135	0.122516	5.148	1.212190	1.034348	8.611296	4.030759	0.57
3818	1.800428	4.349374	160.612387	3.300214	0.003752	0.005080	0.122932	5.075	1.211032	1.038050	9.273704	4.253079	0.64
3819	0.000000	4.345622	160.055672	0.000000	0.003246	0.004972	0.123349	5.002	1.209872	1.041904	9.936111	4.243349	0.67
3820	6.933015	4.342376	158.942241	8.466507	0.006131	0.004915	0.123765	4.929	1.208713	1.045385	10.598518	4.371786	0.62
3821	0.000000	4.336244	158.457154	1.142865	0.000393	0.004279	0.124181	4.856	1.207555	1.047970	11.260926	4.987571	0.61
3822	0.000000	4.335852	157.759221	0.000000	0.004987	0.004237	0.124598	4.783	1.206395	1.049658	11.923333	5.152927	0.58
3823	0.000000	4.330865	156.506611	0.000000	0.004880	0.004498	0.125014	4.710	1.205237	1.050450	12.585741	5.203840	0.56
3824	0.000000	4.325985	155.880306	0.000000	0.004372	0.004314	0.125430	4.637	1.204077	1.050345	13.248148	5.040534	0.56
3825	0.000000	4.321613	155.254001	0.000000	0.005726	0.004453	0.125847	4.564	1.193257	1.049344	13.910555	5.448369	0.56
3826	0.000000	4.315887	154.398320	0.000000	0.004014	0.004355	0.126263	4.491	1.182437	1.047447	14.572963	5.861305	0.54
3827	0.000000	4.311872	154.001392	0.000000	0.003896	0.004140	0.126679	4.418	1.171617	1.044654	15.235370	6.209193	0.52
3828	0.000000	4.307976	152.713987	0.000000	0.004858	0.004250	0.127096	4.345	1.160797	1.040964	15.897778	5.772770	0.52
3829	0.000000	4.303119	151.252610	0.000000	0.004474	0.004373	0.127512	4.272	1.149976	1.036377	16.560185	6.107339	0.51
3830	0.000000	4.298645	151.111899	0.000000	0.006270	0.004387	0.127928	4.199	1.139156	1.030895	17.222592	6.540321	0.50
3831	0.000000	4.292375	150.104384	0.000000	0.004800	0.004831	0.128345	4.126	1.128336	1.024516	17.885000	6.593228	0.49

Getting the data ready with lead and lag factors

The above functions converts the data into timeseries series with customized n_lag and n_lead steps. The output of this function contains data of lag and lead steps as columns with (t-n) or (t+n) timestamps

(3832, 13)

