Table (List) of Functions

Linear Group

Variable

variable

Linear function including only one variable

Linear

linear

Linear function with many variables

Linear Multiple

linearmulti

Set of linear functions (used for building multiple linear constraints)

Nonlinear Group

Polynomial Absolute

polynom_abs

Sum of absolute values of components of a vector, i.e., L1 norm of a vector;

Sum of  powers of absolute values of a vector, e.g., (Lp norm)^p 

CVaR Component Positive

cvar_comp_pos

Average of the largest (1-α)% of components of a vector, where 0≤α≤1

CVaR Component Negative

cvar_comp_neg

Average of the largest (1-α)% of components of a -( vector), where 0≤α≤1

CVaR Component Absolute

cvar_comp_abs

Scaled CVaR Norm = Average of the largest  (1-α)% of absolute values of components of a vector, where 0≤α≤1

VaR Component Positive

var_comp_pos

α*I -th value in the ordered  ascending sequence of components of a I-dimensional vector, where 0≤α≤1; e.g., α=0.8, I=100, we order components of the vector and take the component number α*I = 80  

VaR Component Negative

var_comp_neg

α*I -th value in the ordered  ascending sequence of components of a -(I-dimensional vector), where 0≤α≤1; e.g., α=0.8, I=100,  we order components of the -(vector) and take the element number α*I = 80

Maximum Component Positive

max_comp_pos

Maximal  (largest)  component of a vector

VaR Component Negative

max_comp_neg

Maximal  (largest)  component of -(vector)

Maximum Component Absolute

max_comp_abs

L-infinity norm of a vector, i.e., the largest absolute value of components of the vector

Quadratic Function

quadratic

Quadratic function

Squareroot Quadratic

sqrt_quadratic

Squareroot of Quadratic function

Logarithms Sum

log_sum

Linear combination of logarithms of components of a vector

Cardinality Group

Cardinality Positive

cardn_pos

Number of components of a vector exceeding some thresholds

Cardinality Negative

cardn_neg

Number of components of vector below exceeding some thresholds

Cardinality

cardn_

Number of components of a vector below thresholds_1 or exceeding thresholds_2

Buyin Positive

buyin_pos

Number of components of vector exceeding positive thresholds_1 and below positive thresholds_2

Buyin Negative

buyin_neg

Number of components of vector exceeding negative thresholds_1 and below negative thresholds_2

Buyin

buyin

Number of components of vector exceeding negative thresholds_1 and below negative thresholds_2 or exceeding positive thresholds_3 and below positive  thresholds_4

Fixed Charge Positive

fxchg_pos

Sum of fixed changes for components of a vector exceeding some positive thresholds

Fixed Charge Negative

fxchg_neg

Sum of fixed changes for components of a vector below some negative thresholds

Fixed Charge

fxchg

Sum of fixed changes_1 for components of a vector exceeding some positive thresholds plus Sum of fixed changes_2 for components of a vector below some negative thresholds

Deterministic Norms

L1 Norm

polynom_abs

Sum of absolute values of components of a vector, i.e., L1 norm of a vector

L2 Norm

sqrt_quadratic

Squareroot of sum of squares of components of a vector, i.e., L2 norm of a vector

Lp Norm

lp_norm

p-th root of sum of p-th powers of absolute values of a vector components (for version 2.2 only)

L-Infinity Norm

max_comp_abs

The largest absolute value of components of a vector, i.e., L-Infinity norm of a vector

Scaled CVaR Norm

cvar_comp_abs

Average of the largest  (1-α)%  absolute values of components of a vector, where 0≤α≤1, i.e.,  Scaled CVaR  norm of a vector

Loss

L

Linear Loss scenarios

Gain

- L

Gain scenarios =- (Linear Loss scenarios)

Recourse

recourse

Recourse scenarios obtained by solving LP at the second stage of two-stage stochastic programming problem 

Spline_Sum

spline_sum

Sum of Splines scenarios: factors in linear regression are transformed with splines and summed up for every observation (scenario); this function is a generalization of residual in linear regression defined for every observation (scenario)

Average Group

Average Loss

avg

Average Loss obtained by averaging Linear Loss scenarios, i.e., it is a linear function with coefficients obtained by averaging  coefficients of Linear Loss scenarios.

Average Gain

avg_g

Average Gain obtained by averaging -(Linear Loss ) scenarios, i.e., it is a linear function with coefficients obtained by averaging  coefficients of  -(Linear Loss) scenarios.

Average Max

avg_max_risk

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over Linear Loss functions (over M functions for every scenario).  Average Max is calculated by averaging Maximum Loss scenarios.

Average Max for Gain

avg_max_risk_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Gain scenarios function is calculated by maximizing losses over  -(Linear Loss)  functions for every scenario (over M functions for every scenario).  Average Max for Gain is calculated by averaging Maximum Gain scenarios.

Average Max Deviations

avg_max_dev

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Deviation scenarios function is calculated by maximizing losses over (Linear Loss) -  (Average Linear Loss over scenarios) functions (over M functions for every scenario).  Average Max Deviation is calculated by averaging Maximum Deviation scenarios.

Average Max Deviation for Gain

avg_max_dev_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Gain Deviation scenarios function is calculated by maximizing losses over -(Linear Loss)+  (Average Linear Loss over scenarios) functions (over M functions for every scenario).  Average Max Gain Deviation is calculated by averaging Maximum Gain Deviation scenarios.

Average Recourse

avg(recourse(.))

Average of  Recourse  scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Average Gain Recourse

avg_g(recourse(.))

Average of -(Recourse ) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

CVaR Group

CVaR

cvar_risk

Conditional Value-at-Risk for Linear Loss  scenarios (also called Expected Shortfall and Tail VaR), i.e., the average of largest (1-α)% of Losses.

CVaR for Gain

cvar_risk_g

Conditional Value-at-Risk for -(Linear Loss ) scenarios (also called Expected Shortfall and Tail VaR), i.e., the average of largest (1-α)% of -(Losses).

