Abstract

The researcher analyzes mammal sleep with 62 species in 1976 by using Lasso method (least absolute shrinkage and selection operator)that provides stability, higher selection variables, computational efficiency and higher prediction accuracy. the results of Average Parameter Estimate for using adaptive Lasso in SAS indicates that the position of slow wave and paradoxical sleep is account for 100%, overall danger index is 93%. The distributions of overall danger index and slow wave with paradoxical sleep as wee as gestation time from Refit model shows normal histogram for paradoxical sleep. In partition statement of “glmselect”procedure, ASE value (Average Square Error) of the validation from overall danger index is the minimum of all parameters in the selected model. On the other hand in selection steps for ASE, the adaptive Lasso method seems to have fewer than Lasso; for complicate and large data, elastic net can deal with more parameters than observations and combine one and a couple of groups that are consist of multiple variables by shrinking the coefficients of correlated variables toward each other.

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INTRODUCTION

Lasso (least absolute shrinkage and selection operator) is one of popular regression statistical methods. It is mainly applied some advanced means such as variable selection or regularization to generate and interpret the data models in statistics, so that, deeply analyze and reveal the object data. It is introduced by Tibshirani^[1] originally, Lasso was constructed with least squares models to display necessary amount on the estimators and the correlation each other, such as, the connection of ridge regression and coefficients. If so, users might to get the subset of predictors with minimum prediction error for a response variable. Lasso can perform zeroing out variable and shrinkage to improve the value of prediction that is it can select variables with less bias for variables that ‘really matter’, it can also allow much more parameters numbers than observations (but only include up to variables); discarding non-useful variables, etc. Many years ago, the studies for mammal sleep has been conducted. Giraffe’s sleep at a zoo was tested with paradoxical sleep. To analyze behavioral sleep for mammals, many researchers conducted to regression study such as assessing 152 nights in 5 adults, 2 immatures and one juvenile giraffe at a zoological garden with PS (Paradoxical Sleep). The results showed that “ANOVA factor interval from 18-8 h”^[2]. A scholar used logarithmic transformation to analyze correlation between sleep gestation and rapid eye movement for 79 mammal species^[3]. Some [email protected] potential covariates such as body weight, cage locations were screened with the sleep parameters and regarded as normal distribution^[4].

If two process modes and transition between different numbers of daily mammal sleep by wake were similar with mathematical models. For some mammals, the sleep/wake patterns of suprachiasmatic nucleus of the hypothalamus were simulated to be different values of the modulation parameter a that is between -1 and 1 at different periods^[5]. Gamma distribution was used to analysis of the amplitude-frequency for spindle occurrences, the result showed that the mean frequency of the cluster of spindles shift from 11-13 Hz^[6]. For female rats with asleep, two-way ANOVA was applied to analyze the distance and escape latency of group to test ovariectomized female Wistar rats if they are the normality^[7]. However, these analyses bring into the following questions: they pick a model if a model selection is reasonable the selected model has been affected by outliers. Is the prediction more accurate? Do we deal with more parameters (that is p) than observations (that is n)? if so what statistical methods can we choose? Penalized regression method (adaptive Lasso and Elastic net) can perform the jobs that traditional selection methods such as backward, forward and stepwise selections cannot do more numbers of prediction variables than number of sample sizes and combining the group with multiple variables, etc.

MATERIALS AND METHODS

Allison and Cicchetti^[8] published “Sleep in Mammals: Ecological and Constitutional Correlates” at A. A .A. Science. They thought that “slow-wave sleep was negative associated with a factor related to body size” and “paradoxical sleep was related to a factor with predatory danger,” based on data (See Supplemental Files S1).

It collected 62 mammals. It included the following variables: Species of animal, body weights (Body W) by kilogram; brain weight (Brain W) by grain, Slow Wave (SWS) with nondreaming sleep by (hrs/day), Paradoxical (PS) with dreaming sleep (hrs/day), Total Sleep (TS) with sum of slow wave and paradoxical sleep (hrs/day), maximum life span (LS, years), Gestation Time (GT, days), Predation Index (PI, point 1-5) at which point 1 denoted least likely to be preyed upon and point 5 denoted the most likely preyed upon, sleep exposure index (SEI, point 1-5) at which point 1 denoted the least exposed such as animal sleep in a well-protected den and point 5 denoted the most protected den, overall danger index (ODI, point 1-5) at which point 1 denoted the least danger from other animals and most danger from other animals according to the above two indicated and other information. Also, “-999” expresses missing values.

