DISTINCTION BETWEEN SPEPECTRAL ANALYSIS AND HARMONIC ANALYSIS IN TIME SERIES ANALYSIS: EXPERIENCE FROM THE NUMBER CRUNCHER STATISTICAL SYSTEM (NCSS) SOFTWARE. Authors Okafor Uchenwa Linusa, M.O. Oladejoa, C.O.Uwab, D.T.Chineyob and Jighjigh,A.Tc. a. Department of Mathematical Sciences, Faculty of Science Nigerian Defence Academy, Kaduna b.Department of Computer Science, Faculty of Military Science and Interdisciplinary Studies ,Nigerian Defence Academy, Kaduna. c Department OF Mathematics/Computer Science. Benue State University, Makurdi Abstract This paper shows a clear distinction between Spectral Analysis and Harmonic Analysis and highlights what is common amongst these two terms. The distinction and commonalities are illustrated using the data from sales of cement obtained from a sales record. The work identified the role of spectral analysis as that of showing the periods /wavelengths having containing more energy with the sine and cosine components but without stating whether the components are significant or not. Spectral analysis helps to identify hidden periodicities but does not form the model equation. Harmonic analysis on the other hand makes use of the periods with higher energies identified by the spectral analysis to form a model equation, it also shows whether the frequencies are significant or not by displaying the t-values for each kth harmonic value used to form the model equation. From the giving data, harmonic analysis makes predictions /forecasts, performs model parameter estimation, Analysis of variance, displays the asymptotic correlation matrix of parameters. In harmonic analysis, the graphs of errors against time and against the predicted values are also plotted. Keywords: Harmonic analysis, spectral analysis, period, wavelength, model, forecasting. Corresponding author’s address: email-linusokafor @gmail.com,phone;+234-9022982454 Co-authors address: (1)mikeoladejo2003@yahoo.com.(2)ozocel2014@gmail.com.(3)atamber@bsum.edu.ng.(4)dtchinyio @yahoo.com IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 1 1.0 Introduction Most text book writers take spectral analysis as harmonic analysis and vice versa.Brillinger(2001) considered harmonic analysis as a tool for search of hidden periodicities. But Rebeca (2004) unlike most of the other authors explained that harmonic analysis works on the assumption that the modeler already knows the period, one can easily set up a regression equation and find out the amplitude, the phase and the mean of the signal. In the analysis of time series data, etc, it is often necessary to identify the periodicities in the data by spectral analysis which is concerned with the partition of the variation in a time series between the components at different frequencies or periods. In spectral analysis, if the data is not stationary it should first be detrended by fitting a proper curve to the data set and then using that to remove the trend. But if the data does not show obvious trend, one can then remove the mean and proceed to carry out spectral analysis. Differencing can also be employed to make the data set stationary. The Number Cruncher Statistical System (NCSS) has six portions on carrying out spectral analysis as shown in Table 1. Table 1. Fourier Analysis (p,q,s,t,u) (a) (b) (c (d) (e ) (f) Frequencies Wavelength Period Cosine(�� ) Sine(�� ) Spectrum But we recommend a seven- column portion as given in Table 2. so as to make room for the harmonic number (k) Table 2. Fourier Analysis (p,q,s,t,u) (a) (b) (c (d) (e ) (f) (g) Harmonics (k) Frequencies(f) (f� ) Wavelength L = � � Period Cosine(�� ) Sine(�� ) Spectrum Where, �� = � � ∗ ∑ �� � � ��� ∗ Cos(Ɵ� ) �� = � � ∗ ∑ �� � � ��� ∗ Sin(Ɵ� ) IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 2 The data points are: �� =(�� , �� ,…�� ) K = � � for even number of data points, or ��� � for odd numbered data points. When the data has a clear seasonal length, L, then k = � � The column headed spectrum shows the amount of energy or proportion of the contribution of each of the frequencies to the variance of the series. Spectral analysis also makes the plots of the spectrum against each of the following items: time, wavelength, and, frequency It is the spectral analysis that helps one to begin to think of the frequencies /wavelength that will enter into the model building process. Spectral analysis does not give the model, the phase angels and makes no prediction from the data. In Harmonic analysis, one uses the frequencies identified from spectral analysis to form a model and harmonic analysis gives about eight different specific results as shown through Table 3 to Table 10.Fourier analysis is part of harmonic analysis, but in literature, most authors’ mistake spectral analysis for harmonic analysis and most statistical software’s also fall into the same misconception, but the use of the Number Cruncher Statistical System(NCSS), illustrates the role of either spectral analysis or harmonic analysis. 2. Statement of the problem. In order to carry out a Fourier series analysis of any time series data, one must use the periodic components and time series data are seasonal or periodic and more often than not these periodicities are hidden. To unmask the hidden periodic components spectral analysis needs to be performed on the data set. Results from spectral analysis are then used to perform harmonic analysis through Fourier series analysis. There is therefore the need to make a clear distinction between spectral analysis and harmonic analysis in the Fourier analysis of time series data . 3. Aim The aim of this study is to show the difference between spectral analysis and harmonic analysis in the analysis of time series data through a worked example using the Number Cruncher Statistical System. 4. Data Source and Materials Used. The data set used in this work is the sales of bags of cement obtained from Ogwumabiri Market in Obowo Local Government Area of Imo State ,Nigeria and this is shown graphically in Figure 1 and the data set is shown in Table 1.The software used to do the analysis of date was the Number Cruncher Statistical System (NCCS2019) by Chris Hintz IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 3 Fig.1.Monthly Sales of Cement 2010 to 2018 Table 1.Sales of Cement 2010 TO 2017 Year Month/ 2010 2011 2012 2013 2014 2015 2016 2017 2018 Jan 300 330 310 350 420 480 530 490 653 Feb 345 355 345 368 530 560 590 540 640 Marc 420 450 460 480 590 610 760 690 670 Apr 350 370 390 410 500 520 580 560 690 May 290 350 360 390 480 490 560 530 610 Jun 320 330 345 360 450 460 530 500 570 Jul 280 300 320 340 390 400 500 490 600 Aug 340 360 370 390 490 530 580 560 640 Sep 370 390 410 450 510 590 630 640 662 Oct 390 420 450 470 570 635 690 695 735 Nov 430 470 490 560 630 690 740 760 790 Dec 500 540 690 675 740 730 800 820 830 5. Methodology The data was first subjected to outlier test by the Grubbs test for outliers and the (NCSS 2019) found are no outliers. 51. Spectral Analysis The software (NCSS2019) gave the spectral analysis results in two folds as will be shown in the sequel, it removed the trend as the data displayed in Table 1. depicts trending pattern. Note that if the data has no trend, the mean will be then removed from the data.The spectral plots are displayed in figures 2 and 3. Fig.2 .Plot of Periodogram against Frequency. 