DATA 624 - Project1 - Timeseries Forecast Models

Libraries

library(kableExtra)
library(tidyverse)
library(ggplot2)
library(dplyr)
library(GGally)
library(fpp2)
library(seasonal)
library(grid)
library(gridExtra)
library(forecast)
library(urca)
library(lubridate)
library(imputeTS)
library(xts)

PART A: DataSet - ATM Cash Draw

atmdata <- readxl::read_excel("ATM624Data.xlsx", skip=0)

atmdata %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

## Warning: namespace 'highr' is not available and has been replaced
## by .GlobalEnv when processing object '<unknown>'

DATE	ATM	Cash
39934	ATM1	96.000000
39934	ATM2	107.000000
39935	ATM1	82.000000
39935	ATM2	89.000000
39936	ATM1	85.000000
39936	ATM2	90.000000
39937	ATM1	90.000000
39937	ATM2	55.000000
39938	ATM1	99.000000
39938	ATM2	79.000000
39939	ATM1	88.000000
39939	ATM2	19.000000
39940	ATM1	8.000000
39940	ATM2	2.000000
39941	ATM1	104.000000
39941	ATM2	103.000000
39942	ATM1	87.000000
39942	ATM2	107.000000
39943	ATM1	93.000000
39943	ATM2	118.000000
39944	ATM1	86.000000
39944	ATM2	75.000000
39945	ATM1	111.000000
39945	ATM2	111.000000
39946	ATM1	75.000000
39946	ATM2	25.000000
39947	ATM1	6.000000
39947	ATM2	16.000000
39948	ATM1	102.000000
39948	ATM2	137.000000
39949	ATM1	73.000000
39949	ATM2	95.000000
39950	ATM1	92.000000
39950	ATM2	103.000000
39951	ATM1	82.000000
39951	ATM2	80.000000
39952	ATM1	86.000000
39952	ATM2	118.000000
39953	ATM1	73.000000
39953	ATM2	30.000000
39954	ATM1	20.000000
39954	ATM2	7.000000
39955	ATM1	100.000000
39955	ATM2	118.000000
39956	ATM1	93.000000
39956	ATM2	104.000000
39957	ATM1	90.000000
39957	ATM2	59.000000
39958	ATM1	94.000000
39958	ATM2	40.000000
39959	ATM1	98.000000
39959	ATM2	106.000000
39960	ATM1	73.000000
39960	ATM2	18.000000
39961	ATM1	10.000000
39961	ATM2	9.000000
39962	ATM1	97.000000
39962	ATM2	136.000000
39963	ATM1	102.000000
39963	ATM2	118.000000
39964	ATM1	85.000000
39964	ATM2	64.000000
39965	ATM1	85.000000
39965	ATM2	77.000000
39966	ATM1	108.000000
39966	ATM2	133.000000
39967	ATM1	94.000000
39967	ATM2	45.000000
39968	ATM1	14.000000
39968	ATM2	14.000000
39969	ATM1	3.000000
39969	ATM2	20.000000
39970	ATM1	96.000000
39970	ATM2	147.000000
39971	ATM1	109.000000
39971	ATM2	105.000000
39972	ATM1	96.000000
39972	ATM2	132.000000
39973	ATM1	145.000000
39973	ATM2	93.000000
39974	ATM1	81.000000
39974	ATM2	26.000000
39975	ATM1	16.000000
39975	ATM2	7.000000
39976	ATM1	142.000000
39976	ATM2	112.000000
39977	ATM1	NA
39977	ATM2	91.000000
39978	ATM1	120.000000
39978	ATM2	72.000000
39979	ATM1	106.000000
39979	ATM2	66.000000
39980	ATM1	NA
39980	ATM2	82.000000
39981	ATM1	108.000000
39981	ATM2	24.000000
39982	ATM1	21.000000
39982	ATM2	NA
39983	ATM1	140.000000
39983	ATM2	134.000000
39984	ATM1	110.000000
39984	ATM2	95.000000
39985	ATM1	115.000000
39985	ATM2	82.000000
39986	ATM1	NA
39986	ATM2	90.000000
39987	ATM1	108.000000
39987	ATM2	99.000000
39988	ATM1	66.000000
39988	ATM2	NA
39989	ATM1	13.000000
39989	ATM2	3.000000
39990	ATM1	99.000000
39990	ATM2	117.000000
39991	ATM1	105.000000
39991	ATM2	53.000000
39992	ATM1	104.000000
39992	ATM2	44.000000
39993	ATM1	98.000000
39993	ATM2	56.000000
39994	ATM1	110.000000
39994	ATM2	110.000000
39995	ATM1	79.000000
39995	ATM2	36.000000
39996	ATM1	16.000000
39996	ATM2	12.000000
39997	ATM1	110.000000
39997	ATM2	128.000000
39998	ATM1	96.000000
39998	ATM2	72.000000
39999	ATM1	114.000000
39999	ATM2	122.000000
40000	ATM1	126.000000
40000	ATM2	100.000000
40001	ATM1	126.000000
40001	ATM2	108.000000
40002	ATM1	73.000000
40002	ATM2	25.000000
40003	ATM1	4.000000
40003	ATM2	6.000000
40004	ATM1	19.000000
40004	ATM2	22.000000
40005	ATM1	114.000000
40005	ATM2	135.000000
40006	ATM1	98.000000
40006	ATM2	69.000000
40007	ATM1	97.000000
40007	ATM2	52.000000
40008	ATM1	114.000000
40008	ATM2	81.000000
40009	ATM1	78.000000
40009	ATM2	27.000000
40010	ATM1	19.000000
40010	ATM2	4.000000
40011	ATM1	102.000000
40011	ATM2	147.000000
40012	ATM1	94.000000
40012	ATM2	102.000000
40013	ATM1	108.000000
40013	ATM2	83.000000
40014	ATM1	91.000000
40014	ATM2	55.000000
40015	ATM1	86.000000
40015	ATM2	74.000000
40016	ATM1	78.000000
40016	ATM2	22.000000
40017	ATM1	16.000000
40017	ATM2	4.000000
40018	ATM1	114.000000
40018	ATM2	104.000000
40019	ATM1	115.000000
40019	ATM2	81.000000
40020	ATM1	108.000000
40020	ATM2	61.000000
40021	ATM1	102.000000
40021	ATM2	70.000000
40022	ATM1	129.000000
40022	ATM2	126.000000
40023	ATM1	79.000000
40023	ATM2	33.000000
40024	ATM1	13.000000
40024	ATM2	7.000000
40025	ATM1	103.000000
40025	ATM2	126.000000
40026	ATM1	90.000000
40026	ATM2	92.000000
40027	ATM1	68.000000
40027	ATM2	81.000000
40028	ATM1	85.000000
40028	ATM2	49.000000
40029	ATM1	99.000000
40029	ATM2	146.000000
40030	ATM1	86.000000
40030	ATM2	79.000000
40031	ATM1	13.000000
40031	ATM2	37.000000
40032	ATM1	116.000000
40032	ATM2	136.000000
40033	ATM1	105.000000
40033	ATM2	111.000000
40034	ATM1	123.000000
40034	ATM2	78.000000
40035	ATM1	114.000000
40035	ATM2	57.000000
40036	ATM1	127.000000
40036	ATM2	106.000000
40037	ATM1	111.000000
40037	ATM2	38.000000
40038	ATM1	34.000000
40038	ATM2	15.000000
40039	ATM1	151.000000
40039	ATM2	119.000000
40040	ATM1	110.000000
40040	ATM2	110.000000
40041	ATM1	115.000000
40041	ATM2	68.000000
40042	ATM1	112.000000
40042	ATM2	60.000000
40043	ATM1	132.000000
40043	ATM2	92.000000
40044	ATM1	94.000000
40044	ATM2	22.000000
40045	ATM1	24.000000
40045	ATM2	11.000000
40046	ATM1	122.000000
40046	ATM2	121.000000
40047	ATM1	104.000000
40047	ATM2	89.000000
40048	ATM1	128.000000
40048	ATM2	62.000000
40049	ATM1	120.000000
40049	ATM2	79.000000
40050	ATM1	174.000000
40050	ATM2	83.000000
40051	ATM1	96.000000
40051	ATM2	31.000000
40052	ATM1	13.000000
40052	ATM2	4.000000
40053	ATM1	121.000000
40053	ATM2	96.000000
40054	ATM1	133.000000
40054	ATM2	50.000000
40055	ATM1	118.000000
40055	ATM2	52.000000
40056	ATM1	91.000000
40056	ATM2	56.000000
40057	ATM1	120.000000
40057	ATM2	104.000000
40058	ATM1	88.000000
40058	ATM2	27.000000
40059	ATM1	19.000000
40059	ATM2	13.000000
40060	ATM1	150.000000
40060	ATM2	107.000000
40061	ATM1	144.000000
40061	ATM2	125.000000
40062	ATM1	121.000000
40062	ATM2	103.000000
40063	ATM1	105.000000
40063	ATM2	42.000000
40064	ATM1	133.000000
40064	ATM2	90.000000
40065	ATM1	109.000000
40065	ATM2	29.000000
40066	ATM1	18.000000
40066	ATM2	8.000000
40067	ATM1	1.000000
40067	ATM2	2.000000
40068	ATM1	105.000000
40068	ATM2	84.000000
40069	ATM1	112.000000
40069	ATM2	62.000000
40070	ATM1	82.000000
40070	ATM2	77.000000
40071	ATM1	111.000000
40071	ATM2	78.000000
40072	ATM1	79.000000
40072	ATM2	25.000000
40073	ATM1	13.000000
40073	ATM2	8.000000
40074	ATM1	112.000000
40074	ATM2	113.000000
40075	ATM1	99.000000
40075	ATM2	71.000000
40076	ATM1	140.000000
40076	ATM2	94.000000
40077	ATM1	110.000000
40077	ATM2	59.000000
40078	ATM1	180.000000
40078	ATM2	89.000000
40079	ATM1	73.000000
40079	ATM2	18.000000
40080	ATM1	7.000000
40080	ATM2	6.000000
40081	ATM1	106.000000
40081	ATM2	115.000000
40082	ATM1	103.000000
40082	ATM2	81.000000
40083	ATM1	93.000000
40083	ATM2	61.000000
40084	ATM1	96.000000
40084	ATM2	71.000000
40085	ATM1	117.000000
40085	ATM2	69.000000
40086	ATM1	80.000000
40086	ATM2	36.000000
40087	ATM1	14.000000
40087	ATM2	14.000000
40088	ATM1	120.000000
40088	ATM2	104.000000
40089	ATM1	91.000000
40089	ATM2	73.000000
40090	ATM1	96.000000
40090	ATM2	86.000000
40091	ATM1	74.000000
40091	ATM2	85.000000
40092	ATM1	108.000000
40092	ATM2	126.000000
40093	ATM1	73.000000
40093	ATM2	31.000000
40094	ATM1	13.000000
40094	ATM2	9.000000
40095	ATM1	93.000000
40095	ATM2	114.000000
40096	ATM1	94.000000
40096	ATM2	78.000000
40097	ATM1	76.000000
40097	ATM2	45.000000
40098	ATM1	111.000000
40098	ATM2	60.000000
40099	ATM1	88.000000
40099	ATM2	91.000000
40100	ATM1	76.000000
40100	ATM2	22.000000
40101	ATM1	9.000000
40101	ATM2	7.000000
40102	ATM1	87.000000
40102	ATM2	75.000000
40103	ATM1	105.000000
40103	ATM2	66.000000
40104	ATM1	78.000000
40104	ATM2	64.000000
40105	ATM1	67.000000
40105	ATM2	51.000000
40106	ATM1	90.000000
40106	ATM2	94.000000
40107	ATM1	68.000000
40107	ATM2	23.000000
40108	ATM1	9.000000
40108	ATM2	4.000000
40109	ATM1	78.000000
40109	ATM2	127.000000
40110	ATM1	74.000000
40110	ATM2	61.000000
40111	ATM1	74.000000
40111	ATM2	0.000000
40112	ATM1	60.000000
40112	ATM2	95.000000
40113	ATM1	75.000000
40113	ATM2	79.000000
40114	ATM1	61.000000
40114	ATM2	38.000000
40115	ATM1	9.000000
40115	ATM2	8.000000
40116	ATM1	90.000000
40116	ATM2	119.000000
40117	ATM1	86.000000
40117	ATM2	57.000000
40118	ATM1	86.000000
40118	ATM2	58.000000
40119	ATM1	79.000000
40119	ATM2	80.000000
40120	ATM1	90.000000
40120	ATM2	82.000000
40121	ATM1	80.000000
40121	ATM2	49.000000
40122	ATM1	21.000000
40122	ATM2	16.000000
40123	ATM1	93.000000
40123	ATM2	116.000000
40124	ATM1	104.000000
40124	ATM2	61.000000
40125	ATM1	109.000000
40125	ATM2	59.000000
40126	ATM1	88.000000
40126	ATM2	80.000000
40127	ATM1	96.000000
40127	ATM2	86.000000
40128	ATM1	70.000000
40128	ATM2	23.000000
40129	ATM1	15.000000
40129	ATM2	7.000000
40130	ATM1	73.000000
40130	ATM2	91.000000
40131	ATM1	94.000000
40131	ATM2	57.000000
40132	ATM1	108.000000
40132	ATM2	58.000000
40133	ATM1	73.000000
40133	ATM2	61.000000
40134	ATM1	87.000000
40134	ATM2	77.000000
40135	ATM1	75.000000
40135	ATM2	20.000000
40136	ATM1	10.000000
40136	ATM2	5.000000
40137	ATM1	92.000000
40137	ATM2	132.000000
40138	ATM1	87.000000
40138	ATM2	49.000000
40139	ATM1	74.000000
40139	ATM2	57.000000
40140	ATM1	73.000000
40140	ATM2	68.000000
40141	ATM1	93.000000
40141	ATM2	80.000000
40142	ATM1	66.000000
40142	ATM2	31.000000
40143	ATM1	18.000000
40143	ATM2	3.000000
40144	ATM1	99.000000
40144	ATM2	85.000000
40145	ATM1	94.000000
40145	ATM2	53.000000
40146	ATM1	136.000000
40146	ATM2	46.000000
40147	ATM1	6.000000
40147	ATM2	2.000000
40148	ATM1	140.000000
40148	ATM2	113.000000
40149	ATM1	73.000000
40149	ATM2	22.000000
40150	ATM1	9.000000
40150	ATM2	5.000000
40151	ATM1	140.000000
40151	ATM2	112.000000
40152	ATM1	103.000000
40152	ATM2	59.000000
40153	ATM1	110.000000
40153	ATM2	72.000000
40154	ATM1	90.000000
40154	ATM2	77.000000
40155	ATM1	135.000000
40155	ATM2	85.000000
40156	ATM1	67.000000
40156	ATM2	27.000000
40157	ATM1	12.000000
40157	ATM2	1.000000
40158	ATM1	109.000000
40158	ATM2	91.000000
40159	ATM1	84.000000
40159	ATM2	36.000000
40160	ATM1	92.000000
40160	ATM2	46.000000
40161	ATM1	84.000000
40161	ATM2	100.000000
40162	ATM1	118.000000
40162	ATM2	73.000000
40163	ATM1	68.000000
40163	ATM2	22.000000
40164	ATM1	14.000000
40164	ATM2	9.000000
40165	ATM1	90.000000
40165	ATM2	117.000000
40166	ATM1	92.000000
40166	ATM2	44.000000
40167	ATM1	93.000000
40167	ATM2	44.000000
40168	ATM1	85.000000
40168	ATM2	78.000000
40169	ATM1	93.000000
40169	ATM2	89.000000
40170	ATM1	70.000000
40170	ATM2	33.000000
40171	ATM1	13.000000
40171	ATM2	5.000000
40172	ATM1	90.000000
40172	ATM2	102.000000
40173	ATM1	91.000000
40173	ATM2	68.000000
40174	ATM1	102.000000
40174	ATM2	64.000000
40175	ATM1	97.000000
40175	ATM2	81.000000
40176	ATM1	42.000000
40176	ATM2	9.000000
40177	ATM1	2.000000
40177	ATM2	2.000000
40178	ATM1	14.000000
40178	ATM2	2.000000
40179	ATM1	132.000000
40179	ATM2	117.000000
40180	ATM1	108.000000
40180	ATM2	56.000000
40181	ATM1	120.000000
40181	ATM2	55.000000
40182	ATM1	120.000000
40182	ATM2	112.000000
40183	ATM1	90.000000
40183	ATM2	100.000000
40184	ATM1	48.000000
40184	ATM2	17.000000
40185	ATM1	15.000000
40185	ATM2	7.000000
40186	ATM1	86.000000
40186	ATM2	49.000000
40187	ATM1	109.000000
40187	ATM2	96.000000
40188	ATM1	115.000000
40188	ATM2	47.000000
40189	ATM1	123.000000
40189	ATM2	107.000000
40190	ATM1	123.000000
40190	ATM2	93.000000
40191	ATM1	60.000000
40191	ATM2	27.000000
40192	ATM1	22.000000
40192	ATM2	9.000000
40193	ATM1	81.000000
40193	ATM2	78.000000
40194	ATM1	98.000000
40194	ATM2	47.000000
40195	ATM1	94.000000
40195	ATM2	31.000000
40196	ATM1	76.000000
40196	ATM2	91.000000
40197	ATM1	96.000000
40197	ATM2	77.000000
40198	ATM1	72.000000
40198	ATM2	22.000000
40199	ATM1	12.000000
40199	ATM2	1.000000
40200	ATM1	91.000000
40200	ATM2	125.000000
40201	ATM1	74.000000
40201	ATM2	53.000000
40202	ATM1	104.000000
40202	ATM2	37.000000
40203	ATM1	74.000000
40203	ATM2	69.000000
40204	ATM1	91.000000
40204	ATM2	69.000000
40205	ATM1	66.000000
40205	ATM2	16.000000
40206	ATM1	28.000000
40206	ATM2	2.000000
40207	ATM1	152.000000
40207	ATM2	95.000000
40208	ATM1	97.000000
40208	ATM2	45.000000
40209	ATM1	100.000000
40209	ATM2	61.000000
40210	ATM1	85.000000
40210	ATM2	71.000000
40211	ATM1	123.000000
40211	ATM2	91.000000
40212	ATM1	46.000000
40212	ATM2	28.000000
40213	ATM1	38.000000
40213	ATM2	9.000000
40214	ATM1	138.000000
40214	ATM2	97.000000
40215	ATM1	74.000000
40215	ATM2	60.000000
40216	ATM1	85.000000
40216	ATM2	27.000000
40217	ATM1	78.000000
40217	ATM2	74.000000
40218	ATM1	123.000000
40218	ATM2	45.000000
40219	ATM1	50.000000
40219	ATM2	26.000000
40220	ATM1	28.000000
40220	ATM2	2.000000
40221	ATM1	105.000000
40221	ATM2	73.000000
40222	ATM1	109.000000
40222	ATM2	48.000000
40223	ATM1	26.000000
40223	ATM2	9.000000
40224	ATM1	54.000000
40224	ATM2	25.000000
40225	ATM1	105.000000
40225	ATM2	58.000000
40226	ATM1	179.000000
40226	ATM2	110.000000
40227	ATM1	125.000000
40227	ATM2	94.000000
40228	ATM1	84.000000
40228	ATM2	25.000000
40229	ATM1	99.000000
40229	ATM2	101.000000
40230	ATM1	110.000000
40230	ATM2	77.000000
40231	ATM1	80.000000
40231	ATM2	6.000000
40232	ATM1	1.000000
40232	ATM2	3.000000
40233	ATM1	111.000000
40233	ATM2	99.000000
40234	ATM1	115.000000
40234	ATM2	86.000000
40235	ATM1	108.000000
40235	ATM2	20.000000
40236	ATM1	100.000000
40236	ATM2	72.000000
40237	ATM1	152.000000
40237	ATM2	81.000000
40238	ATM1	90.000000
40238	ATM2	16.000000
40239	ATM1	4.000000
40239	ATM2	2.000000
40240	ATM1	128.000000
40240	ATM2	110.000000
40241	ATM1	87.000000
40241	ATM2	75.000000
40242	ATM1	84.000000
40242	ATM2	35.000000
40243	ATM1	69.000000
40243	ATM2	102.000000
40244	ATM1	112.000000
40244	ATM2	100.000000
40245	ATM1	62.000000
40245	ATM2	9.000000
40246	ATM1	4.000000
40246	ATM2	2.000000
40247	ATM1	94.000000
40247	ATM2	100.000000
40248	ATM1	102.000000
40248	ATM2	85.000000
40249	ATM1	98.000000
40249	ATM2	33.000000
40250	ATM1	88.000000
40250	ATM2	24.000000
40251	ATM1	123.000000
40251	ATM2	111.000000
40252	ATM1	55.000000
40252	ATM2	5.000000
40253	ATM1	4.000000
40253	ATM2	1.000000
40254	ATM1	92.000000
40254	ATM2	106.000000
40255	ATM1	88.000000
40255	ATM2	102.000000
40256	ATM1	92.000000
40256	ATM2	25.000000
40257	ATM1	92.000000
40257	ATM2	64.000000
40258	ATM1	117.000000
40258	ATM2	78.000000
40259	ATM1	73.000000
40259	ATM2	5.000000
40260	ATM1	3.000000
40260	ATM2	1.000000
40261	ATM1	85.000000
40261	ATM2	83.000000
40262	ATM1	88.000000
40262	ATM2	78.000000
40263	ATM1	76.000000
40263	ATM2	66.000000
40264	ATM1	93.000000
40264	ATM2	15.000000
40265	ATM1	116.000000
40265	ATM2	64.000000
40266	ATM1	68.000000
40266	ATM2	3.000000
40267	ATM1	19.000000
40267	ATM2	0.000000
40268	ATM1	127.000000
40268	ATM2	102.000000
40269	ATM1	93.000000
40269	ATM2	99.000000
40270	ATM1	97.000000
40270	ATM2	41.000000
40271	ATM1	102.000000
40271	ATM2	79.000000
40272	ATM1	109.000000
40272	ATM2	71.000000
40273	ATM1	68.000000
40273	ATM2	9.000000
40274	ATM1	4.000000
40274	ATM2	2.000000
40275	ATM1	105.000000
40275	ATM2	103.000000
40276	ATM1	79.000000
40276	ATM2	67.000000
40277	ATM1	90.000000
40277	ATM2	85.000000
40278	ATM1	87.000000
40278	ATM2	78.000000
40279	ATM1	92.000000
40279	ATM2	67.000000
40280	ATM1	63.000000
40280	ATM2	12.000000
40281	ATM1	3.000000
40281	ATM2	1.000000
40282	ATM1	103.000000
40282	ATM2	97.000000
40283	ATM1	80.000000
40283	ATM2	106.000000
40284	ATM1	80.000000
40284	ATM2	74.000000
40285	ATM1	84.000000
40285	ATM2	86.000000
40286	ATM1	81.000000
40286	ATM2	55.000000
40287	ATM1	88.000000
40287	ATM2	13.000000
40288	ATM1	3.000000
40288	ATM2	1.000000
40289	ATM1	100.000000
40289	ATM2	100.000000
40290	ATM1	69.000000
40290	ATM2	103.000000
40291	ATM1	85.000000
40291	ATM2	44.000000
