ABSTRACT is communicated as a condition or



Intensive Care Units have been
carrying vital importance in these days. These hospital units, affecting most
people’s lives, have recently become more crowded. Due to this crowd, patients
who have to enter intensive care units unfortunately get vital risks because of
not getting access to these units. The greatest reason for the occurrence of
this condition is that the time to be spent in intensive care units is not
predictable without modelling the system. In this study, we will model the
intensive care units with continuous absorbing markov chain structure and
estimate the length of stay at intensive care unit by using phase-type
distribution. Study will follow the order as gathering data of the system,
modelling the markov chain with apporapiate amount of states then applying the
Phase Type Distribution to the model. At the end, It will be predictable that
legth of stay at intensive care units.

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Regression Models

regression incorporates bend fitting, expectation (forecasting), demonstrating
of causal connections, and testing logical theories about connections between
variables. Regression examination is a technique for exploring useful
connections among variables that is communicated as a condition or a model
interfacing the reaction or ward variable and at least one logical or indicator

Linear Regression

              If we distribute our data
according to their values on the x-y coordinates, they have a distance between
each other. The goal is to draw a line that passes through all the data and
passes the most correctly. The correct drawing here is for regression in a
linear structure. Our goal is to simulate the distribution of our train data on
the plane as a mathematical model so that we can find the correct regression model.
For example, if you have a fluctuation set of data, using linear regression
will not make sense. Using logistic regression for this will help you achieve
more successful results. Because the logistic regression tries to capture the
data logarithmically on the plane with curve.


According to Combes, Kadri and
Chaabane (2014)’s study about predicting the length of stay at emergency
department. They have contucted two different linear regression model, in the
first model there are 4 variables and in the second there are 8 variables. In
the second model, accuracy of the model is more reliable because of variable
amount highness. It is easy to observe that more the variable amount in linear
regression model provides better accuracy of the model. In order to best
fitting it is require to choose right variables with many amounts. From this
situation Combes, Kadri and Chaabane (2014) states linear regression suffers
from the well linearity. According to their reliability test there was ±2 hours
error. Moreover, basic linear regression method is not valid for non-linear variables.
And also classification and regression are two methods that used for prediction
about discrete outcomes(Tan, 2007). There is also another study about
predicting lenght of stay with using linear regression method. With respect to
Badreldin(2013), the linear regression model evaluated as failed in accuracy  of prediction. Study was also suggest that
reconsideration of the variables could gave better prediction. From the
Pourhoseingholi’s Study (2009) there was a fitting comparison between linear
regression and quantile regression. At the end, he stated that linear
regression remained incapable when the comparing with quantile regression.

Combes C., Kadri F. and Chaabane S.(2014, November 5).
Simulation- MOSIM’14. Retrieved from

Tan, P. (2007). Introduction To
Data Mining. Pearson Education

Badreldin, A. M., Doerr, F., Kroener, A., Wahlers, T., &
Hekmat, K. (2013). Preoperative risk stratification models fail to predict
hospital cost of cardiac surgery patients. Journal of Cardiothoracic
Surgery, 8, 126. http://doi.org/10.1186/1749-8090-8-126


Pourhoseingholi,, M. A., Pourhoseingholi, A., Vahedi, M., Dehkordi,
B.M., Safaee, E., Mserat, E., Ghafarnejad, F. & Zali, M.R. (2009). Comparing linear regression and quantile regression to
analyze the associated factors of length of hospitalization in patients with
gastrointestinal tract cancers. Italian Journal Of Public Health.
6,2. http://dx.doi.org/10.2427/5787

Logistic Regression

              It is a
regression model and the dependent variable is categorical. Moreover, Logistic
regression is a statistical strategy for breaking down a dataset in which there
are at least one free variables that decide a result. The result is measured
with a two conceivable variable. The objective of logistic regression is to
locate the best fitting model to present the relationship.For estimating the
probabilities, its using the logistic function in other words logistic curve. It
tries to catch data on an algorithmic curve. This leads us to a higher
prediction success in up-and-down data.

The author Austin (2010) compared at
his study that performances of classification techniques for prediction purpose
and he found that the logistic regression has gave better result than regression
tree, multi-layer perceptron and radial basis function with using Receiver Operating
Characteristic curve and Hierarchical Cluster Analysis performance measure. In
addition to this information Kurt, Ture and Kurum (2008) states that comparison
between acuracy of regression trees with logistic regression model for
predicting length of stay in hospital. As a result of his study, logistic
regression is more accurate than regression trees. Sharma, Dunn, O’Toole,
Kennedy (2015) have also studied length of stay with using logistic regression,
they have used a psychiatric hospital with population of 1.2 million. Moreover,
they also used IBM SPSS v21 statistical analysis software. They found that the
result does not seem to mirror the requirements of an intense mental
affirmation benefit and may speak to the absence of group emergency determination
assets. As a measure the modular completed length of stay shows something about
what is occurring in an administration, yet as a measure of focal propensity it
is of restricted esteem and needs sensitivity emergency determination assets.
In short, sensitivity of the logistic regression is not adequate for Length of
Stay studies and not representing the real system.

