# Forecast of the minimum temperature in Goleta based on a SUTSE model

This [unedited] guest post is by a student in my PSTAT262MC class (background post).
Please praise/critique/comment on its quality and importance to you.

Gabriele Lillacci says: As explained in Blog 1, the model and the forecasts that are described in the following are based on data of the daily minimum temperature measured at 3 airport locations in Southern California. Two remarks on the data set are in order. (1) Data for January 2010 became available in the past few days, so this information was included as well. (2) For reasons of computational feasibility, only a subset of the data set could be actually used. For the above reasons, the actual inference and forecast process was based on daily observations from 1/1/2008 to 1/31/2010. The modeling was carried out in two stages. First, a univariate dynamic linear model (DLM) was fit for each location. Then, in order to take full advantage of the structure of the data set, the individual models were combined to produce a seemingly unrelated time series equations (SUTSE) model. Each univariate DLM contains a Fourier form seasonal component (with a period of 365 and 2 harmonics) and a local level component. The periodicity of the data seems to be constant over time, therefore the evolution variance of the seasonal component is set to 0. On the other hand, the evolution variance of the local level component and the overall observation variance are parameters to be estimated. The SUSTE model was obtained by combining the three univariate DLMs and by allowing ``crosstalk'' among them. Both evolution covariances in the local level components and observation covariances were allowed. Inference for the SUTSE model was performed by Gibbs sampling. Four Markov chains were run starting from different initial conditions, each for 5000 iterations. The first 1000 iterations of each chain were discarded as burn-in. No lack of convergence was detected using the Gelman-Rubin diagnostic. Mixing was assessed by looking at trace plots and by computing ergodic means. The figure below shows a one-year-ahead forecast of the daily minimum temperature in Goleta based on the total 16000 post burn-in MCMC samples. The thick black central line represents the median, while the dashed lines show the 95% credible interval. The red lines highlight the two chosen forecast dates, 3/5/2010 and 11/30/2010 (see Blog 1 for details). The two forecasts (95% interval) are 41.82 (6.09, 77.56) and 40.42 (-60.78, 141.67) respectively.

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Gabriele Lillacci says: As explained in Blog 1, the model and the forecasts that are described in the following are based on data of the daily minimum temperature measured at 3 airport locations in Southern California. Two remarks on the data set are in order. (1) Data for January 2010 became available in the past few days, so this information was included as well. (2) For reasons of computational feasibility, only a subset of the data set could be actually used. For the above reasons, the actual inference and forecast process was based on daily observations from 1/1/2008 to 1/31/2010. The modeling was carried out in two stages. First, a univariate dynamic linear model (DLM) was fit for each location. Then, in order to take full advantage of the structure of the data set, the individual models were combined to produce a seemingly unrelated time series equations (SUTSE) model. Each univariate DLM contains a Fourier form seasonal component (with a period of 365 and 2 harmonics) and a local level component. The periodicity of the data seems to be constant over time, therefore the evolution variance of the seasonal component is set to 0. On the other hand, the evolution variance of the local level component and the overall observation variance are parameters to be estimated. The SUSTE model was obtained by combining the three univariate DLMs and by allowing ``crosstalk'' among them. Both evolution covariances in the local level components and observation covariances were allowed. Inference for the SUTSE model was performed by Gibbs sampling. Four Markov chains were run starting from different initial conditions, each for 5000 iterations. The first 1000 iterations of each chain were discarded as burn-in. No lack of convergence was detected using the Gelman-Rubin diagnostic. Mixing was assessed by looking at trace plots and by computing ergodic means. The figure below shows a one-year-ahead forecast of the daily minimum temperature in Goleta based on the total 16000 post burn-in MCMC samples. The thick black central line represents the median, while the dashed lines show the 95% credible interval. The red lines highlight the two chosen forecast dates, 3/5/2010 and 11/30/2010 (see Blog 1 for details). The two forecasts (95% interval) are 41.82 (6.09, 77.56) and 40.42 (-60.78, 141.67) respectively.