Two researchers pointed out that the species in the laboratory were not considered some factors such as environmental or ecological influences. For example, the definition of good sleeper and bad sleeper that in general, good sleeper was >8 h per day but for most mammals in the laboratory, they need have more time to be adaptive to the laboratory. Hence, some mammals did not meet the standardized time and sometimes, their sleep were not stable. For example, cat is good sleeper and rabbit is not good sleeper. So, the collection of their sleep time was subject to different sleep time requirement; some variables such as “slow wave sleep” was observed by the electroencephalogram that test behavioral and the acquiescence of autonomic nervous system; “paradoxical sleep” was defined as brief irregular activities of the extremities and facial muscle movement with the dreaming measured by a low-voltage electroencephalogram; “life-span” was calculated by their maximum amounts of time under natural environmental living without diseases and predator’s threat or other dangerous factors; “predation index” was rated by five-point scale that the probabilities to be preyed. For example if some slept in a burrow, den or well-protected position, point 1 were given; “overall danger index” was evaluated that the mammal’s danger to be preyed. For example if some species slept in the maximum exposure positions, then, they obtained a point 5, otherwise if the minimum exposure place, they got a value of 1. However, this analysis was rudimentary and did not have analyze by statistical methods in detail. Hence, i would like to analyze it by using Lasso selections to obtain more accurate estimate.

Statistical analysis: I take advantage of SAS 9.4 that manages analytics more readily to assess and estimate data characteristics. The statistical methods are Lasso selections and some traditional regression such as stepwise selections to analyze the mammal sleep data to further analyze variable correlation each other. In a linear mode, the response variable Y is modeled as a linear combination of the predictor variables, a₁, ..., a_p plus random noise that is Y = β₀+β₁ a_i1+, ..., a_ip+∈_I.

Model selection: Statistical model selection joins and performs the predictive estimation for different models and selects a best model from among the alternatives. However, model selection is not easy to find an approximate best method of the truth and its accuracy of the model prediction. Moreover, it is not necessarily guaranteed the underlying truth. Because lasso can construct stable results for the data and more predictors than the sample size, this method is better than traditional selection such as forward, backward, stepwise selections. Also, it can shrink the regression coefficients, so, variable selection and coefficient estimation are worked same time.

In this study the model consists of Y and X₁-X₁₀: Y is species of animals; dependent X₄ that is PS; Body W, Brain W, SWS, PS, Life-Span, GT, PI, SEI, ODI are X₁-X₇, X₉, X₁₀, respectively.

Lasso selection: This is a specific selection with tuning value z that looks for the solution to the regression constrained minimizing objects:

where, L₁ norm of the regression coefficients is the position of Lasso penalty. It simplifies the sum of their absolute values. In the regression coefficients, the shrinkage is used equally. Beforethe selection, each predictor variable could be standardized. Hence, the GLMSELECT procedure is the best candidate to perform this process and it could generate plots to track the selection process when using the coefficients by the same scale.

Adaptive lasso: Adaptive Lasso is a specific Lasso penalty that weights is used to each parameter to construct the Lasso constraint. These weights from adaptive Lasso manage shrinking more zero coefficients than shrinking the nonzero coefficients^[9]:

The Glmselect procedure performs the adaptive weights by the ordinary least squares estimations of the regression coefficients. Also, it offers some options,for example, if more correlation variables and predictor variables are over the sample sizes, adaptive Lasso might mange stable regression coefficients. This is better than using coefficients of the ordinary least squares.

Elastic net: As i mention proceeding, Lasso cannot deal with the big data when more numbers of selected predictor variables are more than the sample sizes. Elastic net does not only solve this limitation, it does but also combine groups of correlate variables. For example, some objects for data share same specific pathways and build a group but you plan to identify these objects to become this group, then you might to try elastic net. It gets off these limitations that is more numbers of selected predictor variables than number of sample sizes and it combines all variables in formed group without ignoring any members. The following is optimal formula f or elastic net:

Where the penalty of elastic net is the position of L₁ norm and L₂ norm (of the regression coeff icients. L₁ norm part does take variables selected by getting some coefficients as zero. L₂ norm norm performs group selection that helps shrink the coefficients of correlation variables each other. Therefore, we can write equation:

Here, both λ₁ and λ₂ are the tuning parameters.

RESULTS

As i mention preceding section, using model selection and model averagein Lasso, adaptive Lasso or elastic net methods can select prediction candidate models more accurately than other regression techniques. In forecast predictive models, adaptive Lasso method is reasonable to combine many variable selections to construct parsimonious predictive models. A model averaging for high-dimensional regression can participate into combine a couple of all of parameter selection or groups of multiple variables. It describes all of variables selected in the model corresponding to using the model. In the model, parameter estimates become the averages of the estimates for each sample. When a parameter is not chosen by linear model, the estimate value is defined as zero, the effect of shrinking the estimates of rarely picked parameters to be a zero. For example, in Table 1 average parameter Estimates using adaptive Lasso, i use glmselect procedure with “EffectSelectPct” choose of model average tables.

To understand the percentage of samples what the positionof each effect is in the selected model, I use the bar chart of the percentages for each parameter in the model shown in Fig. 1. It graphically describes their positions in the selected model iconically below.is the first one by 100% is the second one by 93%, last one is (about 8%).

To compute the average estimate for a parameter, we can partition the sum of the estimate values for that parameter in each sample by 800 samples. Those parameters of estimate values of zero in the model are not displayed in Fig. 2. But they are listed in another table (not appear in this paper due to the space limitation of the journal). In Fig. 2 we can see that the distributions of the estimates in refit model that are each parameter selected in the refit model.

Table 1:	Average parameter estimate using adaptive Lasso technique

Fig. 1:	The histogram for position of each effect in the model using adaptive Lasso (Effect selection percentage for X4)

Fig. 2(a-d):	The distributions of the estimates with model average refit using Lasso adaptive selection, choose is SBC. X10 is the closest to normal except intercept; Refit parameter estimate distribution for X4 (a) Intercept (b) X7 ( c) X10 and X3*X4

Table 2:	Information for Lasso SAS output

Table 3:	Analysis of variance

Table 4:	Lasso selection summary using Lasso

Because almost each distribution is approximately normal and large number of samples are used (sample size is 800). X₇, X₁₀ and X₃* X₄ display the range between the fifth and 95th percentages of each estimate around (-0.00224, 0.00016), (-0.61116, -0.27128), (-0.00100, -0.00100), respectively.

In the above Table 2 construct a linear model by using “the Partition” statement that 40% of the data as validation data were reserved randomly and 60% as training data, so that, the prediction error of model selection could be estimated. This means that the training set is used to fit the models. For those large data, a validation set is the best method to tune a penalized regression technique. Those observations from the validation one would be used to produce a Lasso solution path and then finding a smallest ASE from the validation data. In addition, we can see out that 356 variables are selected due to classification of Y and as effects with specific levels and possible two-way interaction effects (unlisted). Also, in observation numbers, 26 observations are selected as validation data, 36 of remaining are regarded as training one (Table 3-5).

As you can see the Lasso selection summary, model selection is 42 total steps It shows that there are all the true 35 effects and X₁₀ generates1.390 of the minimum validation ASE value in the step 7 with response variable.

We can try to build a linear model in SAS programming such as GLMSELECT procedure to forecast the level of data. In this study i use first call for Lasso method and the other one is the adaptive Lasso. Both of them were TESTDATA = option for GLMSELECT procedure and the “CHOOSE” is SBC criterion for in the model statement. The partial SAS output are as follows:

For the criteria for AIC, AICC, SBC and Adjust R², they have different implications in the selected model: R² is commonly used to test accuracy in the model. It is a basic matrix to tell us what numbers of variance is in the model, its value reflects variable significance. For example, for 0.78 of R², 78% of the variation in the output variable is measuredby the input variables and adjusted R² is used to compute R² value of those variables that are increased into the model. So, adjust R² value is the statistic based on the independent variables in the model; AIC is an estimator of statistical models in the data. It provides a reference value for the model selection. It evaluates the relative information lost by a specific model that is if the information is lost in a model, then the model is higher

Table 5:	The partial SAS output
*Optimal value of criterion selection stopped because all effects are in the final model

Fig. 3:(a, b)	The criterion plot for AIC, AICC, SBC and adjust R2 selected in the model and b Progression of ASE by Role for X4

quality; AICC is a kind information of AIC in statistical model for correction of small sample sizes; SBC is a very important estimator that selects best predict subsets in the regression models. Both of AIC and SBC have different goals. AIC is a good estimator for distance with the likelihood functions between fitting one and unknown one in the model. If the AIC value is lower, the model will be closed to the truth; SBC is a function estimator for testing if a model is true. When SBC is lower, the model is more likely to the real model. However, we should analyze them based on various assumptions approximations. In Fig. 3a for dependent X4, it has lower value of AIC and AICC at step 9, SBC is the minimum value at step 3, the best value of Adjust R-Square is at step 9. In Fig. 3b, we can see out that the amount of shrinking. In the regression coefficients, this shrinkage is decreased. But the model increase complexity, the ASE on the training data consistently dropping approximation to zero. Also, the prediction error on the test data is decreased by about 0.6 which is the point of minimum ASE in the vertical line of the plot. For test error, the decrease shows that the effects that join the model are important effects for the variation in the response variable before the vertical line. The subsequent increase suggests that the later effects explain the random noise in the training data. Hence, when reaching the minimum ASE value for the test data, the model is selected.

In Fig. 4 plots show that for using SBC criterion of model selection, Lasso and adaptive Lasso have the same set of predictor variables (X₃, X₂, X₁₀), although they are different solution paths. Also, the estimated coefficient values are near the same patterns.

In Table 6 the ASE of the test data form adaptive Lasso (27194) is little lower than one of Lasso (27318). Other values for adaptive Lasso are slightly smaller than corresponding ones for Lasso. Adaptive Lasso has a character to be distinguishing from big data sets. In GLMSELECT package, the former is also less steps than the latter (Fig. 4). Probably because the adaptive Lasso has a relatively higher penalization for zero coefficients and lower penalization for nonzero coefficients. So, it can decrease the estimation bias and advance variable selection accuracy, although Lasso has also an advantage with solution of difficult prediction problems (Table 7).

Fig. 4:(a, b)	Plot of coefficient progression for Lasso with adaptive Lasso (a) Lasso coefficient progression and (b) Adaptive Lasso coefficient progression

Fig. 5:(a, b)	Plot of coefficient progressions of Lasso and Elastic Net. X3, X6, X10 variables enter the model in the elastic net as a group before other candidates such as X5 and X9, etc. But it is not show up the group selection. Additionally, its solution path from elastic net looks more stable and smoother than Lasso path (a) Coefficient progression for X4 and (b) Coefficient progression for X4

Table 6:	Fit statistics for Lasso and Adaptive Lasso

Table 7:	Parameter estimates and fit statistics table for elastic net

A better selection procedure identify a group such as letting and to get together. This is a key process to deal with a complicate and large data, especially for more parameters than observations. For Lasso, one limitation of variable selection is that the predict variables cannot be over sample size and it could limit groups of correlated variables. It takes only one variable for group and getting off remaining variables. However, elastic net method does not have those limitations for selected variable numbers and group selection numbers when generating the groups. I would like to try to use some techniques to explore solving more difficult problems. The following is the plots that apply elastic net method by using the tuning value of as 0.1, taking external cross validation to determine the tuning with cross validation for Lasso, it also cross validation technique as CVMETHOD choose. I would like to test if both have big difference. Lasso coefficient progressionb. Elastic Net coefficient progression.

The following table is parameter estimates of elastic net table with three predictors, X₃, X₁*, X₂*, X₃*, X₄. Here, X₁ X₂*(-0.000000396) and * (-0.000423) are so close to zero, ASE (Test) is 15025 that is significantly less than one of Lasso (27318) and Adaptive Lasso (27194).

DISCUSSION

Lasso technique as a new regression methodis involved penalizing the absolute values of the regression coefficients. It is a very important analysis approach to study and explore mammal research. Lasso selections including adaptive, elastic net and group Lasso are mature technique to help analyze and

solve regression problems of comparative animal physiology. This study conforms some of the consequences based on mammal sleep data in 1976: Paradoxical sleep is correlation coefficient with slow wave Paradoxical sleep is also subject to predatory danger. In addition, this study tells us that body weight and brain weight are correlation with paradoxical sleep (Appendix 1 and 2).

CONCLUSION

I got another conclusion is that using adaptive Lasso and elastic net methods. They help deal with complicate and large data when more parameters than observations in the model. In prediction model, i used its stability, higher prediction accuracy, computational efficiency, higher selection methods of adaptive Lasso to generate GLMSECT procedure that performs model selections. It results in the effect of shrinking the estimates without zero value parameters. It guarantees higher accuracy for prediction models.

The GLMSECT technique is one of powerful model selection procedures. It can provide options and higher graphics to control selection by extensive customization. Also, if we need make partition of big data to training, validation, defining spline effects, selecting individual levels of classification effect, test sets, a couple of fit criterion such as AIC, AICC, SBC or k-fold cross validation to estimate prediction error, it will give powerful support.

ACKNOWLEDGEMENTS

I thank editors and three of anonymous reviewers. They gave constructive comments and suggestions.

APPENDIXES

Appendix1:	S1 the mammal sleep data from 1976 used in main analysis

Appendix 1:	Continue

Appendix 2:	S2 Lasso selection summary

Appendix 2:	Continue
*Optimal value of criterion selection stopped because the selected model is a perfect fit

How to cite this article:

Liming Xie. Statistical Analysis to Mammal Studies Based on Mammal Sleep Data.
DOI: https://doi.org/10.36478/aj.2020.97.106
URL: https://www.makhillpublications.co/view-article/1816-9155/aj.2020.97.106