	var13(t-1)	var1(t)	var2(t)	var3(t)	var4(t)	var5(t)	var6(t)	var7(t)	var8(t)	var9(t)	var10(t)	var11(t)	var12(t)	var13(t)
3433	0.77	11.140963	4.604770	175.539318	12.370481	-0.015000	-0.015471	0.114055	7.306774	1.553723	0.994103	0.000000	3.531847	0.79
3434	0.79	7.092983	4.646600	173.173278	8.446492	-0.015000	-0.018052	0.114097	7.402581	1.564240	0.995003	0.000000	3.790510	0.76
3435	0.76	7.480072	4.676653	172.372999	8.540036	-0.015000	-0.018758	0.114138	7.498387	1.574756	0.996043	0.000000	3.149089	0.78
3436	0.78	6.950440	4.701843	171.920668	7.975220	-0.015000	-0.018959	0.114179	7.594194	1.585272	0.997222	0.000000	3.038312	0.76
3437	0.76	0.000000	4.724019	171.468337	0.000000	-0.015000	-0.019481	0.114221	7.690000	1.595789	0.998541	0.000000	4.016658	0.73
3438	0.73	0.000000	4.742669	170.789783	0.000000	-0.015000	-0.020379	0.114262	7.573000	1.606305	1.000000	0.000000	4.710386	0.73
3439	0.73	0.000000	4.758406	170.424495	0.000000	-0.014733	-0.021182	0.114303	7.456000	1.571215	1.001598	2.042404	5.001716	0.72
3440	0.72	0.000000	4.773139	169.241475	0.000000	-0.013018	-0.021962	0.114345	7.339000	1.536125	1.003336	4.084807	4.935763	0.71
3441	0.71	0.000000	4.786158	168.649965	0.000000	-0.012439	-0.022481	0.114386	7.222000	1.501035	1.005213	6.127211	5.270744	0.70
3442	0.70	0.000000	4.798597	168.058455	0.000000	-0.011227	-0.020505	0.114427	7.105000	1.465945	1.007230	8.169615	5.364919	0.67
3443	0.67	0.000000	4.809824	166.979819	0.000000	-0.010377	-0.017360	0.114469	6.988000	1.430855	1.009387	10.212018	5.166235	0.65
3444	0.65	0.000000	4.820201	167.386588	0.000000	-0.002658	-0.014621	0.114510	6.871000	1.395765	1.011683	12.254422	5.975295	0.63
3445	0.63	0.000000	4.822859	165.901183	0.000000	-0.015000	-0.014077	0.114551	6.754000	1.360675	1.014119	14.296826	6.325639	0.62
3446	0.62	0.000000	4.842611	165.763054	0.000000	-0.005270	-0.012386	0.114593	6.637000	1.325585	1.016694	16.339230	6.388215	0.59
3447	0.59	0.000000	4.847881	164.752958	0.000000	-0.010147	-0.011536	0.114634	6.520000	1.290495	1.019409	18.381633	5.553434	0.59
3448	0.59	0.000000	4.858028	163.535143	0.000000	-0.007505	-0.010713	0.114675	6.403000	1.255405	1.022264	20.424037	5.656065	0.57
3449	0.57	0.000000	4.865532	163.327754	0.000000	-0.005381	-0.009777	0.114717	6.286000	1.202354	1.025258	22.466441	5.889984	0.56
3450	0.56	0.000000	4.870913	162.317328	0.000000	-0.004589	-0.008935	0.114758	6.169000	1.149304	1.028391	24.508844	6.349167	0.55
3451	0.55	0.000000	4.875503	161.169102	0.000000	-0.004188	-0.008109	0.114799	6.052000	1.096253	1.031665	26.551248	7.182276	0.54
3452	0.54	0.000000	4.879691	160.612387	0.000000	-0.002656	-0.007252	0.114841	5.935000	1.043203	1.035077	28.593652	5.982390	0.53
3453	0.53	0.000000	4.882347	160.055672	0.000000	-0.002271	-0.006442	0.114882	5.818000	0.990152	1.038630	30.636055	6.189987	0.52
3454	0.52	1.306480	4.884618	158.942241	3.453240	-0.002794	-0.006455	0.114923	5.701000	0.937102	1.041517	32.678459	6.074026	0.52
3455	0.52	0.000000	4.887412	158.457154	0.000000	-0.001582	-0.004638	0.114965	5.584000	0.884051	1.042935	34.720863	5.854209	0.52
3456	0.52	0.000000	4.888995	157.759221	0.000000	-0.000075	-0.004119	0.115006	5.467000	0.831001	1.042882	36.763266	5.998155	0.51
3457	0.51	0.000000	4.889070	156.506611	0.000000	0.007405	-0.002364	0.115047	5.350000	0.777950	1.041361	38.805670	6.350226	0.50
3458	0.50	0.000000	4.881665	155.880306	0.000000	-0.008308	-0.002444	0.115089	5.233000	0.724900	1.038369	40.848074	6.589590	0.49
3459	0.49	0.000000	4.889973	155.254001	0.000000	-0.000301	-0.001936	0.115130	5.116000	0.685513	1.033907	42.890477	5.823888	0.49
3460	0.49	0.000000	4.890274	154.398320	0.000000	0.001430	-0.001334	0.115171	4.999000	0.646126	1.027976	44.932881	6.191218	0.48
3461	0.48	0.000000	4.888844	154.001392	0.000000	0.001356	-0.000780	0.115213	4.882000	0.606738	1.020575	46.975285	5.938021	0.48
3462	0.48	0.000000	4.887488	152.713987	0.000000	0.001358	-0.000378	0.115254	4.765000	0.567351	1.011705	49.017689	6.331419	0.47
3463	0.47	0.000000	4.886130	151.252610	0.000000	0.001511	0.000000	0.115295	4.648000	0.527964	1.001364	51.060092	6.652679	0.46
3464	0.46	0.000000	4.884618	151.111899	0.000000	0.001741	0.000453	0.115337	4.531000	0.488577	0.989554	53.102496	7.210034	0.45
3465	0.45	0.000000	4.882878	150.104384	0.000000	0.002655	0.000877	0.115378	4.414000	0.449189	0.976274	55.144900	7.138806	0.45
3466	0.45	0.000000	4.880223	149.409657	0.000000	0.001748	0.001060	0.115419	4.297000	0.409802	0.961524	57.187303	6.510536	0.45
3467	0.45	0.000000	4.878474	148.712596	0.000000	0.000761	0.000395	0.115461	4.180000	0.370415	0.945305	59.229707	7.678308	0.44
3468	0.44	0.000000	4.877713	147.320807	0.000000	0.001448	0.001371	0.115502	4.186129	0.331027	0.927616	61.272111	7.203015	0.45
3469	0.45	0.000000	4.876265	146.856294	0.000000	0.001908	0.001592	0.115543	4.192258	0.323979	0.908457	63.486426	7.461562	0.44
3470	0.44	0.000000	4.874357	145.998608	0.000000	0.002754	0.001724	0.115585	4.198387	0.316930	0.887828	65.700742	7.091059	0.44
3471	0.44	0.000000	4.871603	144.676409	0.000000	0.002224	0.001811	0.115626	4.204516	0.309881	0.865730	67.915057	6.737927	0.45
3472	0.45	0.000000	4.869379	144.302931	0.000000	0.002229	0.001898	0.115667	4.210645	0.302833	0.842162	70.129372	6.867434	0.44
3473	0.44	0.000000	4.867150	143.423800	0.000000	0.003160	0.002063	0.115709	4.216774	0.295784	0.817124	72.343688	7.176614	0.42