CVaR Normal Independent

cvar_risk_ni

Special case of the CVaR  when all coefficients in Linear Loss function are independent normally distributed random values.

CVaR for Gain Normal Independent

cvar_risk_ni_g

Special case of the CVaR for Gain when all coefficients in -(Linear Loss ) function are independent  normally distributed random values.

CVaR Normal Dependent

cvar_risk_nd

Special case of the CVaR when all coefficients in Linear Loss function are mutually dependent  normally distributed random values.

CVaR for Gain Normal Dependent

cvar_risk_nd_g

Special case of the CVaR for Gain when all coefficients in -(Linear Loss ) function are mutually dependent normally distributed random values

CVaR Deviation

cvar_dev

Conditional Value-at-Risk for (Linear Loss) - (Average over  Linear Loss  scenarios) , i.e., the average of largest (1-α)% of  (Linear Loss) - (Average over Linear Loss scenarios) scenarios.

CVaR Deviation for Gain

cvar_dev_g

Conditional Value-at-Risk for -(Linear Loss ) + (Average  over scenarios Linear Loss) , i.e., the average of largest (1-α)% of - (Linear Loss) + (Average  over scenarios Linear Loss) scenarios.

CVaR Deviation Normal Independent

cvar_ni_dev

Special case of the CVaR Deviation when all coefficients in Linear Loss function are independent normally distributed random values.

CVaR Deviation for Gain Normal Independent

cvar_ni_dev_g

Special case of the CVaR Deviation for Gain when all coefficients in -(Linear Loss ) function are independent normally distributed random values.

CVaR Deviation  Normal Dependent

cvar_nd_dev

Special case of the CVaR Deviation when all coefficients in Linear Loss function are mutually dependent normally distributed random values

CVaR Deviation for Gain Normal Dependent

cvar_nd_dev_g

Special case of the CVaR Deviation for Gain when all coefficients in -(Linear Loss ) function are mutually dependent normally distributed random values

CVaR for Mixture of Normal Independent

avg_cvar_risk_ni

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Avg_cvar_risk_ni is the CVaR of the mixture of Normally Independent random values.

CVaR for Gain for Mixture of Normal Independent

avg_cvar_risk_ni_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_cvar_risk_ni_g is the CVaR of the mixture of Normally Independent random values.

CVaR Deviation for Mixture of Normal Independent

avg_cvar_ni_dev

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average CVaR Deviation Normal Independent is a weighted sum of CVaR Deviation Normal Independent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Average CVaR Deviation Normal Independent

avg_cvar_ni_dev

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_cvar_ni_dev is the CVaR of the mixture of Normally Independent random values.

CVaR Deviation for Gain for Mixture of Normal Independent

avg_cvar_ni_dev_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_cvar_ni_dev_g is the CVaR of the mixture of Normally Independent random values.

CVaR Max

cvar_max_risk

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over Linear Loss functions (over M functions for every scenario).  CVaR Max is calculated by taking CVaR of the Maximum Loss scenarios.

CVaR Max for Gain

cvar_max_risk_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Gain scenarios function is calculated by maximizing losses over -(Linear Loss) functions (over M functions for every scenario).  CVaR Max for Gain is calculated by taking CVaR of the Maximum Gain scenarios.

CVaR Max Deviation

cvar_max_dev

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over (Linear Loss)-(Expected Linear Loss)  functions (over M functions for every scenario).  CVaR Max Deviation is calculated by taking CVaR of the Maximum Loss scenarios.

CVaR Max Deviation for Gain

cvar_max_dev_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum  Gain scenarios function is calculated by maximizing losses over -(Linear Loss)+(Expected Linear Loss)  functions (over M functions for every scenario).  CVaR Max Deviation for Gain is calculated by taking CVaR of the Maximum  Gain scenarios.

CVaR for Discrete Distribution as Function of Atom Probabilities

pcvar

This function is similar to the standard CVaR function, but decision variables are probabilities of scenarios.

CVaR for Mixture of Normal Distributions as Function of Mixture Weights

wcvar_ni

This function calculates CVaR for a mixture of normal distributions as a function of variable weights in this mixture

CVaR  Recourse

cvar_risk(recourse(.))

CVaR of Recourse scenarios.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

CVaR for Gain Recourse

cvar_risk_g(recourse(.)

CVaR of -(Recourse) scenarios.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

CVaR Deviation Recourse

cvar_dev(recourse(.))

CVaR of (Deviation Recourse) = (Recourse-(Expected Recourse)) scenarios.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

CVaR Deviation for Gain Recourse

cvar_dev_g(recourse(.))

CVaR of -(Deviation Recourse) = (-Recourse+(Expected Recourse)) scenarios.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

VaR Group

VaR

var_risk

Value-at-Risk for Linear Loss  scenarios, i.e., α%  percentile of Linear Loss scenarios.  

VaR for Gain

var_risk_g

Value-at-Risk for -(Linear Loss ) scenarios, i.e., α%  percentile of -(Linear Loss) scenarios.

VaR Normal Independent

var_risk_ni

Special case of the VaR  when all coefficients in Linear Loss function are independent normally distributed random values.

VaR for Gain Normal Independent

var_risk_ni_g

Special case of the VaR for Gain  when all coefficients in Linear Loss function are independent normally distributed random values.

VaR  Normal Dependent

var_risk_nd

Special case of the VaR when all coefficients in Linear Loss function are mutually dependent  normally distributed random values. 

VaR for Gain Normal Dependent

var_risk_nd_g

Special case of the VaR for Gain when all coefficients in Linear Loss function are mutually dependent  normally distributed random values.

VaR Deviation

var_dev

Value-at-Risk for (Linear Loss ) - (Average over Linear Loss scenarios) , i.e., α%  percentile of (Linear Loss) - (Average over Linear Loss scenarios) scenarios.  

