© 2026 Dr. Debashis Chatterjee.
All rights reserved.

This laboratory manual is prepared exclusively for educational use.

1 Lab 0: Software Setup and Data Handling

1.1 Objective

Configure the R environment and explore datasets.

1.2 Dataset Visualization

par(mfrow = c(1,3), mar = c(4,4,3,1))

plot(AirPassengers, col="dodgerblue", lwd=2,
     main="AirPassengers\n(Trend & Seasonality)")

plot(uspop, col="firebrick", lwd=2,
     main="US Population\n(Growth Curve)")

plot(nhtemp, col="darkgreen", lwd=2,
     main="NH Temperature\n(Stationary)")

par(mfrow=c(1,1))

2 Lab 1: Modified Exponential Curve Fitting

2.1 Theory

\[ Y_t = a b^{t} \]

Taking logs:

\[ \ln(Y_t) = \ln(a) + t\ln(b) \]

2.2 Implementation

y_exp <- as.numeric(uspop)[1:10]
t_exp <- 1:length(y_exp)

fit_exp <- lm(log(y_exp) ~ t_exp)

a <- exp(coef(fit_exp)[1])
b <- exp(coef(fit_exp)[2])

fitted_exp <- a * b^t_exp

plot(t_exp, y_exp, type="o", pch=16, col="black",
     main="Modified Exponential Fit (US Pop 1790–1880)",
     xlab="Decade Index", ylab="Population (Millions)")
lines(t_exp, fitted_exp, col="darkorange", lwd=3)

legend("topleft",
       legend=c("Actual","Fitted"),
       col=c("black","darkorange"),
       lty=1, lwd=c(1,3), pch=c(16,NA))

summary(fit_exp)
## 
## Call:
## lm(formula = log(y_exp) ~ t_exp)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.060499 -0.012490  0.008677  0.012272  0.043371 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.111133   0.021312   52.14 2.03e-11 ***
## t_exp       0.286538   0.003435   83.42 4.76e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0312 on 8 degrees of freedom
## Multiple R-squared:  0.9989, Adjusted R-squared:  0.9987 
## F-statistic:  6959 on 1 and 8 DF,  p-value: 4.755e-13

3 Lab 2: Gompertz Curve Fitting

\[ Y_t = K \exp(-A e^{-bt}) \]

y_pop <- as.numeric(uspop)
t_pop <- 1:length(y_pop)

gompertz_model <- nls(
  y_pop ~ K * exp(-A * exp(-b * t_pop)),
  start=list(K=400, A=4, b=0.15)
)

plot(t_pop, y_pop, pch=16,
     main="Gompertz Fit",
     xlab="Decade Index", ylab="Population")
lines(t_pop, predict(gompertz_model),
      col="dodgerblue", lwd=3)

4 Lab 3: Logistic Curve Fitting

\[ Y_t = \frac{K}{1 + A e^{-bt}} \]

logistic_model <- nls(
  y_pop ~ K/(1 + A*exp(-b*t_pop)),
  start=list(K=400, A=100, b=0.25)
)

plot(t_pop, y_pop, pch=16,
     main="Gompertz vs Logistic",
     xlab="Decade Index", ylab="Population")

lines(t_pop, predict(gompertz_model),
      col="blue", lwd=2)

lines(t_pop, predict(logistic_model),
      col="red", lwd=2, lty=2)

legend("topleft",
       legend=c("Actual","Gompertz","Logistic"),
       col=c("black","blue","red"),
       pch=c(16,NA,NA),
       lty=c(NA,1,2), lwd=2)

5 Lab 4: Trend by Moving Averages

trend_ma <- ma(AirPassengers, order=12, centre=TRUE)

plot(AirPassengers, col="gray60",
     main="12-Month Centered Moving Average",
     ylab="Passengers")
lines(trend_ma, col="red", lwd=3)

