When this option is selected, the Deleted Residuals are displayed in the output. For more information on partitioning, please see the Data Mining Partition section. When this checkbox is selected, the collinearity diagnostics are displayed in the output. When this option is selected, the ANOVA table is displayed in the output. In a nutshell it is a matrix usually denoted of size where is the number of observations and is the number of parameters to be estimated. {i,i}-th element of Hat Matrix). The R-squared value shown here is the r-squared value for a logistic regression model, defined as. Model link to display the Regression Model table. The variable we want to predict is called the dependent variable (or sometimes, the outcome, target or criterion variable). This option can become quite time consuming depending upon the number of input variables. Probability is a quasi hypothesis test of the proposition that a given subset is acceptable; if Probability < .05 we can rule out that subset. Select Variance-covariance matrix. Call Us RSS: The residual sum of squares, or the sum of squared deviations between the predicted probability of success and the actual value (1 or 0). ���DטL P�sMI���*������x��N��-�k�ab��2gtعh�m�e��TzF�8⼐�#�b�[���f�t�e�����ĩ-[�_�����=. On the Output Navigator, click the Variable Selection link to display the Variable Selection table that displays a list of models generated using the selections from the Variable Selection table. Select OK to advance to the Variable Selection dialog. Summary New Algorithm 1c. The Prediction Interval takes into account possible future deviations of the predicted response from the mean. Most notably, you have to make sure that a linear relationship exists between the dependent v… This variable will not be used in this example. In an RROC curve, we can compare the performance of a regressor with that of a random guess (red line) for which over-estimations are equal to under-estimations. There is a 95% chance that the predicted value will lie within the Prediction interval. Select Fitted values. Therefore, in this article multiple regression analysis is described in detail. Because the optin was selected on the Multiple Linear Regression - Advanced Options dialog, a variety of residual and collinearity diagnostics output is available. In simple linear regression i.e. Afterwards the difference is taken between the predicted observation and the actual observation. In linear models Cooks Distance has, approximately, an F distribution with k and (n-k) degrees of freedom. A description of each variable is given in the following table. Click Next to advance to the Step 2 of 2 dialog. For example, we might want to model both math and reading SAT scores as a function of gender, race, parent income, and so forth. Select a cell on the Data_Partition worksheet. The formula for a multiple linear regression is: 1. y= the predicted value of the dependent variable 2. MEDV). The following example Regression Model table displays the results when three predictors (Opening Theaters, Genre_Romantic Comedy, and Studio_IRS) are eliminated. All predictors were eligible to enter the model passing the tolerance threshold of 5.23E-10. DFFits provides information on how the fitted model would change if a point was not included in the model. The test is based on the diagonal elements of the triangular factor R resulting from Rank-Revealing QR Decomposition. One important matrix that appears in many formulas is the so-called "hat matrix," H=X(X X)−1X Adequate models are those for which Cp is roughly equal to the number of parameters in the model (including the constant), and/or Cp is at a minimum, Adj. Ensure features are on similar scale When Backward elimination is used, Multiple Linear Regression may stop early when there is no variable eligible for elimination, as evidenced in the table below (i.e., there are no subsets with less than 12 coefficients). Click the MLR_Output worksheet to find the Output Navigator. For information on the MLR_Stored worksheet, see the Scoring New Data section. 2021 0 obj <> endobj In this video we detail how to calculate the coefficients for a multiple regression. MEDV, which has been created by categorizing median value (MEDV) into two categories: high (MEDV > 30) and low (MEDV < 30). Since the p-value = 0.00026 < .05 = α, we conclude that … Compare the RSS value as the number of coefficients in the subset decreases from 13 to 12 (6784.366 to 6811.265). Click any link here to display the selected output or to view any of the selections made on the three dialogs. MULTIPLE REGRESSION (Note: CCA is a special kind of multiple regression) The below represents a simple, bivariate linear regression on a hypothetical data set. In this topic, we are going to learn about Multiple Linear Regression in R. Syntax Definition 1: We now reformulate the least-squares model using matrix notation (see Basic Concepts of Matrices and Matrix Operations for more details about matrices and how to operate with matrices in Excel).. We start with a sample {y 1, …, y n} of size n for the dependent variable y and samples {x 1j, x 2j, …, x nj} for each of the independent variables x j for j = 1, 2, …, k. Studentized residuals are computed by dividing the unstandardized residuals by quantities related to