Quality Metrics for Linear Regression¶
Given a data set \(X = (x_i)\) that contains vectors of input variables \(x_i = (x_{i1}, \ldots, x_{ip})\), respective responses \(z_i = (z_{i1}, \ldots, z_{ik})\) computed at the prediction stage of the linear regression model defined by its coefficients \(\beta_{ht}\), \(h = 1, \ldots, k\), \(t = 1, \ldots, p\), and expected responses \(y_i = (y_{i1}, \ldots, y_{ik})\), \(i = 1, \ldots, n\), the problem is to evaluate the linear regression model by computing the root mean square error, variancecovariance matrix of beta coefficients, various statistics functions, and so on. See Linear Regression for additional details and notations.
For linear regressions, the library computes statistics listed in tables below
for testing insignificance of beta coefficients and one of the following values of QualityMetricsId
:
singleBeta
for a single coefficientgroupOfBetas
for a group of coefficients
For more details, see [Hastie2009].
Details¶
The statistics are computed given the following assumptions about the data distribution:
Responses \(y_{ij}\), \(i = 1, \ldots, n\), are independent and have a constant variance \(\sigma_j^2\), \(j = 1, \ldots, k\)
Conditional expectation of responses \(y_{.j}\), \(j = 1, \ldots, k\), is linear in input variables \(x_{.} = (x_{.1}, \ldots , x_{.p})\)
Deviations of \(y_{ij}\), \(i = 1, \ldots, n\), around the mean of expected responses \(\text{ERM}_j\), \(j = 1, \ldots, k\), are additive and Gaussian.
Testing Insignificance of a Single Beta¶
The library uses the following quality metrics:
Quality Metric 
Definition 

Root Mean Square (RMS) Error 
\(\sqrt{\frac{1}{n} \sum _{i=1}^{n} (y_{ij}  x_{ij})^2}\), \(j = 1, \ldots, k\) 
Vector of variances \(\sigma^2 = (\sigma_1^2, \ldots, \sigma_k^2)\) 
\(\sigma_j^2 = \frac {1}{n  p  1} \sum _{i=1}^{n} (y_{ij}  x_{ij})^2\), \(j = 1, \ldots, k\) 
A set of variancecovariance matrices \(C = C_1, \ldots, C_k\) for vectors of betas \(\beta_{jt}\), \(j = 1, \ldots, k\) 
\(C_j = {(X^T X)}^{1} \sigma_j^2\), \(j = 1, \ldots, k\) 
Zscore statistics used in testing of insignificance of a single coefficient \(\beta_{jt}\) 
\(\text{zscore}_{jt} = \frac {\beta_{jt}}{\sigma_j \sqrt{v_t}}\), \(j = 1, \ldots, k\), \(\sigma_j\) is the \(j\)th element of the vector of variance \(\sigma^2\) and \(ν_t\) is the \(t\)th diagonal element of the matrix \({(X^T X)}^{1}\) 
Confidence interval for \(\beta_{jt}\) 
\((\beta_{jt}  \text{pc}_{1\alpha} \sqrt{v_t}, \beta_{jt} + \text{pc}_{1\alpha} \sqrt{v_t})\), \(j = 1, \ldots, k\), \(\text{pc}_{1\alpha}\) is the \((1\alpha)\) percentile of the Gaussian distribution, \(\sigma_j\) is the \(j\)th element of the vector of variance \(\sigma^2\), \(ν_t\) is the \(t\)th diagonal element of the matrix \({(X^T X)}^{1}\) 
Testing Insignificance of a Group of Betas¶
The library uses the following quality metrics:
Quality Metric 
Definition 

Mean of expected responses, \(\text{ERM} = (\text{ERM}_1, \ldots, \text{ERM}_k)\) 
\(\text{ERM}_j = \frac {1}{n} \sum _{i=1}^{n} y_{ij}\), \(j = 1, \ldots, k\) 
Variance of expected responses, \(\text{ERV} = (\text{ERV}_1, \ldots, \text{ERV}_k)\) 
\(\text{ERV}_j = \frac {1}{n  1} \sum _{i=1}^{n} (y_{ij}  \text{ERM}_j)^2\), \(j = 1, \ldots, k\) 
Regression Sum of Squares \(\text{RegSS} = (\text{RegSS}_1, \ldots, \text{RegSS}_k)\) 
\(\text{RegSS}_j = \frac {1}{n} \sum _{i=1}^{n} (z_{ij}  \text{ERM}_j)^2\), \(j = 1, \ldots, k\) 
Sum of Squares of Residuals \(\text{ResSS} = (\text{ResSS}_1, \ldots, \text{ResSS}_k)\) 
\(\text{ResSS}_j = \sum _{i=1}^{n} (y_{ij}  z_{ij})^2\), \(j = 1, \ldots, k\) 
Total Sum of Squares \(\text{TSS} = (\text{TSS}_1, \ldots, \text{TSS}_k)\) 
\(\text{TTS}_j = \sum _{i=1}^{n} (y_{ij}  \text{ERM}_j)^2\), \(j = 1, \ldots, k\) 
Determination Coefficient \(R^2 = (R_1^2, \ldots, R_k^2)\) 
\(R^2_j = \frac {\text{RegSS}_j}{\text{TTS}_j }\), \(j = 1, \ldots, k\) 
Fstatistics used in testing insignificance of a group of betas \(F = (F_1, \ldots, F_k)\) 
\(F_j = \frac {(\text{ResSS}_{0j}  \text{ResSS}_j)/(p  p_0)} {{\text{ResSS}_j}/(n  p  1)}\), \(j = 1, \ldots, k\), where \(\text{ResSS}_j\) are computed for a model with \(p + 1\) betas and \(\text{ResSS}_{0j}\) are computed for a reduced model with \(p_0 + 1\) betas (\(p  p_0\) betas are set to zero) 
Batch Processing¶
Testing Insignificance of a Single Beta¶
Algorithm Input
The quality metric algorithm for linear regression accepts the input described below.
Pass the Input ID
as a parameter to the methods that provide input for your algorithm.
For more details, see Algorithms.
Input ID 
Input 


