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The goal of this paper is to do some basic proofs for lasso and have a deep understanding of linear regression. In this paper, firstly I give a review of methods in linear regression, and most concerns with the method of lasso. Lasso for ¡®least absolute shrinkage and selection operator¡¯ is a regularized version of method adds a constraint which uses norm less or equal to a given value t. By doing so, some predictor coefficients would be shrank and some others might be set to 0. We can attain good interpretation and prediction accuracy by using lasso method. Secondly, I provide some basic proofs for lasso, which would be very helpful in understanding lasso. Additionally, some geometric graphs are also given and one example is illustrated.

In this paper, we study the influence of the National Telecom Business Volume by the data in 2008 that have been published in China Statistical Yearbook of Statistics. We illustrate the procedure of modeling “National Telecom Business Volume” on the following eight variables, GDP, Consumption Levels, Retail Sales of Social Consumer Goods Total Renovation Investment, the Local Telephone Exchange Capacity, Mobile Telephone Exchange Capacity, Mobile Phone End Users, and the Local Telephone End Users. The testing of heteroscedasticity and multicollinearity for model evaluation is included. We also consider AIC and BIC criterion to select independent variables, and conclude the result of the factors which are the optimal regression model for the amount of telecommunications business and the relation between independent variables and dependent variable. Based on the final results, we propose several recommendations about how to improve telecommunication services and promote the economic development.

<p>I present an application to visualization of GPS data and Linear Correlations and models. A collection of data for each vehicle is used to compute correlations. Deviating correlations can be indicative of a faulty vehicle.</p><p> The correlation values for each vehicle are computed with use linear regression algorithms using up to 4 signals in the data (with varied time window), and display the model parameters in a window next to the GPS map. Multiple measurements (multiple drive routes and multiple model parameters) are displayed at the same time, allowing tracking over time and comparison of different vehicles.</p><p> </p><p> The whole technique is demonstrated on three data which is set on first frame by user. The results are displayed with a java application and Google Map.</p>

I present an application to visualization of GPS data and Linear Correlations and models. A collection of data for each vehicle is used to compute correlations. Deviating correlations can be indicative of a faulty vehicle. The correlation values for each vehicle are computed with use linear regression algorithms using up to 4 signals in the data (with varied time window), and display the model parameters in a window next to the GPS map. Multiple measurements (multiple drive routes and multiple model parameters) are displayed at the same time, allowing tracking over time and comparison of different vehicles. The whole technique is demonstrated on three data which is set on first frame by user. The results are displayed with a java application and Google Map.

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<p> Protein sequences display replacement rate heterogeneity across sites. In an earlier work, half of the causal site-wise variation in replacement rates was explained by a simple linear regression model consisting of terms for the solvent exposure of each residue, distance from the active site, and glycines in unusual main-chain conformations. Replacement rates vary not only across sites, they may also vary over time. In this study, we apply the linear regression model to phylogenies divided into subtrees to see if lineage-specific rate shifts have a structural basis that can be detected by the model. This approach is applied to two different data sets. The first set consists of phylogenies containing two representative structures, divided into subtrees such that one structure is present in each subtree. These structures have little or no obvious functional divergence between them. The model is tested with permutations of subtrees and structures from each subtree. While there is a slight effect of the specific structure on the fit of the model, the specific subtree has a greater effect. The second data set involves homologous structure pairs where the quaternary structure has changed at some point in the phylogeny. These pairs are examined to see how the change in constraint on the new interface sites affect the replacement rate, and its relationship with other structural factors. We find that the unique interfaces are as conserved as the shared ones, and they exhibit a different relationship between replacement rates and indicators of constraint than the shared interfaces or other protein sites. We also find that the unique interfaces display characteristic amino acid preferences that may identify interfaces which are still in the process of stabilizing. </p> / Thesis / Master of Science (MSc)

In this thesis the air infiltration through the envelope of a naturally ventilated stone church located in Bergby (Gävle, Sweden) is studied. The project is focused on Multiple Linear Regression (MLR) modeling the air change rate (ACH) inside the church hall and studying the factors (stack effect and wind effect) that influence the air infiltration. The weather parameters outside the building were recorded in a weather station and the properties of the air inside the church was analyzed with different methods. Infrared thermography techniques and thermistors were used to measure the temperature inside, the tracer gas method to measure the ACH and the blower door technique to measure the tightness of the building envelope. In order to know the pressure coefficients on the church envelope a physical model of the building was studied in a wind tunnel. Firstly, only the values obtained from the weather station were used to calculate the predictors of ACH and see which parameter influence more on its variation:  temperature difference (∆T) indicating the stack effect; and wind speed (WS), the component of wind speed perpendicular to the long-side facades of the church (WS90) and their square values (WS2 and WS902) indicating the wind effect. The data obtained in the wind tunnel were later used to do the MLR study with new predictors for indicating wind effect (∆Cp∙WS, ∆Cp∙WS2, ∆CpOUT-IN·A∙WS, ∆CpOUT-IN·A∙WS2, ∆CpC-H∙WS, ∆CpC-H∙WS2). Better prediction of ACH was obtained with the square of the wind speed (WS2) instead of the magnitude itself (WS). However, the latter (WS) provided better results than the regression with the magnitude of the perpendicular component of the wind (WS90). Although wind speed influences in ACH, it alone seems to be a very poor predictor of ACH since has a negative correlation with ΔT when the data under study include both day and night. However when high wind speed are detected it has quite strong influence. The most significant predictions of ACR were attained with the combined predictors ∆T &amp; WS and ∆T &amp; ∆CpOUT-IN·A∙WS2. The main conclusion taken from the MLR analysis is that the stack effect is the most significant factor influencing the ACH inside the church hall. This leads to suggest that an effective way of reducing ACH could be sealing the floor and ceiling of the church because from those areas the air infiltration has big influence on the ACH inside the church hall, and more in this case that have been noted that the floor is very leaky. Although different assumptions have been done during the analyses that contribute to make the predictions deviate from reality, at the end it would be possible to asses that MLR can be a useful tool for analyzing the relative importance of the driving forces for ACR in churches and similar buildings, as long as the included predictors not are too mutually correlated, and that attained models that are statistically significant also are physically realistic. / Church project

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