Disease Prediction with a Maximum Entropy Method

1 Introduction

Disease prediction is an effective way to assess a person’s health status. Studies have shown that in many cases, there are identifiable indicators or preventable risk factors before the onset of the patient’s disease. These early warnings can effectively reduce the individual’s risk of disease. Theoretically, this can reduce the number of treatments needed and increase the necessary effective interventions. However, the combination of problem factors caused by different diseases and the patient’s past medical history are so complicated that no doctor can fully understand all of this. Currently, doctors can use family and health history and physical examinations to estimate the patient’s risk and guide laboratory tests to further evaluate the patient’s health. However, these sporadic and qualitative ”risk assessments” are usually only for a few diseases, depending on the experience, memory and time of the particular doctor. Therefore, the current medical care is after the fact. Once the symptoms of the disease appear, it is involved, rather than actively treating or eliminating the disease as soon as possible.

Today the prevailing model of prospective heath care is firmly based on the genome revolution. Indeed, technologies ranging from linkage equilibrium and candidate gene association studies to genome wide associations have provided an extensive list of disease-gene associations, offering us detailed information on mutations, SNPs, and the associated likelihood of developing specific disease phenotypes.

The basic assumption behind the research is that once we have classified all disease-related mutations, we can use various molecular biomarkers to predict each individual’s susceptibility to future diseases, thus bringing us into a predictive medicine era. However, these rapid advances have also revealed the limitations of genome-based methods. Considering that the signals provided by most disease-related SNPs or mutations are very weak, it is becoming increasingly clear that the prospect of genome-based methods may not be realized soon.

Does this mean that prospective disease prediction methods must wait until genomics methods are sufficiently mature? Our purpose is to prove that the method based on medical history provides hope for the prospective prediction of disease.

In this paper, we mainly study the disease prediction and comorbidity of diseases. Our approach is distinctly different in that we are trying to build a general predictive system which can utilize a less constrained feature space, i.e. taking into account all available demographics and previous medical history. Moreover, we rely primarily on ICD-10-CM (International Classification of Diseases, Tenth Revision, Clinical Modification) codes (see Section 2) for making predictions to account for the previous medical history, rather than specialized test results.

2 Data

2.1 Source Data and Population

Our database comprises the medical records of 354,552 patients in China with a total of 2,904,257 hospital visits. The data was originally compiled from Insurance claims during 2007 to 2017. Such medical records are highly complete and accurate, and they are frequently used for epidemiological and demographic research.

The input for our methods consists of each patient’s personal information, such as gender, birthday, treatment-date, and diagnosis history, provided per patient’s visit. Each data record consists of a hospital visit, represented by a patient ID and a diagnosis code per visit, as defined by the International Classification of Diseases, Tenth Revision,Clinical Modification(ICD-10-CM). The International Statistical Classification of Diseases and Related Health Problems (ICD) provides codes to classify diseases and a wide variety of signs, symptoms, abnormal findings, social circumstances, and external causes of injury or disease. It is published by the World Health Organization.

Each disease or health condition is given a unique code, and can be up to 6 characters long, such as A01.001. The first character is a letter while the others are digits. ICD-10 codes are hierarchical in nature, so the 6 characters codes can be collapsed to fewer characters identifying a small family of related medical conditions. For instance, code A01.001 is a specific code for typhoid fever. This code can be collapsed to A01.

Moreover, we classify diseases of the same category into one class. For example, A90 is the code for Dengue fever (classical dengue) and A91 is the code for Dengue hemorrhagic fever. We classify them into the same class named F_A90. Thus, the 20 thousand origin ICD-10 codes are classified into 429 classes.

A sample patient medical history is shown in Table 1. Each line represents one hospital visit. Demographic data are also available.

patient_id	gender	treatment_date	code
14532	F	2011-10-15	F_M47
14532	F	2011-11-19	F_N91
14532	F	2012-10-09	F_L20
14532	F	2012-10-19	F_N60
14532	F	2013-05-08	F_B37
14532	F	2013-06-04	F_H10
14532	F	2013-06-15	F_K04
14532	F	2013-08-23	F_L20

Table 1: Medical History Sample

In our medical database, the number of visits per patient ranges from 1 to 491, with a median of 4. Also, the average is 8.19. Table 2 shows the 20 most prevalent diseases in our database.

