Statistical Modelling and Variable Selection in Climate Science
377
0.0685140739126797, 0.00685140739126797,
0.000685140739126797. The values under each value of λ provides the values
of six regression coefficients obtained for ˆ
β lasso . For example, the values in the second columns indicate ˆ
β lasso = ( ˆ
β 1lasso , ˆ
β 2lasso , ˆ
β 3lasso , ˆ
β 4lasso , ˆ
β 5lasso , ˆ
β 6lasso )
T
= (0.595512226, 0.403999981, 0.816642613, 0.000000000, 0.000000000,
0.005371423)
T corresponding to λ = 0.685140739126797. For this particular
set of values, we observe that the last three values (viz., ˆ
β 4lasso = 0.000000000,
ˆ
β 5lasso = 0.000000000 and ˆ
β 6lasso = 0.005371423) are very close to zero. Hence
one can decide to ignore the corresponding variables X 4 , X 5 and X 6 . So one
can conclude that the variables—number of cars per square kilometer in a city,
population density (number of persons per square kilometer) in a city and aluminum
toxicity in soil are not making any significant impact on rainfall, and hence, can be
dropped from the model. The remaining variables, viz., wind speed, precipitation,
and relative humidity are important variables, and thus the model obtained by
considering the observations on these three variables will give a good fitted linear
regression model based on LASSO selection. The decision to choose the variables
in other columns of the outcome, or equivalently using other values of λ is the
choice of the experimenter. Usually, any choice will give a similar conclusion with
a minor change in the subset of selected explanatory variables. For example, instead
of basing the decision on the second column of the output, if third, fourth or fifth
columns corresponding to other values of λ are chosen, one can observe that they
are also giving a similar indication, i.e., to ignore the three variables, viz., number
of cars per square kilometer in a city, population density (number of persons per
square kilometer) in a city and aluminum toxicity in soil.
7 Conclusions
We have considered the multiple linear regression modelling and LASSO technique
for subset selection of important explanatory variables. These are the useful techniques in finding out a statistical model based on a given set of data on independent
and dependent variables. We have given step by step details about how to obtain
the model under usual conditions. When any of the assumptions of linear regression
model, e.g., constant variance, uncorrelated random errors, correlated explanatory
variables, normal distribution of random errors, etc. are violated, then different types
of problems occurs and are tackled in different ways. Such methods to find models
under these issues have not been discussed in this article. Here it is noteworthy to
mention that various types of graphical and analytical procedures are available to test
the validity of such assumptions based on the given sample of data. LASSO has also
been extended in different directions and various versions and extensions of LASSO
like elastic net, group LASSO, sparse group LASSO, overlap group LASSO and
fused LASSO are also now available in the literature and software.
377
0.0685140739126797, 0.00685140739126797,
0.000685140739126797. The values under each value of λ provides the values
of six regression coefficients obtained for ˆ
β lasso . For example, the values in the second columns indicate ˆ
β lasso = ( ˆ
β 1lasso , ˆ
β 2lasso , ˆ
β 3lasso , ˆ
β 4lasso , ˆ
β 5lasso , ˆ
β 6lasso )
T
= (0.595512226, 0.403999981, 0.816642613, 0.000000000, 0.000000000,
0.005371423)
T corresponding to λ = 0.685140739126797. For this particular
set of values, we observe that the last three values (viz., ˆ
β 4lasso = 0.000000000,
ˆ
β 5lasso = 0.000000000 and ˆ
β 6lasso = 0.005371423) are very close to zero. Hence
one can decide to ignore the corresponding variables X 4 , X 5 and X 6 . So one
can conclude that the variables—number of cars per square kilometer in a city,
population density (number of persons per square kilometer) in a city and aluminum
toxicity in soil are not making any significant impact on rainfall, and hence, can be
dropped from the model. The remaining variables, viz., wind speed, precipitation,
and relative humidity are important variables, and thus the model obtained by
considering the observations on these three variables will give a good fitted linear
regression model based on LASSO selection. The decision to choose the variables
in other columns of the outcome, or equivalently using other values of λ is the
choice of the experimenter. Usually, any choice will give a similar conclusion with
a minor change in the subset of selected explanatory variables. For example, instead
of basing the decision on the second column of the output, if third, fourth or fifth
columns corresponding to other values of λ are chosen, one can observe that they
are also giving a similar indication, i.e., to ignore the three variables, viz., number
of cars per square kilometer in a city, population density (number of persons per
square kilometer) in a city and aluminum toxicity in soil.
7 Conclusions
We have considered the multiple linear regression modelling and LASSO technique
for subset selection of important explanatory variables. These are the useful techniques in finding out a statistical model based on a given set of data on independent
and dependent variables. We have given step by step details about how to obtain
the model under usual conditions. When any of the assumptions of linear regression
model, e.g., constant variance, uncorrelated random errors, correlated explanatory
variables, normal distribution of random errors, etc. are violated, then different types
of problems occurs and are tackled in different ways. Such methods to find models
under these issues have not been discussed in this article. Here it is noteworthy to
mention that various types of graphical and analytical procedures are available to test
the validity of such assumptions based on the given sample of data. LASSO has also
been extended in different directions and various versions and extensions of LASSO
like elastic net, group LASSO, sparse group LASSO, overlap group LASSO and
fused LASSO are also now available in the literature and software.