This paper will begin by initially discussing the potential uses and challenges of efficient and accurate traffic forecasting. The data we used includes traffic volume from seven locations on a busy Athens street in April and May of 2000. This data was used as part of a traffic forecasting competition. Our initial observation, was that due to the volatility and oscillating nature of daily traffic volume, simple linear regression models will not perform well in predicting the time-series data. For this we present the Harmonic Time Series model. Such model (assuming all predictors are significant) will include a sinusoidal term for each time index within a period of data. Our assumption is that traffic volumes have a period of one week (which is evidenced by the graphs reproduced in our paper). This leads to a model that has 6,720 sine and cosine terms. This is clearly too many coefficients, so in an effort to avoid over-fitting and having an efficient model, we apply the sub-setting algorithm known as Adaptive Lass.

Quadratic growth curves of 2nd degree polynomial are widely used in longitudinal studies. For a 2nd degree polynomial, the vertex represents the location of the curve in the XY plane. For a quadratic growth curve, we propose an approximate confidence region as well as the confidence interval for x and y-coordinates of the vertex using two methods, the gradient method and the delta method. Under some models, an indirect test on the location of the curve can be based on the intercept and slope parameters, but in other models, a direct test on the vertex is required. We present a quadratic-form statistic for a test of the null hypothesis that there is no shift in the location of the vertex in a linear mixed model. The statistic has an asymptotic chi-squared distribution. For 2nd degree polynomials of two independent samples, we present an approximate confidence region for the difference of vertices of two quadratic growth curves using the modified gradient method and delta method. Another chi-square test statistic is derived for a direct test on the vertex and is compared to an F test statistic for the indirect test. Power functions are derived for both the indirect F test and the direct chi-square test. We calculate the theoretical power and present a simulation study to investigate the power of the tests. We also present a simulation study to assess the influence of sample size, measurement occasions and nature of the random effects. The test statistics will be applied to the Tell Efficacy longitudinal study, in which sound identification scores and language protocol scores for children are modeled as quadratic growth curves for two independent groups, TELL and control curriculum. The interpretation of shift in the location of the vertices is also presented.