time-series-analysis

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Introduction:

Present study is based on the notion of time series analysis, the analysis of time series is useful in administration, planning, evaluation, of socio economic progress as well as for research in various scientific fields including pure sciences, econometrics and humanities. In the present study the significance of difference in net sale, gross profit made by company, Revenue generated across the months and season studied. finding significance of difference with respect to shop making amount of profit in relation to location of shop, number of sales occurred for different months of the year, average sales for different months of the year, Gross profit for different months of the year Total of 366 respondents were studied and data were collected using a pretested questionnaire.

 

Problem definition and business intelligence required.

In this study we try to investigate and tests the hypothesis

Number of sale of a product varies over a period of time, here we tests the hypothesis of whether difference in number of sales, Average sales, Gross profit varies over a period of month, In another hypothesis study is undertaken to test hypothesis of whether difference in number of sales and average sales varies between rainy days and Gross Profit.

Business Intelligence required:

  • To test the significance of difference Analysis of variance and General Linear Model using post Hoc tests for multiple comparisons between categories of month and the season is studied. Bonferroni test is used for multiple comparisons and correlation is used to study relationship between the variables.
  • Bar Chart, Scatter Diagram, box plot is used to visualize the data graphically.

To answer the aforesaid hypothesis following analysis is conducted.

1.0 What are my top selling products?

On the basis of quantity sold following are the top selling products are.

Natural Coconut Milk Icecream

 

 

 

 

Bar Graph Depicting Location wise Net Profit

 

 

3.0 What location in the shop makes the most amount of Profit?

 

Observed that maximum sales were observed from front location. 39074$

 

 

 

 

 

 

Bar Graph Depicting Location Wise Maximum Sales

 

 

4.0 Is there a difference in number of Sales between different months of the year?

Solution:

 

Box plot representing distribution of data median at middle, third quartile at the top of box and first quartile at lower part of the box and dots represents the outliers these are the characteristics of boxplot, looking at the graph it is seen that median values for the month range in the tune of 1000.

 

 

 

 

Multiple comparisons using Post-Hoc Bonferroni test

Multiple Comparisons
Dependent Variable: Number of sales

Bonferroni

(I) Month of the year(J) Month of the year FebruaryMean

Difference

(I-J)

-96.41

-97.72

-112.51

-87.23

47.14

-29.32

-44.15

-24.37

-63.13

-207.76

-98.82

96.41 -1.32

-16.11 9.18

143.54

67.08

Std.

Error

80.748

Sig.

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0 .650

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

95% Confidence

Interval

Lower BoundUpper

Bound

177.93

172.01

159.45

182.50

319.10

240.40

225.58

247.60

206.59 64.20

170.91

370.74

273.02

260.43

283.51

420.08

341.42

January

February

370.74

March79.391

367.45

April80.050

384.48

May79.391

356.95

June80.050

224.82

July79.391

299.05

August79.391

313.88

September80.050

296.33

October79.391

332.86

November80.050

479.73

December79.391

368.54

January80.748

177.93

March80.748

275.65

April81.396

292.65

May80.748

265.16

June81.396

132.99

July80.748

 

 

March

April

52.26

72.04

33.27

-111.36

-2.41

97.72 1.32

-14.79 10.49

144.86

68.40

53.57

73.35

34.59

-110.04 -1.10

112.51

16.11

14.79

25.28

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

207.26326.59

348.58

307.61

165.18

271.93

367.45

275.65

257.17

280.22

416.82

338.12

323.30

345.32

304.31

161.92

268.63

384.48

292.65

286.76

297.25

August80.748

222.08

September81.396

204.50

October80.748

241.07

November81.396

387.90

December80.748

276.75

January79.391

172.01

February80.748

273.02

April80.050

286.76

May79.391

259.23

June80.050

127.10

July79.391

201.33

August79.391

216.16

September80.050

198.61

October79.391

235.14

November80.050

382.01

December79.391

270.82

January80.050

159.45

February81.396

260.43

March80.050

257.17

May80.050

 