40292	ATM1	85.000000
40292	ATM2	61.000000
40293	ATM1	109.000000
40293	ATM2	89.000000
40294	ATM1	74.000000
40294	ATM2	11.000000
40295	ATM1	4.000000
40295	ATM2	2.000000
40296	ATM1	96.000000
40296	ATM2	107.000000
40297	ATM1	82.000000
40297	ATM2	89.000000
40298	ATM1	85.000000
40298	ATM2	90.000000
40299	NA	NA
40300	NA	NA
40301	NA	NA
40302	NA	NA
40303	NA	NA
40304	NA	NA
40305	NA	NA
40306	NA	NA
40307	NA	NA
40308	NA	NA
40309	NA	NA
40310	NA	NA
40311	NA	NA
40312	NA	NA
39934	ATM3	0.000000
39935	ATM3	0.000000
39936	ATM3	0.000000
39937	ATM3	0.000000
39938	ATM3	0.000000
39939	ATM3	0.000000
39940	ATM3	0.000000
39941	ATM3	0.000000
39942	ATM3	0.000000
39943	ATM3	0.000000
39944	ATM3	0.000000
39945	ATM3	0.000000
39946	ATM3	0.000000
39947	ATM3	0.000000
39948	ATM3	0.000000
39949	ATM3	0.000000
39950	ATM3	0.000000
39951	ATM3	0.000000
39952	ATM3	0.000000
39953	ATM3	0.000000
39954	ATM3	0.000000
39955	ATM3	0.000000
39956	ATM3	0.000000
39957	ATM3	0.000000
39958	ATM3	0.000000
39959	ATM3	0.000000
39960	ATM3	0.000000
39961	ATM3	0.000000
39962	ATM3	0.000000
39963	ATM3	0.000000
39964	ATM3	0.000000
39965	ATM3	0.000000
39966	ATM3	0.000000
39967	ATM3	0.000000
39968	ATM3	0.000000
39969	ATM3	0.000000
39970	ATM3	0.000000
39971	ATM3	0.000000
39972	ATM3	0.000000
39973	ATM3	0.000000
39974	ATM3	0.000000
39975	ATM3	0.000000
39976	ATM3	0.000000
39977	ATM3	0.000000
39978	ATM3	0.000000
39979	ATM3	0.000000
39980	ATM3	0.000000
39981	ATM3	0.000000
39982	ATM3	0.000000
39983	ATM3	0.000000
39984	ATM3	0.000000
39985	ATM3	0.000000
39986	ATM3	0.000000
39987	ATM3	0.000000
39988	ATM3	0.000000
39989	ATM3	0.000000
39990	ATM3	0.000000
39991	ATM3	0.000000
39992	ATM3	0.000000
39993	ATM3	0.000000
39994	ATM3	0.000000
39995	ATM3	0.000000
39996	ATM3	0.000000
39997	ATM3	0.000000
39998	ATM3	0.000000
39999	ATM3	0.000000
40000	ATM3	0.000000
40001	ATM3	0.000000
40002	ATM3	0.000000
40003	ATM3	0.000000
40004	ATM3	0.000000
40005	ATM3	0.000000
40006	ATM3	0.000000
40007	ATM3	0.000000
40008	ATM3	0.000000
40009	ATM3	0.000000
40010	ATM3	0.000000
40011	ATM3	0.000000
40012	ATM3	0.000000
40013	ATM3	0.000000
40014	ATM3	0.000000
40015	ATM3	0.000000
40016	ATM3	0.000000
40017	ATM3	0.000000
40018	ATM3	0.000000
40019	ATM3	0.000000
40020	ATM3	0.000000
40021	ATM3	0.000000
40022	ATM3	0.000000
40023	ATM3	0.000000
40024	ATM3	0.000000
40025	ATM3	0.000000
40026	ATM3	0.000000
40027	ATM3	0.000000
40028	ATM3	0.000000
40029	ATM3	0.000000
40030	ATM3	0.000000
40031	ATM3	0.000000
40032	ATM3	0.000000
40033	ATM3	0.000000
40034	ATM3	0.000000
40035	ATM3	0.000000
40036	ATM3	0.000000
40037	ATM3	0.000000
40038	ATM3	0.000000
40039	ATM3	0.000000
40040	ATM3	0.000000
40041	ATM3	0.000000
40042	ATM3	0.000000
40043	ATM3	0.000000
40044	ATM3	0.000000
40045	ATM3	0.000000
40046	ATM3	0.000000
40047	ATM3	0.000000
40048	ATM3	0.000000
40049	ATM3	0.000000
40050	ATM3	0.000000
40051	ATM3	0.000000
40052	ATM3	0.000000
40053	ATM3	0.000000
40054	ATM3	0.000000
40055	ATM3	0.000000
40056	ATM3	0.000000
40057	ATM3	0.000000
40058	ATM3	0.000000
40059	ATM3	0.000000
40060	ATM3	0.000000
40061	ATM3	0.000000
40062	ATM3	0.000000
40063	ATM3	0.000000
40064	ATM3	0.000000
40065	ATM3	0.000000
40066	ATM3	0.000000
40067	ATM3	0.000000
40068	ATM3	0.000000
40069	ATM3	0.000000
40070	ATM3	0.000000
40071	ATM3	0.000000
40072	ATM3	0.000000
40073	ATM3	0.000000
40074	ATM3	0.000000
40075	ATM3	0.000000
40076	ATM3	0.000000
40077	ATM3	0.000000
40078	ATM3	0.000000
40079	ATM3	0.000000
40080	ATM3	0.000000
40081	ATM3	0.000000
40082	ATM3	0.000000
40083	ATM3	0.000000
40084	ATM3	0.000000
40085	ATM3	0.000000
40086	ATM3	0.000000
40087	ATM3	0.000000
40088	ATM3	0.000000
40089	ATM3	0.000000
40090	ATM3	0.000000
40091	ATM3	0.000000
40092	ATM3	0.000000
40093	ATM3	0.000000
40094	ATM3	0.000000
40095	ATM3	0.000000
40096	ATM3	0.000000
40097	ATM3	0.000000
40098	ATM3	0.000000
40099	ATM3	0.000000
40100	ATM3	0.000000
40101	ATM3	0.000000
40102	ATM3	0.000000
40103	ATM3	0.000000
40104	ATM3	0.000000
40105	ATM3	0.000000
40106	ATM3	0.000000
40107	ATM3	0.000000
40108	ATM3	0.000000
40109	ATM3	0.000000
40110	ATM3	0.000000
40111	ATM3	0.000000
40112	ATM3	0.000000
40113	ATM3	0.000000
40114	ATM3	0.000000
40115	ATM3	0.000000
40116	ATM3	0.000000
40117	ATM3	0.000000
40118	ATM3	0.000000
40119	ATM3	0.000000
40120	ATM3	0.000000
40121	ATM3	0.000000
40122	ATM3	0.000000
40123	ATM3	0.000000
40124	ATM3	0.000000
40125	ATM3	0.000000
40126	ATM3	0.000000
40127	ATM3	0.000000
40128	ATM3	0.000000
40129	ATM3	0.000000
40130	ATM3	0.000000
40131	ATM3	0.000000
40132	ATM3	0.000000
40133	ATM3	0.000000
40134	ATM3	0.000000
40135	ATM3	0.000000
40136	ATM3	0.000000
40137	ATM3	0.000000
40138	ATM3	0.000000
40139	ATM3	0.000000
40140	ATM3	0.000000
40141	ATM3	0.000000
40142	ATM3	0.000000
40143	ATM3	0.000000
40144	ATM3	0.000000
40145	ATM3	0.000000
40146	ATM3	0.000000
40147	ATM3	0.000000
40148	ATM3	0.000000
40149	ATM3	0.000000
40150	ATM3	0.000000
40151	ATM3	0.000000
40152	ATM3	0.000000
40153	ATM3	0.000000
40154	ATM3	0.000000
40155	ATM3	0.000000
40156	ATM3	0.000000
40157	ATM3	0.000000
40158	ATM3	0.000000
40159	ATM3	0.000000
40160	ATM3	0.000000
40161	ATM3	0.000000
40162	ATM3	0.000000
40163	ATM3	0.000000
40164	ATM3	0.000000
40165	ATM3	0.000000
40166	ATM3	0.000000
40167	ATM3	0.000000
40168	ATM3	0.000000
40169	ATM3	0.000000
40170	ATM3	0.000000
40171	ATM3	0.000000
40172	ATM3	0.000000
40173	ATM3	0.000000
40174	ATM3	0.000000
40175	ATM3	0.000000
40176	ATM3	0.000000
40177	ATM3	0.000000
40178	ATM3	0.000000
40179	ATM3	0.000000
40180	ATM3	0.000000
40181	ATM3	0.000000
40182	ATM3	0.000000
40183	ATM3	0.000000
40184	ATM3	0.000000
40185	ATM3	0.000000
40186	ATM3	0.000000
40187	ATM3	0.000000
40188	ATM3	0.000000
40189	ATM3	0.000000
40190	ATM3	0.000000
40191	ATM3	0.000000
40192	ATM3	0.000000
40193	ATM3	0.000000
40194	ATM3	0.000000
40195	ATM3	0.000000
40196	ATM3	0.000000
40197	ATM3	0.000000
40198	ATM3	0.000000
40199	ATM3	0.000000
40200	ATM3	0.000000
40201	ATM3	0.000000
40202	ATM3	0.000000
40203	ATM3	0.000000
40204	ATM3	0.000000
40205	ATM3	0.000000
40206	ATM3	0.000000
40207	ATM3	0.000000
40208	ATM3	0.000000
40209	ATM3	0.000000
40210	ATM3	0.000000
40211	ATM3	0.000000
40212	ATM3	0.000000
40213	ATM3	0.000000
40214	ATM3	0.000000
40215	ATM3	0.000000
40216	ATM3	0.000000
40217	ATM3	0.000000
40218	ATM3	0.000000
40219	ATM3	0.000000
40220	ATM3	0.000000
40221	ATM3	0.000000
40222	ATM3	0.000000
40223	ATM3	0.000000
40224	ATM3	0.000000
40225	ATM3	0.000000
40226	ATM3	0.000000
40227	ATM3	0.000000
40228	ATM3	0.000000
40229	ATM3	0.000000
40230	ATM3	0.000000
40231	ATM3	0.000000
40232	ATM3	0.000000
40233	ATM3	0.000000
40234	ATM3	0.000000
40235	ATM3	0.000000
40236	ATM3	0.000000
40237	ATM3	0.000000
40238	ATM3	0.000000
40239	ATM3	0.000000
40240	ATM3	0.000000
40241	ATM3	0.000000
40242	ATM3	0.000000
40243	ATM3	0.000000
40244	ATM3	0.000000
40245	ATM3	0.000000
40246	ATM3	0.000000
40247	ATM3	0.000000
40248	ATM3	0.000000
40249	ATM3	0.000000
40250	ATM3	0.000000
40251	ATM3	0.000000
40252	ATM3	0.000000
40253	ATM3	0.000000
40254	ATM3	0.000000
40255	ATM3	0.000000
40256	ATM3	0.000000
40257	ATM3	0.000000
40258	ATM3	0.000000
40259	ATM3	0.000000
40260	ATM3	0.000000
40261	ATM3	0.000000
40262	ATM3	0.000000
40263	ATM3	0.000000
40264	ATM3	0.000000
40265	ATM3	0.000000
40266	ATM3	0.000000
40267	ATM3	0.000000
40268	ATM3	0.000000
40269	ATM3	0.000000
40270	ATM3	0.000000
40271	ATM3	0.000000
40272	ATM3	0.000000
40273	ATM3	0.000000
40274	ATM3	0.000000
40275	ATM3	0.000000
40276	ATM3	0.000000
40277	ATM3	0.000000
40278	ATM3	0.000000
40279	ATM3	0.000000
40280	ATM3	0.000000
40281	ATM3	0.000000
40282	ATM3	0.000000
40283	ATM3	0.000000
40284	ATM3	0.000000
40285	ATM3	0.000000
40286	ATM3	0.000000
40287	ATM3	0.000000
40288	ATM3	0.000000
40289	ATM3	0.000000
40290	ATM3	0.000000
40291	ATM3	0.000000
40292	ATM3	0.000000
40293	ATM3	0.000000
40294	ATM3	0.000000
40295	ATM3	0.000000
40296	ATM3	96.000000
40297	ATM3	82.000000
40298	ATM3	85.000000
39934	ATM4	776.993423
39935	ATM4	524.417959
39936	ATM4	792.811362
39937	ATM4	908.238457
39938	ATM4	52.832103
39939	ATM4	52.208454
39940	ATM4	55.473609
39941	ATM4	558.503251
39942	ATM4	904.341359
39943	ATM4	879.493588
39944	ATM4	274.022340
39945	ATM4	396.108347
39946	ATM4	274.547188
39947	ATM4	16.321159
39948	ATM4	852.307037
39949	ATM4	379.561703
39950	ATM4	31.284953
39951	ATM4	491.850577
39952	ATM4	83.705480
39953	ATM4	128.653781
39954	ATM4	14.357590
39955	ATM4	815.358321
39956	ATM4	758.218587
39957	ATM4	601.421108
39958	ATM4	906.796873
39959	ATM4	502.907894
39960	ATM4	88.273138
39961	ATM4	35.438336
39962	ATM4	338.459425
39963	ATM4	4.547996
39964	ATM4	122.667745
39965	ATM4	150.234945
39966	ATM4	721.155281
39967	ATM4	443.012568
39968	ATM4	17.151720
39969	ATM4	14.887695
39970	ATM4	740.543155
39971	ATM4	1058.083384
39972	ATM4	576.183701
39973	ATM4	1484.126887
39974	ATM4	193.787327
39975	ATM4	27.054779
39976	ATM4	1190.898921
39977	ATM4	746.495617
39978	ATM4	1220.767882
39979	ATM4	1021.503529
39980	ATM4	372.623610
39981	ATM4	321.272935
39982	ATM4	92.476655
39983	ATM4	116.644133
39984	ATM4	202.413503
39985	ATM4	524.434514
39986	ATM4	80.644354
39987	ATM4	64.273589
39988	ATM4	90.636571
39989	ATM4	48.046061
39990	ATM4	1026.032172
39991	ATM4	423.770971
39992	ATM4	60.626453
39993	ATM4	540.281404
39994	ATM4	173.705568
39995	ATM4	393.239982
39996	ATM4	41.948621
39997	ATM4	310.006527
39998	ATM4	110.051915
39999	ATM4	682.182367
40000	ATM4	54.667904
40001	ATM4	213.989896
40002	ATM4	738.126360
40003	ATM4	16.206081
40004	ATM4	16.223205
40005	ATM4	1050.206528
40006	ATM4	438.477361
40007	ATM4	546.691592
40008	ATM4	858.236420
40009	ATM4	446.733514
40010	ATM4	94.672035
40011	ATM4	644.390623
40012	ATM4	568.815899
40013	ATM4	704.507042
40014	ATM4	571.645285
40015	ATM4	479.875501
40016	ATM4	418.864062
40017	ATM4	27.569130
40018	ATM4	834.834363
40019	ATM4	910.834075
40020	ATM4	468.106927
40021	ATM4	768.121660
40022	ATM4	1089.167762
40023	ATM4	266.898417
40024	ATM4	7.110665
40025	ATM4	704.192012
40026	ATM4	495.350606
40027	ATM4	142.638951
40028	ATM4	428.560174
40029	ATM4	894.969364
40030	ATM4	610.290623
40031	ATM4	70.613731
40032	ATM4	593.570261
40033	ATM4	341.601807
40034	ATM4	735.035708
40035	ATM4	462.773028
40036	ATM4	1156.496108
40037	ATM4	454.091939
40038	ATM4	283.337265
40039	ATM4	571.508226
40040	ATM4	772.179652
40041	ATM4	260.125517
40042	ATM4	357.515267
40043	ATM4	16.157597
40044	ATM4	334.312503
40045	ATM4	25.696404
40046	ATM4	254.887098
40047	ATM4	357.003129
40048	ATM4	1245.594280
40049	ATM4	917.371157
40050	ATM4	592.180082
40051	ATM4	412.474975
40052	ATM4	82.911171
40053	ATM4	996.010226
40054	ATM4	103.910375
40055	ATM4	1116.915044
40056	ATM4	816.847015
40057	ATM4	914.493389
40058	ATM4	648.209198
40059	ATM4	140.515569
40060	ATM4	1495.154775
40061	ATM4	1301.396344
40062	ATM4	779.717193
40063	ATM4	744.262278
40064	ATM4	200.391803
40065	ATM4	854.179084
40066	ATM4	50.719369
40067	ATM4	7.404799
40068	ATM4	1061.192780
40069	ATM4	715.021459
40070	ATM4	35.444668
40071	ATM4	491.884510
40072	ATM4	343.496923
40073	ATM4	20.714186
40074	ATM4	505.717125
40075	ATM4	97.325113
40076	ATM4	473.570486
40077	ATM4	899.773474
40078	ATM4	1712.074986
40079	ATM4	281.388412
40080	ATM4	26.337969
40081	ATM4	328.644988
40082	ATM4	761.371143
40083	ATM4	629.319901
40084	ATM4	235.753411
40085	ATM4	1195.223181
40086	ATM4	782.407474
40087	ATM4	108.331743
40088	ATM4	846.556566
40089	ATM4	576.169344
40090	ATM4	441.883927
40091	ATM4	319.404846
40092	ATM4	154.163484
40093	ATM4	543.239375
40094	ATM4	124.334415
40095	ATM4	449.074910
40096	ATM4	614.654064
40097	ATM4	219.237855
40098	ATM4	945.736417
40099	ATM4	9.691243
40100	ATM4	696.245755
40101	ATM4	8.084283
40102	ATM4	845.463546
40103	ATM4	212.902683
40104	ATM4	9.155135
40105	ATM4	46.831421
40106	ATM4	30.372376
40107	ATM4	400.484974
40108	ATM4	61.338241
40109	ATM4	428.046952
40110	ATM4	1.563260
40111	ATM4	9.018826
40112	ATM4	50.304215
40113	ATM4	313.414359
40114	ATM4	626.687136
40115	ATM4	5.882076
40116	ATM4	77.803024
40117	ATM4	337.894871
40118	ATM4	211.812232
40119	ATM4	690.122884
40120	ATM4	596.381877
40121	ATM4	65.278483
40122	ATM4	77.444454
40123	ATM4	43.721410
40124	ATM4	964.175255
40125	ATM4	834.959599
40126	ATM4	636.962626
40127	ATM4	927.078614
40128	ATM4	75.873055
40129	ATM4	43.690550
40130	ATM4	621.292501
40131	ATM4	312.582041
40132	ATM4	825.635396
40133	ATM4	413.620440
40134	ATM4	194.047355
40135	ATM4	345.695900
40136	ATM4	32.428111
40137	ATM4	655.315200
40138	ATM4	638.136824
40139	ATM4	15.306757
40140	ATM4	299.663010
40141	ATM4	626.662127
40142	ATM4	601.141158
40143	ATM4	64.131249
40144	ATM4	562.763323
40145	ATM4	317.253589
40146	ATM4	1167.264437
40147	ATM4	47.365125
40148	ATM4	993.777470
40149	ATM4	687.196935
40150	ATM4	71.147567
40151	ATM4	1046.971281
40152	ATM4	1009.050230
40153	ATM4	288.649516
40154	ATM4	591.508331
40155	ATM4	230.605915
40156	ATM4	578.431798
40157	ATM4	70.210639
40158	ATM4	580.796406
40159	ATM4	149.150820
40160	ATM4	403.861263
40161	ATM4	124.958721
40162	ATM4	230.148897
40163	ATM4	19.008453
40164	ATM4	128.638640
40165	ATM4	327.965321
40166	ATM4	532.245673
40167	ATM4	877.397771
40168	ATM4	662.112869
40169	ATM4	300.629002
40170	ATM4	667.628261
40171	ATM4	14.573322
40172	ATM4	660.355296
40173	ATM4	510.988823
40174	ATM4	164.462091
40175	ATM4	748.172762
40176	ATM4	174.241405
40177	ATM4	20.192725
40178	ATM4	100.686543
40179	ATM4	985.956884
40180	ATM4	597.058060
40181	ATM4	468.457250
40182	ATM4	856.581158
40183	ATM4	684.774261
40184	ATM4	381.562058
40185	ATM4	152.092362
40186	ATM4	271.934402
40187	ATM4	135.499645
40188	ATM4	1105.018595
40189	ATM4	291.765790
40190	ATM4	1141.426655
40191	ATM4	140.502720
40192	ATM4	8.781528
40193	ATM4	85.260115
40194	ATM4	66.655933
40195	ATM4	709.938568
40196	ATM4	567.549590
40197	ATM4	486.824572
40198	ATM4	17.481238
40199	ATM4	49.208112
40200	ATM4	357.233561
40201	ATM4	179.888632
40202	ATM4	728.992926
40203	ATM4	261.121740
40204	ATM4	628.863047
40205	ATM4	277.044815
40206	ATM4	41.475487
40207	ATM4	1574.779330
40208	ATM4	669.707653
40209	ATM4	979.675944
40210	ATM4	426.406008
40211	ATM4	153.242692
40212	ATM4	274.949921
40213	ATM4	136.276358
40214	ATM4	454.416950
40215	ATM4	458.304270
40216	ATM4	112.030628
40217	ATM4	417.912276
40218	ATM4	10919.761638
40219	ATM4	42.438078
40220	ATM4	280.043427
40221	ATM4	412.318881
40222	ATM4	852.837416
40223	ATM4	179.702084
40224	ATM4	226.047368
40225	ATM4	989.195610
40226	ATM4	824.916781
40227	ATM4	966.610561
40228	ATM4	734.220167
40229	ATM4	121.332622
40230	ATM4	287.931829
40231	ATM4	502.748346
40232	ATM4	9.752776
40233	ATM4	258.138869
40234	ATM4	1170.288228
40235	ATM4	193.079004
40236	ATM4	402.770395
40237	ATM4	1275.968167
40238	ATM4	819.881512
40239	ATM4	26.392615
40240	ATM4	893.884507
40241	ATM4	360.641557
40242	ATM4	859.899389
40243	ATM4	381.473089
40244	ATM4	9.711205
40245	ATM4	601.016953
40246	ATM4	32.454958
40247	ATM4	553.395793
40248	ATM4	572.374702
40249	ATM4	218.831558
40250	ATM4	828.326828
40251	ATM4	630.762176
40252	ATM4	339.418304
40253	ATM4	31.502210
40254	ATM4	486.679944
40255	ATM4	335.363138
40256	ATM4	340.008595
40257	ATM4	291.282267
40258	ATM4	46.029410
40259	ATM4	201.403616
40260	ATM4	9.558300
40261	ATM4	877.966103
40262	ATM4	778.110791
40263	ATM4	707.613494
40264	ATM4	351.457835
40265	ATM4	711.049107
40266	ATM4	502.518192
40267	ATM4	23.337569
40268	ATM4	492.909936
40269	ATM4	405.312584
40270	ATM4	818.394331
40271	ATM4	152.270800
40272	ATM4	281.113528
40273	ATM4	470.425772
40274	ATM4	2.507317
40275	ATM4	414.741223
40276	ATM4	719.426783
40277	ATM4	811.716825
40278	ATM4	889.584209
40279	ATM4	616.127547
40280	ATM4	61.144354
40281	ATM4	27.325671
40282	ATM4	767.778153
40283	ATM4	326.084093
40284	ATM4	825.197816
40285	ATM4	383.815025
40286	ATM4	195.483454
40287	ATM4	711.164653
40288	ATM4	29.869011
40289	ATM4	556.792330
40290	ATM4	386.175335
40291	ATM4	165.294181
40292	ATM4	5.451815
40293	ATM4	542.280602
40294	ATM4	403.839336
40295	ATM4	13.697331
40296	ATM4	348.201061
40297	ATM4	44.245345
40298	ATM4	482.287107