Disease	Prevalence
Acute upper respiratory infection	20.88%
Hypertension and its complications	7.35%
Dermatitis and pruritus	3.75%
Gastritis and duodenitis	3.49%
Chronic bronchitis	3.28%
Pulp, gum, and alveolar ridge diseases	3.15%
Hard tissue disease of teeth	2.73%
Abnormal uterine and vaginal bleeding	2.42%
Chronic rhinitis, nasopharyngitis and pharyngitis	1.99%
Non-infectious gastroenteritis and colitis	1.97%
Chronic ischemic heart disease	1.93%
Inflammation of the vagina and vulva	1.72%
Pneumonia	1.63%
Abnormal thyroid (parathyroid) function	1.62%
Other diabetes	1.56%
Backache	1.54%
Acute lower respiratory infection	1.51%
Cervical disc disease	1.44%
Type II diabetes	1.37%
Female pelvic inflammatory disease	1.18%

Table 2: 20 Most Prevalent Diseases

2.2 Quantifying the Strength of Comorbidity Relationships

In order to measure the correlation from disease comorbidity, we need to quantify the intensity of disease comorbidity by introducing the concept of distance between the two diseases. One difficulty of this method is that there are biases in different statistical measures, which overestimate or underestimate the relationship between rare or epidemic diseases. Given that the number of diagnoses (prevalence) for a particular disease follows a long tail distribution, these biases are important, which means that although most diseases are rarely diagnosed, a small number of diseases have been diagnosed in a large part of the population.

Therefore, quantifying comorbidity usually requires us to compare diseases that affect dozens of patients with diseases that affect millions of patients.

We will use two comorbidity measures to quantify the distance between two diseases: The Absolute Logarithmic Relative Risk (ALRR) and $ϕ$ -correlation( $ϕ$ ).

The Emperical Relative Risk of observing a pair of diseases $i$ and $j$ affecting the same patient is given by

E R R_{i j} = \frac{C_{i j} N}{P_{i} P_{j}},

where $C_{i j}$ is the number of patients affected by both diseases, $N$ is the total number of patients in the population and $P_{i}$ and $P_{j}$ are the prevalences of diseases $i$ and $j$ .

E R R_{i j} = \frac{\frac{C_{i j}}{N}}{\frac{P_{i}}{N} \cdot \frac{P_{j}}{N}} .

Thus, the Relative Risk is defined as

R R_{i j} = \frac{p_{i j}}{p_{i} p_{j}},

Here, $p_{i j}$

is the transition probability from disease

i

to disease

j

and

p_{i}

is the incidence probability of disease

i

. The Absolute Logarithmic Relative Risk is defined as

M_{i j} = ∣ ∣ ∣ log (\frac{p_{i j}}{p_{i} p_{j}}) ∣ ∣ ∣ .

The emperical $ϕ$

-correlation, which is Pearson’s correlation for binary variables, can be expressed mathematically as

\frac{C_{i j} N - P_{i} P_{j}}{\sqrt{P_{i} P_{j} (N - P_{i}) (N - P_{j})}} = \frac{\frac{C_{i j}}{N} - \frac{P_{i}}{N} \cdot \frac{P_{j}}{N}}{\sqrt{\frac{P_{i}}{N} \cdot \frac{P_{j}}{N} (1 - \frac{P_{i}}{N}) (1 - \frac{P_{j}}{N})}} .

Therefore, the $ϕ$ -correlation is defined as

ϕ_{i j} = \frac{p_{i j} - p_{i} p_{j}}{\sqrt{p_{i} p_{j} (1 - p_{i}) (1 - p_{j})}} .

These two comorbidity measures are not completely independent of each other, as they both increase with the number of patients affected by both diseases, yet both measures have their intrinsic biases. For example, RR overestimates relationships involving rare diseases and underestimates the comorbidity between highly prevalent illnesses, whereas $ϕ$ accurately discriminates comorbidities between pairs of diseases of similar prevalence but underestimates the comorbidity between rare and common diseases.

3 Methodology

In this section. We will formulate the maximum entropy method we used to predict disease risk.

3.1 Some notations

Suppose there are $n$ diseases and $N$ records. Let us use $d_{i}$ to denote disease $i$ . A record is a pair of diseases $(d_{i}, d_{j})$ which means that there is a patient with a diagnosis of disease $d_{j}$ simultaneously or after disease $d_{i}$ . Let us use $S_{k}$ to denote record $k$ ( $1 \leq i \leq n, 1 \leq k \leq N$ ).

Assume that $S_{k} = (f (k), g (k))$ . Here, $f$ and $g$ are maps.

f, g : {1, 2, \dots, N} \to {1, 2, \dots, n} .

$f (k)$ is called the first disease and $g (k)$ is called the second disease in record $k$ .

In this paper, we assume that $f$ and $g$ are surjective. If $f$ is not surjective, we can remove the diseases with indexes in ${1, 2, \dots, n} ∖ Range (f)$ from the medical history. Then the surjective assumption can be satisfied for $f$ . The same can be done for $g$ .

Assume that

Denote by

χ_{C} = {\begin{matrix} 1, x \in C 0, x \notin C \end{matrix} .

Let $A = X Y, B = Y X$ . Then

A_{i j} = N \sum k = 1 X_{i k} Y_{k j} = N \sum k = 1 χ_{{f (k) = i, g (k) = j}}

is the number of patients who suffer from disease $i$ before disease $j$ .

B_{k m} = n \sum j = 1 Y_{k j} X_{j m} = n \sum j = 1 χ_{{f (m) = j, g (k) = j}} .