 

May

June

159.65

83.19

68.36

88.15

49.38

-95.25

13.70

87.23 -9.18

-10.49

-25.28

134.37

57.90

43.08

62.86

24.09

-120.53

-11.59

-47.14

-143.54

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

246.68433.84

355.15

340.33

362.33

321.34

178.94

285.66

356.95

265.16

259.23

246.68

406.33

327.63

312.80

334.83

293.82

151.43

258.14

224.82

132.99

June80.703

114.53

July80.050

188.78

August80.050

203.60

September80.703

186.04

October80.050

222.59

November80.703

369.43

December80.050

258.27

January79.391

182.50

February80.748

283.51

March79.391

280.22

April80.050

297.25

June80.050

137.60

July79.391

211.82

August79.391

226.65

September80.050

209.10

October79.391

245.63

November80.050

392.50

December79.391

281.31

January80.050

319.10

February81.396

 

 

July-144.86

-159.65

-134.37

-76.46

-91.29

-71.51

-110.27

-254.90

-145.96

29.32

-67.08

-68.40

-83.19

-57.90

76.46

-14.83 4.96

-33.81

-178.44

-69.49

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0 .114

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

420.08127.10

114.53

137.60

195.50

180.68

202.68

161.69 19.28

126.01

299.05

207.26

201.33

188.78

211.82

348.43

254.90

276.92

235.92 93.53

200.23

March80.050

416.82

April80.703

433.84

May80.050

406.33

July80.050

348.43

August80.050

363.25

September80.703

345.69

October80.050

382.24

November80.703

529.09

December80.050

417.92

January79.391

240.40

February80.748

341.42

March79.391

338.12

April80.050

355.15

May79.391

327.63

June80.050

195.50

August79.391

284.55

September80.050

267.01

October79.391

303.54

November80.050

450.40

December79.391

 

 

August

Septemb

er

44.15

-52.26

-53.57

-68.36

-43.08

91.29

14.83

19.78

-18.98

-163.61

-54.67

24.37

-72.04

-73.35

-88.15

-62.86

71.51 -4.96

-19.78

-38.77

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

339.22313.88

222.08

216.16

203.60

226.65

363.25

284.55

291.75

250.74

108.35

215.06

296.33

204.50

198.61

186.04

209.10

345.69

267.01

252.18

233.20

January79.391

225.58

February80.748

326.59

March79.391

323.30

April80.050

340.33

May79.391

312.80

June80.050

180.68

July79.391

254.90

September80.050

252.18

October79.391

288.71

November80.050

435.58

December79.391

324.39

January80.050

247.60

February81.396

348.58

March80.050

345.32

April80.703

362.33

May80.050

334.83

June80.703

202.68

July80.050

276.92

August80.050

291.75

October80.050

 

 

October

Novemb

er

-183.40

-74.45

63.13

-33.27

-34.59

-49.38

-24.09

110.27

33.81

18.98

38.77

-144.63

-35.68

207.76

111.36

110.04 95.25

120.53

254.90

178.44

163.61

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0 .650

1.00

0

1.00

0

1.00

0

1.00

0 .114

1.00

0

1.00

310.7390.79

197.52

332.86

241.07

235.14

222.59

245.63

382.24

303.54

288.71

310.73

127.34

234.04

479.73

387.90

382.01

369.43

392.50

529.09

450.40

435.58

November80.703

457.58

December80.050

346.41

January79.391

206.59

February80.748

307.61

March79.391

304.31

April80.050

321.34

May79.391

293.82

June80.050

161.69

July79.391

235.92

August79.391

250.74

September80.050

233.20

November80.050

416.59

December79.391

305.41

January80.050-64.20
February81.396

165.18

March80.050

161.92

April80.703

178.94

May80.050

151.43

June80.703-19.28
July80.050-93.53
August80.050

 

 