0 200 400 600 800 1000 1 5 9 131721252933374145495357616569737781858993 Series1 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 4 Fig.2.Plot of Periodogram against Wavelength Table 2.Spectral Analysis Results Sales of Cement (k)Harmonic Frequency Wavelength Period Cosine(a's) Sine(b's) Spectrum 1 0.129536 48.50526 16617.3 32.59697 -33.6977 1.00E+33 2 0.193623 32.4507 43948.08 73.84023 18.99913 1.00E+33 3 0.257709 24.38095 29364.56 -53.4418 32.06624 1.00E+33 4 0.321795 19.52542 38437.91 -71.1648 4.476748 1.00E+33 5 0.385881 16.28269 16091.53 -32.3992 -32.8458 1.00E+33 6 0.449968 13.96364 103884.1 -53.0957 -104.51 1.00E+33 7 0.514054 12.22281 2.15E+07 -1541.93 -681.31 1.00E+33 8 0.57814 10.86792 390103.1 206.1583 95.39722 1.00E+33 9 0.642227 9.78344 254522.4 173.7618 58.94443 1.00E+33 10 0.706313 8.895753 60140.59 -88.3929 11.91382 1.00E+33 11 0.770399 8.155752 613223.8 227.667 -171.124 1.00E+33 12 0.834486 7.529412 1110957 -370.889 -96.9354 1.00E+33 13 0.898572 6.992413 116157.1 -103.743 67.841 1.00E+33 14 0.962658 6.526912 1615692 410.1017 213.3926 1.00E+33 15 1.026744 6.119522 84240.29 96.81339 42.07458 1.00E+33 16 1.090831 5.76 11582.41 -29.9363 25.21726 1.00E+33 17 1.154917 5.440378 1979010 -438.565 -263.514 1.00E+33 18 1.219003 5.154362 134286 106.6879 -79.8794 1.00E+33 19 1.28309 4.896918 617032.6 46.99103 -281.8 1.00E+33 20 1.347176 4.663968 290284.8 -186.796 -59.206 1.00E+33 21 1.411262 4.452174 283300.5 181.0477 68.52771 1.00E+33 22 1.475349 4.25878 787854 197.8401 255.0974 1.00E+33 23 1.539435 4.081488 2601302 338.3845 -479.155 1.00E+33 24 1.603521 3.918367 3221233 -652.678 -10.3855 1.00E+33 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 5 (k)Harmonic Frequency Wavelength Period Cosine(a's) Sine(b's) Spectrum 25 1.667608 3.767784 919057.6 4.082906 -348.646 1.00E+33 26 1.731694 3.628346 1046506 -294.5 -227.374 1.00E+33 27 1.79578 3.498861 10701.19 37.52555 2.713192 1.00E+33 28 1.859866 3.378299 699923.8 -265.652 148.3689 1.00E+33 29 1.923953 3.265769 3044655 107.2807 625.4841 1.00E+33 30 1.988039 3.160494 1704203 435.7885 -188.458 1.00E+33 31 2.052125 3.061794 4084301 626.947 -383.665 1.00E+33 32 2.116212 2.969072 186455.8 -9.93235 -156.733 1.00E+33 33 2.180298 2.881801 72246.95 -97.085 11.45203 1.00E+33 34 2.244384 2.799514 745561.3 -198.049 -243.716 1.00E+33 35 2.308471 2.721796 512814.4 -169.859 -197.438 1.00E+33 35 2.308471 2.721796 512814.4 -169.859 -197.438 1.00E+33 36 2.372557 2.648276 541663.8 -202.375 175.1979 1.00E+33 37 2.436643 2.578623 951056.5 210.9437 285.1424 1.00E+33 38 2.50073 2.512541 455378.2 147.9449 195.828 1.00E+33 39 2.564816 2.449761 895302.3 -246.868 -239.76 1.00E+33 40 C 2.390042 4245736 422.5156 618.9477 1.00E+33 41 2.692988 2.333164 3190907 -113.368 -639.713 1.00E+33 42 2.757075 2.278932 1882224 -413.898 278.6835 1.00E+33 43 2.821161 2.227163 1877624 441.369 231.4324 1.00E+33 44 2.885247 2.177694 1599362 230.5599 -397.997 1.00E+33 45 2.949334 2.130374 485149.8 211.2075 -139.878 1.00E+33 46 3.01342 2.085068 1806633 -459.92 165.6836 1.00E+33 47 3.077506 2.041648 2362332 154.5963 -537.2 1.00E+33 48 3.141593 2 1065080 -375.348 1.17E-10 1.00E+33 The plots of periodogram against frequency and that of periodogram against wavelengths are shown as Fig.2. and Fig.3 and each of them show that there is only one dominant frequency at the harmonic number k =7 ,with a frequency f= 0.514054 and wavelength 12.22281. . Table 2 shows the Fourier analysis result indicating the highest energy content at the 7th harmonic k=7 as 2.15*107 ,the cosine component is -1541.93and the sine component is -681.31 7 0.514054 12.22281 2.15E+07 -1541.93 -681.31 1.00E+33 Spectral analysis results end after identification of wavelength, sine term, cosine term, period, and the spectrum. Using the NCSS,the harmonic numbers are not shown, hence we invoke the use of Excel to supplement the analysis in order to add the harmonic numbers as shown in Colum (a) in Table 1. The use of Harmonic analysis Harmonic analysis is used to obtain the various section as explained here under. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 6 Below are the steps involved in carrying out the harmonic analysis: After performing the spectral analysis as discussed above, when the harmonic section of the NCSS2019 was now employed the following results were obtained: Table 3.Run Summary Section ─── Item Value Item Value Dependent Variable Yt Total Rows 156 Time Variable t Rows with Missing Values 60 R² 0.96 Rows Used 96 Estimated Model Ft =307.273+(4.317)*t-15.1354)*SIN((0.517*t+ 66.998*COS(0.517t)) The amplitude value of 68.68638 is obtained from the squares of the sine and cosine coefficients 0f; 68.68638 = �(−15.1354)� + (66.998)� Table 4.Regression Coefficients Section Regression Standard T- Statistic Upper 95% Independent Coefficient Error to Test Prob Conf. Limit Variable b(i) sb(i) H0: β(i)=0 Level of β(i) Intercept 307 5.59439 54.93 0 318.38 Trend 4.32 0.10026 43.06 0 4.5164 Sin(12.14371) -15 3.90635 -3.87 0.9999 -7.3771 Cos(12.14371) 67 3.93496 17.03 0 74.813 In this calculation, the software uses the wavelength value of 12.44 ,instead of the the frequency value. Wave Length Frequency Amplitude Phase 12.144 0.52 68.68638 0.22218 Table 5.Analysis of Variance Table ─── Sum of Mean F-ratio DF Squares Squares Intercept 1 25546130 25546130 Total (Adjusted) 95 1685138 MSE/MSE Model 4 27163716 MSR=6790928.96 9248.64 Error 92 67552.16 MSE=734.2626 Total 96 27231268 734.2626 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 7 Table 6.Correlation Matrix of Regression Coefficients Intercept Trend Sin(12.14371) Cos(12.14371) Intercept 1 - 0.869221 -0.083666 0.005182 Trend -0.9 1 0.095877 0.003687 Sin(12.14371) -0.1 0.095877 1 -0.000566 Cos(12.14371) 0.01 0.003687 -0.000566 1 Table 1.SALES OF CEMENT 2010 TO 2017 Month/Year 2010 2011 2012 2013 2015 2016 2017 2018 Jan 365 420 410 511 630 630 750 735 Feb 345 380 430 525 650 680 640 743 Marc 290 392 400 480 590 700 690 700 Apr 267 380 390 430 520 650 710 670 May 280 365 360 410 530 560 630 660 Jun 275 320 355 420 500 570 610 625 Jul 280 300 340 490 580 550 680 630 Aug 320 360 390 480 530 580 630 640 Sep 350 390 430 520 640 640 680 675 Oct 390 420 490 540 635 690 725 730 Nov 430 470 512 560 690 730 765 760 Dec 450 480 560 590 700 745 780 800 Table 7.Forecasts SALES OF CEMENT 2010 TO 2017 Month/Year 2010 2011 2012 2013 2015 2016 2017 2018 Jan 362 417.4384 472.2456 526.7375 634.72 688 741 741 Feb 337 393.712 450.1702 506.459 618.43 674 729 729 Marc 306 363.18 420.0715 476.9838 590.76 648 704 704 Apr 279 334.9653 390.9591 447.1585 560.08 617 674 674 May 264 317.5843 371.5844 425.921 535.55 591 646 646 Jun 266 316.7174 368.1497 419.9612 524.72 578 631 631 Jul 285 333.7216 382.6842 431.9696 531.56 582 633 633 Aug 318 365.2758 412.5133 459.9327 555.41 604 652 652 Sep 358 404.2496 450.9583 497.6603 591.16 638 685 685 Oct 394 441.5705 489.0851 536.4061 630.57 677 724 724 Nov 419 468.5985 518.0427 567.1571 664.46 713 761 761 Dec 427 479.3884 531.3807 582.9934 685.09 736 786 786 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 8 Table 3. forecasts Errors of SALES OF CEMENT 2010 TO 2017 Month/Year 2010 2011 2012 2013 2015 2016 2017 2018 Jan 2.67 2.56165 -62.24556 -15.73755 - 4.7183 -58 8.72 - 6.28 Feb 7.89 - 13.71204 -20.1702 18.54102 31.57 5.93 -89 13.6 Marc -16 28.82 -20.0715 3.016177 - 0.7581 52.4 -14 - 4.28 Apr -12 45.03472 - 0.9590976 -17.15847 - 40.076 33.3 36.5 - 3.51 May 16.1 47.41569 -11.58441 -15.92099 - 5.5459 -31 -16 13.7 Jun 9.34 3.282614 -13.14965 0.03875451 -24.72 -7.7 -21 - 5.97 Jul -5.1 - 33.72165 -42.6842 58.03038 48.437 -32 47.4 - 2.59 Aug 1.81 - 5.275815 -22.51331 20.06731 - 25.413 -24 -22 - 11.9 Sep -7.5 - 14.24964 -20.95833 22.33971 48.843 1.99 -5 - 9.96 Oct -3.8 - 21.57052 0.914923 3.593923 4.4325 12.5 0.74 5.74 Nov 11.2 1.401448 -6.042702 -7.157125 25.542 17.3 4.36 - 0.64 Dec 23 0.611624 28.61931 7.006566 14.913 9.42 -5.7 14.3 Fig.3.Predicted and Actual values Values against Time 0 100 200 300 400 500 600 700 800 900 1 7 1319253137434955616773798591 Actual Value Predicted Value IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 9 Fig.4.Graph of Residuals against Time The plot of residuals against time shown in Fig.4 demonstrates that the errors are random in nature, and the plot of actual and predicted values against time shown in Fig.3. is an evidence that that the actual and predicted values are reasonably close to each other. Fig.5.Plot of Autocorrelation Functionsof Errors(ACF). Fig .6. Plot of Partial Autocorrelation Coefficients(PAFC) -100 -50 0 50 100 0 50 100 150 Residual Residual IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 10 Table 4.Autocorrelations of C6 (0,0,0,0,1) ───────────────────────────────────────────────── Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.017623 11 -0.206787 21 0.048251 31 -0.009072 2 -0.198659 12 0.126689 22 -0.023265 32 0.103056 3 -0.033041 13 -0.023120 23 -0.068452 33 0.137523 4 0.037400 14 -0.141602 24 0.078515 34 -0.077645 5 -0.024658 15 -0.041887 25 -0.061904 35 -0.112095 6 0.096889 16 0.149367 26 -0.114473 36 0.028800 7 0.186567 17 -0.122003 27 0.067000 37 0.010588 8 0.027944 18 -0.154647 28 0.087605 38 -0.094329 9 0.019432 19 -0.086806 29 -0.046813 39 0.058022 10 -0.094473 20 -0.017866 30 -0.097414 40 0.082354 Significant if |Correlation|> 0.204124 Table 5.Partial Autocorrelations of Yt (0,0,0,0,1) ──────────────────────────────────────────── Lag Correlation Lag Correlation Lag Correlation Lag Correlation 1 0.017623 11 -0.210698 21 0.062361 31 -0.099679 2 -0.199031 12 0.092510 22 -0.052010 32 0.070056 3 -0.026389 13 -0.178303 23 -0.026525 33 0.106009 4 -0.001082 14 -0.177207 24 0.089137 34 -0.127389 5 -0.038603 15 -0.094584 25 -0.089422 35 0.004287 6 0.108780 16 0.074532 26 -0.020151 36 -0.062159 7 0.181286 17 -0.082308 27 0.110811 37 -0.065611 8 0.067983 18 -0.036400 28 -0.087167 38 -0.049178 9 0.109374 19 -0.129148 29 -0.030008 39 -0.049802 10 -0.075281 20 -0.000654 30 -0.124178 40 -0.092858 Significant if |Correlation|> 0.204124 The results from both Tables 5 and 6 show that that the PAC F,and ACF have absolutes values that are less than the table value of 0.204124 as indicated by the software.This further confirms the adequacy of the adopted model.The plots of ACF and PACF shown in Figures 4 and 5 each did not exhibit spike at any lag,demonstrating that the correct model has been obtained. Conclusion. The paper has explained briefly the difference between spectral analysis and harmonic analysis given the results in each case using an example. It has also shown that to carry out harmonic analysis, spectral analysis must be performed to obtain significant frequency that will be used to get an appropriate model. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 11 References 1.David R. Brillinger ((2 NumberZ001) Time series Data Analysis and Theory.Holden Day.Inc,San Francisco. 2.Chris Hintz( 2019 ) .number Cruncher Statistical System (NCSS 2019) 3. IJO - INTERNATIONAL JOURNAL OF MATHEMATICS Volume 03 |Issue 01 | January 2020 www.ijojournals.com 12 Word Bookmarks 12Partial_Autocorrelations_of_C6_(0,0,0, 12Autocorrelations_of_C6_(0,0,0,0,1)12/1