$B$ is a matrix with entries ranged in ${0, 1}$ . $B_{k m} = 1$ if and only if there is a disease $j$ such that the patient in record $k$ suffer a second disease $j$ and the patient in record $m$ suffer a first disease $j$ . Our task is to evaluate the transition probability from record $k$ to record $m$ .

3.2 Entropy for Markov Chains

Suppose $M$ is a non-negative matrix. If for each $k, m \in {1, 2, \dots, N}$ , there exists $l \geq 1$ such that $(M^{l})_{k, m} > 0$ , then $M$ is said to be irreducible.

Now we define the entropy for Markov chains. A matrix

B

is called a skeleton matrix if its entries are either

0

1

. A non-negative matrix

P

as called a Markov transition matrix if

N \sum m = 1 P_{k, m} = 1, \forall k = 1, 2, \dots, N .

Moreover, if $P_{k, m} > 0 \Leftrightarrow B_{k, m} = 1$ , then $P$ is called the Markov transition matrix associated with the skeleton matrix $B$ .

For a non-negative vector

p = (p_{1}, \dots, p_{N})

, if

N \sum k = 1 p_{k} = 1, p P = p,

then $p$ is called a stationary distribution of $P$ .

For a non-negative matrix $W = (w_{k, m})$ , if

N \sum k, m = 1 w_{k, m} = 1,

and for all $1 \leq k \leq N$ ,

N \sum m = 1 w_{m, k} = N \sum m = 1 w_{k, m} .

Then $W$ is called a Markov weight matrix.

Here are some connections between Markov transition matrix and Markov weight matrix.

For a Markov transition matrix $P$ with stationary distribution $p$ . Define

w_{k, m} = p_{k} P_{k, m} .

Then it is easy to verify that $W = (w_{k, m})$ is a Markov weight matrix.

On the contrary, given a Markov weight matrix $W = (w_{k, m})$ , set

p_{k} = N \sum m = 1 w_{k, m}, P_{k, m} = \frac{w_{k, m}}{p_{k}} .

Then $W$ is the Markov weight matrix associated with $P$ .

Now we define the entropy for a Markov transitin matrix. First, let us consider the chain with length $l$ .

	$H_{l} (P)$	$= - N \sum i_{0} = 1 N \sum i_{1} = 1 \dots N \sum i_{l} = 1 p_{i_{0}} P_{i_{0}, i_{1}} \dots P_{i_{l - 1}, i_{l}} log (p_{i_{0}} P_{i_{0}, i_{1}} \dots P_{i_{l - 1}, i_{l}})$
		$= - N \sum i_{0} = 1 p_{i_{0}} log (p_{i_{0}}) - l N \sum k = 1 N \sum m = 1 p_{k} P_{k, m} log (P_{k, m}) .$

So the entropy for a chain with unit length is defined as

H (P) = lim l \to \infty \frac{H_{l} (P)}{l} = - N \sum k = 1 N \sum m = 1 p_{k} P_{k, m} log (P_{k, m}) .

3.3 Maximum Entropy Theorem

The principle of maximum entropy is a basic principle in information theory(see e.g.

[6]

). It states that the probability distribution which best represents the current state of knowledge is the one with largest entropy. Since the distribution with the maximum entropy is the one that makes the fewest assumptions about the true distribution of data, the principle of maximum entropy can be seen as an application of Occam’s razor(see e.g.

[7]).

Theorem 3.1.

Suppose $B$ is irreducible. $λ$

is the maximum eigenvalue of

B

, and

l = (l_{1}, \dots, l_{N}), r = (r_{1}, \dots, r_{N})^{T}

are the corresponding left and right eigenvectors with

N \sum k = 1 l_{k} r_{k} = 1.

Then the entropy of the Markov chain associated with the skeleton matrix $B$ attains the maximum $log λ$ when

w_{k, m} = \frac{1}{λ} B_{k, m} l_{k} r_{m}, 1 \leq k, m \leq N .

Here, $W = (w_{k, m})$ is the weight matrix for $B$ .

Proof.

see Appendix.

∎

Theorem 3.2.

Suppose $A$ is irreducible. $λ$ is the maximum eigenvalue of $A$ , and $L = (L_{1}, \dots, L_{n}), R = (R_{1}, \dots, R_{n})^{T}$ are the corresponding left and right eigenvectors with

n \sum j = 1 L_{j} R_{j} = 1.

Then the entropy of the Markov chain associated with the skeleton matrix $B$ attains the maximum $log λ$ when

v_{i j} = \frac{1}{λ} A_{i j} L_{i} R_{j}, 1 \leq i, j \leq n .

Here, $V = (v_{i j})$ is the weight matrix for $A$ .

Proof.

see Appendix.

∎

Remark 3.1.