Pointer to the \(n \times k\) numeric table with responses (\(k\) dependent variables) used for training the linear regression model. This table can be an object of any class derived from 

Pointer to the model computed at the training stage of the linear regression algorithm. The model can only be an object of the 

Pointer to the \(n \times k\) numeric table with responses (\(k\) dependent variables) computed at the prediction stage of the linear regression algorithm. This table can be an object of any class derived from 
Algorithm Parameters
The quality metric algorithm for linear regression has the following parameters:
Parameter 
Default Value 
Description 



The floatingpoint type that the algorithm uses for intermediate computations. Can be 


Performanceoriented computation method, the only method supported by the algorithm. 

\(0.05\) 
Significance level used in the computation of confidence intervals for coefficients of the linear regression model. 

\(0.001\) 
Values below this threshold are considered equal to it. 
Algorithm Output
The quality metric algorithm for linear regression calculates the result described below.
Pass the Result ID
as a parameter to the methods that access the results of your algorithm.
For more details, see Algorithms.
Result ID 
Result 


Pointer to the \(1 \times k\) numeric table that contains root mean square errors computed for each response (dependent variable) Note By default, this result is an object of the 

Pointer to the \(1 \times k\) numeric table that contains variances \(\sigma^2_j\), \(j = 1, \ldots, k\) computed for each response (dependent variable). Note By default, this result is an object of the 

Pointer to the DataCollection object that contains \(k\) numeric tables, each with the \(m \times m\) variancecovariance matrix for betas of the jth response (dependent variable), where m is the number of betas in the model (m is equal to p when interceptFlag is set to false at the training stage of the linear regression algorithm; otherwise, m is equal to p + 1 ). The collection can contain objects of any class derived from 

Pointer to the \(k \times m\) numeric table that contains the Zscore statistics used in the testing of insignificance of individual linear regression coefficients,
where \(m\) is the number of betas in the model (\(m\) is equal to \(p\) when Note By default, this result is an object of the 

Pointer to the \(k \times 2 \times m\) numeric table that contains limits of the confidence intervals for linear regression coefficients:
where \(m\) is the number of betas in the model (\(m\) is equal to \(p\) when Note By default, this result is an object of the 

Pointer to the \(m \times m\) numeric table that contains the \({(X^TX)}^{1}\) matrix,
where \(m\) is the number of betas in the model (\(m\) is equal to \(p\) when 
Testing Insignificance of a Group of Betas¶
Algorithm Input
The quality metric algorithm for linear regression accepts the input described below.
Pass the Input ID
as a parameter to the methods that provide input for your algorithm.
For more details, see Algorithms.
Input ID 
Input 


Pointer to the \(n \times k\) numeric table with responses (\(k\) dependent variables) used for training the linear regression model. This table can be an object of any class derived from 

Pointer to the \(n \times k\) numeric table with responses (\(k\) dependent variables) computed at the prediction stage of the linear regression algorithm. This table can be an object of any class derived from 

Pointer to the \(n \times k\) numeric table with responses (\(k\) dependent variables) computed at the prediction stage of the linear regression algorithm using the reduced linear regression model, where \(p  p_0\) out of \(p\) beta coefficients are set to zero. This table can be an object of any class derived from 
Algorithm Parameters
The quality metric algorithm for linear regression has the following parameters:
Parameter 
Default Value 
Description 



The floatingpoint type that the algorithm uses for intermediate computations. Can be 


Performanceoriented computation method, the only method supported by the algorithm. 

\(0\) 
Number of beta coefficients used for prediction. 

\(0\) 
Number of beta coefficients (\(p_0\)) used for prediction with the reduced linear regression model, where \(p  p_0\) out of \(p\) beta coefficients are set to zero. 
Algorithm Output
The quality metric algorithm for linear regression calculates the result described below.
Pass the Result ID
as a parameter to the methods that access the results of your algorithm.
For more details, see Algorithms.
Result ID 
Result 


Pointer to the \(1 \times k\) numeric table that contains the mean of expected responses computed for each dependent variable. 

Pointer to the \(1 \times k\) numeric table that contains the variance of expected responses computed for each dependent variable. 

Pointer to the \(1 \times k\) numeric table that contains the regression sum of squares computed for each dependent variable. 

Pointer to the \(1 \times k\) numeric table that contains the sum of squares of residuals computed for each dependent variable. 

Pointer to the \(1 \times k\) numeric table that contains the total sum of squares computed for each dependent variable. 

Pointer to the \(1 \times k\) numeric table that contains the determination coefficient computed for each dependent variable. 

Pointer to the \(1 \times k\) numeric table that contains the Fstatistics computed for each dependent variable. 
Note
By default, these results are objects of the HomogenNumericTable
class, but you can define the result as an object of any class
derived from NumericTable
, except for PackedTriangularMatrix
, PackedSymmetricMatrix
, and CSRNumericTable
.
Examples¶
Batch Processing: