# Trend Fitting Models

Linear Trend, Quadratic Trend And Exponential Trend:

The component factor of a time series often studied is Trend. If you plot the data (say revenue) on vertical axis and time (say year) on horizontal axis, you will come to now the different trend (linear, quadratic and exponential) present in the series.

Parameters of the trend fitting models are estimated by using method of least square.

Here, we will model the revenue as a function of time based on different trend.

Linear Trend Model:

Ŷi = b0 + b1Xi

Ŷi = Predicted value for revenue at time i
Xi = 0 for the first year, 1 for the second year, 3 for the third year and so on
b0 = Fitted trend value reflecting the predicted mean during the base year or first year (0)
b1 = Average change in Ŷi per year

You can also use the above equation to make forecast of any future vales.

Ŷi+1 = b0 + b1Xi+1

Ŷi = b0 + b1Xi + b2Xi2

Ŷi = Predicted value for revenue at time i
Xi = 0 for the first year, 1 for the second year, 3 for the third year and so on
b0 = Fitted trend value reflecting the predicted mean during the base year or first year (0)
b1 = Average change in Ŷi per year
b2 = Average change in Ŷi due to quadratic trend effect

You can also use the above equation to make forecast of any future vales.

Ŷi+1 = b0 + b1Xi+1 + b2X2i+1

Exponential Trend Model:

Log (Ŷi) = b0 + b1Xi

Here, it includes Log so our parameters (b0, b1) are on log scale. So,

β0 = antilog(b0)
β1 = antilog(b1)

Hence,

Ŷi = (β0) (β1) X
Forecast = Ŷi+1 = (β0) (β1) X+1

In above equations, β1×100% represents the estimated annual compound growth rate (in %)

In [25]:

Trend Fitting Models

import numpy as np
import pandas as pd import matplotlib.pyplot as plt
import seaborn as sns import math

eastmank_revenue = pd.read_excel ("./Data/EASTMANK.XLS", sheet_name = "Data") [[ "Year" , "Real Revenue"]]
eastmank_revenue = eastmank_revenue.rename (columns = { "Real Revenue" : "Real_Revenue" }) eastmank_revenue.head()

Out [25]:

Year Real_Revenue
0 1975 9.3
1 1976 9.5
2 1977 9.9
3 1978 10.7
4 1979 11.0
Implementtion of Trend Fitting Models using Python

In [26]:

# Create coded year, quadratic and exponential variables

eastmank_revenue["coded_year"] = eastmank_revenue.index
eastmank_revenue["coded_year_square"] = eastmank_revenue["coded_year"] ** 2
eastmank_revenue["log_Real_Revenue"] = np.log (eastmank_revenue)["Real_Revenue"]

Out [26]:

Year Real_Revenue coded_year coded_year_square log_Real_Revenue

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