Decemb

er

183.40

144.63

108.95 98.82

2.41

1.10

-13.70 11.59

145.96

69.49

54.67

74.45

35.68 -108.95

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

1.00

0

108.35457.58

416.59

380.91

368.54

276.75

270.82

258.27

281.31

417.92

339.22

324.39

346.41

305.41

163.02

September80.703-90.79
October80.050

127.34

December80.050

163.02

January79.391

170.91

February80.748

271.93

March79.391

268.63

April80.050

285.66

May79.391

258.14

June80.050

126.01

July79.391

200.23

August79.391

215.06

September80.050

197.52

October79.391

234.04

November80.050

380.91

Based on observed means.

The error term is Mean Square(Error) = 97695.752.

Post hoc multiple comparison tests. Once it is determined that differences exist among the means using Analysis of variance, post hoc range tests and pair wise multiple comparisons can determine which means differ. Comparisons are made on unadjusted values. These tests are used for fixed between-subjects factors only. These tests of between-subjects effects help to determine the significance of factor. Hence, from the above table shows that number of sale do not differ significantly for months, P>0.5.

Analysis of Variance (ANOVA).

Source of Variation Sum of Squares df Mean Square F Sig.
Between Groups 1399993.2111272721.3030.221
Within Groups 34584296.435497695.752

 

The differences between these months was not found statistically significant, F(11,354) = 1.303, p = .221 and we conclude that number of sale do not differ significantly with month.

5.0 Is there a difference in number of Average sales between different months of the year?

 

Multiple Comparisons
Dependent Variable: Average_Sale

Bonferroni

(I) Month of the year(J) Month of the yearMean

Difference

(I-J)

Std. ErrorSig.95% Confidence

Interval

Lower BoundUpper Bound
JanuaryFebruary.161.000-3.353.66
March-1.681.0221.000-5.151.80
April-.021.0311.000-3.523.49
May-.211.0221.000-3.693.26
June.261.0221.000-3.223.73
July-.081.0141.000-3.533.36
August-1.591.0141.000-5.041.85
Septemb

er

1.271.0221.000-2.214.74
October-.291.0221.000-3.763.18
Novembe

r

-2.431.0221.000-5.901.04
Decembe

r

-.401.0311.000-3.903.10
FebruaryJanuary-.161.0311.000-3.663.35
March-1.841.0221.000-5.311.64
April-.171.0311.000-3.683.33
May-.371.0221.000-3.843.10
June.101.0221.000-3.373.58

 

July-.241.0141.000-3.693.21
August-1.751.0141.000-5.201.69
Septemb

er

1.111.0221.000-2.364.59
October-.451.0221.000-3.923.03
Novembe

r

-2.591.022.781-6.06.89
Decembe

r

-.561.0311.000-4.062.95
MarchJanuary1.681.0221.000-1.805.15
February1.841.0221.000-1.645.31
April1.661.0221.000-1.815.14
May1.471.0141.000-1.984.91
June1.941.0141.000-1.515.38
July1.601.0061.000-1.825.01
August.081.0061.000-3.333.50
Septemb

er

2.951.014.256-.506.39
October1.391.0141.000-2.064.83
Novembe

r

-.751.000-4.202.69
Decembe

r

1.281.0221.000-2.204.75
AprilJanuary.021.0311.000-3.493.52
February.171.0311.000-3.333.68
March-1.661.0221.000-5.141.81
May-.201.0221.000-3.673.28
June.281.0221.000-3.203.75
July-.071.0141.000-3.513.38
August-1.581.0141.000-5.021.87
Septemb

er

1.291.0221.000-2.194.76
October-.271.0221.000-3.753.20
Novembe

r

-2.411.0221.000-5.891.06
Decembe

r

-.381.0311.000-3.893.12
MayJanuary.211.0221.000-3.263.69

 

February.371.0221.000-3.103.84
March-1.471.0141.000-4.911.98
April.201.0221.000-3.283.67
June.471.0141.000-2.973.92
July.131.0061.000-3.293.55
August-1.381.0061.000-4.802.04
Septemb

er

1.481.0141.000-1.964.93
October-.081.0141.000-3.523.37
Novembe

r

-2.221.0141.000-5.661.23
Decembe

r

-.191.0221.000-3.663.29
JuneJanuary-.261.0221.000-3.733.22
February-.101.0221.000-3.583.37
March-1.941.0141.000-5.381.51
April-.281.0221.000-3.753.20
May-.471.0141.000-3.922.97
July-.341.0061.000-3.763.08
August-1.851.000-5.271.56
Septemb

er

1.011.0141.000-2.434.46
October-.551.0141.000-3.992.90
Novembe

r

-2.691.014.553-6.13.76
Decembe

r

-.661.0221.000-4.132.82
JulyJanuary.081.0141.000-3.363.53
February.241.0141.000-3.213.69
March-1.601.0061.000-5.011.82
April.071.0141.000-3.383.51
May-.131.0061.000-3.553.29
June.341.0061.000-3.083.76
August-1.51.9971.000-4.901.88
Septemb

er

1.351.0061.000-2.074.77
October-.211.0061.000-3.623.21
Novembe-2.351.0061.000-5.761.07

 

r
Decembe

r

-.321.0141.000-3.763.13
AugustJanuary1.591.0141.000-1.855.04
February1.751.0141.000-1.695.20
March-.081.0061.000-3.503.33
April1.581.0141.000-1.875.02
May1.381.0061.000-2.044.80
June1.851.0061.000-1.565.27
July1.51.9971.000-1.884.90
Septemb

er

2.861.006.308-.556.28
October1.301.0061.000-2.114.72
Novembe

r

-.841.0061.000-4.252.58
Decembe

r

1.191.0141.000-2.254.64
SeptemberJanuary-1.271.0221.000-4.742.21
February-1.111.0221.000-4.592.36
March-2.95.256-6.39.50
April-1.291.0221.000-4.762.19
May-1.481.0141.000-4.931.96
June-1.011.0141.000-4.462.43
July-1.351.0061.000-4.772.07
August-2.861.006.308-6.28.55
October-1.561.0141.000-5.001.89
Novembe

r

-3.70*1.014.020-7.14-.25
Decembe

r

-1.671.0221.000-5.141.81
OctoberJanuary.291.0221.000-3.183.76
February.451.0221.000-3.033.92
March-1.391.0141.000-4.832.06
April.271.0221.000-3.203.75
May.081.0141.000-3.373.52
June.551.0141.000-2.903.99
July.211.0061.000-3.213.62
August-1.301.0061.000-4.722.11
Septemb

er

1.561.0141.000-1.895.00
Novembe

r

-2.141.0141.000-5.591.30
Decembe

r

-.111.0221.000-3.583.36
NovemberJanuary2.431.0221.000-1.045.90
February2.591.022.781-.896.06
March.751.0141.000-2.694.20
April2.411.0221.000-1.065.89
May2.221.0141.000-1.235.66
June2.691.014.553-.766.13
July2.351.0061.000-1.075.76
August.841.0061.000-2.584.25
Septemb

er

3.70*1.014.020.257.14
October2.141.0141.000-1.305.59
Decembe

r

2.031.0221.000-1.445.50
DecemberJanuary.401.000-3.103.90
February.561.0311.000-2.954.06
March-1.281.0221.000-4.752.20
April.381.0311.000-3.123.89
May.191.0221.000-3.293.66
June.661.0221.000-2.824.13
July.321.0141.000-3.133.76
August-1.191.0141.000-4.642.25
Septemb

er

1.671.0221.000-1.815.14
October.111.0221.000-3.363.58
Novembe

r

-2.031.0221.000-5.501.44
Based on observed means.

The error term is Mean Square(Error) = 15.416.

*. The mean difference is significant at the 0.05 level.

Analysis of variance shows average sale is significantly different for months, now by using Post hoc multiple comparison Bonferroni tests, pair wise multiple comparisons can determine which means differ. Hence, from the above table shows that average sale do not differ significantly for months, P>0.5.

 

Boxplot showing month-wise distribution of average sale.

 

H1: Average Sale of month differ significantly and unequal.

Analysis of Variance (ANOVA)

Source of Variation Sum of Squares df Mean Square F Sig.
Between Groups 335.6511130.5141.9790.030
Within Groups 5333.83134615.416

The differences in average sale between these months was statistically significant, F(11,346) = 1.979, p = .030 and we conclude that average sale differ significantly with month.

6.0 Is there a correlation between rain days and Gross Profit?

There is no relationship between Rainfall and Gross profit r=0.008, P=0.885 and correlation is found to be insignificant and very weak.

 

RainfallGross Profit
RainFall10.008*
Gross Profit0.008*1

P=0.885

 

Scatter plot between RainFall and Gross Profit

 

 

Multiple Comparisons
Dependent Variable: Net_Sales

Bonferroni

(I)

Season of the year

Summe

r

(J) Mean Std. Sig.

Season Difference Erro of the (I-J) r year

95% Confidence

Interval

Lower BoundUpper Bound
Autumn -34.62 46.2 P>0.0

97 5

-157.44 88.19
AutumnWinter55.0046.2

97

P>0.05

P>0.05

P>0.05

P>0.05

-67.82177.82
Spring -33.65 46.4

23

-156.80 89.50
Summer34.6246.2

97

-88.19157.44
Winter 89.62 46.1 -32.86 212.10
70
Spring.9846.2

97

P>0.05

P>0.05

P>0.05

P>0.05

P>0.05

P>0.05

P>0.05

-121.84123.79
Winter Summer -55.0046.2

97

-177.8267.82
Spring
Autumn-89.6246.1

70

-212.1032.86
Spring -88.65 46.2

97

-211.4734.17
Summer33.6546.4

23

-89.50156.80
Autumn -.98 46.2

97

-123.79121.84
Winter88.6546.2

97

-34.17211.47
Based on observed means.

The error term is Mean Square(Error) = 98056.709.

 

Post hoc multiple comparison test reveal that there is no significant difference in average sale across the categories of months, Bonferoni test is employed for multiple comparison. Statistical significance is tested at 5% level of significance. Results are not statistically significant here we fail to reject the null hypothesis.

 

Boxplot showing distribution of month wise number of sale

 

 

 

Hypothesis:

H0: Mean average sale equal for all groups (i.e., µ1 = µ2 = µ3 = … = µ12)

(ANOVA TABLE)

Source of Variation Sum of Squares df Mean Square F Sig.
Between Groups 855.9403285.3131.1380.334
Within Groups 90773.03362250.754

Results of the analysis of variance reveals that there is statistically significant difference were observed in determining the number of sale across the months, F(3,362) = 1.138, p = .334 and we conclude that number of sale do not differ significantly with month.

Multiple Comparisons
Dependent Variable: Average_Sale

Bonferroni

(I) Season of the year(J) Season of the yearMean

Differen ce (I-J)

Std. ErrorSig.95% Confidence

Interval

Lower BoundUpper Bound
SummerAutumn-.56.6031.00 0-2.161.04
Winter-.40.5981.00 0-1.991.19
Spring-.40.6011.00 0-2.001.19
AutumnSummer.56.6031.00 0-1.042.16
Winter.16.5941.00 0-1.411.74
Spring.16.5971.00 0-1.431.74

1.0 Is there a difference in number of Average sales between different Seasons?

WinterSummer.40.5981.00 0-1.191.99
Autumn-.16.5941.00 0-1.741.41
Spring.00.5931.00 0-1.571.57
SpringSummer.40.6011.00 0-1.192.00
Autumn-.16.5971.00 0-1.741.43
Winter.00.5931.00 0-1.571.57
Based on observed means.

The error term is Mean Square(Error) = 15.973.

 

Multiple comparison using bonferroni post hoc test was conducted the results shows that the average sale do not differ significantly among the season of the year from Jan-Dec all p values are greater than 5%. Hence comparison shows with respect to season shows the insignificant results.

 

Descriptive statistics for season-wise average sale.

Season of

the year

Statistic
Summer Mean18.18
SD3.5
Autumn Mean18.74
SD18.74
Winter Mean18.58
SD5.3
Spring Mean18.58
SD3.9

 

 

Boxplot Showing distribution of average sale for season.

 

 

Source of Variation Sum of Squares Df Mean Square F Sig.
Between Groups 15.14835.0490.3160.814
Within Groups 5654.33435415.973
Total 5669.483357

(ANOVA TABLE)

Using analysis of variance determine whether there are any statistically significant differences between the means of two or more Season. The differences between these season was not statistically significant, F(3,354) = 0.316, p = .814 and we conclude that Average sale do not differ significantly with season.

9.0 Is there a difference in number of Gross Profit between different Seasons?

Post Hoc tests of Multiple comparision.

Multiple Comparisons
Dependent Variable: Gross_Sales

Bonferroni

(I) Season of the year(J)

Season of the year

Mean

Differenc e (I-J)

Std. ErrorSig.95% Confidence Interval
Lower BoundUpper Bound
SummerAutumn-22.9448.08 9P>0.05-150.51104.63
Winter58.7848.08 9P>0.05-68.79186.36
Spring-46.4448.22 0P>0.05-174.3681.48
AutumnSumme

r

22.9448.08 9P>0.05-104.63150.51
Winter81.7247.95 7P>0.05-45.50208.95
Spring-23.5048.08 9P>0.05-151.08104.07
WinterSumme

r

-58.7848.08 9P>0.05-186.3668.79
Autumn-81.7247.95 7P>0.05-208.9545.50
Spring-105.2348.08 9P>0.05-232.8022.35
SpringSumme

r

46.4448.22 0P>0.05-81.48174.36
Autumn23.5048.08 9P>0.05-104.07151.08
Winter105.2348.08 9P>0.05-22.35232.80
Based on observed means.

The error term is Mean Square(Error) = 105796.319.

By using Post hoc tests it is found that gross profit do not differ significantly across the categories of season when multiple comparisons is employed, Bonferoni test is used for multiple comparison.

Statistical significance is tested at 5% level of significance all P Values are found to be greater than 0.5

 

Descriptive statistics for Gross profit of different season.

 

Season of the year Statistic SE
Summer Mean31.42
SD31.67
Autumn Mean19.73
SD16.62
Winter Mean27.64
SD18.72
Springs Mean44.211
SD41.36

(ANOVA TABLE)

Source of Variation Sum of Squares Df Mean Square F Sig.
Between Groups 28591.75739530.58611.4560.000
Within Groups 301149.197362831.904
Total 329740.954365

There is significant difference in mean gross profit across the season, F (3,362) = 11.456, p = .000 and we conclude that Gross profit differ significantly.

 

Boxplot depicts the distribution of data and median values and outliers indicated by asterisks pertaining to gross profit

 

 

Results:

Following hypothesis was tested and these hypotheses along with significant/insignificant findings are stated below.

HypothesisSupported/ Not SupportedP Value
Whether there is a difference in number of sales between different months of the year.

 

Not SupportedP.0.05
Whether a difference in number of average sales between different months of the year.

 

SupportedP<0.05
Whether there is a difference in number of gross profit between different months of the year

 

SupportedP<0.05
To study relationship and correlation between rainy days and Gross Profit

 

Correlation is weakP>0.05

(r = 0.08)

Whether there is a difference in number of sales between different seasons

 

Not SupportedP>0.05
Whether there is a difference in number of average sales between different seasons.

 

Not supportedP>0.05

 

 

 

 

 

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