Factors of bio-rhythms and applications to *PCA

Introduction

The basic content of this paper target upon identifying the fundamental causes those are responsible for the various grades achieved by different students on taking up the tests on different subjects on different time slots. It took us a long time to identify some of the major causes which were responsible for the so-called variation between the predicted marks and marks received on different examinations given on different times during a week's slot. The Bio-rhythm is, as understood, a mental drive which is a gross of four broad divisions depicted below, 1. Physical Bio-rhythm 2. Emotional Bio-rhythm 3. Intellectual Bio-rhythm and 4. Spiritual Bio-rhythm As advanced studies reflect that these main divisions, stated above, have also further subdivisions. It is principally hypothesised that, in general, all these main characteristics are the function of the birth date and more precisely the time of birth. To be more precise all these factors experienced as a gross effect on a given time are associated with the total time elapsed after the birth of the individual. Total number of days that the subject, without any major illness, has passed from the time of birth up to or a day before the test is administered is an important factor. It is accepted that mental behaviour and attitude which are gross outcomes of Bio-rhythms directly reflect the individual performance of the subject. Some of the dominating characteristics are immutable but there exists some more characteristics which by imparting regular instilling pre-designed lessons certainly elevates some features to qualify for better standards on the tests to follow. The content of the first paper on the same line has been and we have focussed total contemplation on collection of primary data on different time frames.

PCA on Bio-rhythmic Components

Principal component analysis (PCA) has been known to be one of the most important processes derived from application of linear algebra. PCA is used abundantly in all forms of analysis -from neuroscience to computer graphics because it is a simple, non-parametric method of extracting relevant information, a simplified version obtained by bypassing confusing statistical routines on primary data sets. PCA provides a structural routine indicating 'how to reduce a complex multi-variate data set to a data of fewer state / variables' and dimension. At this stage it is noteworthy that these new variable do not conform in nature with those of original variables. So it is really left to the discretion of the studied writer to interpret and then to apply the new variables to the original primary data. Thus the reduction of variables is liable to cause more profound work of evaluation and interpretation. We follow the following procedure in order to generate a tabular data.

(1) We take a randomly selected sample of 16 students from a group of 120 students studying in the same standard of one school providing same educational environment. According to the assumption [1] we know their birth dates and approximate birth time.

(2) The students are well informed about the date and the time they will be taking up the examination on a particular subject.

(3) We define a parameter't' as follows. t = Date of examination -Birth Date of student. [ 't' is a real value.] (4) We convert this real value using the following formula, shown below*, associating 't' value in to scores corresponding to four different components. All these scores are in the interval [-1, 1]. [Note: Negative value is an indication that the subject on that time does not exhibit positive attitude on that features and vice versa.]

Emotional Bio -rhythm Score = sin (

Intellectual Bio -rhythm Score = sin

Spiritual Bio -rhythm Score = sin

Tabulated format of these readings is,

Calculation of range conversion for Bio-rhythms from [-1, 1] to [1,100]:

For sake of simplicity let us consider range conversion from [-1, 1] to [1,100] using formula,

With this conversion data is shifted to, The same has been shown below in concise form for further statistical process.

Shifting data with respect to mean value

In order to make calculation simpler we subtract mean of each bio-rhythm from original figures of the same Bio-rhythm; technically saying we find deviations from mean. The deviations have been tabularized below. 𝑛-1

and 𝐶𝑜𝑣(𝑋, 𝑋) = 𝑉𝑎𝑟(𝑋). We extract the entries from above matrix A and on application of the above formula we get the following matrix Cov.(A). We find latent values associated with this matrix and corresponding latent (Eigen) vectors. The Eigen vectors will find some invariant direction and the Eigen values giving the contraction factor for the vector in connection to the one of original matrix. These are found using characteristic equation derived as |𝑐𝑜𝑣(𝐴) -𝜆

𝐶𝑜𝑣(𝐴

Energy associated to Eigen values of Cov. (A)

At this stage we define energy of each Eigen value as its own percentage in the 'less than type of cumulative contribution' on the scale of a centage scale. To achieve this we have two criteria.

(1) Selection of variables will follow decreasing order of Eigen values.

(2) The cut off criteria will be about 70 % in accumulation. As a virtue of these two criteria, we are able to fix up the first two leading variates who dominates in the set of given four Variates; which are X 1 and X 2 .

To define Principal component we will consider Feature vector as, This feature matrix F such defined will transform original data matrix A in 2 dimensional subspace re-oriented Data Matrix A New defined as the regular product of two matrices A and F taken in the order as they appear.

Statistical analysis using Paired t-test

Next goal is to test data statistically with help of paired t-test, for which we will consider marks predicted by bio rhythms and marks predicted by Principal component Analysis. Converting these two set of marks in same range of [1,100]

Comment and Application

Now the next important question arises about the reliability of the marks derived by using the new parameters. To justify this situation we need to apply 'paired 't' test'. We apply it on the data that we have cited above and in order to generalize our conclusion we have verified the effectiveness in many cases by taking different samples and of different sizes. A few of such Cases have been cited in the annexure and even some more in thesis that follows. To the height of that acceptance and rejection cases have been strictly examined at 0.01 level of significance. All these facts are duly equipped in the thesis. We proceed to justify our assumption statistically as follows

Paired t-test on Bio rhythm Marks-I and PCA Marks-II

In the context of our data we have the total routine for application of 'paired't' test. We define the null hypothesis as follows. ). This falls in rejection Region. We reject H 0 .We accept H 1 and conclude that in this case, PCA Marks-II (𝑑 2𝑖 ) is fairly closer to the actual marks obtained by the students. This establishes fairness in associating PCA to contraction of original variables.

Conclusion

All these facts and figures that we have generated by experimental work on different groups of the students on different time interval assert to astounding fact that application of PCA to transferring multiple variables of Bio-rhythmic characteristics, which are basically subtle in context to physical gestures and expressions, play a dominating role. New variables generated during the process prove justifying and apportioned quantum of original variables.

Vision

With some surpassing comments we would like to throw this problem to open mindedness with little reservation on our side. 1. Should we incorporate more bio-rhythms for explicit results? 2. We have a conjecture that 'Do we tend to accuracy as we include more bio-rhythms?'

References

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