INDR $372-$ Spring $2024$ Fikri Karaesmen

HOMEWORK $1,$ Due Date: March $18,2024$

$$ Please work in groups of two or individually and submit one file

for each group with both names.

$$ For this homework please perform all computations in a python notebook. Please don$$t use forecasting packages or functions, you are expected to implement your own forecasts. Please submit one python

notebook file that clearly shows all the computations.

$$ In addition to the the notebook, submit a short typed summary

report that includes the results $($error tables, prediction intervals etc.$)$

of all exercises. Also add a general assessment of the methods $($which

method is the best, which should be avoided etc.$).$ The report is

part of the overall grade and must be written and formatted

clearly.

Exercises

$1.$ Forecasting sales of Renault vehicles. The data file contains the monthly

domestic sales of total monthly sales of Renault brand cars in Turkey

from the beginning of $2013$ to the end of $2022($found in the blackboard

page$).$

$($a$)$ Plot the data and visually assess whether there is significant trend

and seasonality.

$($b$)$ To obtain a benchmark for errors, implement the following naive

forecasts i$)$ Ft $=$ Dt$1$ for t $>1$ and ii$)$ Ft $=$ Dt$12$ for t $>12.$

Report the Mean Absolute Error $($MAE$),$ Mean Absolute Percentage Error $($MAPE$)$ and Root Mean Squared Error $($RMSE$)$

of these forecasts for years $2019$ until the end of $2022.$ These error

measures constitute a simple benchmark for all other approaches

$($i$.$e$.$ hopefully you will obtain lower errors by more sophisticated

methods$).$ Note that different methods require different initialization periods. To be consistent, we start forecasting as early

as possible in $2013$ but start comparing the errors from January

$2019$ onwards.

$1$

$($c$)$ Use a $3-$period moving average to forecast the one month$-$ahead

monthly demand. Report the MAE, MAPE and RMSE of the

forecast for years $2019$ through $2022.$ Report $90$ percent prediction intervals $($using the RMSE estimated in years $2014$ to $2018)$

for the one$-$month ahead forecasts for year $2022.$

$($d$)$ Comment on the residual diagnostics $($i$.$e$.$ independence and normality of residuals$).$ What is the drawback of this forecast with

respect to this data?

$($e$)$ Use exponential smoothing to forecast the one period ahead monthly

demand. Experiment with at least $10$ different smoothing constants $(\backslash$alpha $=0\mathrm{.}1,0\mathrm{.}2,...,1)$ and for the best smoothing constant,

report the MAE, MAPE and RMSE of the forecast for years $2019$

through $2022.$ Report $90$ percent prediction intervals $($using the

RMSE estimated in years $2014$ to $2018)$ for the one$-$month ahead

forecasts for $2022.$ How do these compare with MA$-3$ forecasts?

$($f$)$ Use Double Exponential Smoothing to forecast the one period

ahead monthly demand. Experiment with different values of the

smoothing constants $(\backslash$alpha $=0\mathrm{.}1,0\mathrm{.}2,...,1,\backslash$beta $=0\mathrm{.}1,0\mathrm{.}2,...,1)$ and

for the best smoothing constants report the MAE, MAPE and

RMSE of the forecast for years $2019$ through $2022.$ Report $90$

percent prediction intervals $($using the RMSE estimated in years

$2014$ to $2018)$ for the one$-$month ahead forecasts for $2022.$

$($g$)$ Comment on the resdual diagnostics $($i$.$e$.$ independence and normality of residuals$).$ Comment on the comparison to the previous

forecasts? What is the drawback of this forecast with respect to

this data?

$($h$)$ In part f$,$ you must have found the smoothing constants, $\backslash$alpha

$$ and

$\backslash$beta

$$

that leads to the best error performance. Use $\backslash$alpha

$$ and $\backslash$beta

$$ you

found in part f to find $3-$month and $6-$month ahead forecasts for

year $2022.$ Report the MAE, MAPE and RMSE of these forecasts

for $2022$ and comment on the differences.

$($i$)$ To take into account the effect of seasonality, perform the following data transformation: Ut $=$ Dt $$ Dt$12.$ Plot the transformed series $($Ut$)$ and visually verify whether seasonality is eliminated. Use simple exponential smoothing to find a forecast Gt

for Ut $($make sure to test different values of the smoothing parameter to minimize the error$).$ To obtain a forecast Ft for

Dt

$,$ you can then consider Ft $=$ Gt $+$ Dt$12$ or a smoothed version Ft $=$ Gt $+\backslash$gamma Dt$12+(1\backslash$gamma $)$Ft$12$ for a smoothing constant

$2$

$0<=\backslash$gamma $<=1.$ For the best one$-$month ahead forecast, report the

MAE, MAPE and RMSE of the forecast for years $2019$ through

$2022.$

$($j$)$ Your homework report should include a table similar to the one

below.

Method Spec. RMSE MAPE

Benchmark $1-$

Benchmark $2-$

MA$-3-$

ES $-$

DES $-$

Seasonal

Note that the model specification for exponential smoothing is:

$\backslash$alpha

$=...,\backslash$beta

$=....$

$2.$ Forecasting sales of domestic sales of beer. The data file contains the

total monthly sales of beer in Turkey from the beginning of $2010$ to

the end of $2014($found in the blackboard page$).$

$($a$)$ Plot the data and visually assess whether there is significant trend

and seasonality.

$($b$)$ To obtain a benchmark for errors, implement the following naive

forecasts i$)$ Ft $=$ Dt$1$ for t $>1$ and ii$)$ Ft $=$ Dt$12$ for t $>12.$

Report the Mean Absolute Error $($MAE$),$