#summary(atmdata)

Missing Value Analysis

## Counts of missing data per feature
train_na_df <- data.frame(apply(atmdata, 2, function(x) length(which(is.na(x)))))
train_na_df1 <- data.frame(apply(atmdata, 2,function(x) {sum(is.na(x)) / length(x) * 100}))

train_na_df <- cbind(Feature = rownames(train_na_df), train_na_df, train_na_df1)
colnames(train_na_df) <- c('Feature Name','No. of NA Recocrds','Percentage of NA Records')
rownames(train_na_df) <- NULL


train_na_df%>% filter(`No. of NA Recocrds` != 0) %>% arrange(desc(`No. of NA Recocrds`)) %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="150px")

Feature Name	No. of NA Recocrds	Percentage of NA Records
Cash	19	1.2890095
ATM	14	0.9497965

Data Pre-Processing

Firstly, removing records with missing ‘ATM’ values -

atmdata <- atmdata %>% dplyr::filter(ATM != '')

Converting the ‘DATE’ column from Excel numeric format to proper date format -

atmdata$DATE_Formatted <- as.Date(atmdata$DATE, origin = "1899-12-30")
atmdataDF <- atmdata %>% select(DATE_Formatted,ATM,Cash) 
colnames(atmdataDF) <- c("DATE","ATM","Cash")

Pivoting the dataframe by ATM:

atmdataDF <- atmdataDF %>% pivot_wider(names_from = ATM, values_from = Cash)
atmdataDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

DATE	ATM1	ATM2	ATM3	ATM4
2009-05-01	96	107	0	776.993423
2009-05-02	82	89	0	524.417959
2009-05-03	85	90	0	792.811362
2009-05-04	90	55	0	908.238457
2009-05-05	99	79	0	52.832103
2009-05-06	88	19	0	52.208454
2009-05-07	8	2	0	55.473609
2009-05-08	104	103	0	558.503251
2009-05-09	87	107	0	904.341359
2009-05-10	93	118	0	879.493588
2009-05-11	86	75	0	274.022340
2009-05-12	111	111	0	396.108347
2009-05-13	75	25	0	274.547188
2009-05-14	6	16	0	16.321159
2009-05-15	102	137	0	852.307037
2009-05-16	73	95	0	379.561703
2009-05-17	92	103	0	31.284953
2009-05-18	82	80	0	491.850577
2009-05-19	86	118	0	83.705480
2009-05-20	73	30	0	128.653781
2009-05-21	20	7	0	14.357590
2009-05-22	100	118	0	815.358321
2009-05-23	93	104	0	758.218587
2009-05-24	90	59	0	601.421108
2009-05-25	94	40	0	906.796873
2009-05-26	98	106	0	502.907894
2009-05-27	73	18	0	88.273138
2009-05-28	10	9	0	35.438336
2009-05-29	97	136	0	338.459425
2009-05-30	102	118	0	4.547996
2009-05-31	85	64	0	122.667745
2009-06-01	85	77	0	150.234945
2009-06-02	108	133	0	721.155281
2009-06-03	94	45	0	443.012568
2009-06-04	14	14	0	17.151720
2009-06-05	3	20	0	14.887695
2009-06-06	96	147	0	740.543155
2009-06-07	109	105	0	1058.083384
2009-06-08	96	132	0	576.183701
2009-06-09	145	93	0	1484.126887
2009-06-10	81	26	0	193.787327
2009-06-11	16	7	0	27.054779
2009-06-12	142	112	0	1190.898921
2009-06-13	NA	91	0	746.495617
2009-06-14	120	72	0	1220.767882
2009-06-15	106	66	0	1021.503529
2009-06-16	NA	82	0	372.623610
2009-06-17	108	24	0	321.272935
2009-06-18	21	NA	0	92.476655
2009-06-19	140	134	0	116.644133
2009-06-20	110	95	0	202.413503
2009-06-21	115	82	0	524.434514
2009-06-22	NA	90	0	80.644354
2009-06-23	108	99	0	64.273589
2009-06-24	66	NA	0	90.636571
2009-06-25	13	3	0	48.046061
2009-06-26	99	117	0	1026.032172
2009-06-27	105	53	0	423.770971
2009-06-28	104	44	0	60.626453
2009-06-29	98	56	0	540.281404
2009-06-30	110	110	0	173.705568
2009-07-01	79	36	0	393.239982
2009-07-02	16	12	0	41.948621
2009-07-03	110	128	0	310.006527
2009-07-04	96	72	0	110.051915
2009-07-05	114	122	0	682.182367
2009-07-06	126	100	0	54.667904
2009-07-07	126	108	0	213.989896
2009-07-08	73	25	0	738.126360
2009-07-09	4	6	0	16.206081
2009-07-10	19	22	0	16.223205
2009-07-11	114	135	0	1050.206528
2009-07-12	98	69	0	438.477361
2009-07-13	97	52	0	546.691592
2009-07-14	114	81	0	858.236420
2009-07-15	78	27	0	446.733514
2009-07-16	19	4	0	94.672035
2009-07-17	102	147	0	644.390623
2009-07-18	94	102	0	568.815899
2009-07-19	108	83	0	704.507042
2009-07-20	91	55	0	571.645285
2009-07-21	86	74	0	479.875501
2009-07-22	78	22	0	418.864062
2009-07-23	16	4	0	27.569130
2009-07-24	114	104	0	834.834363
2009-07-25	115	81	0	910.834075
2009-07-26	108	61	0	468.106927
2009-07-27	102	70	0	768.121660
2009-07-28	129	126	0	1089.167762
2009-07-29	79	33	0	266.898417
2009-07-30	13	7	0	7.110665
2009-07-31	103	126	0	704.192012
2009-08-01	90	92	0	495.350606
2009-08-02	68	81	0	142.638951
2009-08-03	85	49	0	428.560174
2009-08-04	99	146	0	894.969364
2009-08-05	86	79	0	610.290623
2009-08-06	13	37	0	70.613731
2009-08-07	116	136	0	593.570261
2009-08-08	105	111	0	341.601807
2009-08-09	123	78	0	735.035708
2009-08-10	114	57	0	462.773028
2009-08-11	127	106	0	1156.496108
2009-08-12	111	38	0	454.091939
2009-08-13	34	15	0	283.337265
2009-08-14	151	119	0	571.508226
2009-08-15	110	110	0	772.179652
2009-08-16	115	68	0	260.125517
2009-08-17	112	60	0	357.515267
2009-08-18	132	92	0	16.157597
2009-08-19	94	22	0	334.312503
2009-08-20	24	11	0	25.696404
2009-08-21	122	121	0	254.887098
2009-08-22	104	89	0	357.003129
2009-08-23	128	62	0	1245.594280
2009-08-24	120	79	0	917.371157
2009-08-25	174	83	0	592.180082
2009-08-26	96	31	0	412.474975
2009-08-27	13	4	0	82.911171
2009-08-28	121	96	0	996.010226
2009-08-29	133	50	0	103.910375
2009-08-30	118	52	0	1116.915044
2009-08-31	91	56	0	816.847015
2009-09-01	120	104	0	914.493389
2009-09-02	88	27	0	648.209198
2009-09-03	19	13	0	140.515569
2009-09-04	150	107	0	1495.154775
2009-09-05	144	125	0	1301.396344
2009-09-06	121	103	0	779.717193
2009-09-07	105	42	0	744.262278
2009-09-08	133	90	0	200.391803
2009-09-09	109	29	0	854.179084
2009-09-10	18	8	0	50.719369
2009-09-11	1	2	0	7.404799
2009-09-12	105	84	0	1061.192780
2009-09-13	112	62	0	715.021459
2009-09-14	82	77	0	35.444668
2009-09-15	111	78	0	491.884510
2009-09-16	79	25	0	343.496923
2009-09-17	13	8	0	20.714186
2009-09-18	112	113	0	505.717125
2009-09-19	99	71	0	97.325113
2009-09-20	140	94	0	473.570486
2009-09-21	110	59	0	899.773474
2009-09-22	180	89	0	1712.074986
2009-09-23	73	18	0	281.388412
2009-09-24	7	6	0	26.337969
2009-09-25	106	115	0	328.644988
2009-09-26	103	81	0	761.371143
2009-09-27	93	61	0	629.319901
2009-09-28	96	71	0	235.753411
2009-09-29	117	69	0	1195.223181
2009-09-30	80	36	0	782.407474
2009-10-01	14	14	0	108.331743
2009-10-02	120	104	0	846.556566
2009-10-03	91	73	0	576.169344
2009-10-04	96	86	0	441.883927
2009-10-05	74	85	0	319.404846
2009-10-06	108	126	0	154.163484
2009-10-07	73	31	0	543.239375
2009-10-08	13	9	0	124.334415
2009-10-09	93	114	0	449.074910
2009-10-10	94	78	0	614.654064
2009-10-11	76	45	0	219.237855
2009-10-12	111	60	0	945.736417
2009-10-13	88	91	0	9.691243
2009-10-14	76	22	0	696.245755
2009-10-15	9	7	0	8.084283
2009-10-16	87	75	0	845.463546
2009-10-17	105	66	0	212.902683
2009-10-18	78	64	0	9.155135
2009-10-19	67	51	0	46.831421
2009-10-20	90	94	0	30.372376
2009-10-21	68	23	0	400.484974
2009-10-22	9	4	0	61.338241
2009-10-23	78	127	0	428.046952
2009-10-24	74	61	0	1.563260
2009-10-25	74	0	0	9.018826
2009-10-26	60	95	0	50.304215
2009-10-27	75	79	0	313.414359
2009-10-28	61	38	0	626.687136
2009-10-29	9	8	0	5.882076
2009-10-30	90	119	0	77.803024
2009-10-31	86	57	0	337.894871
2009-11-01	86	58	0	211.812232
2009-11-02	79	80	0	690.122884
2009-11-03	90	82	0	596.381877
2009-11-04	80	49	0	65.278483
2009-11-05	21	16	0	77.444454
2009-11-06	93	116	0	43.721410
2009-11-07	104	61	0	964.175255
2009-11-08	109	59	0	834.959599
2009-11-09	88	80	0	636.962626
2009-11-10	96	86	0	927.078614
2009-11-11	70	23	0	75.873055
2009-11-12	15	7	0	43.690550
2009-11-13	73	91	0	621.292501
2009-11-14	94	57	0	312.582041
2009-11-15	108	58	0	825.635396
2009-11-16	73	61	0	413.620440
2009-11-17	87	77	0	194.047355
2009-11-18	75	20	0	345.695900
2009-11-19	10	5	0	32.428111
2009-11-20	92	132	0	655.315200
2009-11-21	87	49	0	638.136824
2009-11-22	74	57	0	15.306757
2009-11-23	73	68	0	299.663010
2009-11-24	93	80	0	626.662127
2009-11-25	66	31	0	601.141158
2009-11-26	18	3	0	64.131249
2009-11-27	99	85	0	562.763323
2009-11-28	94	53	0	317.253589
2009-11-29	136	46	0	1167.264437
2009-11-30	6	2	0	47.365125
2009-12-01	140	113	0	993.777470
2009-12-02	73	22	0	687.196935
2009-12-03	9	5	0	71.147567
2009-12-04	140	112	0	1046.971281
2009-12-05	103	59	0	1009.050230
2009-12-06	110	72	0	288.649516
2009-12-07	90	77	0	591.508331
2009-12-08	135	85	0	230.605915
2009-12-09	67	27	0	578.431798
2009-12-10	12	1	0	70.210639
2009-12-11	109	91	0	580.796406
2009-12-12	84	36	0	149.150820
2009-12-13	92	46	0	403.861263
2009-12-14	84	100	0	124.958721
2009-12-15	118	73	0	230.148897
2009-12-16	68	22	0	19.008453
2009-12-17	14	9	0	128.638640
2009-12-18	90	117	0	327.965321
2009-12-19	92	44	0	532.245673
2009-12-20	93	44	0	877.397771
2009-12-21	85	78	0	662.112869
2009-12-22	93	89	0	300.629002
2009-12-23	70	33	0	667.628261
2009-12-24	13	5	0	14.573322
2009-12-25	90	102	0	660.355296
2009-12-26	91	68	0	510.988823
2009-12-27	102	64	0	164.462091
2009-12-28	97	81	0	748.172762
2009-12-29	42	9	0	174.241405
2009-12-30	2	2	0	20.192725
2009-12-31	14	2	0	100.686543
2010-01-01	132	117	0	985.956884
2010-01-02	108	56	0	597.058060
2010-01-03	120	55	0	468.457250
2010-01-04	120	112	0	856.581158
2010-01-05	90	100	0	684.774261
2010-01-06	48	17	0	381.562058
2010-01-07	15	7	0	152.092362
2010-01-08	86	49	0	271.934402
2010-01-09	109	96	0	135.499645
2010-01-10	115	47	0	1105.018595
2010-01-11	123	107	0	291.765790
2010-01-12	123	93	0	1141.426655
2010-01-13	60	27	0	140.502720
2010-01-14	22	9	0	8.781528
2010-01-15	81	78	0	85.260115
2010-01-16	98	47	0	66.655933
2010-01-17	94	31	0	709.938568
2010-01-18	76	91	0	567.549590
2010-01-19	96	77	0	486.824572
2010-01-20	72	22	0	17.481238
2010-01-21	12	1	0	49.208112
2010-01-22	91	125	0	357.233561
2010-01-23	74	53	0	179.888632
2010-01-24	104	37	0	728.992926
2010-01-25	74	69	0	261.121740
2010-01-26	91	69	0	628.863047
2010-01-27	66	16	0	277.044815
2010-01-28	28	2	0	41.475487
2010-01-29	152	95	0	1574.779330
2010-01-30	97	45	0	669.707653
2010-01-31	100	61	0	979.675944
2010-02-01	85	71	0	426.406008
2010-02-02	123	91	0	153.242692
2010-02-03	46	28	0	274.949921
2010-02-04	38	9	0	136.276358
2010-02-05	138	97	0	454.416950
2010-02-06	74	60	0	458.304270
2010-02-07	85	27	0	112.030628
2010-02-08	78	74	0	417.912276
2010-02-09	123	45	0	10919.761638
2010-02-10	50	26	0	42.438078
2010-02-11	28	2	0	280.043427
2010-02-12	105	73	0	412.318881
2010-02-13	109	48	0	852.837416
2010-02-14	26	9	0	179.702084
2010-02-15	54	25	0	226.047368
2010-02-16	105	58	0	989.195610
2010-02-17	179	110	0	824.916781
2010-02-18	125	94	0	966.610561
2010-02-19	84	25	0	734.220167
2010-02-20	99	101	0	121.332622
2010-02-21	110	77	0	287.931829
2010-02-22	80	6	0	502.748346
2010-02-23	1	3	0	9.752776
2010-02-24	111	99	0	258.138869
2010-02-25	115	86	0	1170.288228
2010-02-26	108	20	0	193.079004
2010-02-27	100	72	0	402.770395
2010-02-28	152	81	0	1275.968167
2010-03-01	90	16	0	819.881512
2010-03-02	4	2	0	26.392615
2010-03-03	128	110	0	893.884507
2010-03-04	87	75	0	360.641557
2010-03-05	84	35	0	859.899389
2010-03-06	69	102	0	381.473089
2010-03-07	112	100	0	9.711205
2010-03-08	62	9	0	601.016953
2010-03-09	4	2	0	32.454958
2010-03-10	94	100	0	553.395793
2010-03-11	102	85	0	572.374702
2010-03-12	98	33	0	218.831558
2010-03-13	88	24	0	828.326828
2010-03-14	123	111	0	630.762176
2010-03-15	55	5	0	339.418304
2010-03-16	4	1	0	31.502210
2010-03-17	92	106	0	486.679944
2010-03-18	88	102	0	335.363138
2010-03-19	92	25	0	340.008595
2010-03-20	92	64	0	291.282267
2010-03-21	117	78	0	46.029410
2010-03-22	73	5	0	201.403616
2010-03-23	3	1	0	9.558300
2010-03-24	85	83	0	877.966103
2010-03-25	88	78	0	778.110791
2010-03-26	76	66	0	707.613494
2010-03-27	93	15	0	351.457835
2010-03-28	116	64	0	711.049107
2010-03-29	68	3	0	502.518192
2010-03-30	19	0	0	23.337569
2010-03-31	127	102	0	492.909936
2010-04-01	93	99	0	405.312584
2010-04-02	97	41	0	818.394331
2010-04-03	102	79	0	152.270800
2010-04-04	109	71	0	281.113528
2010-04-05	68	9	0	470.425772
2010-04-06	4	2	0	2.507317
2010-04-07	105	103	0	414.741223
2010-04-08	79	67	0	719.426783
2010-04-09	90	85	0	811.716825
2010-04-10	87	78	0	889.584209
2010-04-11	92	67	0	616.127547
2010-04-12	63	12	0	61.144354
2010-04-13	3	1	0	27.325671
2010-04-14	103	97	0	767.778153
2010-04-15	80	106	0	326.084093
2010-04-16	80	74	0	825.197816
2010-04-17	84	86	0	383.815025
2010-04-18	81	55	0	195.483454
2010-04-19	88	13	0	711.164653
2010-04-20	3	1	0	29.869011
2010-04-21	100	100	0	556.792330
2010-04-22	69	103	0	386.175335
2010-04-23	85	44	0	165.294181
2010-04-24	85	61	0	5.451815
2010-04-25	109	89	0	542.280602
2010-04-26	74	11	0	403.839336
2010-04-27	4	2	0	13.697331
2010-04-28	96	107	96	348.201061
2010-04-29	82	89	82	44.245345
2010-04-30	85	90	85	482.287107

summary(atmdataDF)

##       DATE                 ATM1             ATM2             ATM3        
##  Min.   :2009-05-01   Min.   :  1.00   Min.   :  0.00   Min.   : 0.0000  
##  1st Qu.:2009-07-31   1st Qu.: 73.00   1st Qu.: 25.50   1st Qu.: 0.0000  
##  Median :2009-10-30   Median : 91.00   Median : 67.00   Median : 0.0000  
##  Mean   :2009-10-30   Mean   : 83.89   Mean   : 62.58   Mean   : 0.7206  
##  3rd Qu.:2010-01-29   3rd Qu.:108.00   3rd Qu.: 93.00   3rd Qu.: 0.0000  
##  Max.   :2010-04-30   Max.   :180.00   Max.   :147.00   Max.   :96.0000  
##                       NA's   :3        NA's   :2                         
##       ATM4          
##  Min.   :    1.563  
##  1st Qu.:  124.334  
##  Median :  403.839  
##  Mean   :  474.043  
##  3rd Qu.:  704.507  
##  Max.   :10919.762  
##

It can be observed for ATM1 and ATM2, there are some missing ‘NA’ values present which need to be imputed. For ATM3, we have data from only 3 days which could be due to the fact that ATM might have a delayed operational start date compared to other two ATMs. ATM4 clearly has an outlier.

Outlier handling:

atmdataDF <- atmdataDF %>% mutate(ATM4= ifelse(ATM4== max(ATM4),median(ATM4, na.rm = TRUE),ATM4))

Converting into a Timeseries object:

atm_ts <- atmdataDF %>% select(-DATE) %>% ts(start= c(2009,as.numeric(format(as.Date("2009-05-01"), "%j"))), frequency=365)

#atmxts <- xts(atmdataDF %>% select(-DATE), order.by = atmdataDF$DATE)

Time Plot:

autoplot(atm_ts, facets = TRUE) +
  ggtitle("ATM Cash Withdrawal") +
  xlab("Days") + 
  ylab("Thousands of Dollars")

From the plots above, it is evident that- - ATM1 and ATM2 have some weekly or monthly seasonality present in teh data - ATM3 has very little data available - only 3 days - ATM4 looks more or less stationary apart from one observation which could be an outlier

ATM1:

Data Imputation:

I have used imputeTS package to impute missing data for ATM1.

ggplot_na_distribution(atm_ts[,"ATM1"])

atm1_ts <- na_kalman(atm_ts[,"ATM1"], model = "auto.arima")
ggplot_na_imputations(atm_ts[,"ATM1"], atm1_ts)

Before running any models I will check the ACF and PACF plots, and the ndiffs, nsdiffs, and BoxCox.lambda functions to see what they recommend for differencing and what type of model they suggest might be most appropriate.

