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Anisim Sokolov
Anisim Sokolov

Introduction To Linear Regression Analysis Montgomery Pdf Free 347 PORTABLE


a, Retrospective duration estimates (minutes) as a function of veridical clock duration (minutes) during lockdown (S1; pink) and outside of it (SC; grey). Each dot represents a single participant. The regression lines were estimated from the linear mixed effect model; their 95% CIs are shown with grey shading. b, Relative retrospective duration estimates (unitless) as a function of the stringency index (a.u. between 0 and 100) for all sessions (coloured). The coloured dots are individual data points per participant and per session. The regression line was estimated from the linear model; the 95% CI is shown with grey shading. The more stringent governmental rules were, the shorter retrospective durations were estimated to be. c, Relative retrospective duration estimates (unitless) as a function of the mobility index (percent change relative to baseline, prior to lockdown; see the main text) for all sessions (coloured). Each dot is an individual data point per participant and per session. The black line is a regression line estimated from the linear model; the 95% CI is shown with grey shading. The closer to baseline mobility, the shorter retrospective durations were estimated to be.




Introduction To Linear Regression Analysis Montgomery Pdf Free 347



a, Distribution of VAS rating (0 to 100) counts for passage-of-time judgements as a function of session (colour coded). b, Passage-of-time ratings as a function of subjective confinement (5 to 20). The grey dots are individual data points (per participant, per session, per run). The black dots are the mean passage-of-time ratings binned by subjective confinement. Their size scales with the underlying number of individual data points. The black line is a regression line estimated from the linear mixed effect model; the 95% CI is shown with grey shading. The less lonely the participants felt, the faster the passage of time felt.


Hence, these differences in the pertinent variables of the two regression models highlight the change in the weight of the relationship between the characteristics of the rainy season and the annual rainfall amount. The significance of the changes in the structure of the rainy season is assessed from the performance of each regression model (f 1 and f 2) over the two periods. This assessment helps also to select the most representative regression model over both reference and projection periods with regard to the change in the annual rainfall amount. But, in case of the two regression models are not representative, a new regression model (f) is calibrated from the merging set of the pertinent variables over the two periods. So, if we call \( X_1^* \) the set of the pertinent variables over the reference period and \( X_2^* \) the set of the pertinent variables over the projection period, the merged pertinent variables over the two periods are \( X^* = X_1^* UX_2^* \). The regression model f is then elaborated from \( X^* \). Also, the significance of the contribution of the pertinent variables to the change in the mean annual rainfall amount is assessed from a statistical analysis performed through the regression model f.


As verification, the Bayesian regression method (Chen and Martin 2009) was also used. It selects three pertinent variables for the regression model: the number of rain days, the mean daily rainfall and the maximum daily rainfall with a likelihood of 0.61 from a set of 10,000 iterations of the Markov Chain Monte Carlo (Gilks 1996). These three variables are also found to be dominant from the deterministic method, thus the Bayesian method confirms the relevance of these variables for the annual rainfall amount regression model. So, the multiple linear regression model built with the deterministic method is more suitable for the regression because of its simplicity and its appropriate description of the different changes in the evolution of the annual rainfall amount.


The results found in this analysis of daily rainfall correspond well to those Le Barbé et al. (2002) have obtained with another method (leak distribution model) and over the entire Sahel. This confirms the ability of the multiple linear regression model to describe in more detail the evolution of the mean annual rainfall amount and the different characteristics of the rainy season. This procedure will help us to better describe the different changes in the characteristics of the rainy season projected by the five regional climate models for a warmer climate.


The objective of this section is to estimate the contribution of each of the six characteristics of the rainy season to the change in annual mean rainfall determined in the five RCMs. The pertinent variables for the multiple linear regression model selected by the Stepwise procedure depend on the RCM and the target period (reference or projection). Table 7 presents the contribution of each pertinent variable to the total variance of annual rainfall (Eq. 3). The variances obtained from the regression models (f 1 and f 2) are overall higher than 90 % which fulfills the requirement set for the methodology. The variance distribution of each regression model (Table 7) shows that the mean daily rainfall is the most important variable for HadRM3P, RACMO, RCA and REMO models and account for more than 70 % of the total variance. But for CCLM, the mean daily rainfall dominates only during the reference period; the number of rain days becomes dominant during the projection period. Thus, the dominant variables in the regression model for annual rainfall of the five RCMs are not the same as for the observations (number of rain days is the dominant variable in that case). Table 7 shows also the modifications of the pertinent variables between the regression models for each RCM over the two periods. Only REMO presents the same pertinent variables over the two periods. Even if there is a modification of the pertinent variables from one period to another for a given RCM, two variables, the number of rain days and the mean daily rainfall are always selected and account for more than 75 % of the total variance (Table 7). Thus, the number of rain days and the mean daily rainfall are the main variables in the projection of the annual rainfall amount from the RCMs daily rainfall just as was found for the observations.


In the same way as the analysis of the contribution of each variable to the overall variance done before (Table 9), here also the new regression models (f) reproduce over 94 % of the variance of the annual rainfall of the RCMs. Table 9 presents the contribution of each of the six characteristics (Eq. 3) to the relative difference of annual rainfall. This shows that the decrease in annual rainfall simulated by CCLM comes mainly from a decrease in the number of rain days and the delay of the season onset (Table 9). But, the decrease in the annual rainfall for RCA is explained by the decrease in both number of rain days and the mean daily rainfall amounts. In contrast, the increase of seasonal rainfall in HadRM3P and RACMO originate mainly in an increase of mean daily rainfall. The REMO model, which has no significant change in annual rainfall, is characterized by a positive contribution from the mean daily rainfall and a negative contribution from the number of rain days.


In the following analysis, performed with the regression model f, the contribution of each of the six characteristics to the change in annual mean rainfall is quantified through a random permutation of each characteristic for the projection period. Figure 6 presents the annual rainfall amount projected with different combinations of variables. For CCLM, the annual rainfall decrease is mostly explained by the impact of the change in the number of rain days which is the only variable that lowers significantly the annual rainfall from the level of P1 to the one of P2. For HadRM3P and RACMO, the increase in the annual rainfall amount over the second period can be attributed to changes in the mean daily rainfall. For RCA, the amplitude of the decrease in the annual rainfall can only be reproduced by combining the number of the rain days and mean daily rainfall. Indeed, each of these variables has lowered the annual rainfall projected for the second period (Fig. 6). However, for REMO, the two variables (the number of the rain days and the mean daily rainfall) act on the annual rainfall amounts in opposite directions: while the number of rain days lowers the mean annual rainfall, the mean daily rainfall increases it. Only the combination of the two variables produces the near zero change of total rainfall produced by this model. For all RCMs, the variance of the annual rainfall amount does not change significantly between the two periods as demonstrated by the Fligner-Killeen test.


The structure of the rainy seasons is described in this study through a set of eight characteristics: date of the season onset (Onset), date of the end of season (End), season duration (SDR), number of rain days (NbRD), mean daily rainfall (MDR), maximum daily rainfall (MaxR), annual rainfall amount, and mean dry spell length (DryS). The seven characteristics address the main components of the rainy season over Sahel and allow to address properties of the rainy season more relevant for application as agricultural yields and water resources in the region. The characterization of the interannual variability of the observed and simulated rainfall over Burkina Faso is done with the multiple linear regression based on six characteristics of the rainy season (Onset, End, NbRD, MDR, MaxR, and DryS).


The multiple linear regression models developed in this study produced a representative description of the relationships between the annual rainfall amounts and the characteristics of the rainy season which matter to applications such as agronomic yields and water resources. The methodology allowed confirming that the continuous drought condition (since 1970) over West African Sahel is characterized by a decrease in rainfall frequencies at the core of the rainy seasons (June to August).


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