# How does principal component analysis determine the weights for 0

### How to determine indicator weights using principal component analysis?

In SPSS, principal component analysis is achieved by setting the extraction method in factor analysis, if the extraction method set is principal component, then the calculation is the principal component score, in addition, factor analysis and principal component analysis, although the principle is different, but the calculation of the two composite scores is the same.

Determining the weight of the data is also an important prerequisite for data analysis. SPSS factor analysis can be used to determine the weights. The main steps are:

(1) First standardize the data, this is to take into account the inconsistency of the scale between different data, and thus must be dimensionless.

(2) Factor analysis (principal component approach) is performed on the standardized data, using variance maximizing rotation.

(3) Write down the principal factor scores and the equation contribution of each principal factor.

Fj=β1j*X1+β2j*X2+β3j*X3+……+βnj*Xn; Fj is the principal component (j=1, 2, ……, m), X1, X2, X3, … …, Xn for each indicator, β1j, β2j, β3j, ……, βnj is the coefficient score of each indicator in the principal component Fj, and ej is used to indicate the equation contribution rate of Fj.

(4) Find the indicator weights. ωi=[(m∑j)βij*ej]/[(n∑i)(m∑j)βij*ej],ωi is the weight of indicator Xi.

Factor analysis is applied in the determination of the weights of evaluation indicators, the common factor variance of each indicator obtained by principal component analysis, and the size of its value indicates the contribution of the indicator to the overall variation, by calculating the percentage of each common factor variance to the sum of the common factor variance.

### Principal components to calculate the weights of the full step by step comb!

I. Research Scenarios

Principal Component Analysis is used to condense data information, such as a total of 20 indicator values, whether this 20 items can be condensed into 4 generalized indicators. In addition to this, principal component analysis can be used for weight calculation and comprehensive competitiveness research. That is, there are three practical application scenarios:

SPSSAU operation

SPSSAU left side of the dashboard “Advanced Methods” ¡ú “principal components”;

Three, SPSSAU general steps

The first step: to determine whether to carry out principal component (pca) analysis; judgment criteria for the KMO value is greater than 0.6.

The second step: the principal components of the analytical terms with the judgment of the corresponding relationship.

Special Note: If the purpose of the study is to condense the information and to find out the correspondence between the principal components and the analyzed items, SPSSAU suggests to use Factor Analysis [please refer to the manual of Factor Analysis] instead of Principal Component Analysis. The purpose of principal component analysis is to condense the information (but less concerned about the relationship between the principal components and the analyzed items), the calculation of weights, and the calculation of the composite score.

Sometimes less attention is paid to the correspondence between the principal components and the analyzed items, such as when performing the overall competitiveness calculation, which does not require much attention to the correspondence between the principal components and the analyzed items.

Judgment of correspondence between principal components and analytic terms: assuming that the expected three principal components and 10 analytic terms; principal components and analytic terms cross a total of 30 numbers, which is called the “loading coefficient” (the value of the loading coefficient indicates the degree of correlation between the analytic terms and the principal components); for each principal component, there are 10 “load factors” corresponding to the principal components, which are called “load factors”. For each principal component, there are 10 “loading coefficients”, and for each analyzed item, there are 3 “loading coefficient values” (e.g., 0.765, -0.066, and 0.093), and the one whose absolute value is greater than 0.4 is selected as the one whose absolute value is greater than 0.765, and the one whose absolute value is greater than 0.4 is selected as the one whose absolute value is greater than 0.4, and the other one is selected as the one whose absolute value is greater than 0.4, and the other one is selected as the one whose absolute value is greater than 0.4 (0.765). If it corresponds to principal component 1, it means that the analyzed item should be classified under principal component 1.

There are three kinds of cases to delete the unreasonable analytic items; the first one is: if the value of the common degree (common factor variance) of the analytic items is less than 0.4, the corresponding analytic items should be deleted; the second one is: the absolute value of the “loading coefficients” corresponding to the analytic items is less than 0.4, and the item should be deleted; the second one is: the absolute value of “loading coefficient” corresponding to the analytic items is less than 0.4; the item needs to be deleted; the item needs to be deleted. The third category: if there is a serious deviation between an analytical item and its principal component counterpart (usually called ‘the same as the other’), the analytical item should also be deleted.

Step 3: Naming the Principal Components

After removing the irrational analytic terms in step 2, and confirming that the correspondence between the principal components and the analytic terms is good, then the principal components can be named in conjunction with the principal component-analytic term correspondence.

Four, principal component analysis to calculate the weights

1. Variance Explanation Rate Table

The use of principal component analysis to obtain the variance explanation rate table, the principal component analysis to extract a total of two principal components, the root of the eigenvalue is greater than 1, the variance explanation rate of the two principal components are 54.450%, 7.798%, the cumulative variance explanation rate is 54.450%, 7.798%, and the cumulative variance explanation rate is 54.450%, 7.798%, and the cumulative variance explanation rate is 54.450%, 7.798%. 7.798%, and the cumulative variance explained is 62.248%.

The table of loading coefficients shows the loading coefficients of the analyzed items in the principal components, and the loading coefficients can reflect the information extraction situation of the principal components for the analyzed items.

When calculating the weights of the analyzed items, it is necessary to use the information of the loading coefficients to calculate, which is divided into three steps:

Firstly, calculate the linear combination coefficients matrix, with the formula: loading matrix/Sqrt (characteristic root), that is, loading coefficients are divided by the square root of the corresponding characteristic root.

3. Linear combination coefficients and weight results

When calculating the weights of the analyzed terms, it is necessary to use the information of the loading coefficients and so on, and it is divided into three steps:

The first one is to calculate the linear combination coefficients matrix, and the formula is: loading matrix / Sqrt (characteristic root), i.e., the loading coefficient divided by the square root of the corresponding characteristic root.

Example: Principal component 1:

And so on.

Principal Component 2:

And so on.

Second: calculate the composite score coefficient, the formula is: cumulative (linear combination of coefficients * variance explained) / cumulative variance explained, that is, the linear combination of coefficients were multiplied by the variance explained cumulative, and divided by the cumulative variance explained, that is, to get the composite score coefficient.

Example: (0.287*54.45%)/62.25%+(0.1201*7.80%)/62.25%≈0.2661;

(0.278*54.45%)/62.25%+(0.1201*7.80%)/62.25%≈ 0.2683;

(0.2443*54.45%)/62.25% + (0.5818*7.80%)/62.25% ≈ 0.2866;

(0.2617*54.45%)/62.25% + (0.4385*7.80%)/62.25 % ≈ 0.2839;

and so on.

Third: Calculate the weight, the integrated score coefficients are summed and normalized to obtain the weight value of each indicator.

Sum normalization:

Example: the composite score coefficients and the sum of 3.2671, (0.2661 + 0.2683 + … + 0.2199 = 3.2671).

0.2661/3.2671=8.15%; 0.2683/3.2671=8.21%; 0.2866/3.2671=8.77%; and so on.

Load diagram is a graphical display of the relationship between the components and the rotated load values, which is less frequently used, and it is usually necessary to manually add a ‘circle’ to encircle the factors that are next to each other, to more intuitively display the affiliation of the components to the analyzed items. Due to readability and interpretability issues, the focus is generally only on the first few components with the highest variance explained, and in most cases only 2.

V. Explanation of Other Output Indicators

1. KMO and Bartlett’s test

The use of Principal Component Analysis (PCA) for the study of information concentration, first analyze the research data to see if it is suitable for Principal Component Analysis (PCA), and as can be seen in the table above, the KMO is 0.910, which is greater than 0.6. Meet the prerequisite requirements of principal component analysis, meaning that the data can be used for principal component analysis research. As well as the data through the Bartlett sphericity test (p<0.05), indicating that the research data is suitable for principal component analysis.

2. Component Score Coefficient Matrix

If the purpose of using principal component analysis is to condense the information, the “Component Score Coefficient Matrix” table is ignored. If you use principal component analysis to calculate weights, you need to use the “component score coefficient matrix” to establish the relationship equation between the principal components and the study items (based on the standardized data to establish the relationship expression), as follows:

Component score 1

= 0.104*A1+0.101*A2+…+0.101*D2+0.090*D3;

Component Score 2

=0.115*A1+0.192*A2+…-0.044*D2+0.025 *D3;

3. Fragmentation chart

VI.Difficulties

1. What is the meaning of principal component regression?

After the principal component analysis, check to save the ‘Component Score’, the SPSSAU system will generate a new title for identifying the ‘Component Score’, for example: PcaScore1_1234, continue to use the The ‘Component Score’ is used in the next linear regression analysis, which is called ‘Principal Component Regression’, usually ‘Principal Component Regression’ is used to solve the problem of covariance.

2. How is principal component analysis performed on panel data when SPSSAU?

Panel data can be directly subjected to principal component analysis, panel data format is relatively special, in the analysis of the research indicators directly for the analysis can be.

3. SPSSAU when the component scores are standardized data for?

Component scores are calculated based on standardized data by default.

Summary

In various fields of scientific research, in order to comprehensively and objectively analyze the problem, it is often necessary to conduct a large number of observations on multiple variables reflecting the thing, and if these variables are analyzed one by one, it may result in looking at the thing lopsided, and it is not good to draw consistent conclusions, and the Principal Component Analysis (PCA) is a method that takes into account the interrelationships among the indicators and uses the dimensionality reduction to analyze them. The principal component analysis is to consider the interrelationship between the indicators, using the thinking of dimensionality reduction, converting multiple indicators into fewer unrelated composite indicators, thus making the study simpler. The above is a description of the indicators of principal component analysis.

### How to Determine Indicator Weights with Principal Component Analysis

1Enter the data.

2Click the Analyze drop-down menu and select Factor under DataRection.

3Open FactorAnalysis and select the data variables one by one to enter the Variables dialog box.

4 Click the Descriptive button in the main dialog box to open the FactorAnalysis:Descriptives subdialog box, select UnivariateDescriptives in the Statistics column to request the output of the mean and standard deviation of the variables, and select CorrelationMatrix in the CorrelationMatrix column. In the CorrelationMatrix column, select Coefficients to calculate the correlation coefficient matrix, and click the Continue button to return to the FactorAnalysis main dialog box.

5 Click the Extraction button in the main dialog box to open the FactorAnalysis:Extraction subdialog box shown below. Select the default factor extraction method, PrincipalComponents, in the Method list, select the default CorrelationMatrix item in the Analyze column to request that the principal components be solved from the correlation coefficient matrix, and select the number of factors in the Exact column. NumberofFactors;6 in the Exact column, requesting that the scores and explained variance of all principal components be displayed. Click the Continue button to return to the main FactorAnalysis dialog box.

6 Click the OK button in the main dialog box to output the results.

### How to calculate the weights of hierarchical analysis and principal component analysis?

Hierarchical analysis:

Principal Component Analysis and Hierarchical Analysis both calculate the weight of the different, AHP Hierarchical Analysis is a qualitative and quantitative calculation of the weight of the research method, the use of the two-by-two method of comparison, the establishment of the matrix, the use of the size of the number of the relativity of the number of the larger the more important the number of the principle of the weight will be higher, and ultimately calculated to get the importance of each factor.

Principal Component Analysis

(1) Principle of the Method and Applicable Scenarios

Principal Component Analysis (PCA) is a method of condensing data, condensing multiple indicators into a few generalized indicators that are not related to each other (Principal Components), so as to achieve the purpose of dimensionality reduction. Principal component analysis can calculate both principal component weights and indicator weights.

(2) Steps

Use SPSSAU [Advanced Methods – Principal Component Analysis].

If you calculate the principal component weights, you need to use the variance explained. The specific weighting treatment is: variance explained rate divided by cumulative variance explained rate.

For example, in this case, 5 indicators extracted a total of 2 principal components:

Principal component 1 weight: 45.135%/69.390% = 65.05%

Principal component 2 weight: 24.254%/69.390% = 34.95%

If it is the calculation of the weight of the indicators, you can directly view the “Linear combination of coefficients and weight results table”, SPSSAU automatically output the results of the weight share of each indicator. Its calculation principle is divided into three steps:

First: calculate the linear combination coefficients matrix, the formula is: loading matrix/Sqrt (characteristic root), that is, the loading coefficients divided by the square root of the corresponding characteristic root;

Second: calculate the composite score coefficient, the formula is: cumulative (linear combination coefficients * variance explained rate) / cumulative variance explained rate, that is, obtained in the last step in the Linear combination of coefficients obtained in the previous step were multiplied by the variance interpretation rate and then added, and divided by the cumulative variance interpretation rate;

Third: Calculate the weights, the composite score coefficients are normalized to obtain the weight value of each indicator.

### How to use spss to calculate the weight of each indicator, please expert help, thesis urgent ！！！！

1, first you need to select the analysis – regression analysis – linear regression.

2, the next choice to open one of the dialog box.

3, and then the need to calculate the weight of the variable selected into.

4. Then open the Statistics dialog box, which contains the method for calculating weights.

5. Next, you can select the covariance diagnostic, and the weights can be generated automatically.

6. Click OK to generate the results and get the weight of each indicator.