Recall that we have assumed that $f$ and $g$ are surjective. Therefore, $X$ and $Y$ have rank $n$ . Since $A = X Y, B = Y X$ , we have that

det (λ I_{N} - Y X) = λ^{N - n} det (λ I_{n} - X Y) .

Therefore, the eigenvalues of $A$ and $B$ are the same except for the zeros. In particular, the largest eigenvalue of $A$ and $B$ are the same.

Moreover, $Y A = B Y$ . Suppose $a$ is an eigenvector of $A$ with eigenvalue $μ$ , then

μ Y a = Y A a = B Y a .

Since $a \neq 0$ and $Y$ is injective as a map from $R^{n}$ to $R^{N}$ , $Y a \neq 0$ . Hence, $Y a$ is the eigenvector of $B$ with eigenvalue $μ$ .

3.4 Algorithm for Probability Estimation

Following is the algorithm for estimating the related probability.

Step 1. Compute the matrix

A

. Step 2. Use power method to compute the maximum eigenvalue

λ

A

with the corresponding left and right eigenvectors

L_{0}

and

R_{0}

. Let

L = \frac{L_{0}}{\sqrt{L_{0} \cdot R_{0}}}, R = \frac{R_{0}}{\sqrt{L_{0} \cdot R_{0}}} .

Step 3. Compute the weight matrix

V

as follows.

v_{i j} = \frac{1}{λ} A_{i j} L_{i} R_{j}, 1 \leq i, j \leq n .

Step 4. Compute the transition probability as follows.

p_{i j} = \frac{v_{i j}}{n \sum l = 1 v_{i l}} .

Step 5. Compute the stationary distribution as follows.

p_{i} = L_{i} R_{i} .

Algorithm 1 Probability Estimation

3.5 Method for Disease Prediction

The prediction task is to predict the diseases that a person is most likely to have if we know that he has already suffered from diseases $(d_{i_{1}}, d_{i_{2}}, \dots, d_{i_{T}})$ which are ordered by occurance.

We first calculate the probability by Algorithm 1. Then we construct the following quantity

r_{j} = \frac{1}{T} T \sum l = 1 p_{i_{l}, j} . (1 \leq j \leq n)

Then we sort the $r_{j}$ and choose the top 5 disease as the predicted diseases for a person.

We also make an additional assumption, that is, the latest disease take a highest weight. Thus, some modifications are made. We first construct a decreasing sequence ${a_{n}}_{n \geq 1}$ such that $a_{n} > 0$ . (For example, $a_{n} = 1 / n^{2}$ ). Then we modify $r_{j}$ as

r_{j} = \frac{T \sum l = 1 a_{T + 1 - l} \cdot p_{i_{l}, j}}{T \sum l = 1 a_{l}} .

And we use the modified $r_{j}$ to choose the top 5 disease as the predicted diseases for a person.

4 Experiments

4.1 Data Cleaning

The diseases are classified by F-code as described in section 2, and there are 429 F-coded diseases in total. If someone suffered from the same disease for many times, then we keep the earliest record and remove the others. For example, for the patient with patientid=123770, she suffered from mucopurulent conjunctivitis on 2015-12-09 and 2016-03-08, then the record with 2016-03-08 is removed from the history.

We clean the data and collect the records of the same patient together into one record. The history column recorded the patient’s disease history and the diseases are sorted by time and separated by a comma.

The following table is a sample of the cleaned data.

patient_id	history
123770	F_H10,F_H00
135086	F_M65,F_J00,F_K29,F_K01,F_K04
400195	F_J00,F_K29
3218331	F_J00,F_J40
119151	F_J00,F_N60
102519	F_J00,F_L50,F_E34,F_E01
1503387	F_K29,F_K01,F_I83
7044682	F_J00,F_J20,F_J40
182660	F_E01,F_J00,F_J20
1888934	F_K31,F_K29,F_K22,F_K50,F_J00,F_J40,F_J20,F_M70,F_J30,F_J34

Table 3: Sample of Cleaned Data

4.2 Calculate the Probability

We construct the matrix $A$ as follows.

First we initialize a $429 \times 429$ matrix with entries equal to $0$ . We also contruct a map $φ : F-codes \to {0, 1, 2, \dots, 428}$ to index the diseases. Next, for history $(h_{1}, h_{2}, \dots, h_{T})$ , we set

A_{φ (h_{i}), φ (h_{j})} \leftarrow A_{φ (h_{i}), φ (h_{j})} + 1, 1 \leq i \leq j \leq T .

Thus, we establish the matrix $A$ .

Next, we use the power method to calculate the maximum eigenvalue $λ$ of $A$ and the corresponding left and right eigenvectors $L$ and $R$ .

After that, we derive the Markov weight matrix and the transition probability as described in Algorithm 1.

Finally, we can calculate $r_{j}$ as described in subsection 3.5 and derive the related disease prediction.

4.3 Results of Accuracy

To compare the result, We use a method used previously by the insurance company as the benchmark. This method is called the emperical methods, that is, to calculate the incidence rate of diseases and use the top 5 prevalent diseases as predicition for each person.