6 Lab 5: Seasonal Indices (Ratio-to-Trend)

t_air <- 1:length(AirPassengers)
fit_trend <- lm(log(AirPassengers) ~ t_air)
math_trend <- ts(exp(fitted(fit_trend)),
                 start=start(AirPassengers),
                 frequency=12)

ratio_to_trend <- AirPassengers / math_trend

si_raw <- tapply(ratio_to_trend,
                 cycle(AirPassengers),
                 mean)

si_trend <- si_raw / mean(si_raw)

kable(data.frame(Month=month.abb,
                 Index=round(si_trend,3)))
Month Index
Jan 0.910
Feb 0.891
Mar 1.016
Apr 0.984
May 0.982
Jun 1.109
Jul 1.230
Aug 1.219
Sep 1.054
Oct 0.918
Nov 0.796
Dec 0.892

7 Lab 6: Seasonal Indices (Ratio-to-MA)

ratio_ma <- AirPassengers / trend_ma

si_ma_raw <- tapply(ratio_ma,
                    cycle(AirPassengers),
                    mean, na.rm=TRUE)

si_ma <- si_ma_raw / mean(si_ma_raw)

barplot(si_ma, names.arg=month.abb,
        col="steelblue",
        main="Seasonal Indices (Ratio-to-MA)")
abline(h=1, col="red", lwd=2, lty=2)

9 Lab 8: AR/MA Model Fitting

par(mfrow=c(1,2))
acf(nhtemp, col="blue")
pacf(nhtemp, col="red")

par(mfrow=c(1,1))

fit_arma <- arima(nhtemp, order=c(1,0,1))
kable(as.data.frame(coef(fit_arma)))
coef(fit_arma)
ar1 0.9150693
ma1 -0.7088387
intercept 51.1689524

10 Lab 9: Exponential Smoothing

fit_ses <- ses(nhtemp, h=10)

plot(fit_ses,
     main="Simple Exponential Smoothing",
     ylab="Temperature")

lines(fitted(fit_ses),
      col="red", lty=2)

11 Lab 10: Short-Term Forecasting (Box–Jenkins / Brown)

fit_auto <- auto.arima(AirPassengers)
forecast_auto <- forecast(fit_auto, h=24)

plot(forecast_auto,
     main="Auto ARIMA Forecast",
     ylab="Passengers")

fit_holt <- holt(AirPassengers, h=24)

plot(fit_holt,
     main="Holt's Linear Trend Forecast",
     ylab="Passengers")

12 Session Information

sessionInfo()
## R version 4.4.0 (2024-04-24 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 11 x64 (build 26100)
## 
## Matrix products: default
## 
## 
## locale:
## [1] LC_COLLATE=English_United States.utf8 
## [2] LC_CTYPE=English_United States.utf8   
## [3] LC_MONETARY=English_United States.utf8
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.utf8    
## 
## time zone: Asia/Calcutta
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] knitr_1.51      tseries_0.10-60 forecast_9.0.1 
## 
## loaded via a namespace (and not attached):
##  [1] sass_0.4.10        generics_0.1.4     lattice_0.22-9     digest_0.6.35     
##  [5] magrittr_2.0.3     evaluate_1.0.5     grid_4.4.0         RColorBrewer_1.1-3
##  [9] fastmap_1.2.0      jsonlite_1.8.8     scales_1.4.0       jquerylib_0.1.4   
## [13] cli_3.6.2          rlang_1.1.4        cachem_1.1.0       yaml_2.3.12       
## [17] otel_0.2.0         tools_4.4.0        parallel_4.4.0     dplyr_1.1.4       
## [21] colorspace_2.1-1   ggplot2_4.0.2      curl_5.2.1         vctrs_0.6.5       
## [25] R6_2.6.1           zoo_1.8-15         lifecycle_1.0.5    pkgconfig_2.0.3   
## [29] urca_1.3-4         pillar_1.11.1      bslib_0.10.0       gtable_0.3.6      
## [33] glue_1.7.0         quantmod_0.4.28    Rcpp_1.0.12        xfun_0.56         
## [37] tibble_3.2.1       tidyselect_1.2.1   rstudioapi_0.18.0  dichromat_2.0-0.1 
## [41] farver_2.1.2       htmltools_0.5.9    nlme_3.1-168       rmarkdown_2.30    
## [45] xts_0.14.1         timeDate_4052.112  fracdiff_1.5-3     compiler_4.4.0    
## [49] S7_0.2.0           quadprog_1.5-8     TTR_0.24.4