CN103932733B - A kind of digital measuring analytical equipment of interstitial pulmonary fibrosis based on lungs sound - Google Patents

Abstract

The present invention relates to the digital measuring analytical equipment of a kind of interstitial pulmonary fibrosis based on lungs sound, first the lungs sound data of the certain period of time collected are normalized by this device, then wavelet decomposition is carried out, choose interstitial pulmonary fibrosis crackles rale signal and be mainly distributed the wavelet decomposition detail component of frequency range, calculate the standard deviation of relevant wavelet decomposition component signal.Use this detection device that the lungs sound of interstitial pulmonary fibrosis patient is carried out quantitative analysis, assess the order of severity of the lung interfibrillar substance of patient, on the health of patient without impact, detect safer, can be with repeated detection analysis, it is more objective to analyze, accuracy is high, and low cost, simple to operate, it is suitable for monitoring the severity extent of patients with interstitial pneumonia.

Description

A kind of digital measuring analytical equipment of interstitial pulmonary fibrosis based on lungs sound

Technical field

The present invention relates to biomedical signal analysis field, particularly to the digitized of a kind of interstitial pulmonary fibrosis based on lungs sound Detection analytical equipment.

Background technology

Interstitial pulmonary fibrosis is a kind of common pulmonary disease, and at present, doctor is generally by observing chest x-ray sheet or chest CT sheet assesses the order of severity of lung interfibrillar substance.Owing to X-ray has radioactivity, the health of patient there is is considerable influence, Therefore it is difficult to frequently carry out rabat and breast CT inspection.And cost is the highest.

Doctor's lungs sound by auscultation interstitial pulmonary fibrosis patient, can hear the crackles rale relevant to disease, necessarily Degree understands the state of an illness of patient.Auscultation is simple and easy to do, can operate whenever and wherever possible, on the health of patient without impact, can repeatedly examine Cls analysis, it is simple to doctor grasps conditions of patients in time.But its order of accuarcy relies on the personal experience of doctor, objectivity is poor. And the shortcoming of auscultation that human ear is carried out is to there is non-quantitation and certain subjectivity, can only single people's auscultation, contrast front and back is only Can be the individual behavior of doctor, it is difficult to carry out quantitative analysis.And the restriction of stethoscope performance also can make the Lung Sounds can not be good Ground transmission, and the sensitivity of sound is also varied with each individual by human ear, even specially trained specialist, its subjective sensation Also there is certain limitation with experience, cause diagnostic result to be affected by human factors relatively big, thus cause the accurate of diagnostic result Property is on the low side.

Summary of the invention

It is an object of the invention to provide the digital measuring analytical equipment of a kind of interstitial pulmonary fibrosis based on lungs sound, use This detection analytical equipment carries out quantitative analysis to the lungs sound of interstitial pulmonary fibrosis patient, assesses the lung interfibrillar substance of patient The order of severity, on the health of patient without impact, detects safer, can be with repeated detection analysis, and it is more objective to analyze, accuracy height, And low cost, simple to operate, it is suitable for monitoring and follows the tracks of the severity extent of patients with interstitial pneumonia.

Concrete technical scheme is:

The digital measuring analytical equipment of a kind of interstitial pulmonary fibrosis based on lungs sound, the method for employing includes following step Rapid:

1) choosing the digitized Lung Sounds in the period collected, the sample frequency of this digitized Lung Sounds is f；

2) to step 1) in digitized Lung Sounds be normalized；

3) to step 2) in digitized Lung Sounds after normalized, use Mallat orthogonal wavelet transformation to calculate quickly soon Method carries out wavelet decomposition, selects wavelet basis function and wavelet decomposition number of plies j, Lung Sounds is decomposed into different each of frequency range little Wave Decomposition component signal, each wavelet decomposition component signal is carried out wavelet reconstruction by corresponding coefficient of wavelet decomposition and obtains；

4) respectively calculation procedure 3) in the signal characteristic of each wavelet decomposition component signal, burst according to interstitial pulmonary fibrosis Tone signal is mainly distributed frequency range, it is judged that be mainly distributed corresponding several of frequency range in each wavelet decomposition component signal with crackles rale signal Individual wavelet decomposition component signal, and the signal characteristic of these wavelet decomposition component signals is sued for peace, i.e. obtain characterizing interstitial lung The index of fibrosis；It is 200Hz-1000Hz that interstitial pulmonary fibrosis crackles rale signal is mainly distributed frequency range.

5) index characterizing interstitial pulmonary fibrosis degree is compared with reference to index, interstitial pulmonary fibrosis can be drawn The order of severity.

Step 2) in Lung Sounds be normalized the formula of employing be:Wherein x_i It is the i-th data of lungs sound data, x before normalized_minIt is the minima of lungs sound data, x before normalized_maxIt it is normalizing Change processes the maximum of front lungs sound data, y_iIt is the i-th data of lungs sound data after normalized.

Step 1) described in sample frequency f of digitized Lung Sounds be 8000～10000Hz, if this digitized lungs sound The sample frequency of signal is not 8000～10000Hz, and these lungs sound data can carry out resampling to sample frequency is 9000Hz.

Step 1) described in analysis period of digitized Lung Sounds should comprise 2 or more than 2 breathing cycles.

The time span of described digitized Lung Sounds is 8s-12s.

Step 3) described in wavelet decomposition number of plies j >=5, high-frequency band Dn that n-th layer wavelet decomposition component signal is corresponding is: f/2ⁿ⁺¹～f/2ⁿ, n is 1,2,3 ... j, f are signal sampling frequency.

Step 3) described in wavelet basis function be dbN series, 3≤N≤5.

Step 4) described in the signal characteristic of each wavelet decomposition component signal be standard deviation, if signal is x_i, its standard deviation Computing formula is:

$std=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\Σ}}{(x_i-\mathrm{\μ})}^2}$

Wherein, N is the number of array, and μ is the average of array,

Step 4) described in the signal characteristic of each wavelet decomposition component signal be energy, if signal is x_i, its energy balane Formula is:

$E=\underset{i=1}{\overset{N}{\Σ}}{(x_i-\mathrm{\μ})}^2$

Wherein, N is the number of array, and μ is the average of array,

Applicant, by gathering the lungs sound of interstitial pulmonary fibrosis patient, carries out wavelet analysis to Lung Sounds, carries for doctor For a kind of lungs sound quantitative analysis method, this method belongs to safe detection method, and detection method is simple, low cost, it is simple to repeat Repeated detection, it is simple to doctor treat before and after contrast and the contrast of same time different parts, it is simple to doctor grasps in time Conditions of patients, on the health of patient without impact, and more objectivity, accuracy is high, also has digitized feature.

Accompanying drawing explanation

Fig. 1 is the determination method flow chart of the present invention；

Fig. 2 (a) is the lungs sound before patient A treatment and wavelet decomposition figure, when Fig. 2 (b) is to check after patient A treats half a year Lungs sound and wavelet decomposition figure.

Detailed description of the invention

Seeing Fig. 1, the digital measuring analytical equipment of a kind of interstitial pulmonary fibrosis based on lungs sound, the method for employing includes Following steps:

1) choosing the digitized Lung Sounds in the period collected, the sample frequency of this digitized Lung Sounds is f.The analysis period of described digitized Lung Sounds should comprise 2 or more than 2 breathing cycles.Described digitized Lung Sounds Time span typically take 8s-12s.Use the Lung Sounds of electronic stethoscope acquisition and recording patient.In order to specification digitized divides Analysis method, if the sample range of lungs sound data is not in 8000Hz-10000Hz scope, needs to enter data before wavelet decomposition Row resampling so that it is after resampling, signal sampling frequency is 9000Hz.Lungs sound data resampling can use Matlab's Resample function realizes, or is programmed to voluntarily, and the sample frequency of the lungs sound data after resampling is 9000Hz.

2) to step 1) in digitized Lung Sounds be normalized so that it is numerical range is [-1,1].Due to The stethoscopic amplification of different digitizeds is different, from the amplitude range of the lungs sound that different digital auscultation equipment collects There is bigger difference, for the ease of analyzing the Lung Sounds of multiple different amplitude range, can be first by the Lung Sounds gathered It is normalized.

The formula that Lung Sounds is normalized employing is:Wherein x_iIt is at normalization The i-th data of lungs sound data, x before reason_minIt is the minima of lungs sound data, x before normalized_maxIt it is lung before normalized The maximum of sound data, y_iIt is the i-th data of lungs sound data after normalized.

3) to step 2) in digitized Lung Sounds after normalized, use Mallat orthogonal wavelet transformation to calculate quickly soon Method carries out wavelet decomposition, selects wavelet basis function and wavelet decomposition number of plies j, Lung Sounds is decomposed into different each of frequency range little Wave Decomposition component signal, each wavelet decomposition component signal is carried out wavelet reconstruction by corresponding coefficient of wavelet decomposition and obtains.

This step uses the wavelet transformation of Mallat orthogonal wavelet transformation fast algorithm implementation signal.From the angle of spectrum analysis Degree is seen, wavelet transformation is that signal decomposition becomes low frequency and high frequency two parts, in the decomposition of next layer, low frequency part is implemented again Decompose again.If original input signal is f (x), then A₀=f (x), Mallat algorithm can be described as:

$\left\{\begin{array}{c}{A}_{j+1}\left(k\right)=\underset{n}{\Σ}h(n-2k)A_j\left(n\right)\\ D_{j+1}\left(k\right)=\underset{n}{\Σ}g(n-2k)A_j\left(n\right)\end{array}\right.---\left(1\right)$

Signal f (x) is [A in the decomposition result of jth layer_j,D_j,...,D₁]。

Described wavelet decomposition number of plies j >=5, high-frequency band Dn that n-th layer wavelet decomposition component signal is corresponding is: f/2ⁿ⁺¹ ～f/2ⁿ, n is 1,2,3 ... j, f are signal sampling frequency.If the sample range of lungs sound data is in 8000Hz-10000Hz scope In, described wavelet decomposition number of plies j is that 5 effects are preferred.Described wavelet basis function uses dbN series, 3≤N≤5.Preferably, Described wavelet basis function is db3.

4) respectively calculation procedure 3) in the signal characteristic of each wavelet decomposition component signal, burst according to interstitial pulmonary fibrosis Tone signal is mainly distributed frequency range, it is judged that be mainly distributed corresponding several of frequency range in each wavelet decomposition component signal with crackles rale signal Individual wavelet decomposition component signal, and the signal characteristic of these wavelet decomposition component signals is sued for peace, i.e. obtain characterizing interstitial lung The index of fibrosis；It is 200Hz-1000Hz that interstitial pulmonary fibrosis crackles rale signal is mainly distributed frequency range.Sample frequency is The Lung Sounds of 8000Hz-10000Hz, its crackles rale is concentrated mainly on the D3 of wavelet decomposition, in D4, D5 component.Calculate small echo Decompose signal characteristic and the std (D3+D4+D5) of the details coefficients of the 3rd, 4,5 layers, i.e. obtain characterizing interstitial pulmonary fibrosis degree Index.

Step 4) described in the signal characteristic of each wavelet decomposition component signal be standard deviation, if signal is x_i, its standard deviation Computing formula is:

$std=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\Σ}}{(x_i-\mathrm{\μ})}^2}$

Wherein, N is the number (length of data) of array, and μ is the average of array,For wavelet decomposition Details coefficients, its mean μ=0.

Step 4) described in the signal characteristic of each wavelet decomposition component signal can also be energy.Letter for certain length Number, the energy of signal is the biggest, the biggest to the standard deviation of induction signal.Energy is used not have standard deviation stable as signal characteristic.

If signal is x_i, its energy balane formula is:

$E=\underset{i=1}{\overset{N}{\Σ}}{(x_i-\mathrm{\μ})}^2$

Wherein, N is the number (length of data) of array, and μ is the average of array,For wavelet decomposition Details coefficients, its mean μ=0.

5) index characterizing interstitial pulmonary fibrosis degree is compared with reference to index, interstitial pulmonary fibrosis can be drawn The order of severity.Choosing the multistage digitized Lung Sounds at identical time span, same position, repeat the above steps is the most available The index of multiple sign interstitial pulmonary fibrosis degree and reference index, seek the index of multiple sign interstitial pulmonary fibrosis degree respectively Compare again with reference to after the meansigma methods of index, the order of severity of interstitial pulmonary fibrosis can be drawn so that this method analysis More accurate.

It is being calculated by this method step with reference to Lung Sounds of choosing with reference to index.When using this method, one As choose two digitized Lung Sounds (using identical lungs sound detection equipment to obtain) of the same time span collected, point Cai Yong not carry out wavelet decomposition by Mallat orthogonal wavelet transformation fast algorithm, after continuing each step, show that two characterize interstitial lung The index of fibrosis, compares two indices, can be by the index of detected Lung Sounds and reference index Difference, it is judged that the order of severity of detected object interstitial pulmonary fibrosis.In two indices, an index is detected index, one Individual is with reference to index.In two digitized Lung Sounds, one is detected Lung Sounds, and one is with reference to Lung Sounds.

Use this method can analyze detected object and suffer from the order of severity of interstitial pulmonary fibrosis, and detected right As the situation that is in a bad way of different parts, and the therapeutic effect of detected object.Detected object is analyzed according to this method Therapeutic effect, then select detected object treatment before lungs sound with treatment after lungs sound, drawn two respectively by this method Compare after the index of the degree characterizing interstitial pulmonary fibrosis, can show whether the interstitial pulmonary fibrosis of detected object has Improve.Analyze the situation that is in a bad way of detected object different parts according to this method, then select detected object with for the moment Between the lungs sound of different parts, compare drawn the index of two degree characterizing interstitial pulmonary fibrosis respectively by this method after Relatively, can show that the state of an illness ratio at which position of detected object is more serious.

Gather 2 patients with interstitial pneumonia lungs sound, patient A morbidity during and treatment half a year after lungs sound and patient B The lungs sound of different parts.Digital stethoscope is used to carry out detection and the record of lungs sound.Sample frequency is 9000Hz.All lungs sound numbers According to choosing the time span of 8 seconds, use Matlab programming that all lungs sound digital documents of storage are read out and are analyzed, first Being normalized the lungs sound digital signal read, then carry out wavelet decomposition, small echo selects db3, carries out 5 layers of little wavelength-division Solving, wavelet decomposition is as in figure 2 it is shown, the corresponding different frequency range of different little wave component, D1:2250-4500Hz, D2:1125- 2250Hz, D3:562.5-1125Hz, D4:281.25-562.5Hz, D5:140.625-281.25Hz, A5:0-140.625Hz. Wherein, the wavelet decomposition component that A5, D5, D4, D3, D2, D1 obtain after being by 5 layers of wavelet decomposition respectively.

Seeing Fig. 2, Fig. 2 (a) is the lungs sound before patients with interstitial pneumonia A is treated and wavelet decomposition figure, and Fig. 2 (b) is patient A Lungs sound when treatment was checked after half a year and wavelet decomposition figure.S is the Lung Sounds collected, and A5, D5, D4, D3, D2, D1 are respectively The wavelet decomposition component obtained after being by 5 layers of wavelet decomposition.It can also be seen that patient A suffers from from lungs sound and wavelet decomposition figure Between stadium, lungs sound is deposited cash obvious crackles rale, and crackles rale is concentrated mainly on the D3 of wavelet decomposition, in D4, D5 component.

Calculating respectively within 8 second time period, the standard deviation of 6 wavelet decomposition components, table 1 is the little wavelength-division of patient's A lungs sound Amount standard deviation, (a) is the lungs sound Wavelet Component standard deviation of 5 sections gathered before treatment, and (b) is 5 sections gathered after treatment half a year Lungs sound Wavelet Component standard deviation.Table 2 is the Wavelet Component standard deviation of patient's B lungs sound, and the lungs sound of 5 sections of (a) bottom right back of the body collection is little Wave component standard deviation, the lungs sound Wavelet Component standard deviation of 5 sections of (b) upper right back of the body collection.

Table 1 patient's A lungs sound Wavelet Component standard deviation

Lungs sound before (a) treatment

	A5	D5	D4	D3	D2	D1	D5+D4+D3
								1	0.5054	0.0971	0.1016	0.0608	0.0210	0.0086	0.2595
2	0.5139	0.1044	0.1065	0.0638	0.0195	0.0064	0.2747
								3	0.5684	0.0987	0.0961	0.0598	0.0180	0.0062	0.2546
4	0.6031	0.0960	0.0858	0.0508	0.0159	0.0059	0.2326
								5	0.6257	0.0988	0.0879	0.0496	0.0162	0.0061	0.2363

B () treats lungs sound after half a year

	A5	D5	D4	D3	D2	D1	D5+D4+D3
								1	0.3143	0.0321	0.0218	0.0074	0.0043	0.0042	0.0613
2	0.2724	0.0274	0.0170	0.0063	0.0040	0.0039	0.0507
								3	0.242	0.0281	0.0163	0.0061	0.0040	0.0039	0.0505
4	0.2472	0.0315	0.0202	0.0072	0.0040	0.0039	0.0589
								5	0.2569	0.0333	0.0225	0.0078	0.0041	0.0040	0.0636

Table 2 patient's B lungs sound Wavelet Component standard deviation

A () bottom right memorizes record lungs sound

	A5	D5	D4	D3	D2	D1	D5+D4+D3
								1	0.0805	0.0579	0.0431	0.0194	0.0127	0.0144	0.1204
2	0.0873	0.0608	0.0463	0.0200	0.0127	0.0143	0.1271
								3	0.0945	0.0702	0.0450	0.0199	0.0125	0.0146	0.1351
4	0.0825	0.0596	0.0384	0.0171	0.0111	0.0134	0.1151
								5	0.1107	0.0719	0.0489	0.0212	0.0129	0.0148	0.142

B () upper right memorizes record lungs sound

	A5	D5	D4	D3	D2	D1	D5+D4+D3
								1	0.112	0.0479	0.0242	0.0083	0.0042	0.0040	0.0804
2	0.1096	0.0563	0.0251	0.0087	0.0042	0.0040	0.0901
								3	0.1103	0.0554	0.0244	0.0069	0.0041	0.0039	0.0867
4	0.1041	0.0483	0.0212	0.0065	0.0041	0.0040	0.076
								5	0.1202	0.0545	0.0274	0.0081	0.0043	0.0040	0.09

Due to the wavelet details component D3 of Lung Sounds, the interstitial pulmonary fibrosis that the signal frequency range of D4, D5 comprises is correlated with The main frequency distribution of crackles rale, D3, D4, the wavelet decomposition signal of D5 contains the explosion that interstitial pulmonary fibrosis is relevant The main information of sound, lungs sound Wavelet Component D3, the energy of D4 and D5 and with standard deviation and crackles rale can be characterized to a great extent Power with how many.It is many that lungs sound Wavelet Component D3, the standard deviation of D4 and D5 and std (D3+D4+D5) can characterize crackles rale Few, from table 1 data analysis knowable to, patient A lungs sound during one's sickness and the standard deviation sum of D3, D4 and D5 of lungs sound after treating half a year Meansigma methods is respectively 0.25 and 0.057, it can be clearly seen that patient A treatment half a year after lungs sound D3, D4 and D5 standard deviation and Be far smaller than D3, D4 and D5 of patient A lungs sound during one's sickness standard deviation and, illustrate that patient A interstitial lung after treatment half a year is fine Dimensionization takes an evident turn for the better.And feasibility and the accuracy of this method is also demonstrated by rabat and breast CT inspection.

Patient B is the lungs sound in the same period recording different parts, and the upper right of patient B memorizes the lungs sound D3+D4+D5's of record Standard deviation and average are 0.085, and bottom right memorizes the standard deviation of the lungs sound D3+D4+D5 of record and average is 0.128.The most permissible The order of severity going out pulmonary corresponding to the two position is different.The bottom right of patient B memorize the lungs sound D3+D4+D5 of record energy and Be far longer than the upper right of patient B memorize record lungs sound D3+D4+D5 energy and, illustrate that bottom right lung is tighter compared with the state of an illness of right upper lung Weight.Also showing that from imaging examination the order of severity of pulmonary corresponding to the two position is different, bottom right lung is compared with the state of an illness of right upper lung More serious, thus also demonstrate feasibility and the accuracy of this method.

It follows that the numerical values recited of the standard deviation sum of lungs sound Wavelet Component D3, D4 and D5 and interstitial pulmonary fibrosis is tight Weight degree is correlated with, and the value of lungs sound Wavelet Component D3, the standard deviation of D4 and D5 and std (D3+D4+D5) is the biggest, illustrates and interstitial lung The energy of the crackles rale that fibrosis is relevant is big, and the degree of prompting interstitial pulmonary fibrosis is more serious.Lungs sound Wavelet Component D3, D4 and D5 Standard deviation and can to a certain degree reflect crackles rale number, thus reflect the coincident with severity degree of condition of patients with interstitial pneumonia Difference, the order of severity assessing interstitial pulmonary fibrosis patient for doctor provides indirectly reference.This determination method has Simple to operate, on the health of patient without impact, can be with repeated detection analysis.This determination method has low cost, numeral The feature changed, provides a kind of lungs sound quantitative analysis method for doctor, for assisting doctor to understand the serious journey of interstitial pulmonary fibrosis Degree.

Claims (9)

1. the digital measuring analytical equipment of an interstitial pulmonary fibrosis based on lungs sound, it is characterised in that: include electronic auscultation Device, lungs sound sampling module, normalized module, wavelet analysis module, signal characteristic abstraction module；

Described electronic stethoscope is for the Lung Sounds of acquisition and recording patient；

Described lungs sound sampling module is for choosing the digitized Lung Sounds in the period collected, and this digitized lungs sound is believed Number sample frequency be f；

Described normalized module is for being normalized digitized Lung Sounds；

Described wavelet analysis module, for the digitized Lung Sounds after normalized, uses Mallat orthogonal wavelet transformation Fast algorithm carries out wavelet decomposition, selects wavelet basis function and wavelet decomposition number of plies j, Lung Sounds is decomposed into frequency range different Each wavelet decomposition component signal, each wavelet decomposition component signal is carried out wavelet reconstruction by corresponding coefficient of wavelet decomposition and obtains；

Described signal characteristic abstraction module is for calculating the signal characteristic of each wavelet decomposition component signal respectively, fine according to interstitial lung Dimensionization crackles rale signal is mainly distributed frequency range, it is judged that be mainly distributed frequency range with crackles rale signal in each wavelet decomposition component signal Corresponding several wavelet decomposition component signals, and the signal characteristic of these wavelet decomposition component signals is sued for peace, i.e. obtain table Levy the index of interstitial pulmonary fibrosis degree, the index characterizing interstitial pulmonary fibrosis degree compared with reference to index, Draw the order of severity of interstitial pulmonary fibrosis.

Detection analytical equipment the most according to claim 1, it is characterised in that Lung Sounds is carried out by normalized module The formula that normalized uses is:Wherein x_iIt it is the i-th number of lungs sound data before normalized According to, x_minIt is the minima of lungs sound data, x before normalized_maxIt is the maximum of lungs sound data, y before normalized_iIt is to return The i-th data of lungs sound data after one change process.

Detection analytical equipment the most according to claim 1, it is characterised in that described lungs sound sampling module is used for judging lungs sound In the range of whether the sample frequency of the digitized Lung Sounds of sampling module collection is positioned at 8000～10000Hz, if this numeral Change the sample frequency of Lung Sounds not 8000～10000Hz, then the not number in the range of 8000～10000Hz by sample frequency It is 9000Hz that word Lung Sounds carries out resampling to sample frequency.

Detection analytical equipment the most according to claim 1, it is characterised in that during the analysis of described digitized Lung Sounds Section should comprise 2 or more than 2 breathing cycles.

Detection analytical equipment the most according to claim 4, it is characterised in that the time span of described digitized Lung Sounds For 8s-12s.

Detection analytical equipment the most according to claim 1, it is characterised in that the wavelet decomposition number of plies of described wavelet decomposition J >=5, high-frequency band Dn that n-th layer wavelet decomposition component signal is corresponding is: f/2ⁿ⁺¹～f/2ⁿ, n is 1,2,3 ... j.

Detection analytical equipment the most according to claim 1, it is characterised in that the wavelet basis function of described wavelet decomposition is DbN series, 3≤N≤5.

Detection analytical equipment the most according to claim 1, it is characterised in that the letter of described each wavelet decomposition component signal Number it is characterized as standard deviation, if signal is x_i, its standard deviation computing formula is:

$std=\sqrt{\frac{1}{N}\underset{i=1}{\overset{N}{\Σ}}{(x_i-\mathrm{\μ})}^2}$

Wherein, N is the number of array, and μ is the average of array,

Detection analytical equipment the most according to claim 1, it is characterised in that the letter of described each wavelet decomposition component signal Number it is characterized as energy, if signal is x_i, its energy balane formula is:

$E=\underset{i=1}{\overset{N}{\Σ}}{(x_i-\mathrm{\μ})}^2$

Wherein, N is the number of array, and μ is the average of array,