1. Time Series ka Parichay
Time series ek sequence hai observations ki, jo time ke saath record ki gayi hoti hai. Sales data, temperature readings, stock prices — sab time series hain.
Y = f(T, S, C, I)
T = Trend Component (long-term movement)
S = Seasonal Component (recurring patterns)
C = Cyclical Component (business cycles, >1 year)
I = Irregular Component (random fluctuations)
1.1 Additive Model vs Multiplicative Model
Additive: Y = T + S + C + I (constant variation)
Multiplicative: Y = T × S × C × I (percentage variation, more common)
2. Trend Analysis
2.1 Moving Average Method
Raw data ko smooth karta hai seasonal fluctuations remove karke.
3-year MA at time t = (Yₜ₋₁ + Yₜ + Yₜ₊₁) / 3
4-year Centered MA needs two steps (even period ke liye)
Udaharan: 3-Year Moving Average
| Year | Sales (Y) | 3-yr MA |
|---|
| 2018 | 120 | — |
| 2019 | 130 | (120+130+150)/3 = 133.3 |
| 2020 | 150 | (130+150+140)/3 = 140.0 |
| 2021 | 140 | (150+140+180)/3 = 156.7 |
| 2022 | 180 | — |
2.2 Least Squares Trend Line
Y = a + bt (linear trend)
b = [nΣtY − ΣtΣY] / [nΣt² − (Σt)²]
a = Ȳ − b·t̄
(Same as regression — time t is X variable)
Udaharan: Trend Fitting
Sales data (t=1 to 5): 10, 14, 17, 21, 26
Σt=15, ΣY=88, Σt²=55, ΣtY=293, n=5
b = (5×293 − 15×88)/(5×55 − 225) = (1465−1320)/50 = 2.9
a = 88/5 − 2.9×15/5 = 17.6 − 8.7 = 8.9
Trend: Ŷ = 8.9 + 2.9t
Forecast t=6: Ŷ = 8.9 + 17.4 = 26.3
3. Seasonal Indices
Seasonal Index (SI) = Average of Season / Grand Average × 100
SI > 100 → above average season
SI < 100 → below average season
Σ(quarterly SI) = 400, Σ(monthly SI) = 1200
Udaharan: Quarterly Seasonal Index
Quarterly averages: Q1=80, Q2=120, Q3=100, Q4=100
Grand Average = (80+120+100+100)/4 = 100
SI Q1 = 80/100×100 = 80
SI Q2 = 120/100×100 = 120
Sum = 80+120+100+100 = 400 ✓