# Time plot
atm1_ts %>% ggtsdisplay(main = "Cash Drawn from ATM1"
                         ,xlab = "Days"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

Observations:

From ACF, it seems like there is weekly seasonality in the data is evident with large spikes on lags 7, 14, 21 etc.
Also, PACF shows a high spike at lag=7
There is no clear trend present in data

Hence, due to the presence of strong weekly trend I chaged the frequency of time series to 7.

atm1_weekly_ts <- ts(atm1_ts, frequency=7)

atm1_weekly_ts %>% ggtsdisplay(main = "Cash Drawn from ATM1"
                         ,xlab = "Weeks"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

ndiffs(atm1_weekly_ts)

## [1] 0

nsdiffs(atm1_weekly_ts)

## [1] 1

atm1_lambda <- BoxCox.lambda(atm1_weekly_ts)

cat("Box Cox Transformation factor lambda=",atm1_lambda)

## Box Cox Transformation factor lambda= 0.3240927

For ATM1 no first order differencing is recommended, only a first order seasonal difference and a box-cox transformation with lambda = 0.3240927. Let’s plot the data again after these transformations are performed to see what impact they have.

atm1_weekly_ts %>% BoxCox(atm1_lambda) %>% diff(lag=7) %>% ggtsdisplay(main = "Cash Drawn from ATM1 - w/ Box Cox Transform + Seasonal Differencing"
                         ,xlab = "Weeks"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

The plot above shows stationary timeseries data with most of the seasonality eliminated, although there are still spikes in the ACF plot at lag 7 and in the PACF plot at lags 7, 14, and 21. So I am going to add a first order differencing to eliminate any remaining seasonality -

atm1_weekly_ts %>% BoxCox(atm1_lambda) %>% diff(lag=7) %>% diff() %>% ggtsdisplay(main = "Cash Drawn from ATM1 - w/ Box Cox Transform + Seasonal + 1st Order Differencing"
                         ,xlab = "Weeks"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

#### Model1: Holt-Winters w/ Box Cox

atm1_model_fit1 <- atm1_weekly_ts %>% hw(h=31, seasonal="additive", 
                           damped=TRUE, lambda = atm1_lambda)
autoplot(atm1_model_fit1) + theme(panel.background = element_blank()) +
  ggtitle("Holt-Winters Damped Additive Method w/ Box Cox Transofrm") +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model1 Accuracy

accuracyDF <- data.frame(Model = "Holt-Winter's Additive Method with Box-Cox Transform", accuracy(atm1_model_fit1), row.names = NULL)

accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	2.587001	26.31578	17.17477	-96.05645	115.8894	0.9095107	0.066713

Model1 Residual

checkresiduals(atm1_model_fit1)

## 
##  Ljung-Box test
## 
## data:  Residuals from Damped Holt-Winters' additive method
## Q* = 24.592, df = 3, p-value = 1.879e-05
## 
## Model df: 12.   Total lags used: 15

The residuals plot looks not too bad, but Ljung-Box test has an extremely small p-value indicating that there is still some autocorrelation in our data as we saw in the plot of the transformed data. Our forecast plot looks not too bad either although those confidence intervals extend way past what we have seen historically in the data.

Model2: ETS

atm1_model_fit2 <- atm1_weekly_ts %>% ets(model="ZZZ", lambda = atm1_lambda)

autoplot(atm1_model_fit2) + theme(panel.background = element_blank())

autoplot(forecast(atm1_model_fit2, h=31)) + theme(panel.background = element_blank()) +
  ggtitle("ETS method w/ Box Cox Transofrm") +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model2 Accuracy

accuracyDF <- rbind(accuracyDF,data.frame(Model = "ETS Method with Box-Cox Transform", accuracy(atm1_model_fit2), row.names = NULL))

accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	2.587001	26.31578	17.17477	-96.05645	115.8894	0.9095107	0.0667130
ETS Method with Box-Cox Transform	2.611202	26.31169	17.15176	-95.24957	115.2117	0.9082923	0.0664474

Model2 Residual

checkresiduals(atm1_model_fit2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,N,A)
## Q* = 24.585, df = 5, p-value = 0.0001675
## 
## Model df: 9.   Total lags used: 14

The ETS model produced almost exactly the same results as Holt Winter’s model with only slightly better RMSE and Ljung-Box results.

Model3: ARIMA

atm1_model_fit3 <- auto.arima(atm1_weekly_ts,stepwise=FALSE, approximation=FALSE)

autoplot(forecast(atm1_model_fit3, h=31)) + theme(panel.background = element_blank()) +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model3 Accuracy

accuracyDF <- rbind(accuracyDF,data.frame(Model = "ARIMA(0,0,1)(1,1,1)[7]", accuracy(atm1_model_fit3), row.names = NULL))

accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	2.5870006	26.31578	17.17477	-96.05645	115.8894	0.9095107	0.0667130
ETS Method with Box-Cox Transform	2.6112023	26.31169	17.15176	-95.24957	115.2117	0.9082923	0.0664474
ARIMA(0,0,1)(1,1,1)[7]	-0.0651367	24.94054	15.68764	-107.44696	123.0573	0.8307584	-0.0057811

Model3 Residual

checkresiduals(atm1_model_fit3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,0,1)(1,1,1)[7]
## Q* = 18.903, df = 11, p-value = 0.06286
## 
## Model df: 3.   Total lags used: 14

The ARIMA model resulted in the best fit with the best RMSE and a Ljung-Box p-value that means we cannot reject the null hypothesis that the series is consistent with white noise. The plot of the forecast also looks like a more reasonable estimate of what we can expect based on the historical data.

Results:

atm1_forecast <- forecast(atm1_model_fit3, h=31)

ATM2:

Following the exact same procedure for ATM2 -

ggplot_na_distribution(atm_ts[,"ATM2"])

atm2_ts <- na_kalman(atm_ts[,"ATM2"], model = "auto.arima")
ggplot_na_imputations(atm_ts[,"ATM2"], atm2_ts)

atm2_weekly_ts <- ts(atm2_ts, frequency=7)

# Time plot
atm2_weekly_ts %>% ggtsdisplay(main = "Cash Drawn from ATM2"
                         ,xlab = "Days"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

Observations:

From ACF, it seems like there is weekly seasonality in the data is evident with large spikes on lags 7, 14, 21 etc.
Also, PACF shows a high spike at lag=7
There is no clear trend present in data

ndiffs(atm2_weekly_ts)

## [1] 1

nsdiffs(atm2_weekly_ts)

## [1] 1

atm2_lambda <- BoxCox.lambda(atm2_weekly_ts)

cat("Box Cox Transformation factor lambda=",atm2_lambda)

## Box Cox Transformation factor lambda= 0.7286677

For ATM2 a first order differencing is recommended along with a first order seasonal difference and a box-cox transformation with lambda = 0.7286677. Let’s plot the data again after these transformations are performed to see what impact they have.

atm2_weekly_ts %>% BoxCox(atm2_lambda) %>% diff(lag=7) %>% ggtsdisplay(main = "Cash Drawn from ATM2 - w/ Box Cox Transform + Seasonal Differencing"
                         ,xlab = "Weeks"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

The plot above shows stationary timeseries data with most of the seasonality eliminated, although there are still spikes in the ACF plot at lag 7 and in the PACF plot at lags 1 and 7

Model1: Holt-Winters w/ Box Cox

atm2_model_fit1 <- atm2_weekly_ts %>% hw(h=31, seasonal="additive", 
                           damped=TRUE, lambda = atm2_lambda)
autoplot(atm2_model_fit1) + theme(panel.background = element_blank()) +
  ggtitle("Holt-Winters Damped Additive Method w/ Box Cox Transofrm") +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model1 Accuracy

atm2_accuracyDF <- data.frame(Model = "Holt-Winter's Additive Method with Box-Cox Transform", accuracy(atm2_model_fit1), row.names = NULL)

atm2_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	0.6437591	25.43064	17.93037	-Inf	Inf	0.8611268	0.0193814

Model1 Residual

checkresiduals(atm2_model_fit1)

## 
##  Ljung-Box test
## 
## data:  Residuals from Damped Holt-Winters' additive method
## Q* = 34.089, df = 3, p-value = 1.898e-07
## 
## Model df: 12.   Total lags used: 15

Model2: ETS

atm2_model_fit2 <- atm2_weekly_ts %>% ets(model="ZZZ", lambda = atm2_lambda)

autoplot(atm2_model_fit2) + theme(panel.background = element_blank())

autoplot(forecast(atm2_model_fit2, h=31)) + theme(panel.background = element_blank()) +
  ggtitle("ETS method w/ Box Cox Transofrm") +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model2 Accuracy

atm2_accuracyDF <- rbind(atm2_accuracyDF,data.frame(Model = "ETS Method with Box-Cox Transform", accuracy(atm2_model_fit2), row.names = NULL))

atm2_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	0.6437591	25.43064	17.93037	-Inf	Inf	0.8611268	0.0193814
ETS Method with Box-Cox Transform	0.5087362	25.33260	17.79530	-Inf	Inf	0.8546398	0.0195558

Model2 Residual

checkresiduals(atm2_model_fit2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,N,A)
## Q* = 33.668, df = 5, p-value = 2.773e-06
## 
## Model df: 9.   Total lags used: 14

The ETS model produced almost exactly the same results as Holt Winter’s model.

Model3: ARIMA

atm2_model_fit3 <- auto.arima(atm2_weekly_ts,stepwise=FALSE, approximation=FALSE)

autoplot(forecast(atm2_model_fit3, h=31)) + theme(panel.background = element_blank()) +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model3 Accuracy

atm2_accuracyDF <- rbind(atm2_accuracyDF,data.frame(Model = "ARIMA (2,0,2)(0,1,1)[7]", accuracy(atm2_model_fit3), row.names = NULL))

atm2_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	0.6437591	25.43064	17.93037	-Inf	Inf	0.8611268	0.0193814
ETS Method with Box-Cox Transform	0.5087362	25.33260	17.79530	-Inf	Inf	0.8546398	0.0195558
ARIMA (2,0,2)(0,1,1)[7]	-0.8859274	24.13986	17.04571	-Inf	Inf	0.8186400	-0.0051208

Model3 Residual

checkresiduals(atm2_model_fit3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,0,2)(0,1,1)[7]
## Q* = 10.279, df = 9, p-value = 0.3284
## 
## Model df: 5.   Total lags used: 14

Model4: ARIMA with 1st Order Differencing

Since the ndiffs() function recommended first order differencing but the auto.arima() function did not use differencing in the model, I want to manually adding it to see if we can improve the model.

atm2_model_fit4 <- Arima(diff(atm2_weekly_ts), order=c(2,1,2),seasonal=c(0,1,1), lambda = atm2_lambda)

autoplot(forecast(atm2_model_fit4, h=31)) + theme(panel.background = element_blank()) +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

##### Model4 Accuracy

atm2_accuracyDF <- rbind(atm2_accuracyDF,data.frame(Model = "ARIMA (2,1,2)(0,1,1)[7]", accuracy(atm2_model_fit4), row.names = NULL))

atm2_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	0.6437591	25.43064	17.93037	-Inf	Inf	0.8611268	0.0193814
ETS Method with Box-Cox Transform	0.5087362	25.33260	17.79530	-Inf	Inf	0.8546398	0.0195558
ARIMA (2,0,2)(0,1,1)[7]	-0.8859274	24.13986	17.04571	-Inf	Inf	0.8186400	-0.0051208
ARIMA (2,1,2)(0,1,1)[7]	2.3306931	28.98641	20.53656	NaN	Inf	0.7139819	-0.0411685

Model4 Residual

checkresiduals(atm2_model_fit4)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(2,1,2)(0,1,1)[7]
## Q* = 31.365, df = 9, p-value = 0.0002563
## 
## Model df: 5.   Total lags used: 14

The auto.arima() function gave us the best results so that model will be used for predictions.

Results:

atm2_forecast <- forecast(atm2_model_fit3, h=31)

ATM3:

Sicne there is not enough data for ATM3, I am going to use simple mean value to forecast for month of May -

atm3_ts <- atm_ts[(nrow(atm_ts) - 2):nrow(atm_ts), "ATM3"]
atm3_ts <- ts(atm3_ts, start = 363)
atm_ts[,"ATM3"] -> atm3_ts
atm3_ts[which(atm3_ts == 0)] <- NA


# Time plot
atm3_ts %>% ggtsdisplay(main = "Cash Drawn from ATM3"
                         ,xlab = "Weeks"
                         ,ylab = "Cash in hundreds of dollars")

## Warning: Removed 362 rows containing missing values (geom_point).

#### Mean Forecast Model:

atm3_forecast <- meanf(atm3_ts, 31)

ATM4:

For ATM4, I have followed the exact same procedure as ATM1 and ATM2.

atm4_weekly_ts <- ts(atm_ts[,"ATM4"],frequency = 7)

# Time plot
atm4_weekly_ts %>% ggtsdisplay(main = "Cash Drawn from ATM4"
                         ,xlab = "Days"
                         ,ylab = "Cash in hundreds of dollars")

ndiffs(atm4_weekly_ts)

## [1] 0

nsdiffs(atm4_weekly_ts)

## [1] 0

atm4_lambda <- BoxCox.lambda(atm4_weekly_ts)

cat("Box Cox Transformation factor lambda=",atm4_lambda)

## Box Cox Transformation factor lambda= 0.4525697

Observations:

Weekly seasonal pattern is clearly visible with spikes at lag = 7,14,21 etc..
Here no differencing or seasonal differencing was recommended but a box-cox transformation with lambda = 0.4525697 was.
Looking at the plots however, it’s not clear that the box-cox transformation improved the stationarity of the data. Seasonal spikes are still apparent.

atm4_weekly_ts %>% BoxCox(atm4_lambda) %>% diff(lag=7) %>% ggtsdisplay(main = "Cash Drawn from ATM4 - w/ Box Cox Transform + Seasonal Differencing"
                         ,xlab = "Weeks"
                         ,ylab = "Cash Withdrawn (in hundreds of dollars)")

The plot above shows stationary timeseries data with most of the seasonality eliminated, although there are still spikes in the ACF plot at lag 7 and in the PACF plot at lags 7

Model1: Holt-Winters w/ Box Cox

atm4_model_fit1 <- atm4_weekly_ts %>% hw(h=31, seasonal="additive", 
                           damped=TRUE, lambda = atm1_lambda)
autoplot(atm4_model_fit1) + theme(panel.background = element_blank()) +
  ggtitle("Holt-Winters Damped Additive Method w/ Box Cox Transofrm") +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model1 Accuracy

atm4_accuracyDF <- data.frame(Model = "Holt-Winter's Additive Method with Box-Cox Transform", accuracy(atm4_model_fit1), row.names = NULL)

atm4_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	91.46219	345.9815	264.085	-344.7372	392.3998	0.7623343	0.0982354

Model1 Residual

checkresiduals(atm4_model_fit1)

## 
##  Ljung-Box test
## 
## data:  Residuals from Damped Holt-Winters' additive method
## Q* = 22.128, df = 3, p-value = 6.136e-05
## 
## Model df: 12.   Total lags used: 15

Model2: ETS

atm4_model_fit2 <- atm4_weekly_ts %>% ets(model="ZZZ", lambda = atm4_lambda)

autoplot(atm4_model_fit2) + theme(panel.background = element_blank())

autoplot(forecast(atm4_model_fit2, h=31)) + theme(panel.background = element_blank()) +
  ggtitle("ETS method w/ Box Cox Transofrm") +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model2 Accuracy

atm4_accuracyDF <- rbind(atm4_accuracyDF,data.frame(Model = "ETS Method with Box-Cox Transform", accuracy(atm4_model_fit2), row.names = NULL))

atm4_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	91.46219	345.9815	264.0850	-344.7372	392.3998	0.7623343	0.0982354
ETS Method with Box-Cox Transform	67.23932	338.2241	259.7873	-376.9491	420.7907	0.7499281	0.0973636

Model2 Residual

checkresiduals(atm4_model_fit2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,N,A)
## Q* = 19.488, df = 5, p-value = 0.001559
## 
## Model df: 9.   Total lags used: 14

The ETS model produced a slightly better RMSE than Holt Winter’s model.

Model3: ARIMA

atm4_model_fit3 <- auto.arima(atm4_weekly_ts,stepwise=FALSE, approximation=FALSE)

autoplot(forecast(atm4_model_fit3, h=31)) + theme(panel.background = element_blank()) +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model3 Accuracy

atm4_accuracyDF <- rbind(atm4_accuracyDF,data.frame(Model = "ARIMA (1,0,0)(2,0,0)[7]", accuracy(atm4_model_fit3), row.names = NULL))

atm4_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	91.4621871	345.9815	264.0850	-344.7372	392.3998	0.7623343	0.0982354
ETS Method with Box-Cox Transform	67.2393208	338.2241	259.7873	-376.9491	420.7907	0.7499281	0.0973636
ARIMA (1,0,0)(2,0,0)[7]	-0.1991644	342.2617	282.1983	-525.0011	557.1753	0.8146221	-0.0005572

Model3 Residual

checkresiduals(atm4_model_fit3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,0,0)(2,0,0)[7] with non-zero mean
## Q* = 14.942, df = 10, p-value = 0.1342
## 
## Model df: 4.   Total lags used: 14

For ATM4, the ARIMA model performs poorly as compared to ETS model and slightly better RMSE than the Holt-Winter’s. But since the auto.arima() function did not choose to use any seasonal differencing and some seasonality seems apparent in the plots, a different arima model was tested using first order seasonal differencing until the best performance was attained using the model below.

Model4: ARIMA with Seasonal Differencing

atm4_model_fit4 <- Arima(diff(atm4_weekly_ts), order=c(0,0,1),seasonal=c(14,1,0), lambda = atm4_lambda)

autoplot(forecast(atm4_model_fit4, h=31)) + theme(panel.background = element_blank()) +
  xlab ("Weeks") +
  ylab ("Cash Withdrawn (in hundreds of dollars)")

Model4 Accuracy

atm4_accuracyDF <- rbind(atm4_accuracyDF,data.frame(Model = "ARIMA (0,0,1)(14,1,0)[7]", accuracy(atm4_model_fit4), row.names = NULL))

atm4_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	91.4621871	345.9815	264.0850	-344.7372	392.3998	0.7623343	0.0982354
ETS Method with Box-Cox Transform	67.2393208	338.2241	259.7873	-376.9491	420.7907	0.7499281	0.0973636
ARIMA (1,0,0)(2,0,0)[7]	-0.1991644	342.2617	282.1983	-525.0011	557.1753	0.8146221	-0.0005572
ARIMA (0,0,1)(14,1,0)[7]	10.5447481	396.8201	312.0851	-3078.8560	3429.0398	0.6216631	-0.1830855

Model4 Residual

checkresiduals(atm4_model_fit4)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,0,1)(14,1,0)[7]
## Q* = 14.029, df = 3, p-value = 0.002866
## 
## Model df: 15.   Total lags used: 18

The ETS method gave us the best results so that model will be used for predictions.

Results:

dates <- seq(as.Date("2010-05-01"), length=31, by="days")

atm4_forecast <- forecast(atm4_model_fit2, h=31)

tibble(DATE = rep(max(atmdataDF$DATE) + 1:31, 4),
           ATM = rep(names(atmdataDF)[-1], each = 31),
           Cash = c(atm1_forecast$mean, atm2_forecast$mean,
                   atm3_forecast$mean, atm4_forecast$mean)) %>% 
write_csv("ATM_Forecast.csv")

ATM Forecast File Path:

GitHub File Path

PART B: DataSet - Residential Power Forecast

powerdata <- readxl::read_excel("ResidentialCustomerForecastLoad-624.xlsx", skip=0)

powerdata %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

CaseSequence	YYYY-MMM	KWH
733	1998-Jan	6862583
734	1998-Feb	5838198
735	1998-Mar	5420658
736	1998-Apr	5010364
737	1998-May	4665377
738	1998-Jun	6467147
739	1998-Jul	8914755
740	1998-Aug	8607428
741	1998-Sep	6989888
742	1998-Oct	6345620
743	1998-Nov	4640410
744	1998-Dec	4693479
745	1999-Jan	7183759
746	1999-Feb	5759262
747	1999-Mar	4847656
748	1999-Apr	5306592
749	1999-May	4426794
750	1999-Jun	5500901
751	1999-Jul	7444416
752	1999-Aug	7564391
753	1999-Sep	7899368
754	1999-Oct	5358314
755	1999-Nov	4436269
756	1999-Dec	4419229
757	2000-Jan	7068296
758	2000-Feb	5876083
759	2000-Mar	4807961
760	2000-Apr	4873080
761	2000-May	5050891
762	2000-Jun	7092865
763	2000-Jul	6862662
764	2000-Aug	7517830
765	2000-Sep	8912169
766	2000-Oct	5844352
767	2000-Nov	5041769
768	2000-Dec	6220334
769	2001-Jan	7538529
770	2001-Feb	6602448
771	2001-Mar	5779180
772	2001-Apr	4835210
773	2001-May	4787904
774	2001-Jun	6283324
775	2001-Jul	7855129
776	2001-Aug	8450717
777	2001-Sep	7112069
778	2001-Oct	5242535
779	2001-Nov	4461979
780	2001-Dec	5240995
781	2002-Jan	7099063
782	2002-Feb	6413429
783	2002-Mar	5839514
784	2002-Apr	5371604
785	2002-May	5439166
786	2002-Jun	5850383
787	2002-Jul	7039702
788	2002-Aug	8058748
789	2002-Sep	8245227
790	2002-Oct	5865014
791	2002-Nov	4908979
792	2002-Dec	5779958
793	2003-Jan	7256079
794	2003-Feb	6190517
795	2003-Mar	6120626
796	2003-Apr	4885643
797	2003-May	5296096
798	2003-Jun	6051571
799	2003-Jul	6900676
800	2003-Aug	8476499
801	2003-Sep	7791791
802	2003-Oct	5344613
803	2003-Nov	4913707
804	2003-Dec	5756193
805	2004-Jan	7584596
806	2004-Feb	6560742
807	2004-Mar	6526586
808	2004-Apr	4831688
809	2004-May	4878262
810	2004-Jun	6421614
811	2004-Jul	7307931
812	2004-Aug	7309774
813	2004-Sep	6690366
814	2004-Oct	5444948
815	2004-Nov	4824940
816	2004-Dec	5791208
817	2005-Jan	8225477
818	2005-Feb	6564338
819	2005-Mar	5581725
820	2005-Apr	5563071
821	2005-May	4453983
822	2005-Jun	5900212
823	2005-Jul	8337998
824	2005-Aug	7786659
825	2005-Sep	7057213
826	2005-Oct	6694523
827	2005-Nov	4313019
828	2005-Dec	6181548
829	2006-Jan	7793358
830	2006-Feb	5914945
831	2006-Mar	5819734
832	2006-Apr	5255988
833	2006-May	4740588
834	2006-Jun	7052275
835	2006-Jul	7945564
836	2006-Aug	8241110
837	2006-Sep	7296355
838	2006-Oct	5104799
839	2006-Nov	4458429
840	2006-Dec	6226214
841	2007-Jan	8031295
842	2007-Feb	7928337
843	2007-Mar	6443170
844	2007-Apr	4841979
845	2007-May	4862847
846	2007-Jun	5022647
847	2007-Jul	6426220
848	2007-Aug	7447146
849	2007-Sep	7666970
850	2007-Oct	5785964
851	2007-Nov	4907057
852	2007-Dec	6047292
853	2008-Jan	7964293
854	2008-Feb	7597060
855	2008-Mar	6085644
856	2008-Apr	5352359
857	2008-May	4608528
858	2008-Jun	6548439
859	2008-Jul	7643987
860	2008-Aug	8037137
861	2008-Sep	NA
862	2008-Oct	5101803
863	2008-Nov	4555602
864	2008-Dec	6442746
865	2009-Jan	8072330
866	2009-Feb	6976800
867	2009-Mar	5691452
868	2009-Apr	5531616
869	2009-May	5264439
870	2009-Jun	5804433
871	2009-Jul	7713260
872	2009-Aug	8350517
873	2009-Sep	7583146
874	2009-Oct	5566075
875	2009-Nov	5339890
876	2009-Dec	7089880
877	2010-Jan	9397357
878	2010-Feb	8390677
879	2010-Mar	7347915
880	2010-Apr	5776131
881	2010-May	4919289
882	2010-Jun	6696292
883	2010-Jul	770523
884	2010-Aug	7922701
885	2010-Sep	7819472
886	2010-Oct	5875917
887	2010-Nov	4800733
888	2010-Dec	6152583
889	2011-Jan	8394747
890	2011-Feb	8898062
891	2011-Mar	6356903
892	2011-Apr	5685227
893	2011-May	5506308
894	2011-Jun	8037779
895	2011-Jul	10093343
896	2011-Aug	10308076
897	2011-Sep	8943599
898	2011-Oct	5603920
899	2011-Nov	6154138
900	2011-Dec	8273142
901	2012-Jan	8991267
902	2012-Feb	7952204
903	2012-Mar	6356961
904	2012-Apr	5569828
905	2012-May	5783598
906	2012-Jun	7926956
907	2012-Jul	8886851
908	2012-Aug	9612423
909	2012-Sep	7559148
910	2012-Oct	5576852
911	2012-Nov	5731899
912	2012-Dec	6609694
913	2013-Jan	10655730
914	2013-Feb	7681798
915	2013-Mar	6517514
916	2013-Apr	6105359
917	2013-May	5940475
918	2013-Jun	7920627
919	2013-Jul	8415321
920	2013-Aug	9080226
921	2013-Sep	7968220
922	2013-Oct	5759367
923	2013-Nov	5769083
924	2013-Dec	9606304

summary(powerdata)