VaR Deviation for Gain

var_dev_g

Value-at-Risk for -(Linear Loss ) + (Average over Linear Loss scenarios) , i.e.,  α%  percentile of  -(Linear Loss) + (Average over Linear Loss scenarios) scenarios.  

VaR Deviation Normal Independent

var_ni_dev

Special case of the VaR Deviation when all coefficients in Linear Loss function are independent normally distributed random values.

VaR Deviation for Gain Normal Independent

var_ni_dev_g

Special case of the VaR Deviation for Gain when all coefficients in -(Linear Loss ) function are independent normally distributed random values

VaR Deviation Normal Dependent

var_nd_dev

Special case of the VaR Deviation when all coefficients in Linear Loss function are mutually dependent normally distributed random values

VaR Deviation for Gain Normal Dependent

var_nd_dev_g

Special case of the VaR Deviation for Gain when all coefficients in -(Linear Loss ) function are mutually dependent normally distributed random values

VaR for Mixture of Normal Independent

avg_var_risk_ni

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_var_risk_ni is the VaR of the mixture of Normally Independent random values.

VaR for Gain for Mixture of Normal Independent Normal Independent

avg_var_risk_ni_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_var_risk_ni_g is the VaR of the mixture of Normally Independent random values.

VaR Deviation for Mixture of Normal Independent

avg_var_ni_dev

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_var_ni_dev is the VaR of the mixture of Normally Independent random values.

VaR Deviation for Gain for Mixture of Normal Independent

avg_var_ni_dev_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. avg_var_ni_dev_g is the VaR of the mixture of Normally Independent random values

VaR  Recourse

var_risk(recourse(.))

VaR of Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

VaR for Gain Recourse

var_risk_g(recourse(.))

VaR of -(Recourse) scenarios.  Recourse  scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

VaR Deviation Recourse

var_dev(recourse(.))

VaR of (Deviation Recourse) = (Recourse-(Expected Recourse)) scenarios.  Recourse  scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

VaR Deviation for Gain Recourse

var_dev_g(recourse(.))

VaR of -(Deviation Recourse) = (-Recourse+(Expected Recourse)) scenarios.  Recourse  scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Maximum Group

Maximum

max_risk

Maximum of Linear Loss  scenarios.

Maximum for Gain

max_risk_g

Maximum of -(Linear Loss ) scenarios.

Maximum Deviation

max_dev

Maximum of ((Linear Loss ) - (Average over Linear Loss scenarios)) scenarios.

Maximum Deviation for Gain

max_dev_g

Maximum of (-(Linear Loss ) + (Average over Linear Loss scenarios)) scenarios.

Maximum CVaR

max_cvar_risk

There are M  Linear  Loss  scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new CVaR functions are calculated (for every Loss scenario function).  Maximum CVaR is calculated by taking Maximum over M CVaR functions.

Maximum CVaR for Gain

max_cvar_risk_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new CVaR for Gain functions are calculated (for every -(Loss) scenario function).  Maximum CVaR for Gain is calculated by taking Maximum over M CVaR for Gain  functions (based on -(Loss) scenarios).

Maximum CVaR Deviation

max_cvar_dev

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new CVaR Deviation functions are calculated (for every Loss scenario function).  Maximum CVaR Deviation is calculated by taking Maximum over M CVaR Deviation  functions.

Maximum CVaR Deviation for Gain

max_cvar_dev_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new CVaR Deviation for Gain functions are calculated (for every -(Loss) scenario function).  Maximum CVaR Deviation for Gain is calculated by taking Maximum over M CVaR Deviation for Gain  functions (based on -(Loss) scenarios).

Maximum VaR

max_var_risk

There are M  Linear Loss scenario functions (every Linear Loss  scenario function is defined by a Matrix of Scenarios). M new VaR functions are calculated (for every Loss scenario function).  Maximum VaR is calculated by taking Maximum over M VaR functions.

Maximum VaR for Gain

max_var_risk_g

There are M  Linear Loss scenario functions (every Linear Loss  scenario function is defined by a Matrix of Scenarios). M new VaR for Gain functions are calculated (for every -(Loss) scenario function).  Maximum VaR for Gain is calculated by taking Maximum over M VaR for Gain functions.

Maximum VaR Deviation

max_var_dev

There are M  Linear  Loss  scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new VaR Deviation  functions are calculated (for every Loss scenario function).  Maximum VaR Deviation is calculated by taking Maximum over M VaR Deviation  functions.

Maximum VaR Deviation for Gain

max_var_dev_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). M new CVaR Deviation for Gain functions are calculated (for every -(Loss) scenario function).  Maximum CVaR Deviation for Gain is calculated by taking Maximum over M CVaR Deviation for Gain  functions (based on -(Loss) scenarios).

Maximum Recourse

max_risk(recourse(.))

Maximum over Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Maximum for Gain Recourse

max_risk_g(recourse(.))

Maximum over -(Recourse) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Maximum Deviation Recourse

max_dev(recourse(.))

Maximum over (Recourse)-(Expected Recourse) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Maximum Deviation for Gain Recourse

max_dev_g(recourse(.))

Maximum over -(Recourse)+(Expected Recourse) scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Mean Abs Group

Mean Absolute Error

meanabs_pen

Mean Absolute for Linear Loss scenarios function. Calculated by averaging over scenarios the absolute values of losses .

Mean Absolute Error Normal Independent

meanabs_pen_ni

Mean Absolute of  Linear Loss function with independent normally distributed random coefficients.

Mean Absolute Error Normal Dependent

meanabs_pen_nd

Mean Absolute of  Linear Loss function with mutually dependent normally distributed random coefficients.

Mean Absolute Risk

meanabs_risk

Mean Absolute for Linear Loss scenarios. (Mean Absolute) =Average Loss  + Mean Absolute Deviation.

Mean Absolute Risk for Gain

meanabs_risk_g

Mean Absolute  for Gain for Linear Loss scenarios. (Mean Absolute  for Gain) = -Average Loss + Mean Absolute Deviation.

