Errorlocate uses the linear, categorical and conditional rules from a
rules set formulated with R package `validate`

,
to create a Mixed Integer Problem.

For most users the details of the translation are not relevant and
hidden in `locate_errors`

. Often the number of errors found
and the processing time are much more relevant parameters.

In a few cases, you may run into a problems with your error localization problem:

- The processing time of (some of the records) of
`locate_errors`

is high. `locate_errors`

missed an obvious error.`locate_errors`

indicates that it did not find a valid solution (for some records) .

Problem a. can be addressed by using the parallel argument of
`locate_errors`

(and `replace_errors`

). Problem b
can be due to that `error_locate`

ignores non-linear rules,
and therefore is not able to deduce the errors, because it only takes
linear, categorical and conditional rules into a account.

There may also be problems with your rules set. Problems set may be
mitigated by using the `validatetools`

package that can detect conflicting and redundant rules and has methods
to simplify your rule set.

If you want to dive deep into the mixed integer problem that is
created by `error_locate`

you can use the
`inspect_mip`

function.

In the following sections an example is given of how linear, categorical and conditional rules are written as Mixed Integer Problems. First let’s see how these rules in validator can be formally defined.

Each translatable rule \(r_i(\mathbf{x})\) can be written as a
disjunction of atomic clauses \(C_i^j(x)\): it is a function \(r_i\) that operates on (some of) the values
of record \(\mathbf{x} = (x_1, \ldots,
x_n)\) and is `TRUE`

(valid) or `FALSE`

(not valid)

\[ r_i(\mathbf{x}) = \bigvee_j C_i^j(\mathbf{x}) \]

with each atomic clause:

\[ C_i^j(\mathbf{x}) = \left\{ \begin{array}{l} \mathbf{a}^T\mathbf{x} \leq b \\ \mathbf{a}^T\mathbf{x} = b \\ x_j \in F_{ij} \textrm{with } F_{ij} \subseteq D_j \\ x_j \not\in F_{ij} \textrm{with } F_{ij} \subseteq D_j \\ \end{array} \right. \]

Each linear, categorical or conditional rule \(r_i\) can be written in this form.

```
rules <- validator(example_1 = if (income > 0) age >= 16)
rules$exprs()
#> $example_1
#> income <= 0 | (age - 16 >= -1e-08)
#> attr(,"reference")
#> example_1
#> 1
```

So the rule `if (income > 0) age >= 16`

can be
written as (`income <= 0`

OR `age >=16`

)

```
rules <- validator(example_2 = if (has_house == "yes") income >= 1000)
rules$exprs()
#> $example_2
#> has_house != "yes" | (income - 1000 >= -1e-08)
#> attr(,"reference")
#> example_2
#> 1
```

So the rule `if (has_house == "yes") income >= 1000)`

can be written as (`has_house != "yes"`

OR
`age >=1000`

)

The rules form a system \(R(\mathbf{x})\):

\[ R(\mathbf{x}) = \bigwedge_i r_i \] which means that all rules \(r_i\) must be valid. If \(R(\mathbf{x})\) is true for record \(\mathbf{x}\), then the record is valid, otherwise one (or more) of the rules is violated.

Each rule set \(R(\mathbf{x})\) can be translated into a mip problem and solved.

\[ \begin{array}{r} \textrm{Minimize } f(\mathbf{x}) = 0; \\ \textrm{s.t. }\mathbf{Rx} \leq \mathbf{d} \\ \end{array} \] - \(f(\mathbf{x})\) is the (weighted) number of changed variable: \(\delta_i \in {0,1}\)

\[ f(\mathbf{x}) = \sum_{i=1}^N w_i \delta_i \]

\(\mathbf{R}\) contains rules:

\(\mathbf{R}_H(\mathbf{x}) \leq \mathbf{d}_H\) that were specified with

`validate`

/`validator`

\(\mathbf{R}_0(\mathbf{x}, \mathbf{\delta}) \leq \mathbf{d}_0\) : soft constraints that try fix the current record of \(\mathbf{x}\) to the observed values.