##   CaseSequence     YYYY-MMM              KWH          
##  Min.   :733.0   Length:192         Min.   :  770523  
##  1st Qu.:780.8   Class :character   1st Qu.: 5429912  
##  Median :828.5   Mode  :character   Median : 6283324  
##  Mean   :828.5                      Mean   : 6502475  
##  3rd Qu.:876.2                      3rd Qu.: 7620524  
##  Max.   :924.0                      Max.   :10655730  
##                                     NA's   :1

# Format DATE column
powerdata$`YYYY-MMM` <- paste0(powerdata$`YYYY-MMM`,"-01")
powerdata$DATE <- lubridate::ymd(powerdata$`YYYY-MMM`)

# Plot data
ggplot(powerdata, aes(DATE, KWH)) + geom_line() + 
  labs(title="Residential Power Usage", y="KWH", x="") +
  theme(panel.background = element_blank())

We can clearly see an outlier that is most likely a data error so we will impute the data point with the mean of the other data for the same month.

powerdata$MONTH <- month(powerdata$DATE)

# Remove NA in Sept 2008
powerdata[is.na(powerdata$KWH),]

CaseSequence	YYYY-MMM	KWH	DATE	MONTH
861	2008-Sep-01	NA	2008-09-01	9

powerdata$KWH[is.na(powerdata$KWH)] <- mean(powerdata$KWH[powerdata$MONTH==9], na.rm = TRUE)

# Outlier is in July 2010
powerdata[powerdata$KWH==min(powerdata$KWH),]

CaseSequence	YYYY-MMM	KWH	DATE	MONTH
883	2010-Jul-01	770523	2010-07-01	7

powerdata$KWH[powerdata$KWH==min(powerdata$KWH)] <- mean(powerdata$KWH[powerdata$MONTH==7], na.rm = TRUE)

Convert to a Time Series:

power_ts <- ts(powerdata$KWH, start = c(1998,1), frequency = 12)

# Plot data
autoplot(power_ts) + theme(panel.background = element_blank()) +
  ggtitle("Residential Power Usage") +
  xlab("Time") + 
  ylab("Power Usage (in KWH)")

# Time plot
power_ts %>% ggtsdisplay(main = "Residential Power Usage"
                         ,xlab = "Time"
                         ,ylab = "Power Usage (in KWH)")

ndiffs(power_ts)

## [1] 1

ndiffs(power_ts)

## [1] 1

power_lambda <- BoxCox.lambda(power_ts)

cat("Box Cox Transformation factor lambda=",power_lambda)

## Box Cox Transformation factor lambda= -0.2018638

We can see annual seasonality in the data -

power_ts %>% BoxCox(power_lambda) %>% diff(lag=12) %>% ggtsdisplay(main = "Residential Power Usage - w/ Box Cox Transform + Seasonal Differencing"
                         ,xlab = "Time"
                         ,ylab = "Power Usage (in KWH)")

The series is stationary, so no non-seasonal differencing is needed. The decaying seasonal spikes in the PACF suggests a seasonal AR(1) component, while the very quickly-decaying seasonal spikes in the ACF suggest the possibility of a seasonal MA(1) component. Spikes in the PACF and ACF at k=1 and k=4 suggest non-seasonal AR(1) or AR(4) components, and non-seasonal MA(1) or MA(4) components.

Model1: Holt-Winters w/ Box Cox

power_model_fit1 <- power_ts %>% hw(h=12, seasonal="additive", 
                           damped=TRUE, lambda = power_lambda)
autoplot(power_model_fit1) + theme(panel.background = element_blank()) +
  ggtitle("Holt-Winters Damped Additive Method w/ Box Cox Transofrm") +
  xlab ("Time") +
  ylab ("Power Usage (in KWH)")

Model1 Accuracy

power_accuracyDF <- data.frame(Model = "Holt-Winter's Additive Method with Box-Cox Transform", accuracy(power_model_fit1), row.names = NULL)

power_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	78844.59	620901.7	455088.7	0.477213	6.802764	0.7435267	0.2858823

Model1 Residual

checkresiduals(power_model_fit1)

## 
##  Ljung-Box test
## 
## data:  Residuals from Damped Holt-Winters' additive method
## Q* = 44.62, df = 7, p-value = 1.621e-07
## 
## Model df: 17.   Total lags used: 24

Model2: ETS

power_model_fit2 <- power_ts %>% ets(model="ZZZ", lambda = power_lambda)

autoplot(power_model_fit2) + theme(panel.background = element_blank())

autoplot(forecast(power_model_fit2, h=12)) + theme(panel.background = element_blank()) +
  ggtitle("ETS method w/ Box Cox Transofrm") +
  xlab ("Time") +
  ylab ("Power Usage (in KWH)")

Model2 Accuracy

power_accuracyDF <- rbind(power_accuracyDF,data.frame(Model = "ETS Method with Box-Cox Transform", accuracy(power_model_fit2), row.names = NULL))

power_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	78844.59	620901.7	455088.7	0.4772130	6.802764	0.7435267	0.2858823
ETS Method with Box-Cox Transform	32423.43	613027.6	453744.7	-0.2772083	6.836306	0.7413308	0.2817121

Model2 Residual

checkresiduals(power_model_fit2)

## 
##  Ljung-Box test
## 
## data:  Residuals from ETS(A,A,A)
## Q* = 45.157, df = 8, p-value = 3.436e-07
## 
## Model df: 16.   Total lags used: 24

The ETS model produced a slightly better RMSE than Holt Winter’s model.

Model3: ARIMA

power_model_fit3 <- auto.arima(power_ts,stepwise=FALSE, approximation=FALSE)

autoplot(forecast(power_model_fit3, h=12)) + theme(panel.background = element_blank()) +
  xlab ("Weeks") +
  ylab ("Power Usage (in KWH)")

Model3 Accuracy

power_accuracyDF <- rbind(power_accuracyDF,data.frame(Model = "ARIMA (0,0,3)(2,1,0)[12]", accuracy(power_model_fit3), row.names = NULL))

power_accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
Holt-Winter’s Additive Method with Box-Cox Transform	78844.586	620901.7	455088.7	0.4772130	6.802764	0.7435267	0.2858823
ETS Method with Box-Cox Transform	32423.431	613027.6	453744.7	-0.2772083	6.836306	0.7413308	0.2817121
ARIMA (0,0,3)(2,1,0)[12]	-8220.325	586905.7	423740.1	-0.7944092	6.468297	0.6923092	-0.0111059

Model3 Residual

checkresiduals(power_model_fit3)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(0,0,3)(2,1,0)[12] with drift
## Q* = 15.178, df = 18, p-value = 0.6497
## 
## Model df: 6.   Total lags used: 24

auto.arima() function produced better RMSE than the other two methods. So we will use this model fit to forecast.

Results:

power_forecast <- forecast(power_model_fit3, h=12)

tibble(`YYYY-MMM` = paste0(2014, "-", month.abb),
       KWH = power_forecast$mean) %>% 
write_csv("Power_Forecast.csv")

Power Consumption Forecast File Path:

GitHub File Path

PART C: BONUS - Water Pipeline Datasets

Part C consists of two data sets. These are simple 2 columns sets, however they have different time stamps. Optional assignment is to time-base sequence the data and aggregate based on hour (example of what this looks like, follows). Note for multiple recordings within an hour, take the mean. Then to determine if the data is stationary and can it be forecast. If so, provide a week forward forecast and present results via Rpubs and .rmd and the forecast in an Excel readable file.

# Dataset1
waterdata1 <- readxl::read_excel("Waterflow_Pipe1.xlsx", skip=0)
colnames(waterdata1) <- c("DateTimeNbr","WaterFlow")

waterdata1 %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

DateTimeNbr	WaterFlow
42300.02	23.369599
42300.03	28.002881
42300.04	23.065895
42300.04	29.972809
42300.06	5.997953
42300.06	15.935223
42300.08	26.617330
42300.08	33.282900
42300.08	12.426692
42300.12	21.833494
42300.12	8.483647
42300.14	29.336901
42300.14	19.809146
42300.14	31.019744
42300.16	18.339962
42300.17	16.888527
42300.17	13.664312
42300.20	17.300074
42300.21	23.260984
42300.22	8.152496
42300.22	19.875628
42300.22	32.886499
42300.22	22.260364
42300.23	5.778369
42300.24	32.545557
42300.24	30.687744
42300.24	29.080907
42300.25	30.047784
42300.26	5.752456
42300.27	30.414162
42300.27	26.175716
42300.28	27.155307
42300.28	13.605943
42300.29	11.184568
42300.29	20.383057
42300.32	13.405331
42300.33	14.461091
42300.33	27.234671
42300.34	9.089498
42300.35	29.162384
42300.37	24.123214
42300.38	6.207443
42300.38	27.666923
42300.40	29.781898
42300.42	19.035463
42300.42	14.539055
42300.44	16.304829
42300.46	9.089600
42300.46	24.160478
42300.49	33.013436
42300.50	14.924758
42300.52	20.688466
42300.54	25.396306
42300.56	21.661800
42300.57	23.214093
42300.57	3.811189
42300.59	37.300056
42300.59	26.468501
42300.59	30.532230
42300.60	10.910886
42300.61	31.534557
42300.62	13.552525
42300.64	27.485792
42300.65	19.136787
42300.65	26.371215
42300.66	17.623757
42300.67	26.704584
42300.68	36.172465
42300.69	24.153054
42300.69	27.510917
42300.70	34.642728
42300.70	16.144607
42300.72	18.210857
42300.73	7.485977
42300.73	14.821554
42300.76	15.446337
42300.76	21.084892
42300.76	32.985346
42300.79	13.270245
42300.81	15.531714
42300.83	22.678874
42300.83	19.404605
42300.84	23.996745
42300.85	18.409485
42300.85	21.749392
42300.85	25.759675
42300.86	20.225869
42300.86	16.523979
42300.87	9.328124
42300.87	30.186021
42300.91	15.092513
42300.93	16.576799
42300.93	20.801956
42300.96	33.220434
42300.97	3.468986
42300.98	24.980254
42301.00	14.899085
42301.01	18.882710
42301.01	14.917666
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42307.71	31.123454
42307.71	22.472866
42307.74	19.109189
42307.75	8.965514
42307.76	14.538760
42307.76	6.618785
42307.78	19.410505
42307.78	24.581237
42307.80	19.913452
42307.81	14.319365
42307.82	21.950001
42307.82	29.379711
42307.83	9.750124
42307.83	19.004112
42307.85	37.591036
42307.86	28.464328
42307.87	15.015707
42307.87	22.333341
42307.88	27.741250
42307.88	11.640426
42307.89	15.135784
42307.89	22.123494
42307.89	17.915621
42307.91	23.954026
42307.93	4.021746
42307.93	9.613415
42307.96	13.133139
42307.96	4.952637
42307.97	26.474205
42308.01	4.421879
42308.01	18.253691
42308.03	14.486119
42308.04	31.484788
42308.05	12.772337
42308.07	16.364104
42308.07	35.450027
42308.08	38.479671
42308.08	27.325173
42308.08	12.840285
42308.08	20.807145
42308.12	16.369698
42308.12	28.855674
42308.14	27.944654
42308.15	27.446450
42308.18	21.081974
42308.19	24.238055
42308.21	14.049404
42308.25	26.559184
42308.25	25.242968
42308.25	34.891026
42308.26	30.599200
42308.27	17.366746
42308.28	23.821349
42308.29	11.251934
42308.30	25.670211
42308.31	33.436930
42308.33	36.082648
42308.35	10.927193
42308.36	8.277269
42308.37	15.415538
42308.39	32.181583
42308.41	2.719154
42308.41	21.170975
42308.42	19.835044
42308.44	35.782245
42308.45	20.585977
42308.46	18.806353
42308.47	24.836634
42308.47	19.806278
42308.49	30.785157
42308.49	8.282494
42308.52	19.724971
42308.54	20.268650
42308.54	22.278925
42308.54	12.367646
42308.55	24.372004
42308.58	20.621253
42308.58	14.751192
42308.58	10.408624
42308.60	31.945120
42308.63	8.037941
42308.63	26.541515
42308.65	13.908973
42308.66	14.737002
42308.66	28.771564
42308.68	24.554983
42308.69	15.011016
42308.70	24.000616
42308.73	27.589842
42308.73	35.713119
42308.73	19.495046
42308.74	21.303044
42308.75	21.031095
42308.78	12.594886
42308.78	15.807672
42308.79	9.259209
42308.79	16.782744
42308.79	14.815882
42308.79	11.636250
42308.80	24.213435
42308.81	14.486916
42308.81	32.728819
42308.81	15.302234
42308.81	7.959237
42308.82	31.369263
42308.84	7.349745
42308.85	15.010710
42308.85	27.262566
42308.85	20.517552
42308.86	20.452720
42308.88	27.389651
42308.92	11.400498
42308.93	12.396629
42308.93	11.774302
42308.96	27.606285
42308.97	28.467733
42308.98	31.244255
42308.99	14.237196
42308.99	11.719533
42309.00	15.847017
42309.00	7.076831
42309.02	23.650243
42309.03	30.891522
42309.03	23.070343
42309.05	16.925815
42309.06	26.975276
42309.06	10.376045
42309.07	25.214665
42309.09	14.022721
42309.11	29.029274
42309.14	12.817853
42309.15	27.137068
42309.16	13.715517
42309.17	11.133035
42309.21	22.610985
42309.25	24.388115
42309.26	21.684017
42309.26	4.294392
42309.30	29.918730
42309.32	19.281998
42309.33	19.495424
42309.34	1.335991
42309.34	10.920037
42309.34	19.664501
42309.37	28.550303
42309.43	18.399540
42309.45	12.924725
42309.46	17.521240
42309.46	26.317390
42309.48	26.056817
42309.48	29.616021
42309.53	17.746786
42309.53	7.546854
42309.53	26.363689
42309.54	14.737645
42309.54	29.287755
42309.54	31.996311
42309.54	26.981273
42309.55	16.575896
42309.56	16.491862
42309.57	15.705421
42309.60	17.704946
42309.60	26.155266
42309.60	29.694279
42309.60	24.455547
42309.62	22.425873
42309.63	33.004309
42309.63	4.385275
42309.63	16.915401
42309.64	17.247719
42309.66	25.548663
42309.67	19.692654
42309.69	22.548459
42309.69	18.674775
42309.69	16.977700
42309.70	18.870444
42309.71	24.984427
42309.71	26.191839
42309.76	10.336158
42309.77	28.240343
42309.78	21.630708
42309.80	13.690378
42309.81	21.142271
42309.82	20.366756
42309.82	20.311729
42309.83	19.103980
42309.84	5.537197
42309.84	15.865447
42309.84	10.568377
42309.87	21.558668
42309.87	18.989392
42309.88	10.241452
42309.91	28.110783
42309.92	13.848839
42309.92	21.507690
42309.92	15.281430
42309.92	26.336684
42309.93	29.064266
42309.96	22.844104
42309.98	16.218705
42309.98	21.211707

# Dataset2
waterdata2 <- readxl::read_excel("Waterflow_Pipe2.xlsx", skip=0)
colnames(waterdata2) <- c("DateTimeNbr","WaterFlow")

