Time series analysis numerical

1. TIME SERIES ANALYSIS
NUMERICAL

2. Find the trend in sales.
Estimate the sales for 2018
Year 2012 2013 2014 2015 2016
Sales (Rs. In Lacs) 12 13 15 17 13

3. 𝑇ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑙𝑖𝑛𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦:
𝑦 = 𝑎 + 𝑏𝑥
𝑤ℎ𝑒𝑟𝑒 𝑦 = 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒
𝑎 = 𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝑏 = 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡𝑟𝑒𝑛𝑑 𝑙𝑖𝑛𝑒
𝑥 = 𝑡𝑖𝑚𝑒

4. 𝑇ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑏𝑒𝑙𝑜𝑤:
𝑁𝑜𝑡𝑒: 𝑦𝑒𝑎𝑟 2014 𝑡𝑎𝑘𝑒𝑛 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛
𝑌𝑒𝑎𝑟 𝑇𝑖𝑚𝑒 𝑆𝑐𝑎𝑙𝑒
𝑥
2012 −2
2013 −1
2014 0
2015 1
2016 2

5. 𝑇ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑏𝑒𝑙𝑜𝑤:
𝑁𝑜𝑡𝑒: 𝑦𝑒𝑎𝑟 2014 𝑡𝑎𝑘𝑒𝑛 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛
𝑌𝑒𝑎𝑟 𝑇𝑖𝑚𝑒 𝑆𝑐𝑎𝑙𝑒
𝑥
𝑆𝑎𝑙𝑒𝑠
𝑦
2012 −2 12
2013 −1 13
2014 0 15
2015 1 17
2016 2 13
Σ𝑦 = 70

6. 𝑇ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑏𝑒𝑙𝑜𝑤:
𝑁𝑜𝑡𝑒: 𝑦𝑒𝑎𝑟 2014 𝑡𝑎𝑘𝑒𝑛 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛
𝑌𝑒𝑎𝑟 𝑇𝑖𝑚𝑒 𝑆𝑐𝑎𝑙𝑒
𝑥
𝑆𝑎𝑙𝑒𝑠
𝑦
𝑥2
2012 −2 12 4
2013 −1 13 1
2014 0 15 0
2015 1 17 1
2016 2 13 4
Σ𝑦 = 70 Σ𝑥2
= 10

7. 𝑇ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑏𝑒𝑙𝑜𝑤:
𝑁𝑜𝑡𝑒: 𝑦𝑒𝑎𝑟 2008.5 𝑡𝑎𝑘𝑒𝑛 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛
𝑌𝑒𝑎𝑟 𝑇𝑖𝑚𝑒 𝑆𝑐𝑎𝑙𝑒
𝑥
𝑆𝑎𝑙𝑒𝑠
𝑦
𝑥2
𝑥𝑦
2012 −2 12 4 −24
2013 −1 13 1 −13
2014 0 15 0 0
2015 1 17 1 17
2016 2 13 4 26
Σ𝑦 = 70 Σ𝑥2
= 10 Σx𝑦 = 6

8. 𝑇ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑙𝑖𝑛𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦:
𝑦 = 𝑎 + 𝑏𝑥
𝑤ℎ𝑒𝑟𝑒 𝑦 = 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒
𝑎 = 𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝑏 = 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡𝑟𝑒𝑛𝑑 𝑙𝑖𝑛𝑒
𝑥 = 𝑡𝑖𝑚𝑒
𝑦 =
Σ𝑦
𝑁
=
70
5
= 14
𝑎 = 𝑦 = 14
𝑏 =
Σx𝑦
Σ𝑥2
=
6
10
= 0.6

9. 𝑇ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑏𝑒𝑙𝑜𝑤:
𝑦𝑛 = 𝑎 + 𝑏𝑥 𝑛
𝑇𝑟𝑒𝑛𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛:
𝑦 = 𝑎 + 𝑏𝑥
𝑜𝑟 𝑦 = 14 + 0.6𝑥
∴ 𝑇𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 2012:
𝑦1 = 14 + 0.6𝑥1 = 14 + 0.6 × −2 = 14 − 1.2 = 12.8
𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦,
𝑦2 = 14 + 0.6 × −1 = 13.4
𝑦3 = 14 + 0.6 × 0 = 14
𝑦4 = 14 + 0.6 × 1 = 14.6
𝑦5 = 14 + 0.6 × 2 = 15.2

10. 𝑇ℎ𝑒 𝑡𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑏𝑒𝑙𝑜𝑤:
𝑁𝑜𝑡𝑒: 𝑦𝑒𝑎𝑟 2014 𝑡𝑎𝑘𝑒𝑛 𝑎𝑠 𝑜𝑟𝑖𝑔𝑖𝑛
𝑌𝑒𝑎𝑟 𝑇𝑖𝑚𝑒 𝑆𝑐𝑎𝑙𝑒
𝑥
𝑆𝑎𝑙𝑒𝑠
𝑦
𝑥2
𝑥𝑦 𝑇𝑟𝑒𝑛𝑑 𝑉𝑎𝑙𝑢𝑒
( 𝑦)
2012 −2 12 4 −24 12.8
2013 −1 13 1 −13 13.4
2014 0 15 0 0 14
2015 1 17 1 17 14.6
2016 2 13 4 26 15.2
Σ𝑦 = 70 Σ𝑥2
= 10 Σx𝑦 = 6

11. 𝑇𝑟𝑒𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 2018:
𝑦 = 𝑎 + 𝑏𝑥
𝑎 = 14, 𝑏 = 0.6, 𝑥 = 4
∴ 𝑦 = 14 + 0.6 × 4 = 14 + 2.4 = 16.4
∴ 𝑡ℎ𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑠𝑎𝑙𝑒𝑠 𝑓𝑜𝑟 2018 𝑖𝑠 𝑅𝑠. 16.4 𝐿𝑎𝑐𝑠