Academic Journal of Applied Mathematical Sciences ISSN(e): 2415-2188, ISSN(p): 2415-5225 Vol. 5, Issue. 10, pp: 140-149, 2019 Academic Research Publishing Group URL: https://arpgweb.com/journal/journal/17 DOI: https://doi.org/10.32861/ajams.510.140.149 Original Research Open Access On Two Covariates Cosine and Sine Noisy-Wave Trigonometry Regression of Heartbeats Olanrewaju Rasaki Olawale Department of Statistics, Faculty of Science, University of Ibadan, Ibadan, Oyo State, Nigeria Abstract This paper proposes and describes the acumen on alternate two covariates linear Cosine and Sine regression functions that possessed a noisy-wave or tone frequencies via wave-trend of actualized observations of regressors and responsive variable needed in fitting a wavy equation of trigonometry regression. The method of maximum likelihood was used in estimating parameters associated to the Cosine and Sine alternate functions via vector coefficients as well as their distributional and residual properties. The estimations obtained via the method were enthralled to the noisy-wave mesokurtic observations of babies’ rate of heartbeats exactly an hour after birth (HR 1), two hours after birth (HR2) and three hours after birth (HR3). The implementation and illustrative application was via R using the heartbeat dataset. It was gleaned that the trigonometry equation line of  +1cos  HR 2    2sin  HR1  optimally captured the wave observations and robustly outstripped the alternate Cosine and Sine equation line of  +1sin  HR 2   2cos  HR1  . Keywords: Cosine; Equation line; Noisy-wave; Regression functions; Rates of heartbeat; Sine. CC BY: Creative Commons Attribution License 4.0 1. Introduction Regression analysis is a technique use in modeling the relationship(s) between response variable and predictor(s) or among predictors. This unknown connection could either be a linear or non-linear relationship depending on the transfer function [1, 2]. It is termed “simple regression” if the dependent variable is constrained to only a predictor and “multiple regression” if the formal is subjugated to two or more predictors [3]. The conventional methods of statistic and parameter (regression coefficients, model performance indexes, residual indexes, prediction error indexes, etc.) estimation ranges from Maximum Likelihood (ML), Least Squares (LS), Quasi- Likelihood (QL), Generalized Linear Model (GLM) etc. for parametric approach; method of sieves, difference sequence method, Ordinary differential Equations (ODEs) etc. for non-parametric approach and some amalgamated methods of both parametric and non-parametric that resulted in semi-parametric approach [4-7]. The main purpose of regression modeling is for generalization of studied relationship(s), prediction making, decision-making, diagnosis and to ascertain statistical property of the studied system [8, 9]. According to Hanley [10], a number of extensive studies had been carried-out on different forms of regression estimators to accommodate and recodify the assumptions of normality, independence and attached time factors to covariates. Among the few forms are ridge regression, seasonality regression analysis, Fourier regression, trigonometric series regression analysis, and smoothing splines regression [11-14]. All these mentioned forms are for demonstrating the dummy variables for estimation of seasonal effects in a time series, to penalize estimators in situation where the number of parameters estimated is strictly greater than the sample size, and to free the distributional property of the observations in non-parametric settings [15-17]. Rigdon, et al. [18], propounded a Fourier trigonometric like regression and applied it to uniform time-varying public