Time Series Analysis

Assignment Brief

School of Computer Science and Mathematics

Faculty of Science, Engineering and Computing

Time Series Analysis and Further Inference with Applications Including Insurance Models (ST6300)

Retake Coursework   -   2019-20

Submission deadline of this coursework:

Instructions to Candidates:

Answer Both Questions 1 and 2

1. Q1 and Q2 each carries 100 marks
2. Use 4 decimal places in your calculations

Candidates are required to write their solutions VERY CLEARY, and showing ALL calculations- (Page numbering is also required)

 All of your answers must be incorporated into a single document (Word or Pdf)

1.  The assessment must be submitted electronically via Canvas
2. Any assignment showing evidence of copying may result in a mark of zero

General Guidelines to Candidates

• As this is a coursework assessment there is no extra time allocated for students with a SOSN.
• If you experience technical difficulties, e.g. access and upload issues, or identify a potential error in a question please email the module leader n.saebi@kingston.ac.uk .
• This is an open book assessment so you may consult your notes, textbooks and the Internet.
• You must not collaborate with someone else on this assessment, it should be wholly your own   work. Your work will be checked for evidence of plagiarism and/or collusion using Turnitin.
• If you include graphics in your answer, please embed these into the Word document (e.g. a photograph of a hand-drawn graphic). The source of any copied and pasted figures should be cited. If you have embedded any scanned/photographed diagrams and graphs, then your submission file cannot exceed 100MB. The model, the relevant algorithm, formulae involved and useful explanation of your calculations

Total number of pages including this page: 4

N Saebi

1: The monthly sales of personal computers by an electric outlet for the past 8 years are tabulated below: 

(The data is read column-wise)

Plot the time series and comment on your visual observation of the graph. Then, by applying the following two methods, determine the outlet’s sales forecasts of computers for the months of 2012-2019, where applicable.

1: The Multiplicative Decomposition Method.

2: The Single Exponential Smoothing Method (SES).

In each of the above two methods you need to include at least the following set of guidelines:

1. The model, the relevant algorithm, formulae involved and useful explanation of your calculations.
2. Your Excel worksheets compactly presented with suitable titles and column headings.
3. Calculations of the optimum values (where applicable, using the Excel Solver) for the parameter involved.
4. The sales forecasts for periods 2012-2019, where applicable, and the corresponding forecast errors.  
5. The Mean Squared Error (MSE) and the Mean Absolute Percentage Error (MAPE), for the test period of November 2012to December 2019 (inclusively).                                          

In addition, the following needs to be provided:

• The comparative assessment table for the above two forecasting techniques using the MSE and the MAPE. 
• For each method, illustrate its forecast plot individually. Furthermore, in a single diagram, also show all  forecast plots and critically compare them with the time series plot of the data. Hence, using both assessment criteria table and the forecast plots, decide on the optimum forecasting method for the sales of personal computers. 
• Forecast of the monthly sales of personal computers by the electric outlet for the year 2020, using your optimal model. 

Retake Coursework Continued

2: A random sample  is drawn from a distribution with probability density function with mean and variance and, respectively.   

An estimator of given by for a constant and the sample mean, is being considered.                                                                     

End of Retake Coursework

Sample Answer

Time Series Analysis of Monthly Sales of Personal Computers (2012–2019)

Introduction

In this report, we analyse the monthly sales data of personal computers for an electric outlet over the period 2012 to 2019. The aim is to forecast future sales using two prominent time series forecasting methods: Multiplicative Decomposition and Single Exponential Smoothing (SES). Forecast accuracy will be compared using Mean Squared Error (MSE) and Mean Absolute Percentage Error (MAPE) to recommend the best forecasting method for 2020.

Visual Analysis of Time Series Data

Upon plotting the monthly sales data, a clear seasonal pattern is visible, with sales peaking during the December holiday season and dipping in spring and summer months, particularly in May and June. Additionally, there is a slight upward trend in sales over the 8-year period, suggesting growth in demand for personal computers. The repeating seasonal pattern indicates that the data is suitable for decomposition analysis.

Method 1: Multiplicative Decomposition

In this approach, we break down the time series data into three key components:

Observed Value (Yₜ) = Trend (Tₜ) × Seasonality (Sₜ) × Irregular Variation (eₜ)

Where:

• Yₜ is the actual sales value at time t

• Tₜ represents the long-term trend or direction of the data

• Sₜ captures regular seasonal patterns (e.g., monthly or quarterly changes)

• eₜ accounts for random, unpredictable factors (irregular variation)

This method helps us understand how each component influences the overall sales and allows for more accurate forecasting by isolating and analysing these effects.

Steps for Analysis:

1. Calculate the centred moving average (CMA) to estimate the trend.

2. Determine seasonal indices by dividing actual values by CMA.

3. Deseasonalise the data by dividing actual sales by seasonal indices.

4. Fit a linear trend using least squares method.

5. Forecast future sales by multiplying trend and seasonal factors.

Continued...