waterdata2 %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

DateTimeNbr	WaterFlow
42300.04	18.810791
42300.08	43.087025
42300.12	37.987705
42300.17	36.120379
42300.21	31.851259
42300.25	28.238090
42300.29	9.863582
42300.33	26.679610
42300.38	55.773785
42300.42	54.156889
42300.46	68.374904
42300.50	55.710359
42300.54	56.968260
42300.58	17.206276
42300.62	35.093275
42300.67	44.424928
42300.71	57.322408
42300.75	37.344924
42300.79	11.483011
42300.83	32.117940
42300.88	49.081861
42300.92	49.133546
42300.96	42.064648
42301.00	58.380027
42301.04	53.408031
42301.08	42.332775
42301.12	45.922190
42301.17	32.741215
42301.21	47.879252
42301.25	47.460516
42301.29	52.264966
42301.33	35.389582
42301.38	14.435578
42301.42	45.038179
42301.46	38.896592
42301.50	39.991833
42301.54	56.883595
42301.58	62.728660
42301.62	67.382484
42301.67	29.645108
42301.71	51.586668
42301.75	61.987103
42301.79	46.394571
42301.83	32.838673
42301.88	53.416554
42301.92	70.723677
42301.96	21.570847
42302.00	66.762107
42302.04	36.322123
42302.08	45.114342
42302.12	53.473562
42302.17	27.209341
42302.21	44.193703
42302.25	59.296910
42302.29	42.253496
42302.33	44.747970
42302.38	50.182502
42302.42	32.303380
42302.46	44.413697
42302.50	42.844556
42302.54	56.282628
42302.58	54.936958
42302.62	20.406510
42302.67	60.954459
42302.71	55.811767
42302.75	68.008012
42302.79	20.964754
42302.83	50.112292
42302.88	24.172808
42302.92	17.131600
42302.96	41.938311
42303.00	70.376636
42303.04	34.387617
42303.08	30.377663
42303.12	41.567192
42303.17	72.172881
42303.21	45.745729
42303.25	42.294146
42303.29	28.883164
42303.33	55.506566
42303.38	19.228599
42303.42	39.454913
42303.46	28.521848
42303.50	22.614615
42303.54	8.513215
42303.58	38.228365
42303.62	23.388172
42303.67	59.546335
42303.71	20.342739
42303.75	40.198573
42303.79	42.636429
42303.83	40.250584
42303.88	33.603564
42303.92	33.921165
42303.96	6.574838
42304.00	68.451384
42304.04	14.722223
42304.08	40.885201
42304.12	37.432232
42304.17	22.449903
42304.21	49.813460
42304.25	38.930568
42304.29	22.183581
42304.33	17.156385
42304.38	55.683814
42304.42	4.681630
42304.46	30.998919
42304.50	11.136842
42304.54	30.650987
42304.58	56.688098
42304.62	32.445492
42304.67	41.242405
42304.71	44.380749
42304.75	55.343373
42304.79	14.175375
42304.83	51.636056
42304.88	49.038738
42304.92	33.515478
42304.96	35.479944
42305.00	54.753657
42305.04	42.308140
42305.08	30.650448
42305.12	38.951353
42305.17	23.330517
42305.21	26.130716
42305.25	68.184102
42305.29	17.891049
42305.33	7.689945
42305.38	34.479795
42305.42	5.768535
42305.46	55.996156
42305.50	52.156262
42305.54	48.276526
42305.58	29.068677
42305.62	51.055828
42305.67	50.142903
42305.71	35.699715
42305.75	41.009140
42305.79	70.126446
42305.83	22.225432
42305.88	54.659647
42305.92	50.467770
42305.96	35.307784
42306.00	44.072312
42306.04	36.433573
42306.08	65.174990
42306.12	39.027232
42306.17	21.882484
42306.21	16.475724
42306.25	29.737078
42306.29	44.677987
42306.33	9.726528
42306.38	32.252834
42306.42	36.463706
42306.46	18.570584
42306.50	47.179197
42306.54	16.814399
42306.58	21.411535
42306.62	78.303208
42306.67	37.809087
42306.71	46.143853
42306.75	58.575561
42306.79	10.668517
42306.83	40.423676
42306.88	42.632654
42306.92	17.024037
42306.96	38.288825
42307.00	61.935956
42307.04	16.267685
42307.08	50.454336
42307.12	64.303302
42307.17	36.859830
42307.21	19.045695
42307.25	51.206781
42307.29	43.157475
42307.33	66.533505
42307.38	46.407488
42307.42	31.593622
42307.46	32.880155
42307.50	32.330595
42307.54	44.101568
42307.58	30.998232
42307.62	69.575830
42307.67	28.656834
42307.71	40.347359
42307.75	73.769908
42307.79	30.711824
42307.83	39.900346
42307.88	41.396253
42307.92	14.918891
42307.96	48.476683
42308.00	54.804861
42308.04	74.841654
42308.08	31.581531
42308.12	60.288192
42308.17	22.568788
42308.21	42.800695
42308.25	41.309206
42308.29	14.657249
42308.33	35.299757
42308.38	23.238057
42308.42	36.344364
42308.46	34.088694
42308.50	41.491756
42308.54	42.307821
42308.58	46.513619
42308.62	59.169215
42308.67	49.514779
42308.71	9.835715
42308.75	30.380895
42308.79	40.075967
42308.83	53.075152
42308.88	26.581510
42308.92	37.963427
42308.96	67.098804
42309.00	9.541255
42309.04	50.560327
42309.08	48.675529
42309.12	50.790777
42309.17	39.647840
42309.21	71.594723
42309.25	45.642905
42309.29	22.403748
42309.33	42.583314
42309.38	48.973803
42309.42	36.391814
42309.46	39.451300
42309.50	48.231945
42309.54	23.238733
42309.58	29.103701
42309.62	24.890495
42309.67	39.501753
42309.71	3.553891
42309.75	41.188599
42309.79	41.303353
42309.83	17.858629
42309.88	28.382995
42309.92	51.696846
42309.96	60.225725
42310.00	18.710074
42310.04	69.677403
42310.08	70.643756
42310.12	32.130700
42310.17	43.286488
42310.21	46.519110
42310.25	49.418788
42310.29	22.603133
42310.33	17.204039
42310.38	36.098841
42310.42	37.426039
42310.46	68.958258
42310.50	56.355300
42310.54	27.421826
42310.58	21.977013
42310.62	31.650012
42310.67	19.103725
42310.71	40.573976
42310.75	34.139365
42310.79	31.604146
42310.83	53.455162
42310.88	50.932862
42310.92	21.382117
42310.96	28.707991
42311.00	28.266119
42311.04	56.643025
42311.08	65.491297
42311.12	45.264900
42311.17	32.083226
42311.21	20.034940
42311.25	59.361131
42311.29	63.062853
42311.33	36.311024
42311.38	28.786016
42311.42	41.529467
42311.46	15.911946
42311.50	50.859685
42311.54	43.005399
42311.58	34.650963
42311.62	53.490976
42311.67	30.670295
42311.71	11.915951
42311.75	24.722517
42311.79	38.936218
42311.83	35.450478
42311.88	45.970258
42311.92	56.407256
42311.96	49.554872
42312.00	48.358182
42312.04	16.816164
42312.08	19.084841
42312.12	49.944728
42312.17	21.617885
42312.21	26.981540
42312.25	51.754010
42312.29	35.729160
42312.33	71.273732
42312.38	28.418581
42312.42	16.964691
42312.46	47.892948
42312.50	50.257181
42312.54	48.595430
42312.58	49.996359
42312.62	48.920110
42312.67	68.161543
42312.71	30.529780
42312.75	55.694453
42312.79	47.145238
42312.83	50.778857
42312.88	47.658571
42312.92	26.331098
42312.96	67.761091
42313.00	51.085154
42313.04	41.117956
42313.08	9.068800
42313.12	61.260012
42313.17	20.384767
42313.21	23.371558
42313.25	54.130721
42313.29	47.173422
42313.33	17.330480
42313.38	27.900997
42313.42	60.693928
42313.46	55.750439
42313.50	31.980729
42313.54	46.026815
42313.58	51.486720
42313.62	35.083882
42313.67	29.222731
42313.71	38.054577
42313.75	32.803461
42313.79	34.590651
42313.83	7.041455
42313.88	43.879486
42313.92	54.260075
42313.96	49.607998
42314.00	51.166843
42314.04	60.332261
42314.08	54.000615
42314.12	48.402953
42314.17	71.293996
42314.21	25.864377
42314.25	43.534985
42314.29	10.294754
42314.33	27.211552
42314.38	68.771180
42314.42	40.635575
42314.46	48.547279
42314.50	51.211695
42314.54	62.323132
42314.58	36.042205
42314.62	65.867425
42314.67	43.091257
42314.71	49.999283
42314.75	42.753244
42314.79	43.280302
42314.83	12.215143
42314.88	30.790950
42314.92	43.146264
42314.96	24.862962
42315.00	60.419541
42315.04	2.829191
42315.08	41.358591
42315.12	32.518025
42315.17	32.936246
42315.21	53.903692
42315.25	54.349240
42315.29	48.667338
42315.33	43.697246
42315.38	64.391019
42315.42	54.642566
42315.46	31.579795
42315.50	10.697792
42315.54	69.408304
42315.58	57.739228
42315.62	54.043036
42315.67	21.378107
42315.71	68.557246
42315.75	49.990255
42315.79	60.870852
42315.83	15.231194
42315.88	35.271872
42315.92	47.834724
42315.96	39.042589
42316.00	31.282543
42316.04	34.318646
42316.08	45.695737
42316.12	34.761724
42316.17	41.324282
42316.21	22.815585
42316.25	63.250742
42316.29	13.839852
42316.33	46.006283
42316.38	44.493733
42316.42	63.378401
42316.46	42.817629
42316.50	45.072082
42316.54	46.040137
42316.58	11.573340
42316.62	39.922730
42316.67	40.153338
42316.71	63.037254
42316.75	48.776710
42316.79	53.306946
42316.83	41.484211
42316.88	45.348702
42316.92	54.969466
42316.96	47.595072
42317.00	40.591619
42317.04	43.363619
42317.08	65.789153
42317.12	29.012179
42317.17	38.944844
42317.21	26.349575
42317.25	33.638500
42317.29	8.524681
42317.33	7.047517
42317.38	71.864778
42317.42	33.775195
42317.46	34.524480
42317.50	13.277970
42317.54	20.951475
42317.58	42.715256
42317.62	51.101163
42317.67	8.718182
42317.71	23.884311
42317.75	44.208632
42317.79	47.094925
42317.83	25.140574
42317.88	33.723718
42317.92	23.366525
42317.96	41.851025
42318.00	28.565955
42318.04	49.010418
42318.08	76.953288
42318.12	33.641144
42318.17	61.127699
42318.21	61.822340
42318.25	22.690029
42318.29	52.496753
42318.33	26.962412
42318.38	55.912366
42318.42	38.827366
42318.46	51.170992
42318.50	38.982250
42318.54	36.403632
42318.58	45.511136
42318.62	35.450764
42318.67	12.781466
42318.71	34.596144
42318.75	19.863000
42318.79	61.422724
42318.83	37.871542
42318.88	44.714030
42318.92	52.438546
42318.96	21.870034
42319.00	42.775527
42319.04	37.075660
42319.08	55.922491
42319.12	21.725370
42319.17	27.383226
42319.21	26.474453
42319.25	24.992206
42319.29	29.420236
42319.33	18.593189
42319.38	19.401002
42319.42	30.797664
42319.46	53.005290
42319.50	29.785165
42319.54	48.478284
42319.58	41.000063
42319.62	55.985426
42319.67	23.593107
42319.71	16.929397
42319.75	26.105809
42319.79	27.278099
42319.83	35.966696
42319.88	62.321017
42319.92	22.551005
42319.96	32.326498
42320.00	36.768426
42320.04	16.655779
42320.08	50.903035
42320.12	32.744233
42320.17	43.446611
42320.21	12.872882
42320.25	43.626821
42320.29	17.932564
42320.33	46.430782
42320.38	47.555630
42320.42	23.164628
42320.46	36.320166
42320.50	41.829277
42320.54	65.063811
42320.58	53.996669
42320.62	53.065490
42320.67	36.048012
42320.71	23.164822
42320.75	38.652909
42320.79	48.085267
42320.83	39.547030
42320.88	36.281582
42320.92	22.304580
42320.96	39.054497
42321.00	62.456375
42321.04	21.058794
42321.08	24.986703
42321.12	47.340037
42321.17	42.103710
42321.21	46.277421
42321.25	41.346360
42321.29	8.198421
42321.33	49.820480
42321.38	52.630520
42321.42	14.320846
42321.46	29.429210
42321.50	46.610070
42321.54	27.718158
42321.58	43.646591
42321.62	58.249277
42321.67	37.955810
42321.71	53.167141
42321.75	17.003180
42321.79	36.987767
42321.83	69.421289
42321.88	42.282380
42321.92	60.299183
42321.96	77.388036
42322.00	38.187514
42322.04	73.920364
42322.08	23.427619
42322.12	74.049763
42322.17	71.025842
42322.21	36.310935
42322.25	69.727846
42322.29	47.318911
42322.33	54.552009
42322.38	22.047066
42322.42	33.286940
42322.46	44.805901
42322.50	30.968865
42322.54	32.250332
42322.58	60.449541
42322.62	17.006485
42322.67	42.845801
42322.71	47.528852
42322.75	39.715687
42322.79	23.905351
42322.83	60.468321
42322.88	16.016156
42322.92	24.913911
42322.96	52.127222
42323.00	31.771907
42323.04	33.992850
42323.08	12.282765
42323.12	50.596037
42323.17	31.542007
42323.21	20.514348
42323.25	27.990267
42323.29	70.663541
42323.33	44.674916
42323.38	22.821463
42323.42	28.171757
42323.46	38.060556
42323.50	28.251189
42323.54	20.650803
42323.58	42.038281
42323.62	33.657499
42323.67	24.628537
42323.71	62.740169
42323.75	37.227280
42323.79	66.834372
42323.83	54.885094
42323.88	38.741754
42323.92	52.565861
42323.96	42.011579
42324.00	1.884618
42324.04	33.697009
42324.08	35.136376
42324.12	52.342981
42324.17	56.699116
42324.21	55.563435
42324.25	45.758364
42324.29	50.659673
42324.33	40.169564
42324.38	24.979220
42324.42	47.796589
42324.46	35.726957
42324.50	54.028989
42324.54	45.341975
42324.58	48.476311
42324.62	33.640512
42324.67	30.864809
42324.71	20.844211
42324.75	19.803964
42324.79	36.264543
42324.83	31.291821
42324.88	10.358942
42324.92	31.271942
42324.96	39.143974
42325.00	40.599863
42325.04	4.404232
42325.08	60.532204
42325.12	42.654711
42325.17	11.865801
42325.21	22.798883
42325.25	32.524959
42325.29	65.957563
42325.33	26.382528
42325.38	33.062656
42325.42	18.365385
42325.46	59.433134
42325.50	38.719453
42325.54	70.619444
42325.58	51.069993
42325.62	29.489031
42325.67	15.808655
42325.71	39.223626
42325.75	47.080228
42325.79	41.980568
42325.83	61.214361
42325.88	23.836742
42325.92	35.213413
42325.96	66.174825
42326.00	11.472738
42326.04	40.944768
42326.08	73.297982
42326.12	21.705112
42326.17	13.445279
42326.21	33.855644
42326.25	38.425269
42326.29	7.564169
42326.33	20.792792
42326.38	47.126813
42326.42	38.557200
42326.46	30.619750
42326.50	27.548411
42326.54	34.697505
42326.58	36.676699
42326.62	30.025806
42326.67	44.290001
42326.71	31.476714
42326.75	16.554329
42326.79	28.467637
42326.83	60.756805
42326.88	53.156311
42326.92	43.736182
42326.96	44.678285
42327.00	32.500159
42327.04	53.096718
42327.08	50.450171
42327.12	20.671851
42327.17	48.500930
42327.21	55.872094
42327.25	16.955515
42327.29	47.782032
42327.33	64.182162
42327.38	19.123758
42327.42	26.630040
42327.46	24.693264
42327.50	14.882576
42327.54	54.702633
42327.58	28.045472
42327.62	44.237443
42327.67	53.989477
42327.71	44.208531
42327.75	44.145634
42327.79	40.174149
42327.83	75.699763
42327.88	33.688595
42327.92	60.315099
42327.96	37.811607
42328.00	62.216296
42328.04	21.923579
42328.08	50.856824
42328.12	9.506527
42328.17	39.037833
42328.21	47.392675
42328.25	39.774791
42328.29	48.835634
42328.33	46.283958
42328.38	48.833873
42328.42	41.321219
42328.46	49.126166
42328.50	26.070307
42328.54	37.334013
42328.58	44.013403
42328.62	63.076857
42328.67	44.009289
42328.71	53.765517
42328.75	6.907394
42328.79	51.625765
42328.83	64.845969
42328.88	23.791188
42328.92	41.551253
42328.96	45.032033
42329.00	30.722590
42329.04	37.643845
42329.08	33.978619
42329.12	59.975566
42329.17	24.821922
42329.21	24.426441
42329.25	58.768110
42329.29	47.487387
42329.33	41.046481
42329.38	32.077990
42329.42	61.332384
42329.46	33.779878
42329.50	68.860907
42329.54	28.228687
42329.58	41.392049
42329.62	55.493874
42329.67	41.741668
42329.71	54.743921
42329.75	51.711642
42329.79	21.725879
42329.83	45.572460
42329.88	29.984989
42329.92	31.946214
42329.96	74.120935
42330.00	39.799580
42330.04	28.923704
42330.08	34.125230
42330.12	47.261910
42330.17	31.905987
42330.21	53.219995
42330.25	26.944213
42330.29	54.151730
42330.33	31.557127
42330.38	47.108037
42330.42	16.939418
42330.46	56.867922
42330.50	56.924140
42330.54	7.143941
42330.58	49.925868
42330.62	58.971000
42330.67	34.281085
42330.71	42.682790
42330.75	36.855911
42330.79	22.575510
42330.83	43.106407
42330.88	57.423548
42330.92	60.982311
42330.96	19.184503
42331.00	27.031110
42331.04	24.985017
42331.08	41.859799
42331.12	7.230717
42331.17	24.225717
42331.21	46.453707
42331.25	51.230189
42331.29	44.442608
42331.33	49.115951
42331.38	32.865960
42331.42	23.082460
42331.46	34.718627
42331.50	66.163243
42331.54	68.177732
42331.58	42.731288
42331.62	65.160220
42331.67	18.855938
42331.71	28.938320
42331.75	4.388469
42331.79	44.976017
42331.83	35.722734
42331.88	45.347123
42331.92	22.781328
42331.96	43.114767
42332.00	24.184121
42332.04	43.356235
42332.08	35.089344
42332.12	42.129951
42332.17	65.064061
42332.21	31.054652
42332.25	43.142107
42332.29	33.909621
42332.33	25.199164
42332.38	26.900095
42332.42	54.820399
42332.46	36.714245
42332.50	58.612063
42332.54	48.918951
42332.58	55.173613
42332.62	34.770766
42332.67	4.595560
42332.71	17.717682
42332.75	63.682387
42332.79	16.459204
42332.83	39.174167
42332.88	5.544962
42332.92	51.408889
42332.96	71.206396
42333.00	20.604607
42333.04	45.903558
42333.08	47.996068
42333.12	51.515777
42333.17	27.643937
42333.21	19.950425
42333.25	42.934676
42333.29	33.758964
42333.33	60.198257
42333.38	64.822648
42333.42	26.160919
42333.46	44.729294
42333.50	28.503025
42333.54	75.771776
42333.58	27.112065
42333.62	49.403191
42333.67	38.397854
42333.71	38.091870
42333.75	43.485536
42333.79	13.170863
42333.83	52.137698
42333.88	43.204897
42333.92	37.660492
42333.96	31.197636
42334.00	34.074007
42334.04	64.403364
42334.08	61.575249
42334.12	32.426134
42334.17	30.168290
42334.21	37.227840
42334.25	18.700339
42334.29	27.573602
42334.33	48.217705
42334.38	45.173003
42334.42	18.890009
42334.46	27.141646
42334.50	14.635885
42334.54	30.364299
42334.58	8.753475
42334.62	18.768489
42334.67	39.735482
42334.71	13.730282
42334.75	44.824872
42334.79	39.809994
42334.83	28.565875
42334.88	43.661477
42334.92	55.906271
42334.96	31.129181
42335.00	19.962453
42335.04	42.683700
42335.08	59.089857
42335.12	23.640381
42335.17	26.785668
42335.21	40.640046
42335.25	35.918847
42335.29	48.929382
42335.33	25.118402
42335.38	34.981362
42335.42	8.636697
42335.46	59.959267
42335.50	18.643766
42335.54	17.155756
42335.58	5.500306
42335.62	14.505197
42335.67	24.883430
42335.71	54.744878
42335.75	55.735616
42335.79	32.256413
42335.83	38.956010
42335.88	60.020956
42335.92	69.425402
42335.96	8.688084
42336.00	42.970294
42336.04	27.727545
42336.08	38.962854
42336.12	13.365612
42336.17	31.819558
42336.21	33.103063
42336.25	30.924940
42336.29	26.308201
42336.33	9.565704
42336.38	13.830709
42336.42	48.527078
42336.46	29.341300
42336.50	56.402469
42336.54	24.985439
42336.58	36.644183
42336.62	26.098598
42336.67	24.020212
42336.71	50.831756
42336.75	56.770995
42336.79	47.023474
42336.83	59.280305
42336.88	37.453182
42336.92	72.102666
42336.96	30.348483
42337.00	53.805047
42337.04	22.456437
42337.08	60.534383
42337.12	16.088490
42337.17	36.382437
42337.21	59.246121
42337.25	45.077214
42337.29	45.120941
42337.33	38.232433
42337.38	19.465618
42337.42	38.230367
42337.46	18.690932
42337.50	20.769636
42337.54	61.135055
42337.58	27.915517
42337.62	30.520160
42337.67	28.824846
42337.71	69.571855
42337.75	37.361398
42337.79	38.282275
42337.83	46.356586
42337.88	24.056497
42337.92	53.198139
42337.96	35.252730
42338.00	43.607553
42338.04	60.174882
42338.08	17.935532
42338.12	68.517692
42338.17	29.962754
42338.21	61.862104
42338.25	20.989968
42338.29	60.410201
42338.33	59.574523
42338.38	56.018735
42338.42	24.071468
42338.46	49.377007
42338.50	38.429695
42338.54	26.261928
42338.58	66.245234
42338.62	32.917481
42338.67	25.103363
42338.71	27.791581
42338.75	46.185805
42338.79	33.197581
42338.83	45.298259
42338.88	15.820063
42338.92	19.494634
42338.96	39.350548
42339.00	39.319314
42339.04	2.474597
42339.08	41.595853
42339.12	15.030481
42339.17	71.055191
42339.21	52.047501
42339.25	72.879618
42339.29	37.068471
42339.33	34.298235
42339.38	41.579046
42339.42	40.660273
42339.46	61.635744
42339.50	43.405853
42339.54	18.588931
42339.58	47.898820
42339.62	33.526542
42339.67	53.297195
42339.71	29.570191
42339.75	18.813022
42339.79	32.032767
42339.83	43.580802
42339.88	29.927978
42339.92	65.725895
42339.96	14.997650
42340.00	21.467634
42340.04	35.542969
42340.08	67.452311
42340.12	57.589935
42340.17	35.194673
42340.21	46.114270
42340.25	35.049139
42340.29	34.855851
42340.33	31.369865
42340.38	43.440536
42340.42	5.292336
42340.46	15.588859
42340.50	45.724622
42340.54	46.292510
42340.58	38.483379
42340.62	51.287088
42340.67	35.419347
42340.71	38.717686
42340.75	53.184185
42340.79	66.245006
42340.83	40.550695
42340.88	57.073648
42340.92	49.285885
42340.96	22.874043
42341.00	46.236585
42341.04	77.101826
42341.08	49.775351
42341.12	71.134530
42341.17	58.448860
42341.21	58.534300
42341.25	28.555859
42341.29	39.062662
42341.33	16.662519
42341.38	27.000278
42341.42	44.246493
42341.46	72.966772
42341.50	31.483105
42341.54	66.816731
42341.58	42.936656
42341.62	33.401326
42341.67	66.681471

Data Pre-Processing:

In order to use the two datasets together, the readings for pipeline1 must be converted to - - Separate date & hour components of readings - Convert hour to hour-ending to match pipeline2 - Get average reading for each date & hour - Convert back to DateTime and drop separate date/hour columns

waterdata1$DateTime <-  as.POSIXct(waterdata1$DateTimeNbr*(60*60*24),origin="1899-12-30", tz="GMT")

waterdata1 <- waterdata1 %>% mutate(DATE = date(DateTime),HOUR = hour(DateTime) + 1) %>% 
  group_by(DATE, HOUR) %>% summarise(WaterFlow = mean(WaterFlow)) %>% 
  ungroup() %>% mutate(DateTime = ymd_h(paste(DATE, HOUR))) %>% 
  select(DateTime, WaterFlow)

## `summarise()` regrouping output by 'DATE' (override with `.groups` argument)

waterdata2 <- waterdata2 %>% mutate(DateTime = round(as.POSIXct(DateTimeNbr*(60*60*24),origin="1899-12-30", tz="GMT"),"hours"))  %>% select(DateTime, WaterFlow)

Now, I have joined the two datasets together and converted to a time series object -

Join based on common DateTime column
- sum the WaterFlow to calculate total flow by hour

# create df with both observations for each hour
waterdata <- full_join(waterdata1, waterdata2, by = "DateTime", suffix = c("_1", "_2")) %>%
  mutate(WaterFlow_1 = ifelse(is.na(WaterFlow_1), 0, WaterFlow_1)) %>% 
  mutate(WaterFlow = WaterFlow_1 + WaterFlow_2) %>% 
  select(DateTime, WaterFlow)

waterdata %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")