health surveillance disease data with the assertion of normality assumption, seasonality, and independence ascertained as well as the stationarity of the first and second order- Fourier regression like model. This paper presents a conspectus diversify approach by considering noisy-wave or tone frequencies observations of covariates without seasonality, uniform time varying (unequal spaced time intervals of unordered sequence of set of observations) of recording observations via a Gaussian density function. A two alternate Cosine and Sine linear equation functions (a trigonometry regression approach) will be formulated such the parametric method of maximum likelihood will adopted in estimating the Cosine and Sine alternate equations vector coefficient noisy-wave mesokurtic observations as well as its distributional and residual traits. 2. Material and Method 2.1. The Two Covariates Alternate Cosine and Sine Function Trigonometry Regression Given a linear regression model function with random error variables Y  f ( X  )  i i ; (1)  For i are uncorrelated noisy-wave standardized random variables with mean zero and unity variance. 140 Academic Journal of Applied Mathematical Sciences And, f ( X  )    1 cos X i1   2 sin X i 2 (2) Then, yi    1 cos X i1   2 sin X i 2  i (3) Alternatively; yi    1 sin X i1   2 cos X i 2  i Where, yi  Is a n by 1 vector of responses. X i1 , X i 2  Is a n by p  1 is full rank design matrix of the model.    , 1 ,  2  Is a p by 1 vector of coefficients. i  Is a n by 1 vector of random errors. Cos & Sin  Are the trigonometry noisy-waves for the two covariates. i  y    1 cos X i1   2 sin X i 2 In matrix form; Y  HB   (4) B Is the column vector of parameter to be estimated; H is the coefficient matrix of a square matrix; where the error term  0, I   2  2.2. Maximum Likelihood Estimation Method of the Two Covariates Alternate Cosine and Sine Function (i ) Considering error random variables that are assumed independent and normally distributed with zero mean and unity variance, adopting the maximum likelihood estimation gives 1 2  1 f ( )  2 2 exp   2 2     (5) Then, f ( )  1 2 2 exp  1  2 2 The maximum likelihood gives,  L   1 2  2 2  yi    1 cos X i1   2 sin X i 2    L  f (1 )  f (2 )  (6)  f (n ) n n  2  1  exp   2  yi    1 cos X i1   2 sin X i 2    i 1  2  Taking the log of equation 1 1 n   y    1 cos X i1   2 sin X i 2   ln L  n ln   ln  2     i   2 2 i 1   (7) 2 (8)  ln L n   yi    1 cos X i1   2 sin X i 2        i 1   Equating to zero gives, n  y i 1 i    1 cos X i1   2 sin X i 2   0 (9) n  y     cos X   sin X   ln L i1 i2   1 2  cos X i1   i  1  i 1   Equating to zero gives, 141 Academic Journal of Applied Mathematical Sciences n  cos X  y i1 i 1 i    1 cos X i1   2 sin X i 2   0 (10)   y    1 cos X i1   2 sin X i 2    ln L  sin X i 2   i    2 i 1   n Equating to zero gives, n  sin X  y i2 i 1    1 cos X i1   2 sin X i 2   0 i (11) Expanding equations (7), (8) and (9) gives the system of equations; n n n  yi  n  1  cos X i1   2  sin X i 2  0 i 1 i 1 i 1 n n n i 1 i 1 i 1   n   yi cos X i1      cos X i1   1  cos 2 X i1   2   cos X i1  sin X i 2   0 i 1   y sin X      sin X      sin X  cos X      sin n i 1 n i2 i n i 1 i2 1 n i2 i 1 i1 2 2 i 1  X i2  0 Re-arranging and converting to matrix form gives, n    n  yi    i 1       n    n cos y X    i i 1     1     cos X i1  i 1     2   i 1 n    n sin y X   i2   i    sin X i 2   i 1   i 1 n  sin X i 1   cos n 2 X i1 i 1   i 1  n    cos X i1  sin X i 2   (12) i 1  n  2   sin X i 2   i 1 n  cos X i1  n   sin X  cos X  i2 i1 i 1 i2 Where,  n       n B   1      cos X i1     i 1  2   n       sin X i 2   i 1  B  HT H  1   cos X i1  sin X i 2  i 1 i 1  n n 2   cos X i1    cos X i1  sin X i 2   i 1 i 1  n n 2  sin X cos X sin X   i 2  i1    i2   i 1 i 1 n n   yi    i 1    n     yi cos X i1   13  i 1   n     yi sin X i 2    i 1  (14) n  cos X varies from sine of   cos n i 1 2 X i1  n   sin X  cos X  i2 i 1   i 1  n    cos X i1  sin X i 2   i 1  n  2   sin X i 2   i 1 n  sin X i1 i 1  n   yi    i 1   n   H T y     yi cos X i1    i 1   n     yi sin X i 2    i 1  xi Since cosine of 1 HT y  n   1  n T H H     cos X i1   i 1  n    sin X i 2   i 1 Where,  n i1 1 i2 (15) (16) xi , that is cos( xi )  sin( xi ) 142 Academic Journal of Applied Mathematical Sciences yi    1 sin X i1   2 cos X i 2  i i  yi    1 sin X i1   2 cos X i 2 yi    1 sin X i1   2 cos X i 2  i , the estimates of equation (13) becomes equation (17) below; When  n       n B   1      sin X i1     i 1  2   n       cos X i 2   i 1   i 1 i 1  n n   sin 2 X i1    sin X i1  cos X i 2   i 1 i 1  n n 2    cos X i 2  sin X i1    cos X i 2   i 1 i 1 n n  sin X i1  cos X i 2 1 n    yi   i 1   n      yi sin X i1   17   i 1   n     yi cos X i 2    i 1  2.3. Distributional Properties of the Two Covariates Trigonometric Regression From equation (14)  B  HT H  1 HT y y  HB   Recall from equation (4),  B  HT H So,   HT H  1 H T  HB     H H  B  H 1 T T   B  H T (18) Taking expectation gives, E ( B)  E ( B)  H T E ( ) H E ( )  0 , Recall (19) E ( B)  B So, mean of the estimate B is nothing but B Subtracting " B " from both sides of equation (14) gives B  B  H T But, y  HB   B  B  H T  y  HB   B  B  H T y  H T HB  H T H  1 H 2.4. The Dispersion Matrix of B T V ( B)  E  B  B   B  B      1  E  H T H H            1 1  E  H T H H  T H H T H    1 1   H T H HE ( T ) H H T H     Recall E ( T )   2  ;    HT H   V ( B)  H T H   1  2  1  HH H T H  1  2   (20) 2 Where is the variance of the error term 143 Academic Journal of Applied Mathematical Sciences B This implies that,  B,  H H    1 T  n    n V ( B)     cos X i1   i 1  n    sin X i 2   i 1 Such tha For 2  n  sin X i 2 i 1   cos n i 1 2 X i1  n   sin X  cos X  i2 i 1 1   i 1  n    cos X i1  sin X i 2    2 i 1  n  2   sin X i 2   i 1 n  cos X i1 i1 (21) yi    1 cos X i1   2 sin X i 2  i While  n   n V ( B )     sin X i1   i 1  n   cos X   i2  i 1 For 1   sin X i1  cos X i 2  i 1 i 1  n n 2   sin X i1    sin X i1  cos X i 2    2 i 1 i 1  n n 2  cos sin cos X X X      i2 i1 i2   i 1 i 1 n n yi    1 sin X i1   2 cos X i 2  i (22) 2.5. Variance of the Error Term for the Two the Two Covariates Trigonometric Regression E T 2 y T y  BH T y    n p n p " p " is the number of parameter to be estimated. Where " n " is the number of observations and   n   yi    i 1   n   y T y   , 1 ,  2     yi cos X i1    i 1   n  yi sin X i 2      2  i 1    n p For yi    1 cos X i1   2 sin X i 2  i n   yi    i 1    n  T y y   , 1 ,  2     yi sin X i1    i 1   n     yi cos X i 2   2  i 1    n p For yi    1 sin X i1   2 cos X i 2  i (23) (24) 2.6. Coefficient of Determination for the two Covariates Trigonometric Regression The coefficient of determination being denoted by; 144 Academic Journal of Applied Mathematical Sciences R  2 BH T y  n y yT y  n y 2 2 n   yi    i 1    n  2  , 1 ,  2     yi cos X i1    n y  i 1   n     yi sin X i 2    i 1  R2  2 T y y  ny For yi    1 cos X i1   2 sin X i 2  i n    yi   i 1   n 2   , 1 ,  2     yi sin X i1    n y  i 1   n     yi cos X i 2    i 1  R2  2 T y y  ny For yi    1 sin X i1   2 cos X i 2  i (25) (26) 3. Results The secondary dataset used in validating the obtained estimations above was the readings of rate of heartbeats of newly born babies in Lagos University Teaching Hospital (LUTH), a federal government owned hospital in Lagos state, Nigeria. These rate of heartbeats’ readings variability were recorded in three different time-frames (in hours); rate of heartbeats exactly after an hour