Mean Absolute Risk Normal Independent

meanabs_risk_ni

Mean Absolute when all coefficients in Linear Loss function are independent normally distributed random values. (Mean Absolute Normal Independent) =Average Loss  + Mean Absolute Deviation.

Mean Absolute Risk for Gain Normal Independent

meanabs_risk_ni_g

Mean Absolute for Gain when all coefficients in Linear Loss function are independent normally distributed random values.. (Mean Absolute for Gain Normal Independent) = - Average Loss  + Mean Absolute Deviation.

Mean Absolute Risk Normal Dependent

meanabs_risk_nd

Mean Absolute when all coefficients in Linear Loss function are mutually dependent normally distributed random values.. (Mean Absolute Normal Dependent) =Average Loss  + Mean Absolute Deviation.

Mean Absolute Risk for Gain Normal Dependent

meanabs_risk_nd_g

Mean Absolute for Gain when all coefficients in Linear Loss function are mutually dependent normally distributed random values. (Mean Absolute for Gain Normal Dependent) = - Average Loss  + Mean Absolute Deviation.

Mean Absolute Deviation

meanabs_dev

Mean Absolute Deviation for Linear Loss scenarios.

Mean Absolute Deviation Normal Independent

meanabs_ni_dev

Mean Absolute Deviation for Linear Loss function with independent normally distributed random coefficients.

Mean Absolute Deviation Normal Dependent

meanabs_nd_dev

Mean Absolute Deviation for Linear Loss function with mutually dependent normally distributed random coefficients.

Mean Absolute Error for Recourse

meanabs_pen(recourse(.))

Mean Absolute for Recourse  scenarios function. Calculated by averaging over scenarios the absolute values of Recourse function. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Mean Absolute Risk Recourse

meanabs_risk(recourse(.))

Mean Absolute for Recourse scenarios. (Mean Absolute Recourse) = Average Recourse  + Mean Absolute Deviation of Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Mean Absolute Risk for Gain Recourse

meanabs_risk_g(recourse(.))

Mean Absolute for Gain for Recourse scenarios. (Mean Absolute for Gain Recourse) = -(Average Recourse)  + (Mean Absolute Deviation of Recourse scenarios). Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Mean Absolute Deviation Recourse

meanabs_dev(recourse(.))

Mean Absolute Deviation for Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment Group

Partial Moment

pm_pen

Expected  access of  Linear Loss over some fixed threshold.

Partial Moment for Gain

pm_pen_g

Expected  access of  -( Loss ) over some fixed threshold.

Partial Moment Normal Independent

pm_pen_ni

Expected  access of  Linear Loss over some fixed threshold  for the Loss with independent normally distributed random coefficients.

Partial Moment for Gain Normal Independent

pm_pen_ni_g

Expected  access of  - (Loss ) over some fixed threshold  for the Loss with independent normally distributed random coefficients.

Partial Moment Normal Dependent

pm_pen_nd

Expected  access of Loss over some fixed threshold  for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment for Gain Normal Dependent

pm_pen_nd_g

Expected  access of  - (Loss ) over some fixed threshold  for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment  Deviation

pm_dev

Expected  access of  ( (Loss ) - (Average Loss )) over some fixed threshold.

Partial Moment Gain Deviation

pm_dev_g

Expected  access of  (- (Loss ) + (Average Loss )) over some fixed threshold.

Partial Moment  Deviation Normal Independent

pm_ni_dev

Expected  access of  ( (Loss ) - (Average Loss )) over some fixed threshold for the Loss with independent normally distributed random coefficients.

Partial Moment Gain Deviation Normal Independent

pm_ni_dev_g

Expected  access of  ( - (Loss ) + (Average Loss )) over some fixed threshold for the Loss with independent normally distributed random coefficients.

Partial Moment  Deviation Normal Dependent

pm_nd_dev

Expected  access of ( (Loss ) - (Average Loss ))  over some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Gain Deviation Normal Dependent

pm_nd_dev_g

Expected  access of   (- (Loss ) + (Average Loss ))  over some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Average Partial Moment  Normal Independent

avg_pm_pen_ni

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Partial Moment Normal Independent is a weighted sum of Partial Moment  Normal Independent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Average Partial Moment  for Gain Normal Independent

avg_pm_pen_ni_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Partial Moment for Gain Normal Independent is a weighted sum of Partial Moment for Gain Normal Independent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Average Partial Moment Deviation Normal Independent

avg_pm_ni_dev

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Partial Moment Deviation Normal Independent is a weighted sum of Partial Moment  Deviation Normal Independent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Average Partial Moment Gain Deviation Normal Independent

avg_pm_ni_dev_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Partial Moment Gain Deviation Normal Independent is a weighted sum of Partial Moment Gain Deviation Normal Dependent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Partial Moment Two

pm2_pen

Expected  squared Linear Loss in access of of some fixed threshold.

Partial Moment Twofor Gain

pm2_pen_g

Expected  squared Linear -(Loss ) in access of of some fixed threshold.

Partial Moment Two  Normal Independent

pm2_pen_ni

Expected  squared Linear Loss in access of of some fixed threshold for Loss with independent normally distributed random coefficients.

Partial Moment Two for Gain Normal Independent

pm2_pen_ni_g

Expected  squared Linear -(Loss)  in access of of some fixed threshold for Loss with independent normally distributed random coefficients.

Partial Moment Two Normal Dependent

pm2_pen_nd

Expected  squared Linear Loss in access of of some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Two for Gain Normal Dependent

pm2_pen_nd_g

Expected  squared Linear -(Loss ) in access of of some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Two Deviation for Loss

pm2_dev

Expected  squared access of  ((Loss ) - (Average Loss )) over some fixed threshold.

Partial Moment Two Deviation for Gain

pm2_dev_g

Expected  squared access of  (-(Loss ) + (Average Loss )) over some fixed threshold.