DateTime	WaterFlow
2015-10-23 01:00:00	44.913587
2015-10-23 02:00:00	61.939045
2015-10-23 03:00:00	53.146275
2015-10-23 04:00:00	59.199235
2015-10-23 05:00:00	47.333452
2015-10-23 06:00:00	50.963484
2015-10-23 07:00:00	30.453456
2015-10-23 08:00:00	45.046642
2015-10-23 09:00:00	76.565484
2015-10-23 10:00:00	75.375644
2015-10-23 11:00:00	85.001352
2015-10-23 12:00:00	77.798197
2015-10-23 13:00:00	77.304770
2015-10-23 14:00:00	33.435304
2015-10-23 15:00:00	60.143067
2015-10-23 16:00:00	67.079317
2015-10-23 17:00:00	84.877133
2015-10-23 18:00:00	50.851054
2015-10-23 19:00:00	32.179716
2015-10-23 20:00:00	51.323004
2015-10-23 21:00:00	69.854272
2015-10-23 22:00:00	64.226059
2015-10-23 23:00:00	60.754026
2015-10-24 00:00:00	78.936585
2015-10-24 01:00:00	67.728550
2015-10-24 02:00:00	60.410369
2015-10-24 03:00:00	74.352764
2015-10-24 04:00:00	51.297994
2015-10-24 05:00:00	74.440559
2015-10-24 06:00:00	67.717533
2015-10-24 07:00:00	69.119660
2015-10-24 08:00:00	61.369718
2015-10-24 09:00:00	33.100503
2015-10-24 10:00:00	71.722499
2015-10-24 11:00:00	48.754761
2015-10-24 12:00:00	53.565474
2015-10-24 13:00:00	71.160189
2015-10-24 14:00:00	79.889938
2015-10-24 15:00:00	79.227245
2015-10-24 16:00:00	43.333733
2015-10-24 17:00:00	71.814457
2015-10-24 18:00:00	77.291741
2015-10-24 19:00:00	62.655845
2015-10-24 20:00:00	55.611404
2015-10-24 21:00:00	72.157100
2015-10-24 22:00:00	86.106605
2015-10-24 23:00:00	40.611043
2015-10-25 00:00:00	80.704487
2015-10-25 01:00:00	60.833827
2015-10-25 02:00:00	68.508260
2015-10-25 03:00:00	76.796333
2015-10-25 04:00:00	45.899307
2015-10-25 05:00:00	63.981689
2015-10-25 06:00:00	83.291955
2015-10-25 07:00:00	59.127044
2015-10-25 08:00:00	59.835947
2015-10-25 09:00:00	76.036122
2015-10-25 10:00:00	57.815561
2015-10-25 11:00:00	60.237266
2015-10-25 12:00:00	60.734361
2015-10-25 13:00:00	77.455250
2015-10-25 14:00:00	75.811548
2015-10-25 15:00:00	40.876445
2015-10-25 16:00:00	81.056366
2015-10-25 17:00:00	69.088860
2015-10-25 18:00:00	97.982368
2015-10-25 19:00:00	44.859779
2015-10-25 20:00:00	64.793522
2015-10-25 21:00:00	36.393138
2015-10-25 22:00:00	36.035442
2015-10-25 23:00:00	61.172974
2015-10-26 00:00:00	90.128783
2015-10-26 01:00:00	59.978722
2015-10-26 02:00:00	51.838174
2015-10-26 03:00:00	61.708271
2015-10-26 04:00:00	85.435500
2015-10-26 05:00:00	60.967194
2015-10-26 06:00:00	57.025956
2015-10-26 07:00:00	47.530751
2015-10-26 08:00:00	73.834294
2015-10-26 09:00:00	45.541056
2015-10-26 10:00:00	60.242044
2015-10-26 11:00:00	46.003931
2015-10-26 12:00:00	39.598331
2015-10-26 13:00:00	30.103218
2015-10-26 14:00:00	55.476486
2015-10-26 15:00:00	41.884060
2015-10-26 16:00:00	79.630356
2015-10-26 17:00:00	43.319021
2015-10-26 18:00:00	59.978169
2015-10-26 19:00:00	63.490688
2015-10-26 20:00:00	65.405400
2015-10-26 21:00:00	45.702978
2015-10-26 22:00:00	52.443374
2015-10-26 23:00:00	30.368000
2015-10-27 00:00:00	90.434675
2015-10-27 01:00:00	33.569063
2015-10-27 02:00:00	64.741111
2015-10-27 03:00:00	60.745175
2015-10-27 04:00:00	40.558862
2015-10-27 05:00:00	74.921669
2015-10-27 06:00:00	54.531095
2015-10-27 07:00:00	39.797211
2015-10-27 08:00:00	36.160630
2015-10-27 09:00:00	69.383589
2015-10-27 10:00:00	18.010664
2015-10-27 11:00:00	46.219085
2015-10-27 12:00:00	34.842914
2015-10-27 13:00:00	53.375684
2015-10-27 14:00:00	76.009184
2015-10-27 15:00:00	53.658201
2015-10-27 16:00:00	64.920602
2015-10-27 17:00:00	70.311094
2015-10-27 19:00:00	33.244571
2015-10-27 20:00:00	71.845024
2015-10-27 21:00:00	70.239627
2015-10-27 22:00:00	52.819041
2015-10-27 23:00:00	45.773119
2015-10-28 00:00:00	68.205262
2015-10-28 02:00:00	54.795891
2015-10-28 03:00:00	63.724657
2015-10-28 04:00:00	38.574063
2015-10-28 05:00:00	52.376021
2015-10-28 06:00:00	86.912476
2015-10-28 07:00:00	34.940857
2015-10-28 08:00:00	24.949979
2015-10-28 09:00:00	49.298940
2015-10-28 10:00:00	25.364046
2015-10-28 11:00:00	76.554073
2015-10-28 12:00:00	70.602271
2015-10-28 13:00:00	63.360553
2015-10-28 14:00:00	48.890440
2015-10-28 15:00:00	70.607185
2015-10-28 16:00:00	76.747527
2015-10-28 17:00:00	53.883569
2015-10-28 18:00:00	63.932021
2015-10-28 19:00:00	90.777018
2015-10-28 20:00:00	39.159989
2015-10-28 21:00:00	81.518601
2015-10-28 22:00:00	69.265276
2015-10-28 23:00:00	57.428065
2015-10-29 00:00:00	67.930103
2015-10-29 01:00:00	54.912230
2015-10-29 02:00:00	86.921997
2015-10-29 03:00:00	49.888050
2015-10-29 04:00:00	34.560341
2015-10-29 05:00:00	40.619639
2015-10-29 06:00:00	46.912486
2015-10-29 07:00:00	66.122124
2015-10-29 08:00:00	29.913065
2015-10-29 09:00:00	48.112761
2015-10-29 10:00:00	51.028002
2015-10-29 11:00:00	31.690544
2015-10-29 12:00:00	69.479133
2015-10-29 13:00:00	39.693073
2015-10-29 14:00:00	38.841494
2015-10-29 15:00:00	106.498661
2015-10-29 16:00:00	52.419169
2015-10-29 17:00:00	65.718025
2015-10-29 18:00:00	83.210785
2015-10-29 19:00:00	27.200803
2015-10-29 20:00:00	66.631265
2015-10-29 21:00:00	64.137879
2015-10-29 22:00:00	32.504478
2015-10-29 23:00:00	59.285917
2015-10-30 00:00:00	82.212442
2015-10-30 01:00:00	43.909347
2015-10-30 02:00:00	69.783034
2015-10-30 03:00:00	79.742385
2015-10-30 04:00:00	56.357222
2015-10-30 05:00:00	38.185796
2015-10-30 06:00:00	72.786180
2015-10-30 07:00:00	71.810647
2015-10-30 08:00:00	85.095304
2015-10-30 09:00:00	64.522958
2015-10-30 10:00:00	49.350691
2015-10-30 11:00:00	52.421844
2015-10-30 12:00:00	53.780605
2015-10-30 13:00:00	64.310550
2015-10-30 14:00:00	61.204439
2015-10-30 15:00:00	92.421152
2015-10-30 16:00:00	59.725705
2015-10-30 17:00:00	65.648968
2015-10-30 18:00:00	97.334852
2015-10-30 19:00:00	45.534784
2015-10-30 20:00:00	58.953140
2015-10-30 21:00:00	67.247357
2015-10-30 22:00:00	34.670658
2015-10-30 23:00:00	57.399450
2015-10-31 00:00:00	70.518282
2015-10-31 01:00:00	92.003273
2015-10-31 02:00:00	55.453464
2015-10-31 03:00:00	82.299031
2015-10-31 04:00:00	50.264340
2015-10-31 05:00:00	65.460709
2015-10-31 06:00:00	61.613500
2015-10-31 07:00:00	38.519453
2015-10-31 08:00:00	67.029686
2015-10-31 09:00:00	34.778057
2015-10-31 10:00:00	55.034934
2015-10-31 11:00:00	59.489782
2015-10-31 12:00:00	61.995140
2015-10-31 13:00:00	62.304632
2015-10-31 14:00:00	65.391823
2015-10-31 15:00:00	80.346087
2015-10-31 16:00:00	67.914178
2015-10-31 17:00:00	31.024587
2015-10-31 18:00:00	55.407324
2015-10-31 19:00:00	53.558740
2015-10-31 20:00:00	74.085136
2015-10-31 21:00:00	44.700169
2015-10-31 22:00:00	65.353078
2015-10-31 23:00:00	82.893232
2015-11-01 00:00:00	29.844402
2015-11-01 01:00:00	71.732562
2015-11-01 02:00:00	68.548480
2015-11-01 03:00:00	72.316774
2015-11-01 04:00:00	57.537986
2015-11-01 05:00:00	88.466733
2015-11-01 07:00:00	39.192589
2015-11-01 08:00:00	65.482031
2015-11-01 09:00:00	64.091511
2015-11-01 11:00:00	55.113433
2015-11-01 12:00:00	73.109812
2015-11-01 13:00:00	39.837476
2015-11-01 14:00:00	51.943454
2015-11-01 15:00:00	48.977677
2015-11-01 16:00:00	58.922026
2015-11-01 17:00:00	23.845301
2015-11-01 18:00:00	67.380438
2015-11-01 19:00:00	61.372422
2015-11-01 20:00:00	36.736413
2015-11-01 21:00:00	43.653505
2015-11-01 22:00:00	70.872964
2015-11-01 23:00:00	81.433507
2015-11-02 00:00:00	38.801579
2015-10-27 18:00:00	55.343373
2015-10-28 01:00:00	42.308140
2015-11-01 06:00:00	45.642905
2015-11-01 10:00:00	36.391814
2015-11-02 01:00:00	69.677403
2015-11-02 02:00:00	70.643756
2015-11-02 03:00:00	32.130700
2015-11-02 04:00:00	43.286488
2015-11-02 05:00:00	46.519110
2015-11-02 06:00:00	49.418788
2015-11-02 07:00:00	22.603133
2015-11-02 08:00:00	17.204039
2015-11-02 09:00:00	36.098841
2015-11-02 10:00:00	37.426039
2015-11-02 11:00:00	68.958258
2015-11-02 12:00:00	56.355300
2015-11-02 13:00:00	27.421826
2015-11-02 14:00:00	21.977013
2015-11-02 15:00:00	31.650012
2015-11-02 16:00:00	19.103725
2015-11-02 17:00:00	40.573976
2015-11-02 18:00:00	34.139365
2015-11-02 19:00:00	31.604146
2015-11-02 20:00:00	53.455162
2015-11-02 21:00:00	50.932862
2015-11-02 22:00:00	21.382117
2015-11-02 23:00:00	28.707991
2015-11-03 00:00:00	28.266119
2015-11-03 01:00:00	56.643025
2015-11-03 02:00:00	65.491297
2015-11-03 03:00:00	45.264900
2015-11-03 04:00:00	32.083226
2015-11-03 05:00:00	20.034940
2015-11-03 06:00:00	59.361131
2015-11-03 07:00:00	63.062853
2015-11-03 08:00:00	36.311024
2015-11-03 09:00:00	28.786016
2015-11-03 10:00:00	41.529467
2015-11-03 11:00:00	15.911946
2015-11-03 12:00:00	50.859685
2015-11-03 13:00:00	43.005399
2015-11-03 14:00:00	34.650963
2015-11-03 15:00:00	53.490976
2015-11-03 16:00:00	30.670295
2015-11-03 17:00:00	11.915951
2015-11-03 18:00:00	24.722517
2015-11-03 19:00:00	38.936218
2015-11-03 20:00:00	35.450478
2015-11-03 21:00:00	45.970258
2015-11-03 22:00:00	56.407256
2015-11-03 23:00:00	49.554872
2015-11-04 00:00:00	48.358182
2015-11-04 01:00:00	16.816164
2015-11-04 02:00:00	19.084841
2015-11-04 03:00:00	49.944728
2015-11-04 04:00:00	21.617885
2015-11-04 05:00:00	26.981540
2015-11-04 06:00:00	51.754010
2015-11-04 07:00:00	35.729160
2015-11-04 08:00:00	71.273732
2015-11-04 09:00:00	28.418581
2015-11-04 10:00:00	16.964691
2015-11-04 11:00:00	47.892948
2015-11-04 12:00:00	50.257181
2015-11-04 13:00:00	48.595430
2015-11-04 14:00:00	49.996359
2015-11-04 15:00:00	48.920110
2015-11-04 16:00:00	68.161543
2015-11-04 17:00:00	30.529780
2015-11-04 18:00:00	55.694453
2015-11-04 19:00:00	47.145238
2015-11-04 20:00:00	50.778857
2015-11-04 21:00:00	47.658571
2015-11-04 22:00:00	26.331098
2015-11-04 23:00:00	67.761091
2015-11-05 00:00:00	51.085154
2015-11-05 01:00:00	41.117956
2015-11-05 02:00:00	9.068800
2015-11-05 03:00:00	61.260012
2015-11-05 04:00:00	20.384767
2015-11-05 05:00:00	23.371558
2015-11-05 06:00:00	54.130721
2015-11-05 07:00:00	47.173422
2015-11-05 08:00:00	17.330480
2015-11-05 09:00:00	27.900997
2015-11-05 10:00:00	60.693928
2015-11-05 11:00:00	55.750439
2015-11-05 12:00:00	31.980729
2015-11-05 13:00:00	46.026815
2015-11-05 14:00:00	51.486720
2015-11-05 15:00:00	35.083882
2015-11-05 16:00:00	29.222731
2015-11-05 17:00:00	38.054577
2015-11-05 18:00:00	32.803461
2015-11-05 19:00:00	34.590651
2015-11-05 20:00:00	7.041455
2015-11-05 21:00:00	43.879486
2015-11-05 22:00:00	54.260075
2015-11-05 23:00:00	49.607998
2015-11-06 00:00:00	51.166843
2015-11-06 01:00:00	60.332261
2015-11-06 02:00:00	54.000615
2015-11-06 03:00:00	48.402953
2015-11-06 04:00:00	71.293996
2015-11-06 05:00:00	25.864377
2015-11-06 06:00:00	43.534985
2015-11-06 07:00:00	10.294754
2015-11-06 08:00:00	27.211552
2015-11-06 09:00:00	68.771180
2015-11-06 10:00:00	40.635575
2015-11-06 11:00:00	48.547279
2015-11-06 12:00:00	51.211695
2015-11-06 13:00:00	62.323132
2015-11-06 14:00:00	36.042205
2015-11-06 15:00:00	65.867425
2015-11-06 16:00:00	43.091257
2015-11-06 17:00:00	49.999283
2015-11-06 18:00:00	42.753244
2015-11-06 19:00:00	43.280302
2015-11-06 20:00:00	12.215143
2015-11-06 21:00:00	30.790950
2015-11-06 22:00:00	43.146264
2015-11-06 23:00:00	24.862962
2015-11-07 00:00:00	60.419541
2015-11-07 01:00:00	2.829191
2015-11-07 02:00:00	41.358591
2015-11-07 03:00:00	32.518025
2015-11-07 04:00:00	32.936246
2015-11-07 05:00:00	53.903692
2015-11-07 06:00:00	54.349240
2015-11-07 07:00:00	48.667338
2015-11-07 08:00:00	43.697246
2015-11-07 09:00:00	64.391019
2015-11-07 10:00:00	54.642566
2015-11-07 11:00:00	31.579795
2015-11-07 12:00:00	10.697792
2015-11-07 13:00:00	69.408304
2015-11-07 14:00:00	57.739228
2015-11-07 15:00:00	54.043036
2015-11-07 16:00:00	21.378107
2015-11-07 17:00:00	68.557246
2015-11-07 18:00:00	49.990255
2015-11-07 19:00:00	60.870852
2015-11-07 20:00:00	15.231194
2015-11-07 21:00:00	35.271872
2015-11-07 22:00:00	47.834724
2015-11-07 23:00:00	39.042589
2015-11-08 00:00:00	31.282543
2015-11-08 01:00:00	34.318646
2015-11-08 02:00:00	45.695737
2015-11-08 03:00:00	34.761724
2015-11-08 04:00:00	41.324282
2015-11-08 05:00:00	22.815585
2015-11-08 06:00:00	63.250742
2015-11-08 07:00:00	13.839852
2015-11-08 08:00:00	46.006283
2015-11-08 09:00:00	44.493733
2015-11-08 10:00:00	63.378401
2015-11-08 11:00:00	42.817629
2015-11-08 12:00:00	45.072082
2015-11-08 13:00:00	46.040137
2015-11-08 14:00:00	11.573340
2015-11-08 15:00:00	39.922730
2015-11-08 16:00:00	40.153338
2015-11-08 17:00:00	63.037254
2015-11-08 18:00:00	48.776710
2015-11-08 19:00:00	53.306946
2015-11-08 20:00:00	41.484211
2015-11-08 21:00:00	45.348702
2015-11-08 22:00:00	54.969466
2015-11-08 23:00:00	47.595072
2015-11-09 00:00:00	40.591619
2015-11-09 01:00:00	43.363619
2015-11-09 02:00:00	65.789153
2015-11-09 03:00:00	29.012179
2015-11-09 04:00:00	38.944844
2015-11-09 05:00:00	26.349575
2015-11-09 06:00:00	33.638500
2015-11-09 07:00:00	8.524681
2015-11-09 08:00:00	7.047517
2015-11-09 09:00:00	71.864778
2015-11-09 10:00:00	33.775195
2015-11-09 11:00:00	34.524480
2015-11-09 12:00:00	13.277970
2015-11-09 13:00:00	20.951475
2015-11-09 14:00:00	42.715256
2015-11-09 15:00:00	51.101163
2015-11-09 16:00:00	8.718182
2015-11-09 17:00:00	23.884311
2015-11-09 18:00:00	44.208632
2015-11-09 19:00:00	47.094925
2015-11-09 20:00:00	25.140574
2015-11-09 21:00:00	33.723718
2015-11-09 22:00:00	23.366525
2015-11-09 23:00:00	41.851025
2015-11-10 00:00:00	28.565955
2015-11-10 01:00:00	49.010418
2015-11-10 02:00:00	76.953288
2015-11-10 03:00:00	33.641144
2015-11-10 04:00:00	61.127699
2015-11-10 05:00:00	61.822340
2015-11-10 06:00:00	22.690029
2015-11-10 07:00:00	52.496753
2015-11-10 08:00:00	26.962412
2015-11-10 09:00:00	55.912366
2015-11-10 10:00:00	38.827366
2015-11-10 11:00:00	51.170992
2015-11-10 12:00:00	38.982250
2015-11-10 13:00:00	36.403632
2015-11-10 14:00:00	45.511136
2015-11-10 15:00:00	35.450764
2015-11-10 16:00:00	12.781466
2015-11-10 17:00:00	34.596144
2015-11-10 18:00:00	19.863000
2015-11-10 19:00:00	61.422724
2015-11-10 20:00:00	37.871542
2015-11-10 21:00:00	44.714030
2015-11-10 22:00:00	52.438546
2015-11-10 23:00:00	21.870034
2015-11-11 00:00:00	42.775527
2015-11-11 01:00:00	37.075660
2015-11-11 02:00:00	55.922491
2015-11-11 03:00:00	21.725370
2015-11-11 04:00:00	27.383226
2015-11-11 05:00:00	26.474453
2015-11-11 06:00:00	24.992206
2015-11-11 07:00:00	29.420236
2015-11-11 08:00:00	18.593189
2015-11-11 09:00:00	19.401002
2015-11-11 10:00:00	30.797664
2015-11-11 11:00:00	53.005290
2015-11-11 12:00:00	29.785165
2015-11-11 13:00:00	48.478284
2015-11-11 14:00:00	41.000063
2015-11-11 15:00:00	55.985426
2015-11-11 16:00:00	23.593107
2015-11-11 17:00:00	16.929397
2015-11-11 18:00:00	26.105809
2015-11-11 19:00:00	27.278099
2015-11-11 20:00:00	35.966696
2015-11-11 21:00:00	62.321017
2015-11-11 22:00:00	22.551005
2015-11-11 23:00:00	32.326498
2015-11-12 00:00:00	36.768426
2015-11-12 01:00:00	16.655779
2015-11-12 02:00:00	50.903035
2015-11-12 03:00:00	32.744233
2015-11-12 04:00:00	43.446611
2015-11-12 05:00:00	12.872882
2015-11-12 06:00:00	43.626821
2015-11-12 07:00:00	17.932564
2015-11-12 08:00:00	46.430782
2015-11-12 09:00:00	47.555630
2015-11-12 10:00:00	23.164628
2015-11-12 11:00:00	36.320166
2015-11-12 12:00:00	41.829277
2015-11-12 13:00:00	65.063811
2015-11-12 14:00:00	53.996669
2015-11-12 15:00:00	53.065490
2015-11-12 16:00:00	36.048012
2015-11-12 17:00:00	23.164822
2015-11-12 18:00:00	38.652909
2015-11-12 19:00:00	48.085267
2015-11-12 20:00:00	39.547030
2015-11-12 21:00:00	36.281582
2015-11-12 22:00:00	22.304580
2015-11-12 23:00:00	39.054497
2015-11-13 00:00:00	62.456375
2015-11-13 01:00:00	21.058794
2015-11-13 02:00:00	24.986703
2015-11-13 03:00:00	47.340037
2015-11-13 04:00:00	42.103710
2015-11-13 05:00:00	46.277421
2015-11-13 06:00:00	41.346360
2015-11-13 07:00:00	8.198421
2015-11-13 08:00:00	49.820480
2015-11-13 09:00:00	52.630520
2015-11-13 10:00:00	14.320846
2015-11-13 11:00:00	29.429210
2015-11-13 12:00:00	46.610070
2015-11-13 13:00:00	27.718158
2015-11-13 14:00:00	43.646591
2015-11-13 15:00:00	58.249277
2015-11-13 16:00:00	37.955810
2015-11-13 17:00:00	53.167141
2015-11-13 18:00:00	17.003180
2015-11-13 19:00:00	36.987767
2015-11-13 20:00:00	69.421289
2015-11-13 21:00:00	42.282380
2015-11-13 22:00:00	60.299183
2015-11-13 23:00:00	77.388036
2015-11-14 00:00:00	38.187514
2015-11-14 01:00:00	73.920364
2015-11-14 02:00:00	23.427619
2015-11-14 03:00:00	74.049763
2015-11-14 04:00:00	71.025842
2015-11-14 05:00:00	36.310935
2015-11-14 06:00:00	69.727846
2015-11-14 07:00:00	47.318911
2015-11-14 08:00:00	54.552009
2015-11-14 09:00:00	22.047066
2015-11-14 10:00:00	33.286940
2015-11-14 11:00:00	44.805901
2015-11-14 12:00:00	30.968865
2015-11-14 13:00:00	32.250332
2015-11-14 14:00:00	60.449541
2015-11-14 15:00:00	17.006485
2015-11-14 16:00:00	42.845801
2015-11-14 17:00:00	47.528852
2015-11-14 18:00:00	39.715687
2015-11-14 19:00:00	23.905351
2015-11-14 20:00:00	60.468321
2015-11-14 21:00:00	16.016156
2015-11-14 22:00:00	24.913911
2015-11-14 23:00:00	52.127222
2015-11-15 00:00:00	31.771907
2015-11-15 01:00:00	33.992850
2015-11-15 02:00:00	12.282765
2015-11-15 03:00:00	50.596037
2015-11-15 04:00:00	31.542007
2015-11-15 05:00:00	20.514348
2015-11-15 06:00:00	27.990267
2015-11-15 07:00:00	70.663541
2015-11-15 08:00:00	44.674916
2015-11-15 09:00:00	22.821463
2015-11-15 10:00:00	28.171757
2015-11-15 11:00:00	38.060556
2015-11-15 12:00:00	28.251189
2015-11-15 13:00:00	20.650803
2015-11-15 14:00:00	42.038281
2015-11-15 15:00:00	33.657499
2015-11-15 16:00:00	24.628537
2015-11-15 17:00:00	62.740169
2015-11-15 18:00:00	37.227280
2015-11-15 19:00:00	66.834372
2015-11-15 20:00:00	54.885094
2015-11-15 21:00:00	38.741754
2015-11-15 22:00:00	52.565861
2015-11-15 23:00:00	42.011579
2015-11-16 00:00:00	1.884618
2015-11-16 01:00:00	33.697009
2015-11-16 02:00:00	35.136376
2015-11-16 03:00:00	52.342981
2015-11-16 04:00:00	56.699116
2015-11-16 05:00:00	55.563435
2015-11-16 06:00:00	45.758364
2015-11-16 07:00:00	50.659673
2015-11-16 08:00:00	40.169564
2015-11-16 09:00:00	24.979220
2015-11-16 10:00:00	47.796589
2015-11-16 11:00:00	35.726957
2015-11-16 12:00:00	54.028989
2015-11-16 13:00:00	45.341975
2015-11-16 14:00:00	48.476311
2015-11-16 15:00:00	33.640512
2015-11-16 16:00:00	30.864809
2015-11-16 17:00:00	20.844211
2015-11-16 18:00:00	19.803964
2015-11-16 19:00:00	36.264543
2015-11-16 20:00:00	31.291821
2015-11-16 21:00:00	10.358942
2015-11-16 22:00:00	31.271942
2015-11-16 23:00:00	39.143974
2015-11-17 00:00:00	40.599863
2015-11-17 01:00:00	4.404232
2015-11-17 02:00:00	60.532204
2015-11-17 03:00:00	42.654711
2015-11-17 04:00:00	11.865801
2015-11-17 05:00:00	22.798883
2015-11-17 06:00:00	32.524959
2015-11-17 07:00:00	65.957563
2015-11-17 08:00:00	26.382528
2015-11-17 09:00:00	33.062656
2015-11-17 10:00:00	18.365385
2015-11-17 11:00:00	59.433134
2015-11-17 12:00:00	38.719453
2015-11-17 13:00:00	70.619444
2015-11-17 14:00:00	51.069993
2015-11-17 15:00:00	29.489031
2015-11-17 16:00:00	15.808655
2015-11-17 17:00:00	39.223626
2015-11-17 18:00:00	47.080228
2015-11-17 19:00:00	41.980568
2015-11-17 20:00:00	61.214361
2015-11-17 21:00:00	23.836742
2015-11-17 22:00:00	35.213413
2015-11-17 23:00:00	66.174825
2015-11-18 00:00:00	11.472738
2015-11-18 01:00:00	40.944768
2015-11-18 02:00:00	73.297982
2015-11-18 03:00:00	21.705112
2015-11-18 04:00:00	13.445279
2015-11-18 05:00:00	33.855644
2015-11-18 06:00:00	38.425269
2015-11-18 07:00:00	7.564169
2015-11-18 08:00:00	20.792792
2015-11-18 09:00:00	47.126813
2015-11-18 10:00:00	38.557200
2015-11-18 11:00:00	30.619750
2015-11-18 12:00:00	27.548411
2015-11-18 13:00:00	34.697505
2015-11-18 14:00:00	36.676699
2015-11-18 15:00:00	30.025806
2015-11-18 16:00:00	44.290001
2015-11-18 17:00:00	31.476714
2015-11-18 18:00:00	16.554329
2015-11-18 19:00:00	28.467637
2015-11-18 20:00:00	60.756805
2015-11-18 21:00:00	53.156311
2015-11-18 22:00:00	43.736182
2015-11-18 23:00:00	44.678285
2015-11-19 00:00:00	32.500159
2015-11-19 01:00:00	53.096718
2015-11-19 02:00:00	50.450171
2015-11-19 03:00:00	20.671851
2015-11-19 04:00:00	48.500930
2015-11-19 05:00:00	55.872094
2015-11-19 06:00:00	16.955515
2015-11-19 07:00:00	47.782032
2015-11-19 08:00:00	64.182162
2015-11-19 09:00:00	19.123758
2015-11-19 10:00:00	26.630040
2015-11-19 11:00:00	24.693264
2015-11-19 12:00:00	14.882576
2015-11-19 13:00:00	54.702633
2015-11-19 14:00:00	28.045472
2015-11-19 15:00:00	44.237443
2015-11-19 16:00:00	53.989477
2015-11-19 17:00:00	44.208531
2015-11-19 18:00:00	44.145634
2015-11-19 19:00:00	40.174149
2015-11-19 20:00:00	75.699763
2015-11-19 21:00:00	33.688595
2015-11-19 22:00:00	60.315099
2015-11-19 23:00:00	37.811607
2015-11-20 00:00:00	62.216296
2015-11-20 01:00:00	21.923579
2015-11-20 02:00:00	50.856824
2015-11-20 03:00:00	9.506527
2015-11-20 04:00:00	39.037833
2015-11-20 05:00:00	47.392675
2015-11-20 06:00:00	39.774791
2015-11-20 07:00:00	48.835634
2015-11-20 08:00:00	46.283958
2015-11-20 09:00:00	48.833873
2015-11-20 10:00:00	41.321219
2015-11-20 11:00:00	49.126166
2015-11-20 12:00:00	26.070307
2015-11-20 13:00:00	37.334013
2015-11-20 14:00:00	44.013403
2015-11-20 15:00:00	63.076857
2015-11-20 16:00:00	44.009289
2015-11-20 17:00:00	53.765517
2015-11-20 18:00:00	6.907394
2015-11-20 19:00:00	51.625765
2015-11-20 20:00:00	64.845969
2015-11-20 21:00:00	23.791188
2015-11-20 22:00:00	41.551253
2015-11-20 23:00:00	45.032033
2015-11-21 00:00:00	30.722590
2015-11-21 01:00:00	37.643845
2015-11-21 02:00:00	33.978619
2015-11-21 03:00:00	59.975566
2015-11-21 04:00:00	24.821922
2015-11-21 05:00:00	24.426441
2015-11-21 06:00:00	58.768110
2015-11-21 07:00:00	47.487387
2015-11-21 08:00:00	41.046481
2015-11-21 09:00:00	32.077990
2015-11-21 10:00:00	61.332384
2015-11-21 11:00:00	33.779878
2015-11-21 12:00:00	68.860907
2015-11-21 13:00:00	28.228687
2015-11-21 14:00:00	41.392049
2015-11-21 15:00:00	55.493874
2015-11-21 16:00:00	41.741668
2015-11-21 17:00:00	54.743921
2015-11-21 18:00:00	51.711642
2015-11-21 19:00:00	21.725879
2015-11-21 20:00:00	45.572460
2015-11-21 21:00:00	29.984989
2015-11-21 22:00:00	31.946214
2015-11-21 23:00:00	74.120935
2015-11-22 00:00:00	39.799580
2015-11-22 01:00:00	28.923704
2015-11-22 02:00:00	34.125230
2015-11-22 03:00:00	47.261910
2015-11-22 04:00:00	31.905987
2015-11-22 05:00:00	53.219995
2015-11-22 06:00:00	26.944213
2015-11-22 07:00:00	54.151730
2015-11-22 08:00:00	31.557127
2015-11-22 09:00:00	47.108037
2015-11-22 10:00:00	16.939418
2015-11-22 11:00:00	56.867922
2015-11-22 12:00:00	56.924140
2015-11-22 13:00:00	7.143941
2015-11-22 14:00:00	49.925868
2015-11-22 15:00:00	58.971000
2015-11-22 16:00:00	34.281085
2015-11-22 17:00:00	42.682790
2015-11-22 18:00:00	36.855911
2015-11-22 19:00:00	22.575510
2015-11-22 20:00:00	43.106407
2015-11-22 21:00:00	57.423548
2015-11-22 22:00:00	60.982311
2015-11-22 23:00:00	19.184503
2015-11-23 00:00:00	27.031110
2015-11-23 01:00:00	24.985017
2015-11-23 02:00:00	41.859799
2015-11-23 03:00:00	7.230717
2015-11-23 04:00:00	24.225717
2015-11-23 05:00:00	46.453707
2015-11-23 06:00:00	51.230189
2015-11-23 07:00:00	44.442608
2015-11-23 08:00:00	49.115951
2015-11-23 09:00:00	32.865960
2015-11-23 10:00:00	23.082460
2015-11-23 11:00:00	34.718627
2015-11-23 12:00:00	66.163243
2015-11-23 13:00:00	68.177732
2015-11-23 14:00:00	42.731288
2015-11-23 15:00:00	65.160220
2015-11-23 16:00:00	18.855938
2015-11-23 17:00:00	28.938320
2015-11-23 18:00:00	4.388469
2015-11-23 19:00:00	44.976017
2015-11-23 20:00:00	35.722734
2015-11-23 21:00:00	45.347123
2015-11-23 22:00:00	22.781328
2015-11-23 23:00:00	43.114767
2015-11-24 00:00:00	24.184121
2015-11-24 01:00:00	43.356235
2015-11-24 02:00:00	35.089344
2015-11-24 03:00:00	42.129951
2015-11-24 04:00:00	65.064061
2015-11-24 05:00:00	31.054652
2015-11-24 06:00:00	43.142107
2015-11-24 07:00:00	33.909621
2015-11-24 08:00:00	25.199164
2015-11-24 09:00:00	26.900095
2015-11-24 10:00:00	54.820399
2015-11-24 11:00:00	36.714245
2015-11-24 12:00:00	58.612063
2015-11-24 13:00:00	48.918951
2015-11-24 14:00:00	55.173613
2015-11-24 15:00:00	34.770766
2015-11-24 16:00:00	4.595560
2015-11-24 17:00:00	17.717682
2015-11-24 18:00:00	63.682387
2015-11-24 19:00:00	16.459204
2015-11-24 20:00:00	39.174167
2015-11-24 21:00:00	5.544962
2015-11-24 22:00:00	51.408889
2015-11-24 23:00:00	71.206396
2015-11-25 00:00:00	20.604607
2015-11-25 01:00:00	45.903558
2015-11-25 02:00:00	47.996068
2015-11-25 03:00:00	51.515777
2015-11-25 04:00:00	27.643937
2015-11-25 05:00:00	19.950425
2015-11-25 06:00:00	42.934676
2015-11-25 07:00:00	33.758964
2015-11-25 08:00:00	60.198257
2015-11-25 09:00:00	64.822648
2015-11-25 10:00:00	26.160919
2015-11-25 11:00:00	44.729294
2015-11-25 12:00:00	28.503025
2015-11-25 13:00:00	75.771776
2015-11-25 14:00:00	27.112065
2015-11-25 15:00:00	49.403191
2015-11-25 16:00:00	38.397854
2015-11-25 17:00:00	38.091870
2015-11-25 18:00:00	43.485536
2015-11-25 19:00:00	13.170863
2015-11-25 20:00:00	52.137698
2015-11-25 21:00:00	43.204897
2015-11-25 22:00:00	37.660492
2015-11-25 23:00:00	31.197636
2015-11-26 00:00:00	34.074007
2015-11-26 01:00:00	64.403364
2015-11-26 02:00:00	61.575249
2015-11-26 03:00:00	32.426134
2015-11-26 04:00:00	30.168290
2015-11-26 05:00:00	37.227840
2015-11-26 06:00:00	18.700339
2015-11-26 07:00:00	27.573602
2015-11-26 08:00:00	48.217705
2015-11-26 09:00:00	45.173003
2015-11-26 10:00:00	18.890009
2015-11-26 11:00:00	27.141646
2015-11-26 12:00:00	14.635885
2015-11-26 13:00:00	30.364299
2015-11-26 14:00:00	8.753475
2015-11-26 15:00:00	18.768489
2015-11-26 16:00:00	39.735482
2015-11-26 17:00:00	13.730282
2015-11-26 18:00:00	44.824872
2015-11-26 19:00:00	39.809994
2015-11-26 20:00:00	28.565875
2015-11-26 21:00:00	43.661477
2015-11-26 22:00:00	55.906271
2015-11-26 23:00:00	31.129181
2015-11-27 00:00:00	19.962453
2015-11-27 01:00:00	42.683700
2015-11-27 02:00:00	59.089857
2015-11-27 03:00:00	23.640381
2015-11-27 04:00:00	26.785668
2015-11-27 05:00:00	40.640046
2015-11-27 06:00:00	35.918847
2015-11-27 07:00:00	48.929382
2015-11-27 08:00:00	25.118402
2015-11-27 09:00:00	34.981362
2015-11-27 10:00:00	8.636697
2015-11-27 11:00:00	59.959267
2015-11-27 12:00:00	18.643766
2015-11-27 13:00:00	17.155756
2015-11-27 14:00:00	5.500306
2015-11-27 15:00:00	14.505197
2015-11-27 16:00:00	24.883430
2015-11-27 17:00:00	54.744878
2015-11-27 18:00:00	55.735616
2015-11-27 19:00:00	32.256413
2015-11-27 20:00:00	38.956010
2015-11-27 21:00:00	60.020956
2015-11-27 22:00:00	69.425402
2015-11-27 23:00:00	8.688084
2015-11-28 00:00:00	42.970294
2015-11-28 01:00:00	27.727545
2015-11-28 02:00:00	38.962854
2015-11-28 03:00:00	13.365612
2015-11-28 04:00:00	31.819558
2015-11-28 05:00:00	33.103063
2015-11-28 06:00:00	30.924940
2015-11-28 07:00:00	26.308201
2015-11-28 08:00:00	9.565704
2015-11-28 09:00:00	13.830709
2015-11-28 10:00:00	48.527078
2015-11-28 11:00:00	29.341300
2015-11-28 12:00:00	56.402469
2015-11-28 13:00:00	24.985439
2015-11-28 14:00:00	36.644183
2015-11-28 15:00:00	26.098598
2015-11-28 16:00:00	24.020212
2015-11-28 17:00:00	50.831756
2015-11-28 18:00:00	56.770995
2015-11-28 19:00:00	47.023474
2015-11-28 20:00:00	59.280305
2015-11-28 21:00:00	37.453182
2015-11-28 22:00:00	72.102666
2015-11-28 23:00:00	30.348483
2015-11-29 00:00:00	53.805047
2015-11-29 01:00:00	22.456437
2015-11-29 02:00:00	60.534383
2015-11-29 03:00:00	16.088490
2015-11-29 04:00:00	36.382437
2015-11-29 05:00:00	59.246121
2015-11-29 06:00:00	45.077214
2015-11-29 07:00:00	45.120941
2015-11-29 08:00:00	38.232433
2015-11-29 09:00:00	19.465618
2015-11-29 10:00:00	38.230367
2015-11-29 11:00:00	18.690932
2015-11-29 12:00:00	20.769636
2015-11-29 13:00:00	61.135055
2015-11-29 14:00:00	27.915517
2015-11-29 15:00:00	30.520160
2015-11-29 16:00:00	28.824846
2015-11-29 17:00:00	69.571855
2015-11-29 18:00:00	37.361398
2015-11-29 19:00:00	38.282275
2015-11-29 20:00:00	46.356586
2015-11-29 21:00:00	24.056497
2015-11-29 22:00:00	53.198139
2015-11-29 23:00:00	35.252730
2015-11-30 00:00:00	43.607553
2015-11-30 01:00:00	60.174882
2015-11-30 02:00:00	17.935532
2015-11-30 03:00:00	68.517692
2015-11-30 04:00:00	29.962754
2015-11-30 05:00:00	61.862104
2015-11-30 06:00:00	20.989968
2015-11-30 07:00:00	60.410201
2015-11-30 08:00:00	59.574523
2015-11-30 09:00:00	56.018735
2015-11-30 10:00:00	24.071468
2015-11-30 11:00:00	49.377007
2015-11-30 12:00:00	38.429695
2015-11-30 13:00:00	26.261928
2015-11-30 14:00:00	66.245234
2015-11-30 15:00:00	32.917481
2015-11-30 16:00:00	25.103363
2015-11-30 17:00:00	27.791581
2015-11-30 18:00:00	46.185805
2015-11-30 19:00:00	33.197581
2015-11-30 20:00:00	45.298259
2015-11-30 21:00:00	15.820063
2015-11-30 22:00:00	19.494634
2015-11-30 23:00:00	39.350548
2015-12-01 00:00:00	39.319314
2015-12-01 01:00:00	2.474597
2015-12-01 02:00:00	41.595853
2015-12-01 03:00:00	15.030481
2015-12-01 04:00:00	71.055191
2015-12-01 05:00:00	52.047501
2015-12-01 06:00:00	72.879618
2015-12-01 07:00:00	37.068471
2015-12-01 08:00:00	34.298235
2015-12-01 09:00:00	41.579046
2015-12-01 10:00:00	40.660273
2015-12-01 11:00:00	61.635744
2015-12-01 12:00:00	43.405853
2015-12-01 13:00:00	18.588931
2015-12-01 14:00:00	47.898820
2015-12-01 15:00:00	33.526542
2015-12-01 16:00:00	53.297195
2015-12-01 17:00:00	29.570191
2015-12-01 18:00:00	18.813022
2015-12-01 19:00:00	32.032767
2015-12-01 20:00:00	43.580802
2015-12-01 21:00:00	29.927978
2015-12-01 22:00:00	65.725895
2015-12-01 23:00:00	14.997650
2015-12-02 00:00:00	21.467634
2015-12-02 01:00:00	35.542969
2015-12-02 02:00:00	67.452311
2015-12-02 03:00:00	57.589935
2015-12-02 04:00:00	35.194673
2015-12-02 05:00:00	46.114270
2015-12-02 06:00:00	35.049139
2015-12-02 07:00:00	34.855851
2015-12-02 08:00:00	31.369865
2015-12-02 09:00:00	43.440536
2015-12-02 10:00:00	5.292336
2015-12-02 11:00:00	15.588859
2015-12-02 12:00:00	45.724622
2015-12-02 13:00:00	46.292510
2015-12-02 14:00:00	38.483379
2015-12-02 15:00:00	51.287088
2015-12-02 16:00:00	35.419347
2015-12-02 17:00:00	38.717686
2015-12-02 18:00:00	53.184185
2015-12-02 19:00:00	66.245006
2015-12-02 20:00:00	40.550695
2015-12-02 21:00:00	57.073648
2015-12-02 22:00:00	49.285885
2015-12-02 23:00:00	22.874043
2015-12-03 00:00:00	46.236585
2015-12-03 01:00:00	77.101826
2015-12-03 02:00:00	49.775351
2015-12-03 03:00:00	71.134530
2015-12-03 04:00:00	58.448860
2015-12-03 05:00:00	58.534300
2015-12-03 06:00:00	28.555859
2015-12-03 07:00:00	39.062662
2015-12-03 08:00:00	16.662519
2015-12-03 09:00:00	27.000278
2015-12-03 10:00:00	44.246493
2015-12-03 11:00:00	72.966772
2015-12-03 12:00:00	31.483105
2015-12-03 13:00:00	66.816731
2015-12-03 14:00:00	42.936656
2015-12-03 15:00:00	33.401326
2015-12-03 16:00:00	66.681471