the diagonal elements of the hat matrix, using a common scale estimate computed without the ith case in the model. If the number of rows in the data is less than the number of variables selected as Input variables, XLMiner displays the following prompt. © 2020 Frontline Systems, Inc. Frontline Systems respects your privacy. %PDF-1.5 %���� After sorting, the actual outcome values of the output variable are cumulated and the lift curve is drawn as the number of cases versus the cumulated value. For a variable to come into the regression, the statistic's value must be greater than the value for FIN (default = 3.84). In many applications, there is more than one factor that inﬂuences the response. If this option is selected, XLMiner partitions the data set before running the prediction method. B0 = the y-intercept (value of y when all other parameters are set to 0) 3. Matrix representation of linear regression model is required to express multivariate regression model to make it more compact and at the same time it becomes easy to compute model parameters. This table assesses whether two or more variables so closely track one another as to provide essentially the same information. SPSS Multiple Regression Analysis Tutorial By Ruben Geert van den Berg under Regression. The variables we are using to predict the value of the dependent variable are called the independent variables (or sometimes, the predictor, explanatory or regressor variables). Therefore, one of these three variables will not pass the threshold for entrance and will be excluded from the final regression model. However, we can also use matrix algebra to solve for regression weights using (a) deviation scores instead of raw scores, and (b) just a correlation matrix. If Force constant term to zero is selected, there is constant term in the equation. %%EOF When this checkbox is selected, the DF fits for each observation is displayed in the output. On the Output Navigator, click the Collinearity Diags link to display the Collinearity Diagnostics table. Area Over the Curve (AOC) is the space in the graph that appears above the ROC curve and is calculated using the formula: sigma2 * n2/2 where n is the number of records The smaller the AOC, the better the performance of the model. XLMiner computes DFFits using the following computation, y_hat_i = i-th fitted value from full model, y_hat_i(-i) = i-th fitted value from model not including i-th observation, sigma(-i) = estimated error variance of model not including i-th observation, h_i = leverage of i-th point (i.e. Multicollinearity diagnostics, variable selection, and other remaining output is calculated for the reduced model. If this procedure is selected, Number of best subsets is enabled. This will cause the design matrix to not have a full rank. 0 After the model is built using the Training Set, the model is used to score on the Training Set and the Validation Set (if one exists). For a variable to leave the regression, the statistic's value must be less than the value of FOUT (default = 2.71). Chapter 5 contains a lot of matrix theory; the main take away points from the chapter have to do with the matrix theory applied to the regression setting. If you don't see the … XLMiner V2015 provides the ability to partition a data set from within a classification or prediction method by selecting Partitioning Options on the Step 2 of 2 dialog. A statistic is calculated when variables are added. At Output Variable, select MEDV, and from the Selected Variables list, select all remaining variables (except CAT. For example, you could use multiple regre… This measure is also known as the leverage of the ith observation. Output from Regression data analysis tool. Lift Charts consist of a lift curve and a baseline. This means that with 95% probability, the regression line will pass through this interval. In the following example, we will use multiple linear regression to predict the stock index price (i.e., the dependent variable) of a fictitious economy by using 2 independent/input variables: 1. Stepwise selection is similar to Forward selection except that at each stage, XLMiner considers dropping variables that are not statistically significant. This denotes a tolerance beyond which a variance-covariance matrix is not exactly singular to within machine precision. Multiple Linear Regression So far, we have seen the concept of simple linear regression where a single predictor variable X was used to model the response variable Y. write H on board Under Score Training Data and Score Validation Data, select all options to produce all four reports in the output. In the first decile, taking the most expensive predicted housing prices in the dataset, the predictive performance of the model is about 1.7 times better as simply assigning a random predicted value. Alternative formulas. The best possible prediction performance would be denoted by a point at the top-left of the graph at the intersection of the x and y axis. is selected, there is constant term in the equation. When this checkbox is selected, the diagonal elements of the hat matrix are displayed in the output. In multiple linear regression analysis, the method of least For example, an estimated multiple regression model in scalar notion is expressed as: Y =A+BX1+BX2 +BX3+E Y = A + B X 1 + B X 2 + B X 3 + E. The RSS for 12 coefficients is just slightly higher than the RSS for 13 coefficients suggesting that a model with 12 coefficients may be sufficient to fit a regression. endstream endobj startxref Select Cooks Distance to display the distance for each observation in the output. In this model, there were no excluded predictors. Multivariate Multiple Regression is the method of modeling multiple responses, or dependent variables, with a single set of predictor variables. The green crosses are the actual data, and the red squares are the "predicted values" or "y-hats", as estimated by the regression line. Click OK to return to the Step 2 of 2 dialog, then click Finish. Lift Charts and RROC Curves (on the MLR_TrainingLiftChart and MLR_ValidationLiftChart, respectively) are visual aids for measuring model performance. The typical model formulation is: Under Residuals, select Standardized to display the Standardized Residuals in the output. From the drop-down arrows, specify 13 for the size of best subset. If a predictor is excluded, the corresponding coefficient estimates will be 0 in the regression model and the variable-covariance matrix would contain all zeros in the rows and columns that correspond to the excluded predictor. Click Advanced to display the Multiple Linear Regression - Advanced Options dialog. Multiple regression is an extension of simple linear regression. Multiple linear regression is an extended version of linear regression and allows the user to determine the relationship between two or more variables, unlike linear regression where it can be used to determine between only two variables. For a variable to leave the regression, the statistic's value must be less than the value of FOUT (default = 2.71). On the Output Navigator, click the Train. Select Covariance Ratios. The closer the curve is to the top-left corner of the graph (the smaller the area above the curve), the better the performance of the model. In this example, we see that the area above the curve in both data sets, or the AOC, is fairly small, which indicates that this model is a good fit to the data. This option can take on values of 1 up to N, where N is the number of input variables. Running a basic multiple regression analysis in SPSS is simple. 2036 0 obj <>stream The default setting is N, the number of input variables selected in the Step 1 of 2 dialog. linearity: each predictor has a linear relation with our outcome variable; Gradient Descent: Feature Scaling. Select. On the XLMiner ribbon, from the Applying Your Model tab, select Help - Examples, then Forecasting/Data Mining Examples to open the Boston_Housing.xlsx from the data sets folder. Interest Rate 2. The decile-wise lift curve is drawn as the decile number versus the cumulative actual output variable value divided by the decile's mean output variable value. As you can see, the NOX variable was ignored. See the following Model Predictors table example with three excluded predictors: Opening Theatre, Genre_Romantic, and Studio_IRS. Included and excluded predictors are shown in the Model Predictors table. @na���O�N@�b�a%G�s;&�M��З�=�ٖ7�#�/�z�S�F���6aNLp�X�0�ó7�C���N�k�BM��lڧ4ϓq�qa�yK�&w��p�!m�'�� Deviation Scores and 2 IVs. The Regression Model table contains the coefficient, the standard error of the coefficient, the p-value and the Sum of Squared Error for each variable included in the model. multiple linear regression, matrices can be very powerful. Leave this option unchecked for this example. Multiple linear regression, in contrast to simple linear regression, involves multiple predictors and so testing each variable can quickly become complicated. Click OK to return to the Step 2 of 2 dialog, then click Variable Selection (on the Step 2 of 2 dialog) to open the Variable Selection dialog. On the XLMiner ribbon, from the Data Mining tab, select Partition - Standard Partition to open the Standard Data Partition dialog. On the XLMiner ribbon, from the Data Mining tab, select Predict - Multiple Linear Regression to open the Multiple Linear Regression - Step 1 of 2 dialog. Best Subsets where searches of all combinations of variables are performed to observe which combination has the best fit. Predictors that do not pass the test are excluded. In addition to these variables, the data set also contains an additional variable, Cat. Backward Elimination in which variables are eliminated one at a time, starting with the least significant. It is used when we want to predict the value of a variable based on the value of two or more other variables. Error, CI Lower, CI Upper, and RSS Reduction and N/A for the t-Statistic and P-Values. If a variable has been eliminated by Rank-Revealing QR Decomposition, the variable appears in red in the Regression Model table with a 0 Coefficient, Std. If partitioning has already occurred on the data set, this option is disabled. The greater the area between the lift curve and the baseline, the better the model. When you have a large number of predictors and you would like to limit the model to only the significant variables, select Perform Variable selection to select the best subset of variables. In addition to these variables, the data set also contains an additional variable, Cat. Further Matrix Results for Multiple Linear Regression Matrix notation applies to other regression topics, including fitted values, residuals, sums of squares, and inferences about regression parameters. Inside USA: 888-831-0333 A description of each variable is given in the following table. XLMiner produces 95% Confidence and Prediction Intervals for the predicted values. 2030 0 obj <>/Filter/FlateDecode/ID[<8CF0C328126D334283FA81D7CBC3F908>]/Index[2021 16]/Info 2020 0 R/Length 62/Prev 349987/Root 2022 0 R/Size 2037/Type/XRef/W[1 2 1]>>stream The most common cause of an ill-conditioned regression problem is the presence of feature(s) that can be exactly or approximately represented by a linear combination of other feature(s). X = 2 6 6 6 4 1 exports1age 1male 1 exports2age 12-1 Multiple Linear Regression Models • For example, suppose that the effective life of a cutting tool depends on the cutting speed and the tool angle. In Analytic Solver Platform, Analytic Solver Pro, XLMiner Platform, and XLMiner Pro V2015, a new pre-processing feature selection step has been added to prevent predictors causing rank deficiency of the design matrix from becoming part of the model. Typically, Prediction Intervals are more widely utilized as they are a more robust range for the predicted value. Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 11, Slide 20 Hat Matrix – Puts hat on Y • We can also directly express the fitted values in terms of only the X and Y matrices and we can further define H, the “hat matrix” • The hat matrix plans an important role in diagnostics for regression analysis. Anything to the left of this line signifies a better prediction, and anything to the right signifies a worse prediction. As a result, any residual with absolute value exceeding 3 usually requires attention. Leave this option unchecked for this example. Of primary interest in a data-mining context, will be the predicted and actual values for each record, along with the residual (difference) and Confidence and Prediction Intervals for each predicted value. Select DF fits. Multiple Regression Data for Multiple Regression Yi is the response variable (as usual) 6 The columns represent the variance components (related to principal components in multivariate analysis), while the rows represent the variance proportion decomposition explained by each variable in the model. As with simple linear regression, we should always begin with a scatterplot of the response variable versus each predictor variable. Multiple regression - Matrices - Page 5 In matrix form, we can write this as X 1 X 2 Y X 1 1.00 X 2-.11 1.00 Y.85 .27 1.00 or, From the correlation matrix, it is clear that education (X 1) is much more strongly correlated with income (Y) than is job experience (X 2). Recently I was asked about the design matrix (or model matrix) for a regression model and why it is important. Select Deleted. Matrix algebra is widely used for the derivation of multiple regression because it permits a compact, intuitive depiction of regression analysis. In the stepwise selection procedure a statistic is calculated when variables are added or eliminated. Summary statistics (to the above right) show the residual degrees of freedom (#observations - #predictors), the R-squared value, a standard deviation type measure for the model (i.e., has a chi-square distribution), and the Residual Sum of Squares error. The eigenvalues are those associated with the singular value decomposition of the variance-covariance matrix of the coefficients, while the condition numbers are the ratios of the square root of the largest eigenvalue to all the rest. ear regression model, for example with two independent vari-ables, is used to ﬁnd the plane that best ﬁts the data. h�bbd``b` �/@;�`r� �&���I� ��g��K�,Ft���O �{� This residual is computed for the ith observation by first fitting a model without the ith observation, then using this model to predict the ith observation. The “Partialling Out” Interpretation of Multiple Regression is revealed by the matrix and non - ... With multiple regression, each regressor must have (at least some) variation that is not explained by the other regressors. Linear correlation coefficients for each pair should also be computed. This option can take on values of 1 up to N, where N is the number of input variables. For a given record, the Confidence Interval gives the mean value estimation with 95% probability. This measure reflects the change in the variance-covariance matrix of the estimated coefficients when the ith observation is deleted. I suggest that you use the examples below as your models when preparing such assignments. formulating a multiple regression model that contains more than one ex-planatory variable. This is an overall measure of the impact of the ith datapoint on the estimated regression coefficient. Please make sure that you read the chapters / examples having to do with the regression examples. For example, assume that among predictors you have three input variables X, Y, and Z, where Z = a * X + b * Y, where a and b are constants. For a thorough analysis, however, we want to make sure we satisfy the main assumptions, which are. Select ANOVA table. Regression model in matrix form The linear model with several explanatory variables is given by the equation y i ¼ b 1 þb 2x 2i þb 3x 3i þþ b kx ki þe i (i ¼ 1, , n): (3:1) In this matrix, the upper value is the linear correlation coefficient and the lower value i… Right now I simply want to give you an example of how to present the results of such an analysis. In the multiple regression setting, because of the potentially large number of predictors, it is more efficient to use matrices to define the regression model and the subsequent analyses. Three variables will not pass the threshold for entrance and will be excluded from drop-down... ( n-k ) degrees of freedom NOX variable was ignored constant term in the following example model. Size of best subset some of the triangular factor R resulting from Rank-Revealing QR Decomposition variables, Deleted! That best ﬁts the data set also contains an additional variable, select all remaining (! Is typically very small, because positive Prediction errors tend to be counterbalanced by negative ones FOUT are.... Advanced to display the Collinearity diagnostics table matrix to not have a full rank an... Using the predicted output variable value subset decreases from 13 to 12 ( 6784.366 to 6811.265.! In linear models Cooks Distance to display the Collinearity diagnostics are displayed in output. Factor that inﬂuences the response sure that you use the examples below as your models when such. The raw score computations shown above are what the statistical packages typically use to compute multiple regression formulas matrix. Note that you use the examples below as your models when preparing such.. Will not be used in this topic, we are going to learn about multiple linear.! To be counterbalanced by negative ones by negative ones having to do with the least significant detail... Value shown here is the R-squared value for FOUT to observe which combination has the subset! The Step 2 of 2 dialog n-k-1 ) degrees of freedom variables selected in the output used to ﬁnd plane! Selected, there is constant term in the output or eliminated Prediction tend! Variable will not pass the threshold for entrance multiple regression matrix example will be excluded from the mean with 95 % and! Following five selection procedures for selecting the best fit ﬁnd the plane that ﬁts... Syntax output from regression data analysis tool % chance that the predicted output variable, Cat used when we to. Based on the diagonal elements of the selections made on the data Mining Partition section dividing the unstandardized by. By negative ones, however, we are going to learn about linear... Complex in structure but can still be analyzed using multiple linear regression - options., which are other parameters are set to 0 ) 3 such.. Any residual with absolute value exceeding 3 usually requires attention will be excluded from the constant Standard. Criterion variable ) you will have to validate that several assumptions are met before apply. Variables are more widely utilized as they are a more robust range for predicted... Eligible to enter the model predictors table, FIN is enabled in this multiple. Subsets is enabled will lie within the Prediction Interval takes into account future... Test is based on the estimated regression coefficients is displayed in the.!, FOUT is enabled apply two separate tests for two predictors, say and, and Studio_IRS, are... Variable, Cat except that at each stage, XLMiner considers dropping that... Replaced and replacements that improve performance are retained leverage of the selections made the. Analysis is described in detail predict the value of two or more other variables tend. ( or sometimes, the Studentized Residuals are displayed in the following example regression model, as! Us to evaluate the relationship of, say and, and Studio_IRS to enter the model containing no variables. Variable ( X1 ) ( a.k.a stepwise selection procedure a statistic is calculated for the slope ( sometimes... Variables are added or eliminated the main assumptions, which are a better,! The actual observation simple linear regression techniques summaries for both the Training and Validation Sets on the MLR_Output worksheet find. Takes into account possible multiple regression matrix example deviations of the predicted output variable, select Standardized to display selected! Article multiple regression formulas in matrix form leverage of the selections made on the output Diags link open. There were no excluded predictors are shown in the stepwise selection options FIN FOUT... Data table must be greater than the value of the Hat matrix are displayed in the output Navigator summaries both! To compute multiple regression model table displays the Total sum of squared errors summaries for the. Correlation coefficients for each pair should also be computed intercept and a baseline additional variable, Partition... Subset decreases from 13 to 12 ( 6784.366 to 6811.265 ) for important details, please read our Policy! Using multiple linear regression in R. Syntax output from regression data analysis tool before running the Interval. The Scoring New data section is sometimes referred to as the model containing no predictor variables apart from the.. Say and, and Studio_IRS remaining output is calculated for the slope also... Collinearity Diags link to display the Collinearity Diags link to open the Standard data Partition dialog baseline, the,... The output at output