`inspect_mip`

:Most users will use the function `locate_errors`

to find
errors. The function `inspect_mip`

works exactly same, except
that it operates on just one record in stead of a whole
`data.frame`

. The result of `inspect_mip`

is a mip
object, that is not yet executed and can be inspected.

```
rules <- validator( r1 = age >= 18
, r2 = income >= 0
)
data <- data.frame(age = c(12, 35), income = c(2000, -1000))
data
```

age | income |
---|---|

12 | 2000 |

35 | -1000 |

So we detect two errors in the dataset:

name | items | passes | fails | nNA | error | warning | expression |
---|---|---|---|---|---|---|---|

r1 | 2 | 1 | 1 | 0 | FALSE | FALSE | age - 18 >= -1e-08 |

r2 | 2 | 1 | 1 | 0 | FALSE | FALSE | income - 0 >= -1e-08 |

Lets inspect the first record

The `mip`

object contains the mip problem before it is
executed. We can inspect the lp problem, prior to solving it with
`lpSolveApi`

with

```
Model name: errorlocate
age income .delta_age .delta_income
Minimize 0 0 1.10298728745 1.088278376264
r1 -1 0 0 0 <= -18
r2 0 -1 0 0 <= 0
age_ub 1 0 -1e+07 0 <= 12
income_ub 0 1 0 -1e+07 <= 2000
age_lb -1 0 -1e+07 0 <= -12
income_lb 0 -1 0 -1e+07 <= -2000
Kind Std Std Std Std
Type Real Real Int Int
Upper Inf Inf 1 1
Lower -Inf -Inf 0 0
```

Validator rules `r1`

and `r2`

are encoded in
two lines of the model. The values of the current record are encoded as
soft constraints in `age_ub`

, `age_lb`

,
`income_lb`

and `income_ub`

. These constraints try
to fix the values of `age`

at 12 and `income`

at
2000, but can be violated, setting `.delta_age`

or
`.delta_income`

to 1.

For large problems the lp problem can be written to disk for inspection

Once we execute the mip project, the lp solver is executed on the problem:

Extra arguments are passed through to `lpSolveAPI`

. The
result object contains several properties:

`res$solution`

indicates of a solution was found

`res$s`

indicates the `lpSolveAPI`

status, what
kind of solution was found.

`res$errors`

indicates which fields/values are deemed
erroneous:

`res$values`

contains the values for the valid solution
that has been found by the lpsolver:

Note that the solver has found that setting `age`

from 12
to 18 gives a valid solution. `.delta_age = 1`

indicates that
`age`

contained an error.

The result object `res`

also contains an `lp`

object after optimization. This object can be further investigated using
`lpSolveAPI`

functions.

```
Model name: errorlocate
age income .delta_age .delta_income
Minimize 0 0 1.10298728745 1.088278376264
age_ub 1 0 -1e+07 0 <= 12
income_ub 0 1 0 -1e+07 <= 2000
income_lb 0 -1 0 -1e+07 <= -2000
Kind Std Std Std Std
Type Real Real Int Int
Upper Inf Inf 1 1
Lower 18 0 0 0
```

Note that the lp problem has been simplified. For example the single
variable constraints,the lp problem/object after solving shows that the
solver has optimized some of the rules: it has moved rule
`r1`

and `r2`

into the `Lower boundary`

conditions. It also removed `age_lb`

because that was
superfluous with respect to the boundary conditions.

In categorical rules, each category is coded in a separate column/mip
variable: e.g. if we have a `working`

variable, with two
categories (“job”, “retired”), the mip problem is encoded as
follows:

working |
---|

? |

```
Model name: errorlocate
working:? working:job working:retired .delta_working
Minimize 0 0 0 1.384103718214
r1 0 1 1 0 = 1
working 1 0 0 1 = 1
Kind SOS SOS SOS Std
Type Int Int Int Int
Upper 1 1 1 1
Lower 0 0 0 0
```

Row `r1`

indicates that either `working:job`

or
`working:retired`

must be true. The `Kind`

row
(`SOS`

) indicates that these variables share the same switch,
only one of them can be set.

With categorical variables it is also possible to specify
`if-then`

rules. These are encoded as one mip rule:

```
rules <- validator( r1 = if (voted == TRUE) adult == TRUE)
data <- data.frame(voted = TRUE, adult = FALSE)
```

voted | adult |
---|---|

TRUE | FALSE |

```
Model name: errorlocate
adult voted .delta_adult .delta_voted
Minimize 0 0 1.495953047416 1.358809254132
r1 -1 1 0 0 <= 0
voted 0 1 0 1 = 1
adult -1 0 1 0 = 0
Kind Std Std Std Std
Type Int Int Int Int
Upper 1 1 1 1
Lower 0 0 0 0
```

age | income |
---|---|

12 | 2000 |

`errorlocate`

encodes this rule into multiple rules (as
noted in the theoretical section above), so rule `r1`

is
chopped into 1 rule + 2 sub rules:

`r1: if (income > 0) age >= 16`

:

`r1._lin1: if (r1._lin1 == FALSE) income <= 0`

`r1._lin2: if (r1._lin2 == FALSE) age >= 16`

`r1: r1._lin1 == FALSE | r1._lin2 == FALSE`

This can be seen with:

```
mip$mip_rules()
#> [[1]]
#> r1: r1._lin1 + r1._lin2 <= 1
#> [[2]]
#> r1._lin1: income - 1e+07*r1._lin1 <= 0
#> [[3]]
#> r1._lin2: -age - 1e+07*r1._lin2 <= -16
#> [[4]]
#> income_ub: income - 1e+07*.delta_income <= 2000
#> [[5]]
#> age_ub: age - 1e+07*.delta_age <= 12
#> [[6]]
#> income_lb: -income - 1e+07*.delta_income <= -2000
#> [[7]]
#> age_lb: -age - 1e+07*.delta_age <= -12
```

The resulting lp model is:

```
Model name: errorlocate
age income .delta_age .delta_income r1._lin1 r1._lin2
Minimize 0 0 1.38872261066 1.190017589718 0 0
r1 0 0 0 0 1 1 <= 1
r1._lin1 0 1 0 0 -1e+07 0 <= 0
r1._lin2 -1 0 0 0 0 -1e+07 <= -16
income_ub 0 1 0 -1e+07 0 0 <= 2000
age_ub 1 0 -1e+07 0 0 0 <= 12
income_lb 0 -1 0 -1e+07 0 0 <= -2000
age_lb -1 0 -1e+07 0 0 0 <= -12
Kind Std Std Std Std Std Std
Type Real Real Int Int Int Int
Upper Inf Inf 1 1 1 1
Lower -Inf -Inf 0 0 0 0
```

This works together with categorical, linear and conditional rules.

The weights for each variable are normally set to 1, and
`errorlocate`

adds some random remainder to the weights: so
the solutions are unique and reproducible (using
`set.seed`

).

```
set.seed(42)
rules <- validator( r1 = if (voted == TRUE) adult == TRUE)
data <- data.frame(voted = TRUE, adult = FALSE)
mip <- inspect_mip(data, rules, weight = c(voted = 3, adult=1))
```

`$objective`

contains the generated weights:

These are assigned to the `delta`

variables in the
objective function of the mip.

```
Model name: errorlocate
adult voted .delta_adult .delta_voted
Minimize 0 0 1.468537706648 3.457403021748
r1 -1 1 0 0 <= 0
voted 0 1 0 1 = 1
adult -1 0 1 0 = 0
Kind Std Std Std Std
Type Int Int Int Int
Upper 1 1 1 1
Lower 0 0 0 0
```