waterdata_ts <- xts(waterdata %>% select(-DateTime), order.by = waterdata$DateTime)

Time plot

 waterdata_ts %>% ggtsdisplay(main = "Hourly waterflow Pipeline1 + Pipeline2"
                         ,xlab = "Days"
                         ,ylab = "Total waterflow")

Observations:

There is an initial downward trend in the data uptill Day 10, but after that there is no observable trend.
No apparent seasonality present in data
The variance is more or less constant across time range.
No apparent anomalies or outliers.

From the initial plot data doesn’t look completely stationary even though absence of seasonality and constant variance are good signs. Hence I will perform Box Cox transformation and evaluate 1st order differencing needs to convert the data to stationary.

ndiffs(waterdata_ts)

## [1] 1

water_lambda <- BoxCox.lambda(waterdata_ts)

cat("Box Cox Transformation factor lambda=",water_lambda)

## Box Cox Transformation factor lambda= 0.8353232

 waterdata_ts %>% BoxCox(water_lambda) %>% diff() %>% ggtsdisplay(main = "Hourly Waterflow w/ Box Cox + Differencing"
                         ,xlab = "Days"
                         ,ylab = "Total Waterflow")

## Warning: Removed 1 rows containing missing values (geom_point).

With the Box Cox transformation and 1st order differencing applied, data does appear stationary. Hence we can apply non-season ARIMA with d=1. Also, since ACF and PACF plots both have largest spike at lag=1, so we can assume AR(1) (p=1) and MA(1) (q=1) etc. So we can use an ARIMA (1,1,1) model to forecast the waterflow for the upcoming week.

ARIMA Model:

water_model_fit <- Arima(waterdata_ts, order = c(1, 1, 1), lambda = water_lambda)

autoplot(forecast(water_model_fit, h=7*24)) + theme(panel.background = element_blank()) +
  xlab ("Days") +
  ylab ("Total Waterflow")

Model Accuracy

accuracyDF <- data.frame(Model = "ARIMA (1,1,1)", accuracy(water_model_fit), row.names = NULL)

accuracyDF %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="200px")

Model	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
ARIMA (1,1,1)	0.283184	16.38391	13.36601	-27.89199	50.16992	0.3020555	-0.0032977

Model Residual

checkresiduals(water_model_fit)

## 
##  Ljung-Box test
## 
## data:  Residuals from ARIMA(1,1,1)
## Q* = 2.7066, df = 8, p-value = 0.9514
## 
## Model df: 2.   Total lags used: 10

From Ljung-Box test, the p-Value is 0.9514 which is sufficiently higher than 0.05. So it can be safe to reject the null hypothesis that residuals are not independent and hence the model meets the assumption and generate quality forecast.

Results:

water_forecast <- forecast(water_model_fit, h=7*24)

tibble(`DateTime` = seq(as.POSIXct("2015-12-03 17:00:00",origin="1899-12-30", tz="GMT"), length=7*24, by="hours"),
       WaterFlow = water_forecast$mean) %>% 
write_csv("Water_Forecast.csv")

Waterflow Forecast File Path:

GitHub File Path

DATA 624 - Project1 - Timeseries Forecast Models

Soumya Ghosh

October 20, 2020

Libraries

PART A: DataSet - ATM Cash Draw

Missing Value Analysis

Data Pre-Processing

Time Plot:

ATM1:

Data Imputation:

Observations:

Model1 Accuracy

Model1 Residual

Model2: ETS

Model2 Accuracy

Model2 Residual

Model3: ARIMA

Model3 Accuracy

Model3 Residual

Results:

ATM2:

Observations:

Model1: Holt-Winters w/ Box Cox

Model1 Accuracy

Model1 Residual

Model2: ETS

Model2 Accuracy

Model2 Residual

Model3: ARIMA

Model3 Accuracy

Model3 Residual

Model4: ARIMA with 1st Order Differencing

Model4 Residual

Results:

ATM3:

ATM4:

Observations:

Model1: Holt-Winters w/ Box Cox

Model1 Accuracy

Model1 Residual

Model2: ETS

Model2 Accuracy

Model2 Residual

Model3: ARIMA

Model3 Accuracy

Model3 Residual

Model4: ARIMA with Seasonal Differencing

Model4 Accuracy

Model4 Residual

Results:

PART B: DataSet - Residential Power Forecast

Model1: Holt-Winters w/ Box Cox

Model1 Accuracy

Model1 Residual

Model2: ETS

Model2 Accuracy

Model2 Residual

Model3: ARIMA

Model3 Accuracy

Model3 Residual

Results:

PART C: BONUS - Water Pipeline Datasets

Data Pre-Processing:

Time plot

Observations:

ARIMA Model:

Model Accuracy

Model Residual

Results: