On Two Covariates Cosine and Sine Noisy-Wave Trigonometry Regression of Heartbeats

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 (HR1), 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     1 2 2 1 + cos HR sin HR     optimally captured the wave observations and robustly outstripped the alternate Cosine and Sine equation line of     1 2 2 1 + sin HR cos HR     .

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][5][6][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][12][13][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][16][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.

The Two Covariates Alternate Cosine and Sine Function Trigonometry Regression
Given a linear regression model function with random error variables i For i  are uncorrelated noisy-wave standardized random variables with mean zero and unity variance.  (7), (8) and (9) gives the system of equations; Re-arranging and converting to matrix form gives,

Coefficient of Determination for the two Covariates Trigonometric Regression
The coefficient of determination being denoted by; sin cos

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 (HR 3 ). 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 HR 2 are considered the two covariates (independent variables) because of the fact that the responses of HR 3 rely solely upon the improved heartbeats of the first two hours after birth. 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, HR 1 and HR 2 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 HR 1 , HR 2 and HR 3 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.   , which was the estimated coefficient of the rate of heartbeats exactly after an hour after birth (HR 1 ) 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 (HR 2 ) with P-value=0.0244 that was greater than the P-value=0.0057 of the latter.  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.