We use 300,000 people’s records to calculate $p_{i j}$ and also the top 5 diseases. For another 10,000 people, we use their records from 2007-2014 to calculate the diseases with the highest $r_{j}$ which is described in the previous section and choose the 5 diseases with highest $r_{j}$ -score as prediction, which is known as the maximum entropy method.

Then we examine the diseases they suffer from during 2015-2017 to see how many diseases is accurately predicted by these two methods.

The measurement we use is called the hit rate. It is defined as follows.

H = \frac{| A \cap B |}{| B |},

where A is the disease set predicted by the model and B is the disease set that a person suffer from during 2015-2017.

If we predict 5 diseases using the maximum entropy method, the hit rate is 31.89%. As a contrast, the hit rate is 16.55% for the empirical method.

We also compare the hit rate with 1/2/3/4 predictions for the two methods. The following table summarize the result.

number of predictions	maximum entropy method	empirical method
1	15.01%	7.54%
2	20.50%	10.67%
3	25.21%	13.55%
4	29.01%	14.92%
5	31.89%	16.55%

Table 4: Comparisons of Hit Rate

We can see from the table that the hit rate of the maximum entropy method is approximately twice that of the empirical method.

4.4 Comorbidity Analysis

We first study the ALRR. Recall that $M$ is calculated as follows.

M_{i j} = ∣ ∣ ∣ log (\frac{p_{i j}}{p_{i} p_{j}}) ∣ ∣ ∣ .

If disease $i$ and disease $j$ are independent, then $M_{i j}$ is close to $0$ . So if $M_{i j}$ is large, then disease $i$ and disease $j$ are highly correlated.

If disease $i$ is high blood pressure and disease $j$ is type II diabetes. Then

M_{i j} = 2.17, M_{j i} = 2.39.

Here is list of diseases with high LRR.

disease

i

disease

j

M_{i j}

Type II diabetes

Type I diabetes

3.40

Pulmonary heart disease

Acute ischemic heart disease

3.35

Type II diabetes

atherosclerosis

3.18

Diseases of lip, tongue and oral mucosa

Malignant tumors of the lip,

mouth and pharynx

2.98

heart failure

Arrhythmia

2.96

Metabolic disorders

renal failure

2.84

emphysema

asthma

2.80

pneumonia

Bronchiectasis

2.67

high blood pressure

Type II diabetes

2.47

high blood pressure

renal failure

2.46

alopecia

Seborrheic keratosis

2.41

heart failure

Anal and rectal disorders

2.41

Type II diabetes

high blood pressure

2.39

heart failure

Peptic ulcer

2.38

high blood pressure

atherosclerosis

2.36

Alzheimer disease

Sleep disorders

2.33

high blood pressure

Cerebral hemorrhage or infarction

and its sequelae

2.33

Over nutrition

Other diabetes

2.27

Pulmonary heart disease

Arrhythmia

2.22

high blood pressure

heart failure

2.21

Pituitary hyperfunction

Joint disorder

2.12

Pulmonary heart disease

arthritis

2.08

Table 5: ALRR of maximum entropy method

The following table is a list of diseases such that $M_{i j}$ differs from $M_{j i}$ .

disease

i

disease

j

M_{i j}

M_{j i}

Female pelvic inflammatory disease

Trichomoniasis

1.13

3.40

nephritic nephrotic syndrome

heart failure

1.51

3.35

Metabolic disorders

Malignant tumor of skin

1.37

3.19

Esophageal diseases

Splenic diseases

3.40

1.58

anemia

hypotension

2.89

1.30

Other diabetes

Central nervous system diseases

1.26

2.84

Benign tumor of uterus

Tumors with undetermined or unknown

endocrine gland dynamics

2.89

1.31

Arrhythmia

Mental and behavioral disorders

2.58

1.03

Arthrosis

epilepsy

0.71

1.89

Arrhythmia

Diseases of autonomic nervous system

2.63

1.47

Headache syndrome

Other diseases of arteries and arterioles

1.58

2.74

Acute pancreatitis and other

diseases of pancreas

Type II diabetes

2.78

1.70

asthma

emphysema

1.73

2.80

Ankylosis and other spondylosis

Hypopituitarism

1.87

0.80

Arthrosis

Myasthenia and primary muscle diseases

2.71

1.65

Malignant tumors of digestive organs

Hemangioma and lymphangioma

3.40

2.33

Refractive and accommodative disorders

glaucoma

2.41

1.35

Other diabetes

Optic neuropathy

1.71

2.74

Chronic ischemic heart disease

Pericardial disease

1.26

2.28

Type II diabetes

Over nutrition

0.82

1.83

Table 6: Asymmetric ALRR of maximum entropy method

Next, we consider the $ϕ$ -correlation. Recall that

ϕ_{i j} = \frac{p_{i j} - p_{i} p_{j}}{\sqrt{p_{i} p_{j} (1 - p_{i}) (1 - p_{j})}} .