after birth (HR 1), rate of heartbeats exactly after two hours after birth (HR 2) and rate of heartbeats exactly after three hours after birth (HR3). These readings were recorded for nine hundred and fifty (950) babies in the year 2017. These readings were examined and recorded via Electrocardiogram (ECG). HR 1 and HR2 are considered the two covariates (independent variables) because of the fact that the responses of HR3 rely solely upon the improved heartbeats of the first two hours after birth. Figure-1. The Noisy-Wave of Heartbeats of HR1 and HR2 Figure-2. The Partial Cosine and Sine Wave Trend of HR1 and HR2 145 Academic Journal of Applied Mathematical Sciences From Figure 1 and 2, the trend of the actual readings of the rate of heartbeats exactly one and two hours after birth, that is, HR1 and HR2 for the same level of four mesokurtics (Normal bell-curves) nature possessed. The Sine and Cosine plots of the two readings (the two covariates) revealed and actualized the possessed noisy-wave (Sine and Cosine waves) of the two examined observations of babies’ heartbeats. This suggested a wave particle duality of the heartbeats. In other words, the HR1, HR2 and HR3 heartbeats are noise or tone frequencies, that is, noisy data (noisy-wave) that requested a trigonometry (Cosine and Sine) transformation or Fourier transformation as an alternative to smoothing process or modeling. Table-1. Fitted Cosine and Sine equation of Parameter  1 2  Estimate 58.6749 -0.4261 -0.0692 1.2220  +1cos  HR 2    2sin  HR1  Std. Error 0.1112 0.1539 0.1525 0.0229 t-value 527.647 -2.769 -0.454 53.27 Pr.(>|t|) < 0.0021 0.0057 0.6501 <0.0021 Global Deviance: 17.802 AIC: 5025.802 SBC: 5045.228 log Lik: -2508.901 The Maximum Likelihood estimator is 281.132 HR 3 =58.6749 0.4261cos  HR 2   0.0692 sin  HR1  Such that,  (1.2220 , 0.00052) 146 Academic Journal of Applied Mathematical Sciences Table-2. Fitted Cosine and Sine equation of Parameter  1 2  Estimate 58.6837 0.0325 -0.3650 1.2235  +1sin  HR 2    2cos  HR1  Std. Error 0.1105 0.1586 0.1619 0.0229 t-value 531.246 0.205 -2.255 53.33 Pr.(>|t|) < 0.0021 0.8378 0.0244 <2e-16 Global Deviance: 20.719 AIC: 5028.719 SBC: 5048.144 log Lik: -2510.359 The Maximum Likelihood estimator is 43.47365 HR 3 =58.6837 0.0325 sin  HR2   0.3650 cos HR1  Such that  Table.1 was subjected to the fitted function of (1.2235 , 0.00052)  +1cos  HR 2   2sin  HR1   +1sin  HR 2    2cos  HR1  , while table. 2 was based on . It was deduced that the formal fitted equation robustly accommodated the wave like nature with improved model performance of (AIC: 5025.802; SBC: 5045.228) compare to a less model performance of (AIC: 5028.719; SBC: 5048.144) by the latter. Furthermore, the global aberrances from normal nontrigonometry fitted line of the two alternate Cosine and Sine equations were relatively miniature in the two fitted equations, with a lesser miniature of global deviance of 17.802 in fitted compare to a global deviance of 17.802 in fitted  +1cos  HR 2   2sin  HR1   +1sin  HR 2    2cos  HR1  1 , which was the estimated . In collaboration with the coefficient of the rate of heartbeats exactly after an stated claims, the coefficient of hour after birth (HR1) in the formal equation hinted to be the most significant co-variate in the contributing factor to the next stability of heartbeats of babies in the next three hours and more after birth. This is due to its P-  value=0.0057 being strictly far away from the 5% chance of error. In the latter, it was the coefficient of 2 for rate of heartbeats exactly after an hour after birth (HR2) with P-value=0.0244 that was greater than the P-value=0.0057 of the latter. Figure-3. The Residual Deviance  +1sin  HR 2    2 cos  HR1  of the fitted Equations of  +1cos  HR 2    2sin  HR1  and It was noted that the two alternate fitted functions of Cosine and Sine yielded the same residual indexes in terms of the estimated quantiles density, QQ-plot and approximately the same the observed and estimated frequencies. 147 Academic Journal of Applied Mathematical Sciences Table-3. Centile of the Cosine and Sine alternate equation lines  +1cos  HR 2   Centiles 2sin  HR1  % of cases below % of cases below % of cases below % of cases below % of cases below 0.4 centile is 10 centile is 50 centile is 90 centile is 99.6 centile is 0 6 54.4211 92.5263 97.1579  +1sin  HR 2   2cos  HR1  0 6.1053 54.42105 92.8421 96.9474 Centile otherwise known as percentile has been one of the values of a statistical variable that divides the distribution of the variable into 100 groups having equal frequencies. The 99.6 percent of the values in the function  +1cos  HR 2   2sin  HR1  lies at 97.1579 centile, capturing and explaining the wave nature of the  + sin  HR 2    2cos  HR1  1 covariates above the 99.6 percent of the values in the function 96.9474 centile in capturing and explaining the wave nature of the system. that lies at Table-4. Summary of the Quantile Residuals Cosine and Sine alternate equation Functions Keys: FCC=Filliben Correlation Coefficient Table. 4 divulged the approximately equivalence of the residual variance of the two alternate wave nature of Cosine and Sine equations as well as the same residual location parameter of positive effect by  +1cos  HR 2   2sin  HR1   + sin  HR    cos  HR  1 2 2 1 , whereas adopted the negative effect of location parameter. In addition, the two alternate equations were not affected by skewedness (outliers), since their skewedness coefficients of 1.5899 and 1.5633 respectively are < 3. The Filliben Correlation Coefficient (FCC), which is use as test statistic for normal probability correlation coefficient of composite hypothesized for normality (non-normal) test; since its coefficient r  0.940 for the two equations, it implies the noisy data indicated a length of lower tail (symmetric shorter-tailed) of 94% with 5% level of significant as maintained by Filliben [19]. The CoxSnell residual and Cragg-Uhler coefficients of (0.85 and 0.54) and (0.85 and 0.54) respectively for assessing the goodness-of-fit for the heartbeats’ regression hinted the formal fitted function accommodated the wave nature of the heartbeat of the babies to 85% and the latter was able to explain to 54% . The two residual indexes are alternate 2 index for the Pseudo- R . 4. Conclusion From the anteceding, it is necessary to do ascertain the wavy trend of regressors (covariates) to throe the level of noisy and frequency tone. By doing so, it allows a clear-cut whether the distributional property of the noise is a trigonometry (Cosine, Sine) of two or more covariates linear regression. The rate of heartbeats exactly after an hour, exactly after two hours and exactly three hours after birth followed a noisy Cosine and Sine wave nature trigonometry regression. The alternate Cosine and Sine two covariates was subjected to the heartbeats’ observations such that the fitted equation of  +1cos  HR 2   2sin  HR1   + sin  HR    cos  HR  captured the wave nature than the alternate 1 2 2 1 function of . It is to be noted that this research could be extended to more than two co-variates linear trigonometry regression, that is generalized multiple trigonometry regression. References [1] [2] Dette, H., Neumeyer, N., and Pilz, K. F., 2006. "A simple non-parametric estimator of a monotone regression function." Bernoulli, vol. 12, pp. 469– 490. Prahutama, A., Suparti, D., and Utami, T. W., 2018. 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