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.
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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,
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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 )
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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
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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;
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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
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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)
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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.
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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.
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