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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:

Ŷ_i = b₀ + b₁X_i

Ŷ_i = Predicted revenue
X_i = Time coding
b₀ = Base year trend
b₁ = Annual change

Forecast future values using the same formula.

Ŷ_i+1 = b₀ + b₁X_i+1

Quadratic Trend Model:

Ŷ_i = b₀ + b₁X_i + b₂X_i²

Ŷ_i, X_i, X_i², b₀, b₁, b₂ explained here

Use the equation to forecast future values.

Ŷ_i+1 = b₀ + b₁X_i+1 + b₂X²_i+1

Exponential Trend Model:

Log (Ŷ_i) = b₀ + b₁X_i

Includes logarithmic transformation of the model

β₀ = antilog(b₀)

β₁ = antilog(b₁)

Ŷ_i = (β₀) (β₁)^X

Forecast = Ŷ_i+1 = (β₀) (β₁)^X+1

β₁×100% = Annual compound growth rate

Implementation of Trend Fitting Models using Python:

In [25]:

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

# Load the input files

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

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"])

eastmank_revenue.head()

Out [26]:

Year	Real_Revenue	coded_year	coded_year_square	log_Real_Revenue