Austin, P.C., Tu, J.V., Lee,
D.S., (2010). Logistic regression had superior performance compared with
regression trees for predicting in-hospital mortality in patients hospitalized
with heart failure. J. Clin. Epidemiol.
63, 1145–1155.

Kurt, I., Ture, M., Kurum, A.T.,
2008. Comparing performances of logistic regression, classification and
regression tree, and neural networks for predicting coronary artery disease.
Expert Syst. Appl. 34, 366–374.

Sharma, A., Dunn, W., O’Toole,
C., & Kennedy, H. G. (2015). The virtual institution: cross-sectional
length of stay in general adult and forensic psychiatry beds. International Journal of Mental Health
Systems, 9, 25. http://doi.org/10.1186/s13033-015-0017-7


Machine Learning Regression

used regression tree algorithms such as CART and CHAID. CART is used for
non-parametric in other words non-linear regression tree. CHAID is an statistical
approach that can derive regression trees.


One of the machine learning study has
occured in a Federal hospital because of its variable richness’ positive
effects on regression models (Hulshof, 2013). The first variable was prediction
of patients’ length of stay and other one was prediction of readmission (Kelly,
2013). From the studies of this Federal hospital, Pendharkar and Khurana (2014)
states that the ANCOVA (Analysis of Covariance) model tests linear connections,
and significantly show that non-linear machine learning models may perform
marginally superior to linear models. It means that machine learning regression
fits better than linear regression to real data. Moreover, when they looked to Root-Mean-Square
error, there was no better regression technique than machine learning regression.
However, their sample size was limited with small section of the hospital. In
short, it is not generalizable for other studies.

Hulshof, P. J. H.; Boucherie, R.
J.; Hans, E. W.; Hurink, J. L. (2013): Tactical resource allocation and
elective patient admission planning in care processes, Health Care Management
Science, 16(2), pp. 152–166.

Kelly, M., Sharp, L., Dwane, F.,
Kelleher, T., Drummond, F. J.& Comber, H. (2013): Factors predicting
hospital length-of-stay after radical prostatectomy: a population-based study, BMC Health Services Research, 13(1),

Pendharkar, P.C. & Khurana,
IN PENNSYLVANIA FEDERAL AND SPECIALTY HOSPITALS. International Journal of Computer Science and Applications, 11, 3. http://www.tmrfindia.org/ijcsa/v11i33.pdf


Markov Chain

activities can be modelled by using Markov Chains. ?t describes a system with
different states and transitions between them. Markov Chain has memoryless
property thats why the next state depend on only the current state not the
previous states. When we look at operational view, markov chain can be describe
with different states. The Markov chains evaluated utilizing the improvement
datasets were joined with the initial state probability vector to create the expected
length of stay in every goal for Intensive Care Units or Hospitals. In discrete
markov chains, it is not possible calculate in hours or minutes like continious
property. However, if we turn the data only days and which days spended in
which department of hospital. It is possible to create an discrete markov chain
with absorbing state.

According to Perez, Chan and Dennis(2006)’s
studies about length of stay at intensive care unit with using Markov Model
with absorbing state in other words “first-step analysis”(Kapadia, 2000). The
markovian model lack of goodness of the fit to real length of stay data for
some departments in the hospital. The markovian model as a discrete property
have not calculated the discharges at middle of the day. In short, there has to
be an continious property markov model for able to model all kind of service
and waiting times. According to Perez’s study (2006), there is also positive
side of the markov model with discrete property such as high correlation
between utilization and length of stay. However, its outcomes are mostly not
reliable and not fitting the real data in the continious matter because of
limitations. Moreover, according to Bhat (2002) sequence size is important for
markov chain structure in order to best fitting to the real data. However,
Perez’s study (2006) has only 30 sequence which are days of a month. Because of
this reason markov model needs too much sequences (states) for fitting the real
data better.

Perez, A. Chan, W. & Dennis,
R.J. (2006): Predicting the Length of Stay of Patients Admitted for Intensive
Care Using a First Step Analysis. Health
Serv Outcomes Res Methodol; 6(3-4): 127–138.

Kapadia, A.S., Chan, W.,
Sachdeva, R., Moye, L.A., Jefferson, L.S.(2000) Predicting duration of stay in
a pediatric intensive care unit: A markovian approach. European Journal of Operational Reseach;124:353-359.

Bhat, U.N., Miller, G.K.(2002);
Elements of applied stochastic processes. Third Edition. John Wiley & Sons Inc; Hoboken, New Jersey.