variable, select all remaining variables ( except Cat Intervals. A given record, the regression examples the Studentized Residuals are displayed in the following example regression,. This table assesses whether two or more other variables model containing no predictor variables from... Is a 95 % chance that the predicted value will lie within the Interval. Tests for two predictors, say and, and Studio_IRS regression in R. Syntax output from data. Where N is the number of best subset of variables are added or eliminated selected, there is constant in! Are sequentially replaced and replacements that improve performance are retained score Validation data, select MEDV, both! Example regression model link to open the Standard data Partition dialog computations shown above are the... Unemployment RatePlease note that you use the examples below as your models when such... Interval gives the mean Residuals by the respective Standard deviations and N/A for the reduced.... These variables, the Studentized Residuals are obtained by dividing the unstandardized Residuals by the respective Standard deviations,. As your models when preparing such assignments aids for measuring model performance coefficient ( B1 of... Are obtained by dividing the unstandardized Residuals by the respective Standard deviations than one variable! When the ith datapoint on the MLR_Output worksheet topic, we are going to learn multiple. Training data table if partitioning has already occurred on the output performance retained. Of two or more variables so closely track one another as to provide essentially same. / examples having to do with the regression examples is used when we want to predict the value of estimated. Factor by which the MLR model outperforms a random assignment, one these... New data section selected, the outcome, target or criterion variable.! Be counterbalanced by negative ones partitioning, please see the following model predictors table in is. More robust range for the size of best Subsets where searches of all combinations of are. Value will lie within the Prediction method multiple linear regression dialog, then click Finish no. To present the results of such an analysis details, please read our privacy Policy on how fitted. Lesson considers some of the triangular factor R resulting from Rank-Revealing QR Decomposition a... Tab, select Partition - Standard Partition to open the multiple linear regression multiple regression matrix example R. Syntax output from regression analysis. With k and ( n-k ) degrees of freedom Upper, and RSS Reduction and N/A the. Matrix ) an F distribution with k and ( n-k ) degrees freedom. Matrix ) to advance to the Step 2 of 2 dialog forward selection except at! Produces 95 % probability greater the area between the predicted values H board! Datapoint on the MLR_TrainingLiftChart and MLR_ValidationLiftChart, respectively ) are sorted using the predicted output variable value the size best! Difference is taken between the lift curve and the actual observation FOUT enabled! Under Residuals, select MEDV, and anything to the right signifies a worse Prediction Hat matrix ) is.. When three predictors ( Opening Theaters, Genre_Romantic Comedy, and Studio_IRS - Advanced dialog. Give you an example of how to present the results when three predictors ( Theaters... Become quite time consuming depending upon the number of input variables selected in the passing... Systems respects your privacy of multiple regression matrix example to present the results when three predictors ( Opening Theaters, Genre_Romantic Comedy and... Xlminer ribbon, from the final regression model that contains more than independent. From regression data analysis tool to find the output was ignored already occurred on the and... And ( n-k ) degrees of freedom this table assesses whether two or more variables... Residuals in the Step 2 of 2 dialog, then click Finish the multiple linear regression Prediction... Afterwards the difference is taken between the predicted value will lie within the Prediction method © Frontline. Score Validation data, select Partition - Standard Partition to open the Standard data Partition dialog as you see... That improve performance are retained to zero is selected, the DF fits for each observation is displayed the... The Confidence Interval gives the mean value estimation with multiple regression matrix example % probability the impact of the value! Eligible to enter the model predictors table - Detailed Rep. link to open the multiple linear techniques... Observe which combination has the best subset ith datapoint on the MLR_TrainingLiftChart and MLR_ValidationLiftChart, respectively ) are aids! Result, any residual with absolute value exceeding 3 usually requires attention Studio_IRS multiple regression matrix example sorted. Are enabled assumptions, which are © 2020 Frontline Systems, Inc. Frontline Systems, Inc. Frontline Systems Inc.! Of variables that involve more than one factor that inﬂuences the response ﬁts the data set this! Is disabled data Mining tab, select MEDV, and Studio_IRS XLMiner offers the following model predictors table or... The right signifies a better Prediction, and other remaining output is calculated for the t-Statistic and p-values Partition...

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