Partial Moment Two Deviation Normal Independent

pm2_ni_dev

Expected  squared access of  ((Loss ) - (Average Loss )) over some fixed threshold for the Loss with independent normally distributed random coefficients.

Partial Moment Two Deviation for Gain Normal Independent

pm2_ni_dev_g

Expected  squared access of  (-(Loss ) + (Average Loss )) over some fixed threshold for the Loss with independent normally distributed random coefficients.

Partial Moment Two Deviation  Normal Dependent

pm2_nd_dev

Expected  squared access of  ((Loss ) - (Average Loss )) over some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Two Deviation for Gain Normal Dependent

pm2_nd_dev_g

Expected  squared access of  (-(Loss ) + (Average Loss )) over some fixed threshold for the Loss with mutually dependent normally distributed random coefficients.

Partial Moment Two Max

pm2_max_pen

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over  Linear Loss functions (over M functions for every scenario). Partial Moment Two Max is calculated by taking Partial Moment Two of the Maximum Loss scenarios.

Partial Moment Two Max  for Gain

pm2_max_pen_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Gain scenarios function is calculated by maximizing losses over Linear -(Loss) functions (over M functions for every scenario). Partial Moment Two Max for Gain is calculated by taking Partial Moment Two of the Maximum Gain scenarios.

Partial Moment Two Max Deviation

pm2_max_dev

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Deviation scenarios function is calculated by maximizing losses over (Linear Loss)-(Expected Linear Loss)  functions (over M functions for every scenario). Partial Moment Two Max Deviation is calculated by taking Partial Moment Two of the Maximum Deviation scenarios.

Partial Moment Two Max Deviation for Gain

pm2_max_dev_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Deviation for Gain scenarios function is calculated by maximizing losses over -(Linear Loss)+(Expected Linear Loss)  functions (over M functions for every scenario). Partial Moment Two Max Deviation for Gain is calculated by taking Partial Moment Two of the Maximum Deviation for Gain scenarios.

Partial Moment Recourse

pm_pen(recourse(.))

Expected  access of Recourse scenarios over some fixed threshold.. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment for Gain Recourse

pm_pen_g(recourse(.))

Expected  access of -(Recourse) scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment  Deviation Recourse

pm_dev(recourse(.))

Expected  access of (Recourse)-(Expected Recourse) scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment Gain Deviation Recourse

pm_dev_g(recourse(.))

Expected  access of -(Recourse)+(Expected Recourse) scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment Two Recourse

pm2_pen_g(recourse(.))

Expected  squared access of Recourse scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment Two for Gain Recourse

pm2_pen_g(recourse(.))

Expected squared access of -(Recourse) scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment Two Deviation Recourse

pm2_dev(recourse(.))

Expected squared access of (Recourse)-(Expected Recourse) scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Partial Moment Two Deviation for Gain Recourse

pm2_dev_g(recourse(.))

Expected  squared access of -(Recourse)+(Expected Recourse) scenarios over some fixed threshold. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem.

CDaR Group

CDaR

cdar_dev

For every time moment, j=1,...J ,  portfolio  drawdown = d(j) = maxn ((uncompounded cumulative portfolio return at time moment n ) - (uncompounded cumulative portfolio return at time moment j )). CDaR  = CVaR Component Positive of vector (d(1), ..., d(J)) = average of the largest (1-α)%  components of the vector (d(1), ..., d(J)), where 0≤α≤1 .

CDaR for Gain

cdar_dev_g

For every time moment, j=1,...J ,  portfolio  -drawdown = -d(j) = maxn (-(uncompounded cumulative portfolio return at time moment n ) + (uncompounded cumulative portfolio return at time moment j )). CDaR  for Gain = CVaR Component Positive of vector (-d(1), ...,- d(J)) = average of the largest (1-α)%  components of the vector (-d(1), ..., -d(J)), where 0≤α≤1 .

Drawdown  Maximum

drawdown_dev_max

For every time moment, j=1,...J ,  portfolio drawdown = d(j) =  maxn ((uncompounded cumulative portfolio return at time moment n ) - (uncompounded cumulative portfolio return at time moment  j )). Drawdown Maximum = Maximum Component Positive of vector (d(1), ..., d(J)) =  largest  component of the vector (d(1), ..., d(J)).

Drawdown  Maximum for Gain

drawdown_dev_max_g

For every time moment, j=1,...J ,  portfolio -drawdown = -d(j) =  maxn (-(uncompounded cumulative portfolio return at time moment n ) + (uncompounded cumulative portfolio return at time moment  j )). Drawdown Maximum for Gain =  Maximum Component Positive of vector (-d(1), ...,- d(J)) =  largest  component of the vector (-d(1), ..., -d(J)).

Drawdown  Average

drawdown_dev_avg

For every time moment, j=1,...J ,  portfolio drawdown = d(j) =  maxn ((uncompounded cumulative portfolio return at time moment n ) - (uncompounded cumulative portfolio return at time moment  j )). Drawdown Deviation  = average of components of the vector (d(1), ..., d(J)) .

Drawdown  Average for Gain

drawdown_dev_avg_g

For every time moment, j=1,...J ,  portfolio -(drawdown) = -d(j) =  maxn (-(uncompounded cumulative portfolio return at time moment n ) + (uncompounded cumulative portfolio return at time moment  j )). Drawdown  Average for Gain =  average of components of the vector (-d(1), ...,- d(J)).

CDaR Multiple

cdarmulti_dev

Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio drawdown = d(k,j) = maxn ((uncompounded cumulative portfolio return at time moment n on sample-path k ) - (uncompounded cumulative portfolio return at time moment j on sample-path k )). CDaR Multiple = CVaR Component Positive of the vector (d(1,1), ..., d(K,J)) = average of the largest (1-α)%  components of the vector  (d(1,1), ..., d(K,J)), where 0≤α≤1 .

