Fetal health is a very important aspect in this day and age. The ability to predict the health status based on preliminary examinations could help a lot in saving lives, especially if it is done by an artificial intelligence system, which would streamline the whole process. The issue is complicated by the multitude of diferent algorithms that vary in accuracy and execution time that can be used for such predictions. This project compares three popular artificial intelligence systems: k-nearest neighbor algorithm, naive Bayes classifier, soft sets to show which one of them will prove to be the best. The final results are shown in comparison graphs. In this article, the mathematical issues related to these algorithms and examples of their application in this task are presented.
care problems. All mentioned solutions use diferent
approaches but the incoming data are analyzed mainly in
The reduction of child mortality is one of the key indi- two ways - taking the data, or only the extracted features.
cators of human progress. It is expected that by 2030, Such solutions were analyzed and discussed in terms of
countries end preventable deaths of newborns and chil- data analysis in [
Cardiotocographs (CTGs) is a simple and cost-accessible option to assess fetal health, allowing healthcare professionals to take action in order to prevent child and maternal mortality. The equipment itself works by sending ultrasound pulses and reading its response, thus shedding to check which one of them will obtain the highest light on fetal heart rate (FHR), fetal movements, uterine accuracy. contractions, and more.
Artificial intelligence is very important in all in all the
aspects of modern life [
In [
(1) is the value from the sample vector. id 1 baseline value 120.0 · · · · · · hist._tend
1.0 7–11 (3) (4) 2.1. A calculation example for our base Assuming that our sample vector is shown in Tab. 2.
And having a training set, the Minkowski metric was used to calculate the distance between the sample vector and the analyzed vectors of the training set: 2(, ) = ︃( 21 ∑︁ | − |2 =1
)︃1/2 where is the value from the column of the sample vector, and is the value of the column of the currently analyzed vector from the training set. In project was assumed the value m = 2. After analyzing the entire training set, the distances between the sample vector and the training set vectors are sorted. Next, a neighborhood is selected for the specified k nearest neighbors based on the previously calculated distances, and the most frequent label is selected.
A naive Bayes classifier based on Bayes’ theorem: (|) = (|) ()
() is particularly suitable for problems with very many dimensions in the input. Despite the simplicity of the method, it often performs better than other very complex classifier methods. The classifier learns by analyzing a set of learning data for which the correct classes are given.
The model includes (| )-the probability of observations for diferent class labels and ( )-a prior probability, which is the probability calculated before the random experiment is performed. This is the classical probability calculated in the same way as the overall probability. Bayes’ rule is used to determine ( |)-the conditional probability of a class for a given observation.
The label for which the probability is highest is chosen.
The model is called naive because it assumes a very strong simplification that says that for a fixed class label all are features are independent of each other. 2.1.1. A calculation example for this base Classify the object X = (133.0, ..., 0.0) for this table (decision means 1.0 - Normal, 2.0 - Suspect, 3.0 - Pathological) what is shown in Tab. 2.
After assigning a column to a 1.0/2.0/3.0 group, it is needed to determine which probability is higher.
• ( = 1|) e.g. P(decision = 1.0| base. val. =
133.0, ..., hist._tend .= 0.0), • ( = 2|) e.g. P(decision = 2.0| base. val. =
133.0, ..., hist._tend .= 0.0), • ( = 3|) e.g. P(decision = 3.0| base. val. = 133.0, ..., hist._tend .= 0.0)
So there is the need to calculate ( = 1) · ( = 1|), ( = 2) · ( = 2|) and ( = 3) · ( = 3|) and compare the results.
1. ( = ) = 13 2. ( = []| = ) = [] where under i insert the number of the next column, under [] the value of that column, and under [probability] the calculated probability value, ∈ {1, 2, 3}.
Then it is needed to substitute the calculated values into the formula:
21 ( = · ∏︁ ( = | = )) = [ ] (5)
=1
It is checked which result (probability value) is greater which determines which class the object X should be classified in.
• Soft sets are defined by a set of properties which, however, are not completely precise (e.g. instead of saying that it is 15∘ C outside one can say that it is warm). The general theory of soft sets is very similar to that of fuzzy sets but much simplified.
Thanks to the fact that there are no restrictions on the approximate description of the object, it is very easy to apply this method in practice.
Mathematical assumptions: U - the class of all elements considered. E - parameter class P(U) - set of all subsets of U ⊆ - components considered So the pair (F,A) is called a soft set, where F is an assignment: : → ( )
(6)
− −
E - a set of parameters that determine the health of the fetus
A = {baseline value, ..., hist._tend.} − The class consists of three health states. U = {Normal, Suspect, Pathological} E = {1, · · · , 21 } What follows is the definition of subsets specifying the characteristics that health states satisfy given assumptions , where ∈ 1 : 21: F(1) = {Normal} F(2) = {Normal, Suspect} ...
F(20) = {Pathological} F(21) = {Normal} Based on this, a soft set table can be created (see Tab. 3).
Due to the simplicity of this system, subsets of a soft set can be determined very easily. E.g. (F, A) is a subset of(G,B)if ⊂ and ∀∈ ∈ 1 and (∈) and (∈) are the same. Two soft sets will be equal when the previous poset condition is satisfied and when ( G,B) is also a poset of (F, A).
The project was used a ready-made database, which can be downloaded at [20].
The database consists of one table with 22 columns. 21 columns are numerical data defining the CTG study and the last text column defines the fetal status.
Description of the columns: • ’baseline value’- FHR baseline (beats per minute). • ’accelerations’ - Number of accelerations per second. • ’fetal_movement’- Number of fetal movements per second. • ’uterine_contractions’ - Number of uterine contractions per second. • ’light_decelerations’- Number of light decelerations per second. • ’severe_decelerations’ - Number of severe decelerations per second. • ’prolongued_decelerations’- Number of prolonged decelerations per second. • ’abnormal_short_term_variability’- Percentage of time with abnormal short term variability. • ’mean_value_of_short_term_variability’ - Mean value of short term variability. • ’percentage_of_time_with_abnormal_long_term_ variability’ - Percentage of time with abnormal long term variability. • ’mean_value_of_long_term_variability’- Mean value of long term variability. • ’histogram_width’ - Width of FHR histogram • ’histogram_min’- Minimum (low frequency) of
FHR histogram.
10 tests were performed for each system and extracted the average values, obtaining the following results: • ’histogram_max’ - Maximum (high frequency) of For all k achieved average precision values were around
FHR histogram. 90, which is considered to be a very good result, and it • ’histogram_number_of_peaks’ - Number of his- is concluded that this algorithm performs very well for togram peaks. the database. Unfortunately, a big disadvantage for this • ’histogram_number_of_zeroes’ - Number of his- algorithm is the execution time against a large database.
togram zeros. In each iteration, for each validation sample, distances • ’histogram_mode’ - Histogram mode. to nearest neighbors are recalculated and sorted anew, hence the execution time for the database, for one itera• ’histogram_mean’- Histogram mean. tion was about 173 seconds. • ’histogram_median’- Histogram median. In the naive Bayes classifier algorithm, the final results • ’histogram_variance’- Histogram variance. are not as satisfactory as in the k-nearest neighbor clus• ’histogram_tendency’ - Histogram tendency. tering algorithm. The average precision obtained for 10 iterations was about 82%. This is not a bad result, but compared to the previous algorithm, a significant diference can be seen. It was also checked if the algorithm would improve its final result when splitting the data 80:20 for training and validation data. Unfortunately, the precision did not improve. The advantage of this ∙ A k-nearest neighbours clustering algorithm: algorithm is the speed of its execution, oscillating around 0.3 seconds. − for = 1 : 90.61% For classification using soft sets, an average obtained − for = 2 : 90.24% accuracy was oscillating 82%. The execution time of − for = 3 : 90.39% the algorithm for 1 iteration was about 4 seconds. In − for = 3 : 90.39% this method, quarterlies were used due to the strongly ∙∙ SNoafitvseeBtsa:y8es2-.n7e7a%rest-neighbour classifier: 82.43% oovfeGthrrleaappaplhgisnogrciotchmhmaprsaarucitsneegrdistihtniecosauocrfcputhrroeajcCeycTtaGanrdseuesrxhveoecywusntaioinnnalFytiizgme.de1. and 2.
For each of them, a certain number of tests were performed and extracted average values to better specify the ifnal result.
In the k-nearest neighbor clustering algorithm using the Minkowski metric, 10 tests were performed each for diferent k, where k was assumed to be 1, 2, 3, and 4.
After an accuracy comparison, the k-nearest-neighbor clustering algorithm proved to be the best. However, its execution time was incomparably longer than in the other cases. So if we look at the overall optimality of the performance, the Bayes algorithm with a slightly worse