from tensorflow.keras.layers import Normalization

Normalization by StandardScaler

This code helps in dropping the future Y (at t+n) while training the models. Once we drop the future Y and we have the reframed data, its as simple as training the LSTM for a regression problem.

(3802, 402) (3802, 1)

(3665, 1, 402) (137, 1, 402)

Creating a LSTM model

The first LSTM layer contains estimators for 3 years. The length of the test targets is 300, thus 10 months.

3660

3665 1 402

Once the model is ready we can train the model on the train data and test it on the test. We add some training checkpoints which can be used to efficiently train and earlystop a model.

3665 1 402

predictions for our test data

Once model is trained , we can get the predictions for our test data

5/5 [==============================] - 1s 19ms/step

array([[0.09541333],
       [0.09553236],
       [0.09676951]], dtype=float32)

(137, 1, 402) (137, 1)

array([0.14965787, 0.14949162, 0.15430685, 0.15563299, 0.16076282])

1. We have to reverse the normalization
2. next we'll reverse the transformation of the logarithms of the flow rates

(137, 1) (137, 1)

tf.math.exp( test_X[:,0,1] )

history.history# .loss

array([[1.1001134],
       [1.1002445]], dtype=float32)

array([[107.73059],
       [107.73747]], dtype=float32)

(137, 402)

var2_t is the 10 last feature col

array([4.45560468, 4.94125655, 4.45560468, 4.45560468, 4.45560468,
       4.45560468, 4.68121901, 4.45560468, 4.45560468, 4.45560468,
       4.45560468, 4.45560468, 4.45560468, 4.69108166, 4.69663514,
       4.80012583, 4.45560468, 5.59022869, 5.79308962, 5.02732022])

α1_OK & α10

batchsize=366, predictions = 137

4.9648447

α1

batchsize=183, predictions = 300

MeanAbsolutePercentageError

the predictions over 300 days vs. the observed outflows.

9.235593

973.3673

values = Lupa[["Rainfall_Terni" ,"log_Flow", "Infilt_M6", 'α10','Neradebit', 'smian', 'DroughtIndex', 'Deficit', 'PET_hg']] MeanAbsolutePercentageError 4.4716454

array([-0.32206752,  0.17360827, -0.26424101, ...,  0.04898375,
        1.4423285 , -0.84989194])