CDaR for Gain Multiple

cdarmulti_dev_g

Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio -drawdown = -d(k,j) = maxn (-(uncompounded cumulative portfolio return at time moment n on sample-path k ) + (uncompounded cumulative portfolio return at time moment j on sample-path k )). CDaR for Gain Multiple = CVaR Component Positive of the vector (-d(1,1), ..., -d(K,J)) = average of the largest (1-α)%  components of the vector  (-d(1,1), ..., -d(K,J)), where 0≤α≤1 .

Drawdown  Maximum Multiple

drawdownmulti_dev_max

Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio drawdown = d(k,j) = maxn ((uncompounded cumulative portfolio return at time moment n on sample-path k ) - (uncompounded cumulative portfolio return at time moment j on sample-path k )). Drawdown Maximum Multiple = maximum of components of the vector (d(1,1), ..., d(K,J))

Drawdown  Maximum for Gain Multiple

drawdownmulti_dev_max_g

Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio -drawdown = -d(k,j) =  maxn (-(uncompounded cumulative portfolio return at time moment n on sample-path k ) + (uncompounded cumulative portfolio return at time moment  j on sample-path k )). Drawdown Maximum for Gain Multiple = maximum of components of the vector (-d(1,1), ..., -d(K,J)) . 

Drawdown  Average Multiple

drawdownmulti_dev_avg

Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio drawdown = d(k,j) =  maxn ((uncompounded cumulative portfolio return at time moment n on sample-path k ) - (uncompounded cumulative portfolio return at time moment  j on sample-path k )). Drawdown Average Multiple = average of components of the vector (d(1,1), ..., d(K,J))

Drawdown  Average for Gain Multiple

drawdownmulti_dev_avg_g

Suppose we have k=1,..., K portfolio return sample-paths. For every sample-path k,  and every time moment, j=1,...J ,  portfolio -drawdown = -d(k,j) =  maxn (-(uncompounded cumulative portfolio return at time moment n on sample-path k ) + (uncompounded cumulative portfolio return at time moment  j on sample-path k )). Drawdown Average for Gain Multiple = average of components of the vector (-d(1,1), ..., -d(K,J)) . 

Probability Group

Probability of Exceedance

pr_pen

Probability that Linear Loss  exceeds some fixed threshold.

Probability of Exceedance for Gain

pr_pen_g

Probability that Linear -(Loss ) exceeds some fixed threshold.

Probability of Exceedance Normal Independent

pr_pen_ni

Probability that Linear Loss  exceeds some fixed threshold  for the Loss with independent normally distributed random coefficients.

Probability of Exceedance  for Gain Normal Independent

pr_pen_ni_g

Probability that Linear -(Loss ) exceeds some fixed threshold  for the Loss with independent normally distributed random coefficients.

Probability of Exceedance  for Loss Normal Dependent

pr_pen_nd

Probability that Linear Loss  exceeds some fixed threshold  for the Loss with mutually dependent normally distributed random coefficients.

Probability of Exceedance for Gain Normal Dependent

pr_pen_nd_g

Probability that Linear -(Loss ) exceeds some fixed threshold  for the Loss with mutually dependent normally distributed random coefficients

Probability of Exceedance Deviation

pr_dev

Probability that (Loss)-(Average Loss) exceeds some fixed threshold.

Probability of Exceedance Deviation for Gain

pr_dev_g

Probability that -(Loss)+(Average Loss) exceeds some fixed threshold.

Probability of Exceedance Deviation  Normal Independent

pr_ni_dev

Probability that  (Loss)-(Average Loss) exceeds some fixed threshold  for the Loss with independent normally distributed random coefficients.

Probability of Exceedance Deviation  Normal Dependent

pr_nd_dev

Probability that  (Loss)-(Average Loss) exceeds some fixed threshold  for the Loss with mutually dependent normally distributed random coefficients

Probability of Exceedance Deviation for Gain Normal Dependent

pr_nd_dev_g

Probability that  -(Loss)+(Average Loss) exceeds some fixed threshold  for the Loss with mutually dependent normally distributed random coefficients

Probability of Exceedance Deviation for Gain Normal Independent

pr_ni_dev_g

Probability that -(Loss)+(Average Loss) exceeds some fixed threshold  for the Loss with independent normally distributed random coefficients.

Average Probability of Exceedance Normal Independent

avg_pr_pen_ni

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Probability of Exceedance Normal Independent is a weighted sum of Probability of Exceedance  Normal functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Average Probability of Exceedance  for Gain Normal Independent

avg_pr_pen_ni_g

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Probability of Exceedance for Gain Normal Independent is a weighted sum of Probability of Exceedance for Gain Normal Independent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Average Probability of Exceedance Deviation  Normal Independent

avg_pr_ni_dev

Consider a mixture of (random) Linear Loss functions with positive weights summing up to one. Coefficients in all  Linear Loss functions are independent normally distributed random values. Average Probability of Exceedance Deviation Normal Independent is a weighted sum of Probability of Exceedance Deviation  Normal Independent functions over all Loss functions in the mixture. The weighs in the sum are taken from the mixture of Loss functions.

Probability of Exceedance Multiple

prmulti_pen

There are M  Linear Loss scenario functions (every Linear Loss  scenario function is defined by a Matrix of Scenarios). A new Maximum Loss scenarios function is calculated by maximizing losses over  Linear Loss functions (over M functions for every scenario).  Probability of Exceedance Multiple is the Probability of Exceedance of the Maximum Loss scenarios. (Probability of Exceedance Multiple) = 1-(Probability that all M  Linear Loss functions are below the threshold).

Probability of Exceedance for Gain Multiple

prmulti_pen_g

There are M  Linear Loss scenario functions (every Linear Loss  scenario function is defined by a Matrix of Scenarios). A new Maximum -Loss scenarios function is calculated by maximizing losses over -Loss functions (over M functions for every scenario).  Probability of Exceedance for Gain Multiple is Probability of Exceedance  of the Maximum -Loss scenarios. (Probability of Exceedance for Gain Multiple) = 1-(Probability that all M  Linear -(Loss) functions are below the threshold).

Probability of Exceedance Multiple Normal Independent

prmulti_pen_ni

There are M Linear Loss scenario functions with independent normally distributed random coefficients.

Probability of Exceedance Multiple Normal Independent = 1-(Probability that all M Linear Loss functions are below the threshold).

Probability of Exceedance for Gain Multiple Normal Independent

prmulti_pen_ni_g

There are M Linear Loss scenario functions with independent normally distributed random coefficients.

Probability of Exceedance for Gain Multiple Normal Independent = 1-(Probability that all M -(Loss) functions are below the threshold).

Probability of Exceedance Multiple Normal Dependent

prmulti_pen_nd

There are M Linear Loss scenario functions with mutually dependent normally distributed random coefficients. Probability of Exceedance Multiple Normal Dependent = 1-(Probability that all M Linear Loss functions are below the threshold).

Probability of Exceedance for Gain Multiple Normal Dependent

prmulti_pen_nd_g

There are M Linear Loss scenario functions with mutually dependent normally distributed random coefficients. Probability of Exceedance for Gain Multiple Normal Dependent = 1-(Probability that all M -(Loss) functions are below the threshold).

Probability of Exceedance Deviation  Multiple

prmulti_dev

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Deviation Multiple scenarios function is calculated by maximizing losses over (Loss)-(Average Loss) functions (over M functions for every scenario). Probability of Exceedance Deviation Multiple is the Probability of Exceedance of the Maximum Maximum Deviation Multiple scenarios. (Probability of Exceedance Deviaiont Multiple) = 1-(Probability that all M (Loss)-(Average Loss)  functions are below the threshold).

Probability of Exceedance Deviation for Gain Multiple

prmulti_dev_g

There are M Linear Loss scenario functions (every Linear Loss scenario function is defined by a Matrix of Scenarios). A new Maximum Deviation for Gain Multiple scenarios function is calculated by maximizing losses over -(Loss)+(Average Loss) functions (over M functions for every scenario). Probability of Exceedance Deviation for Gain Multiple is the Probability of Exceedance of the Maximum Maximum Deviation for Gain Multiple scenarios. (Probability of Exceedance Penalty for Gain Multiple) = 1-(Probability that all M -(Loss)+(Average Loss)  functions are below the threshold).

Probability of Exceedance Deviation  Multiple Normal Independent

prmulti_ni_dev

There are M Linear Loss scenario functions with independent normally distributed random coefficients.

Probability of Exceedance Deviation Multiple Normal Independent = 1-(Probability that all M (Loss)-(Average Loss) functions are below the threshold).

Probability of Exceedance Deviation  Multiple Normal Dependent

prmulti_nd_dev

There are M Linear Loss scenario functions with mutually dependent normally distributed random coefficients. Probability of Exceedance Deviation Multiple Normal Dependent = 1-(Probability that all M (Loss)-(Average Loss) functions are below the threshold).

Probability of Exceedance Recourse

pr_pen(recourse(.))

Probability that Recourse scenarios function exceeds some fixed threshold.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Probability of Exceedance  for Gain Recourse

pr_pen_g(recourse(.))

Probability that -(Recourse) scenarios function exceeds some fixed threshold.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Probability of Exceedance Deviation  Recourse

pr_dev(recourse(.))

Probability that (Recourse)-(Average Recourse) scenarios function exceeds some fixed threshold.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Probability of Exceedance Deviation for Gain Recourse

pr_dev_g(recourse(.))

Probability that -(Recourse)+(Average Recourse) scenarios function exceeds some fixed threshold.  Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Standard Group

Root Mean Squared Error

st_pen

Root Squared Error of Linear Loss scenarios calculated with Matrix of Scenarios or Expected Matrix of Products. By definition, it is an average of squared  loss scenarios.

Root Mean Squared Error Normal Independent

st_pen_ni

Root Squared Error of  Linear Loss with independent normally distributed random coefficients. It is calculated with Matrix of Means (one row matrix) and Matrix of Variances (one row matrix).

Root Mean Squared Error Normal Dependent

st_pen_nd

Root Squared Error of  Linear Loss with mutually dependent normally distributed random coefficients. It is calculated with  Matrix of Means (one row matrix) and Covariance Symmetric Matrix.

Standard Risk

st_risk

(Standard Deviation of Linear Loss scenarios)+(Average of Linear Loss scenarios).  It is calculated with Matrix of Scenarios.

Standard Gain

st_risk_g

(Standard Deviation of Linear Loss scenarios)-(Average of Linear Loss scenarios).  It is calculated with Matrix of Scenarios.

Standard Risk Normal Independent

st_risk_ni

(Standard Deviation of Linear Loss)+(Average of Linear Loss).  It is calculated with Matrix of Means (one raw matrix) and Matrix of Variances (one row matrix).

Standard Gain Normal Independent 

st_risk_ni_g

(Standard Deviation of Linear Loss)-(Average of Linear Loss).  It is calculated with Matrix of Means (one raw matrix) and Matrix of Variances (one row matrix).

Standard Risk Normal Dependent

st_risk_nd

(Standard Deviation of Linear Loss)+(Average of Linear Loss). It is calculated with Matrix of Means (one row matrix) and Covariance Symmetric Matrix.

Standard Gain Normal Dependent

st_risk_nd_g

(Standard Deviation of Linear Loss)-(Average of Linear Loss). It is calculated with Matrix of Means (one row matrix) and Covariance Symmetric Matrix.

Standard Deviation 

st_dev

Standard Deviation of Linear Loss scenarios calculated with Matrix of Scenarios.

Mean Square Error 

meansquare, meansquare_err

Mean Square of Linear Loss scenarios calculated with Matrix of Scenarios or Expected Matrix of Products.

Mean Square Error Normal Independent

meansquare_ni

Mean Square error of Linear  Loss with independent normally distributed random coefficients.   It is calculated with Matrix of Means (one row matrix) and Matrix of Variances (one row matrix).

Mean Square Error Normal Dependent

meansquare_nd

Mean Square error of Linear  Loss with mutually dependent normally distributed random coefficients. It is calculated with Matrix of Means (one row matrix) and Covariance Symmetric Matrix.

Variance

variance

Variance of Linear Loss scenarios calculated with Matrix of Scenarios.

Root Squared Error Recourse

st_pen(recourse(.))

Root Squared Error of  Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Standard Risk Recourse

st_risk(recourse(.))

(Standard Deviation of  Recourse scenarios)+(Average of Recourse scenarios). Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Standard Gain Recourse

st_risk_g(recourse(.))

(Standard Deviation of  Recourse scenarios)-(Average of Recourse scenarios). Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Standard Deviation Recourse

st_dev(recourse(.))

Standard Deviation of  Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Meansquare Error Recourse

meansquare(recourse(.))

Meansquare error of Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Variance Recourse

variance(recourse(.))

Variance of Recourse scenarios. Recourse scenarios are obtained by solving LP at the second stage of two-stage stochastic programming problem for every scenario.

Utilities Group

Exponential Utility

exp_eut

Exponential Utility function for Linear Loss scenarios calculated with Matrix of Scenarios.

Exponential Utility Normal Independent

exp_eut_ni

Exponential Utility function for  Linear Loss with independent normally distributed random coefficients.

Exponential Utility Normal Dependent

exp_eut_nd

Exponential Utility function for  Linear Loss with mutually dependent normally distributed random coefficients.

Logarithmic Utility

log_eut

Log Utility function for Linear Loss scenarios calculated with Matrix of Scenarios.

Power Utility

pow_eut

Power Utility function for Linear Loss scenarios calculated with Matrix of Scenarios.

Error Group

Mean Absolute Error

meanabs_pen

Mean Absolute for Linear Loss scenarios function. Calculated by averaging over scenarios the absolute values of losses .

Mean Absolute Error Normal Independent

meanabs_pen_ni

Mean Absolute of  Linear Loss function with independent normally distributed random coefficients.

Mean Absolute Error Normal Dependent

meanabs_pen_nd

Mean Absolute of  Linear Loss function with mutually dependent normally distributed random coefficients.

Mean Square Error

meansquare, meansquare_err

Mean Square of Linear Loss scenarios calculated with Matrix of Scenarios or Expected Matrix of Products.

Mean Square Error Normal Independent

meansquare_ni

Mean Square error of Linear  Loss with independent normally distributed random coefficients.   It is calculated with Matrix of Means (one row matrix) and Matrix of Variances (one row matrix).

Mean Square Error Normal Dependent

meansquare_nd

Mean Square error of Linear  Loss with mutually dependent normally distributed random coefficients. It is calculated with Matrix of Means (one row matrix) and Covariance Symmetric Matrix.

Root Mean Squared Error

st_pen

Root Squared Error of Linear Loss scenarios calculated with Matrix of Scenarios or Expected Matrix of Products. By definition, it is an average of squared  loss scenarios.

Root Mean Squared Error Normal Independent

st_pen_ni

Root Squared Error of  Linear Loss with independent normally distributed random coefficients. It is calculated with Matrix of Means (one row matrix) and Matrix of Variances (one row matrix).

Root Mean Squared Error Normal Dependent

st_pen_nd

Root Squared Error of  Linear Loss with mutually dependent normally distributed random coefficients. It is calculated with  Matrix of Means (one row matrix) and Covariance Symmetric Matrix.

Koenker and Basset Error

kb_err

Koenker and Bassett error of Linear Loss scenarios calculated with Matrix of Scenarios. Used for estimation of Value-at-Risk (i.e., percentile) in Linear Regression.

Rockafellar Error

ro_err

Rockafellar error of Linear Loss scenarios calculated with Matrix of Scenarios. Used for estimation of Mixed Value-at-Risk in Linear Regression. Conditional Value-at-Risk approximately equals the discrete Mixed Value-at-Risk and it can be estimated using  Rockafellar error in Linear Regression.

Distance between Distributions Group

Kolmogorov-Smirnov Distance between Two Distributions

ksm_max

Kolmogorov-Smirnov distance between  a discrete distribution with variable probabilities of atoms and a fixed discrete distribution (as function of variable probabilities). This distance is calculated by maximizing deference between two distributions.

Kolmogorov-Smirnov Distance from a Discrete Distribution to a Mixture of Normal Independent Distributions

ksm_max_ni

Kolmogorov-Smirnov distance between a mixture of normal distributions with variable coefficients and a fixed discrete distribution  (as function of variable coefficients). This distance is calculated by maximizing deference between two distributions.

CVaR Kolmogorov-Smirnov Distance between Two Distributions

ksm_CVaR

CVaR Kolmogorov-Smirnov distance between a discrete distribution with variable probabilities of atoms and a fixed discrete distribution  (as function of variable  probabilities). This distance is calculated by taking CVaR of absolute values of differences between two distributions at atoms of the two discrete distributions with probabilities proportional to the distances between atoms.

CVaR Kolmogorov-Smirnov Distance from a Discrete Distribution to a Mixture of Normal Independent Distributions

ksm_cvar_ni

CVaR Kolmogorov-Smirnov distance between a mixture of normal distributions with variable coefficients and a fixed discrete distribution  (as function of variable coefficients). This distance is calculated by taking CVaR of absolute values of differences between two distributions at atoms of the fixed discrete distribution.

Average Kolmogorov-Smirnov Distance between Two Distributions

ksm_avg

Kantorovich  (or Average Kolmogorov-Smirnov) distance  between a discrete distribution with variable probabilities of atoms and a fixed discrete distribution. This distance is calculated by taking the average of absolute values of differences between two distributions at atoms of the two discrete distributions with probabilities proportional to the distances between atoms.

Relative Entropy

entropyr

Relative entropy where probabilities are variables

Log-Likelihood Group

Logarithms Exponents Sum

logexp_sum

function in Logistic Regression