Next table display 20 disease pairs with high $ϕ$ -correlation.

disease

i

disease

j

ϕ

Mania, bipolar, depression, and

anxiety disorders

sleep disorder

68.23

Type II diabetes

Hypertension and its complications

67.75

Headache syndrome

Pulp, gums and edentulous alveolar

ridge diseases

60.72

Arrhythmia

Hypertension and its complications

58.55

Muscle disorders

Backache

57.28

Shingles

Dermatitis and pruritus

55.30

Headache syndrome

Backache

53.33

Benign uterine tumor

Abnormal uterine and vaginal bleeding

46.75

Upper respiratory tract diseases such as chronic

laryngitis and laryngotracheitis

Chronic rhinitis, nasopharyngitis

and pharyngitis

42.54

Other disorders of kidney and ureter

Other disorders of the urinary system

42.30

anemia

Pulp, gums and edentulous alveolar

ridge diseases

40.85

cellulitis

Dermatophytes and other superficial

fungal diseases

40.74

Other disorders of male reproductive organs

Prostatic hyperplasia and prostatitis

40.69

Urethral disorders

Other disorders of the urinary system

39.66

Other disorders of bone

Osteoporosis without pathological fracture

34.96

Upper respiratory tract diseases such as chronic

laryngitis and laryngotracheitis

Chronic bronchitis

33.39

Other diseases of the digestive system

Gastritis and duodenitis

29.68

Type II diabetes

Metabolic disorders

27.51

Type II diabetes

Dermatitis and pruritus

27.16

Arrhythmia

sleep disorder

27.11

Table 7:

ϕ

-correlation of maximum entropy method

We can see from table 5 many disease pairs with large $M_{i j}$ , such as type II diabetes and hypertension and its complications, which imply that such diseases have intrinsic relations. Tedesco [3] have mentioned that Hypertension is frequently associated with diabetes mellitus and its prevalence doubles in diabetics compared to the general population. This high prevalence is associated with increased stiffness of large arteries. Our result is consistent with their medical research.

5 Conclusions

In this paper, we propose a maximum entropy method for predicting disease risks. It is based on a patient’s medical history with diseases coded in ICD-10 which can be used in various cases. The complete algorithm with strict mathematical derivation is given. We also present experimental results on a medical dataset, demonstrating that our method performs well in predicting future disease risks and achieves an accuracy rate twice that of the traditional method. We also perform a comorbidity analysis to reveal the intrinsic relation of diseases.

Acknowledgement

We would thank Franco Mueller, Jonathan Brezin and Matt Grayson for their collaboration on an early version of this research.

Appendix

Proof of Theorem 3.1.

Suppose $W = (w_{k, m})$ is the weight matrix. Then the entropy of the Markov chain can be rewritten as

	$H (w)$	$= - N \sum k = 1 N \sum m = 1 w_{k, m} (log (w_{k, m}) - log (N \sum m = 1 w_{k, m}))$
		$= - N \sum k = 1 N \sum m = 1 w_{k, m} log (w_{k, m}) + N \sum k = 1 (N \sum m = 1 w_{k, m}) log (N \sum m = 1 w_{k, m}) .$

Let us construct the Lagrangian

	$L$	$= - N \sum k = 1 N \sum m = 1 w_{k, m} log (w_{k, m}) + N \sum k = 1 (N \sum m = 1 w_{k, m}) log (N \sum m = 1 w_{k, m})$
		$+ N \sum k = 1 h_{k} (N \sum m = 1 w_{m, k} - N \sum m = 1 w_{k, m}) + μ (1 - N \sum k = 1 N \sum m = 1 w_{k, m}) .$

If $w_{k, m} \neq 0$ , we have that

\frac{\partial L}{\partial w_{k, m}} = - 1 - log (w_{k, m}) + log (N \sum m = 1 w_{k, m}) + 1 - h_{k} + h_{m} - μ = 0.

\frac{w_{k, m}}{N \sum m = 1 w_{k, m}} = e^{- μ} \frac{e^{h_{m}}}{e^{h_{k}}} .

If $w_{k, m} = 0$ , then $B_{k, m} = 0$ . Therefore,

\frac{w_{k, m}}{N \sum m = 1 w_{k, m}} = e^{- μ} \frac{B_{k, m} e^{h_{m}}}{e^{h_{k}}} .

Set $λ = e^{μ}$ , then

N \sum m = 1 B_{k, m} e^{h_{m}} = λ e^{h_{k}} .

By the Perron-Frobenius theorem(see e.g. [11]), There are no nonnegative eigenvectors for $B$ other than the Perron vector $r$ and its positive multiples. Hence,

\frac{e^{h_{m}}}{e^{h_{k}}} = \frac{r_{m}}{r_{k}} .

And

P_{k, m} = \frac{w_{k, m}}{N \sum m = 1 w_{k, m}} = \frac{1}{λ} B_{k, m} \frac{r_{m}}{r_{k}} .

On the other hand,

λ \frac{w_{k, m}}{r_{m}} = B_{k, m} \frac{p_{k}}{r_{k}} \Rightarrow λ \frac{}{p_{m}} r_{m} = N \sum k = 1 B_{k, m} \frac{p_{k}}{r_{k}} .

Therefore, there exists $t > 0$ such that

\frac{p_{k}}{r_{k}} = t l_{k} .

N \sum k = 1 l_{k} r_{k} = 1 \Rightarrow t = 1.

Hence,

p_{k} = l_{k} r_{k}, w_{k, m} = \frac{1}{λ} l_{k} B_{k, m} r_{m} .

Recall that

H = - N \sum k = 1 N \sum m = 1 p_{k} P_{k, m} log (P_{k, m}),

and $0 \cdot log 0 = lim x \to 0 + x log x = 0$ . It follows that

	$H$	$= - N \sum k = 1 N \sum m = 1 l_{k} r_{k} \frac{1}{λ} B_{k, m} \frac{r_{m}}{r_{k}} log (\frac{1}{λ} B_{k, m} \frac{r_{m}}{r_{k}})$
		$= - N \sum k = 1 N \sum m = 1 l_{k} \frac{1}{λ} B_{k, m} r_{m} (log B_{k, m} + log r_{m} - log r_{k} - log λ)$

Since $B_{k, m} \in {0, 1}$ , $B_{k, m} log B_{k, m} = 0$ ,

	$H$	$= - N \sum k = 1 N \sum m = 1 l_{k} \frac{1}{λ} B_{k, m} r_{m} (log r_{m} - log r_{k}) + log λ$
		$= - N \sum m = 1 l_{m} r_{m} log r_{m} + N \sum k = 1 l_{k} r_{k} log r_{k} + log λ$
		$= log λ .$

∎

Next, we will prove Theorem 3.2. We first prove an auxiliary lemma.

Lemma 1.

Suppose $λ$ is the maximum eigenvalue of $B$ and $X, Y$ are matrices in section 3.1. $l = (l_{1}, \dots, l_{N}), r = (r_{1}, \dots, r_{N})^{T}$ are the corresponding left and right eigenvectors with

N \sum k = 1 l_{k} r_{k} = 1.

Then

⎛ ⎜ ⎜ ⎝ \begin{matrix} (l Y)_{1} ⋱ (l Y)_{n} \end{matrix} ⎞ ⎟ ⎟ ⎠ A ⎛ ⎜ ⎜ ⎝ \begin{matrix} (X r)_{1} ⋱ (X r)_{n} \end{matrix} ⎞ ⎟ ⎟ ⎠ = λ^{2} X ⎛ ⎜ ⎜ ⎝ \begin{matrix} l_{1} r_{1} ⋱ l_{N} r_{N} \end{matrix} ⎞ ⎟ ⎟ ⎠ Y .

(1)

Proof.

Recall that

f, g : {1, 2, \dots, N} \to {1, 2, \dots, n},

and

X_{i, k} = χ_{{f (k) = i}}, Y_{k, j} = χ_{{g (k) = j}} .

A_{i, j} = N \sum k = 1 X_{i, k} Y_{k, j} = \sum k : f (k) = i, g (k) = j 1 = | f^{- 1} (i) \cap g^{- 1} (j) | .

Here, $| S |$ denote the number of elements in $S$ .

B_{k, m} = n \sum j = 1 Y_{k, j} X_{j, m} = n \sum j = 1 χ_{{g (k) = j, f (m) = j}} = χ_{{g (k) = f (m)}} .

Since $B r = λ r$ , we have that

N \sum m = 1 B_{k, m} r_{m} = λ r_{k} \Rightarrow N \sum m = 1 χ_{{g (k) = f (m)}} r_{m} = λ r_{k} .

Since $l B = λ l$ , we have that

N \sum k = 1 B_{k, m} l_{k} = λ l_{m} \Rightarrow N \sum k = 1 χ_{{g (k) = f (m)}} l_{k} = λ l_{m} .

The $(i, j)$ element of the left hand side in (1) is

	$(l Y)_{i} A_{i, j} (X r)_{j}$	$= \| f^{- 1} (i) \cap g^{- 1} (j) \| (N \sum k = 1 l_{k} Y_{k, i}) (N \sum m = 1 r_{m} X_{j, m})$
		$= \| f^{- 1} (i) \cap g^{- 1} (j) \| (N \sum k = 1 l_{k} χ_{{g (k) = i}}) (N \sum m = 1 r_{m} χ_{{f (m) = j}})$

Assume that

f^{- 1} (i) \cap g^{- 1} (j) = {u_{1}, u_{2}, \dots, u_{q}} .

Then

f (u_{1}) = f (u_{2}) = \dots = f (u_{q}) = i, g (u_{1}) = g (u_{2}) = \dots = g (u_{q}) = j .

	$(l Y)_{i} A_{i, j} (X r)_{j}$	$= q (N \sum k = 1 l_{k} χ_{{g (k) = i}}) (N \sum m = 1 r_{m} χ_{{f (m) = j}})$
		$= q \sum t = 1 (N \sum k = 1 l_{k} χ_{{g (k) = f (u_{t})}}) (N \sum m = 1 r_{m} χ_{{f (m) = g (u_{t})}})$
		$= λ^{2} q \sum t = 1 l_{u_{t}} r_{u_{t}} .$

The $(i, j)$ element of the right hand side in (1) is

	$λ^{2} N \sum k = 1 X_{i, k} l_{k} r_{k} Y_{k, j}$	$= λ^{2} N \sum k = 1 l_{k} r_{k} χ_{{f (k) = i, g (k) = j}} = λ^{2} \sum k \in f^{- 1} (i) \cap g^{- 1} (j) l_{k} r_{k}$
		$= λ^{2} q \sum t = 1 l_{u_{t}} r_{u_{t}} .$

Thus, we complete the proof. ∎

Proof of Theorem 3.2.

Suppose $V$ is the weight matrix of $A$ , then

V = X ⎛ ⎜ ⎜ ⎝ \begin{matrix} l_{1} r_{1} ⋱ l_{N} r_{N} \end{matrix} ⎞ ⎟ ⎟ ⎠ Y .

By the above lemma, we have that

V = \frac{1}{λ^{2}} ⎛ ⎜ ⎜ ⎝ \begin{matrix} (l Y)_{1} ⋱ (l Y)_{n} \end{matrix} ⎞ ⎟ ⎟ ⎠ A ⎛ ⎜ ⎜ ⎝ \begin{matrix} (X r)_{1} ⋱ (X r)_{n} \end{matrix} ⎞ ⎟ ⎟ ⎠

Set

L = \frac{l Y}{\sqrt{λ}}, R = \frac{X r}{\sqrt{λ}} .

Then $L, R$ are the left and right eigenvectors of $A$ corresponding to the eigenvalue $λ$ . And

n \sum j = 1 L_{j} R_{j} = L R = \frac{1}{λ} l Y X r = l r = 1.

Hence,

v_{i, j} = \frac{1}{λ} L_{i} A_{i, j} R_{j} .

∎

References

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[11] Carl D. Meyer, Matrix analysis and applied linear algebra. With solutions to problems, 2001, SIAM.

Disease Prediction with a Maximum Entropy Method

Interpretable Disease Prediction based on Reinforcement Path Reasoning over Knowledge Graphs

DSR: A Collection for the Evaluation of Graded Disease-Symptom Relations

Common human diseases prediction using machine learning based on survey data

Study on the numerical value and the distance of symptoms on the Medical Decision Support System

Comparing learning algorithms in neural network for diagnosing cardiovascular disease

Precision disease networks (PDN)

A Tractable Inference Algorithm for Diagnosing Multiple Diseases

1 Introduction

2 Data

2.1 Source Data and Population

2.2 Quantifying the Strength of Comorbidity Relationships

3 Methodology

3.1 Some notations

3.2 Entropy for Markov Chains

3.3 Maximum Entropy Theorem

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

Remark 3.1.

3.4 Algorithm for Probability Estimation

3.5 Method for Disease Prediction

4 Experiments

4.1 Data Cleaning

4.2 Calculate the Probability

4.3 Results of Accuracy

4.4 Comorbidity Analysis

5 Conclusions

Acknowledgement

Appendix

Proof of Theorem 3.1.

Lemma 1.

Proof.

Proof of Theorem 3.2.

References

Disease Prediction with a Maximum Entropy Method

Related Research

Interpretable Disease Prediction based on Reinforcement Path Reasoning over Knowledge Graphs

DSR: A Collection for the Evaluation of Graded Disease-Symptom Relations

Common human diseases prediction using machine learning based on survey data

Study on the numerical value and the distance of symptoms on the Medical Decision Support System

Comparing learning algorithms in neural network for diagnosing cardiovascular disease

Precision disease networks (PDN)

A Tractable Inference Algorithm for Diagnosing Multiple Diseases

1 Introduction

2 Data

2.1 Source Data and Population

2.2 Quantifying the Strength of Comorbidity Relationships

3 Methodology

3.1 Some notations

3.2 Entropy for Markov Chains

3.3 Maximum Entropy Theorem

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

Remark 3.1.

3.4 Algorithm for Probability Estimation

3.5 Method for Disease Prediction

4 Experiments

4.1 Data Cleaning

4.2 Calculate the Probability

4.3 Results of Accuracy

4.4 Comorbidity Analysis

5 Conclusions

Acknowledgement

Appendix

Proof of Theorem 3.1.

Lemma 1.

Proof.

Proof of Theorem 3.2.

References