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Principles of Data Science

Quantitative Problems

Principles of Data ScienceQuantitative Problems
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1.
A U.S. company located in New Hampshire records weekly sales of its main product for a period of 3 years. If a seasonal variation in sales exists, what would be the most likely period of the seasonal variation?
2.
Consider the time series given in the table as shown.
Month Value
1 880.7
2 727.2
3 798.5
4 504.1
5 888.4
6 725.8
7 793.4
8 499.0
9 891.7
10 722.0
11 789.1
12 501.6
Table 5.9 Time Series
a.
Identify the period of the seasonal component.
b.
Using a centered simple moving average (SMA) of window size equal to the period you found in part a, identify a trend-cycle component, t n , in the data.
c.
Detrend the data by subtracting the SMA you found in part b.
d.
Identify the seasonal component, s ^ n , using the method of averaging.
e.
Find the seasonally adjusted time series.
3.
In Collecting and Preparing Data, the following data was given, as provided in the table, showing daily new cases of COVID-19. Since new cases were not reported on Saturdays and Sundays, those weekend cases were added to the Monday cases. One way to deal with the missing data is to use a centered simple moving average to smooth the time series.

Date Weekday New Case
10/18/2021 Monday 3115
10/19/2021 Tuesday 4849
10/20/2021 Wednesday 3940
10/21/2021 Thursday 4821
10/22/2021 Friday 4357
10/23/2021 Saturday 0
10/24/2021 Sunday 0
10/25/2021 Monday 8572
10/26/2021 Tuesday 4463
10/27/2021 Wednesday 5323
10/28/2021 Thursday 5012
10/29/2021 Friday 4710
10/30/2021 Saturday 0
10/31/2021 Sunday 0
11/1/2021 Monday 10415
11/2/2021 Tuesday 5096
11/3/2021 Wednesday 6882
11/4/2021 Thursday 5400
11/5/2021 Friday 6759
11/6/2021 Saturday 0
11/7/2021 Sunday 0
11/8/2021 Monday 10069
11/9/2021 Tuesday 5297
Table 5.10 Sample of COVID-19 Data Cases within 23 Days (source: https://data.cdc.gov/Case-Surveillance)
a.
What is the most appropriate window size to use for centered SMA to address the issue of missing data in their analysis of COVID-19 data from the CDC?
b.
Perform the SMA and illustrate the results by a graph.
4.
Consider the data set USATemps1961-2023.csv.
a.
Use an ACF plot to determine the period of a potential seasonal component.
b.
Use STL to decompose the time series into trend-cycle, seasonal, and noise components.
5.
Use the recursive EMA formula with α = 0.6 to smooth the time series found in USATemps1961-2023.csv, and then use the EMA model to forecast the next value of the series.
6.
A very small time series is given in the table, along with a model.

x n x ^ n
122 117
108 135
172 152
190 169
167 186
Table 5.12 Time Series
a.
Find error measures MAE, RMSE, MAPE, and sMAPE for the model.
b.
Which of the error measures are scale-dependent? What would this imply about a similar time series and model in which all values are multiplied by 10?
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