Fourier Analysis of Time Series †Free Samples to Students

Question: Discuss about the Fourier Analysis of Time Series. Answer: Introduction Registration of a vehicle signifies the legal possession of the vehicle by the owner. It is the proof of ownership of the vehicle and the transfer of responsibility which comes along with owning a vehicle, to the owner. Here in this report I am going to put my focus on registrations of cars because this is a category of vehicle which is most common and is used by the public of New Zealand in larger numbers as compared to other road vehicles. The plots in the following section will commensurate the fact that the number of car registrations in New Zealand has increased since 2006 till end of 2016. A major factor for this is the population of the country. Due to immigration and rising economics of the country the population is increasing which demands more of cars on roads for commutation. With increasing income levels and standards of living in New Zealand people tend to buy their own cars and the need for public transport pushes the sales and hence the registrations of cars. With this investigation, I would like to see the relation between population growth and the increase in the car registration in New Zealand. It is important to establish the research question for any type of research study. For this research study, we want to study the time series analysis regarding the process of car registration over the time period of last ten years. We want to analyze some key parameters of the time series analysis by using the data of car registrations in the New Zealand. We have established the following research questions for this time series analysis. Is there any significant increase in the total number of registrations of cars since last ten years? Is there any significant trend found in the car registrations over the past ten years? Is there a continuous growth in the process of car registrations? For answering these questions we have to use the time series analysis for the data set regarding the car registrations in the New Zealand. It is important to draw the time series plot for the given data for understanding the nature of time series data. By plotting the given time series we understand the different types of trends appeared in the time series data, variation pattern along with time parameter T, seasonal variations, etc. Time series plot plays an important role in understanding the particular pattern of the different variables under study. This time series analysis gives an idea for the future prediction. For the given time series data for car registrations from the year 2006 to 2016, the time series plot is given as below: The above time series plot shows the upward increasing movement. There is a continuous increment in the process of car registrations with time parameter. Also, some small seasonal variation appeared in the given data. For the year 2009, there is downward movement of the time series. The reasons for this downward movement could be identified by studying different associated social and economic parameters for the year 2009. Although there is an appearance of one time downward movement of the time series, but it shows continuous upward movement. Here in the above chart the dependent variable is the number of car registrations made and the independent variable is the period 2006- 2016 each year broken in quarters. The number of cars registered is plotted against quarterly periods from 2006-2016. From the above chart, we see an overall rising trend in the number of car registrations owing to increasing number of cars bought by the people. However there have been two major decline periods in the car registrations from January 2008 to October 2008 and July 2016 to October 2016. From 2006 to 2016 there has been nearly a hike of 23% in the number of cars registered from 2200000 in 2006 to 2700000 in 2016, approximately. The dip may have been due to increasing number of road accidents which may have decreased the incentives for the public to buy own cars. Other reasons could have been increase in the prices of the cars or their complementary products such as gas or may be the cost of obtaining a registration may have surged during these periods. The general increase can be owned to increasing population of New Zealand and rising income levels which make people buy their own cars instead of using public transport for commuting. Time series features - description of seasonality On the left, we have a graph of individual seasonal effects and on the right, is the estimated seasonal effects. The seasonal plot for cars shows for each month the trend in each year. We can see that there are troughs in the July and August months especially in 2009, 2015 and 2016. Also, there was a falling pattern in the early years 2006 2010 in the month of February. The line for the year 2010 shows a stark falling scenario from the month of March which is not the general trend in the other years. In the later part of the year, each year we see a sudden spike in the car registrations. This may be because of the festive season in December and the New Year which make people buy new cars for themselves and therefore increase in the number of registrations. Therefore, we can conclude that the number of car registrations fall in the first half of the year after which it starts to rise and achieves a very high level of registration monthly during the end of each year. From the estimate d seasonal effect graph the number of car registrations fall below the average in the month of April and again go above the average during the November month. The overall effect of the surge in November and December is higher than the combined fall during the February to July period. In the upper part of the chart we see that the green line represents the submission of trend line and the seasonal variation. It shows the expected pattern when the trend line is adjusted for the seasonality in the number of car registrations. The residuals demonstrate the difference between the trend and seasonal line and the original raw data. It is the component of variations which is not explained by the trend line. The green line also represents the expected values if the seasonal patterns remain the same every year. From the thumb rule of finding the outliers in the graph we see that the dip in the October 2008 quarter lies outside the boundary of 50000. This abnormal dip may be due to the rise in the prices of cars, car products and/or the registration fee or increase in the number of accidents. The average car registrations rise from over 2200000 to approximately 2700000. There is nearly a rise of 23% overall. Towards the extremes of the residuals we see that there are greate r fluctuations as compared to the middle section. This suggests that the rise in the car registrations from the 2015 2016 period has been greater than the trend and there is little explanation of that surge captured by the seasonality and trend. Forecast - forecast plot and table The method of Holt-Winters for the prediction purpose deals with the both trend and seasonal variations. There are additive and multiplicative versions for this method. The use of additive or multiplicative methods is depends on the characteristics of the particular time series. The Holt-Winter methods for analysis of time series is nothing but the extension of the single exponential smoothing and double exponential smoothing. We know that the moving average methods are not applicable at all time and we need to use advanced analysis of given time series. Holt-Winters additive time series plot is given as below: In the above graph, we see the predictions and the bandwidth of the car registrations expected during the following years. The dark red line shows the predicted values of the number of car registrations whereas the zone shows the range within which the actual registration is expected to limit itself. Any value within the band is explainable whereas outside the band would be majorly be due to some reason not captured in seasonality and trend. Based on the predictions I believe that the values of car registrations will be in the range of 2600000 to 2900000 approximately. From the chart, we can infer that the fit of the model is poor in the quarters during the years 2008-2010 and again during 2015-2016 because of significant differences between the fitted line and the raw data line. On the contrary the fitted model accurately captures the actual during the 2006-2007 and end 2013 till early 2015 periods. Conclusion From the model fitting and the time series analysis we can infer an increasing trend in the number of car registrations with the years. It is very logical that the with increasing population of New Zealand the number of car registrations also increase owing to a greater demand for cars and greater demand for better standard of living with increasing employment and income levels. We also conclude that the major chunk of the rise in car registrations take place in the end of a year when there is greater increase in the car registrations than the fall experienced during the mid-quarters. With knowledge of increasing demand for cars the car industry and other related industries can plan their production levels and pricings to meet the demand and increase the supply. This information will help in better planning both at corporate and government levels. The government of New Zealand basis this analysis can prepare well for the future rises in the number of cars on road and provide better r oad infrastructure. References Bloomfield, P. (1976). Fourier analysis of time series: An introduction. New York: Wiley Bollerslev, T., Engle, R. F. and Nelson, D. B. (1994). ARCH models. In Handbook of Econometrics, Vol. IV. Ed. by Engle, R. F. and McFadden, D. L., Elsevier, Amsterdam, 29593038. Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control, Revised Edition, Holden-Day, San Francisco. Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd Edition, Springer-Verlag, New York. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica, 50, 9871007. Shumway, R. H. (1988). Applied statistical time series analysis. Englewood Cliffs, NJ: Prentice Hall