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# Importing the data
XUVData = as.data.frame(XUV_company_annual_profit)
head(XUVData)

NA

str(XUVData)

'data.frame':   13 obs. of  2 variables:
 $ Year                     : num  2006 2007 2008 2009 2010 ...
 $ Annual Profit (in Crores): num  88 92 95 95 90 85 98 95 108 118 ...

# assigning the data
t = seq(2006,2018,1)
t

 [1] 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018

#computing n
n = length(xt)
n

[1] 13

#shifting the origin and scale
xt = (t-2012)/1
xt

 [1] -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6

#Assigning profit data
yt=XUVData$`Annual Profit (in Crores)`
yt

 [1]  88  92  95  95  90  85  98  95 108 118 119 117 130

#Fitting linear trend to the data
linear_model<-lm(formula = yt~xt)
linear_model

Call:
lm(formula = yt ~ xt)

Coefficients:
(Intercept)           xt  
    102.308        3.231  

#summary of the fitting linear model
summary(linear_model)

Call:
lm(formula = yt ~ xt)

Residuals:
    Min      1Q  Median      3Q     Max 
-14.077  -4.308   2.385   5.615   8.308 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 102.3077     2.0158  50.753 2.14e-14 ***
xt            3.2308     0.5387   5.997 8.97e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.268 on 11 degrees of freedom
Multiple R-squared:  0.7658,    Adjusted R-squared:  0.7445 
F-statistic: 35.96 on 1 and 11 DF,  p-value: 8.967e-05

#Extracting the estimates for linear trend 
lm(yt~xt)[['coefficients']]

(Intercept)          xt 
 102.307692    3.230769 

#computing the fitted values for linear trend 
fittedlinyt=lm(yt~xt)[['fitted.values']]
fittedlinyt

        1         2         3         4         5         6         7         8         9 
 82.92308  86.15385  89.38462  92.61538  95.84615  99.07692 102.30769 105.53846 108.76923 
       10        11        12        13 
112.00000 115.23077 118.46154 121.69231 

#Fitting quadratic trend to the data
quadratic_model<-lm(yt~xt+I(xt^2))
quadratic_model

Call:
lm(formula = yt ~ xt + I(xt^2))

Coefficients:
(Intercept)           xt      I(xt^2)  
    96.9510       3.2308       0.3826  

#summary of the fitting linear model
summary(quadratic_model)

Call:
lm(formula = yt ~ xt + I(xt^2))

Residuals:
   Min     1Q Median     3Q    Max 
-9.103 -3.341  1.049  3.057  7.913 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  96.9510     2.2438  43.208 1.06e-12 ***
xt            3.2308     0.3978   8.122 1.03e-05 ***
I(xt^2)       0.3826     0.1199   3.190  0.00965 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.366 on 10 degrees of freedom
Multiple R-squared:  0.8839,    Adjusted R-squared:  0.8607 
F-statistic: 38.07 on 2 and 10 DF,  p-value: 2.108e-05

#Extracting the estimates for quadratic trend 
lm(yt~xt+I(xt^2))[['coefficients']]

(Intercept)          xt     I(xt^2) 
 96.9510490   3.2307692   0.3826174 

#computing the fitted values for quadratic trend 
fittedquadyt=lm(yt~xt+I(xt^2))[['fitted.values']]
fittedquadyt

        1         2         3         4         5         6         7         8         9 
 91.34066  90.36264  90.14985  90.70230  92.01998  94.10290  96.95105 100.56444 104.94306 
       10        11        12        13 
110.08691 115.99600 122.67033 130.10989 

#appropriate trend for this data. 
#The appropriate trend for this data is quadratic trends because its fitting trend has shown a high value of Adjusted R-squared than linear trend which indicates the best fit.

#plotting the actual values with the fitted values
plot(t,yt,type='l', lty=1, col='black',xlim=c(2006,2018),
     lwd=2,main = 'Fitting ofLinear and quadratic trend')
lines(t,fittedlinyt, type = 'l',lty=2, col='blue', lwd=2)

lines(t,fittedquadyt, type = 'l',lty=2, col='red', lwd=2)
legend('topleft', lty=c(1,2),
       col = c('black','blue','red'),c('original','Linear fitted','Quadratic fitted'))

#The fitted values corresponding to the appropriate trend onlyfittedquadyt=lm(yt~xt+I(xt^2))[['fitted.values']]
fittedquadyt

        1         2         3         4         5         6         7         8         9 
 91.34066  90.36264  90.14985  90.70230  92.01998  94.10290  96.95105 100.56444 104.94306 
       10        11        12        13 
110.08691 115.99600 122.67033 130.10989 

#changing the origin and scale 2019
Quad_projxt = (2019-2012)/1
Quad_projxt

[1] 7

# Computing projected profit for year 2019
projected_profit= 96.9510490+(3.2307692*Quad_projxt)+(0.3826174*Quad_projxt^2)
projected_profit

[1] 138.3147

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