n and m = 1, 2, 3, … are the current numbers of weather stations from a sample of N or M ≤ N in volume

ϕn and λn are the latitude and longitude of the nth weather station

tn is the long-term average temperature at weather stations ordered by latitude and number n

sn and cn are the average and standard deviation of astronomical insolation at the latitude of the nth weather station for the period under study

\($}} }_} = \sum\limits_ = 0}^P _}s_^} \) is the latitudinal trend of long-term average temperature at the latitude of the nth weather station, represented by an insolation polynomial of degree P

ap is the coefficient of insolation polynomial at degree p = 0, 1, 2, …, P

\(_} = _} - $}} }_}\) are the regression residuals

\(\begin ^} = 1 - \\ ^} } \mathord^} } _} - }(_})} \right]} }}^}}}} \right. \kern-0em} _} - }(_})} \right]} }}^}}} \\ \end \) is the coefficient of determination of the trend

\(\bar _^\) is the average coefficient of determination of the polynomial of degree P for a sample

\(^}($}} }_}) = ($}} }}_})} \mathord ($}} }}_})} (_})}}} \right. \kern-0em} (_})}}\) is the relative dispersion of the trend

\(^}(_}) = (_})} \mathord (_})} (_})}}} \right. \kern-0em} (_})}}\) is the relative dispersion of regression residuals

enn is the majorant of regression residuals

h is the coefficient at stdev(εn) for calculating the majorant of the regression residuals

Built-in Mathcad functions calculate: average value (mean); standard deviation (stdev); variance (var); correlation coefficient (corr); linear regression coefficients (line); polynomial regression coefficients (linfit). To determine the critical values of the Fisher distribution, the built-in Exel function was used.

#1. Data matrix input:

$$\left\| _},_},_},_},_}} \right\|;\,\,\,\,n \in [1,N].$$

#2. Ordering the data matrix by latitude:

$$\left\| _},_},_},_},_}} \right\|;\,\,\,\,_ + 1}}} > _};\,\,\,\,n = 1,\,\,2,\,\,3, \ldots ,\,\,N.$$

#3. Calculation of the latitudinal trend, predictors \(s_^\):

$$}\left( _}} s_^,_}} \right) \to $}} }_}.$$

#4. Calculation of regression residuals:

$$_}: = t_\, - $}} }_}.$$

#5. Calculation of the coefficient of determination:

#6. Calculation of majorants of residuals enn from the envelope en\(}_}\), predictor cn:

$$\begin }\left( _},\left| _}} \right|} \right) \to en}_}, \\ e_}: = en}_} - }(en}_}) + h}(_}). \\ \end $$

#7. Selection of weather stations:

$$_}: = }\left( _} < e_} \cup _} > - e_},_},z} \right).$$

#8. Excluding weather stations marked with “z”, and their renumbering: set 1, 2, 3, …, N replaced by 1, 2, 3, …, M; M ≤ N.

#9. Calculation of the latitudinal trend by the polynomial of degree P = 1 from the latitudinal trend \($}} }_}\) calculated in #3:

$$}_} + _}$}} }_},_}} \to $}} }_}.$$

#10. Calculation of regression residuals:

$$_}: = t_\, - $}} }_}.$$

#11. Calculation of the coefficient of determination:

#12. End.

Note. The number with # indicates the line number in the algorithm.