Estimates of Parameters of GMM: The Expectation Maximization (EM) Algorithm We observe n data points x1,…,xn in Rd . We wish to maximize the GMM likelihood with respect to the parameter set θ={p1,…,pK,μ(1),…,μ(K),σ21,…,σ2K} . Maximizing the log-likelihood log(∏ni=1p(x(i)|θ)) is not tractable in the setting of GMMs. There is no closed-form solution to finding the parameter set θ that maximizes the likelihood. The EM algorithm is an iterative algorithm that finds a locally optimal solution θ^ to the GMM likelihood maximization problem.

To save energy, the air supply to a 2000 ft office has been shut off overnight and the room temperature has dropped to 40°f. in the morning, the thermostat is reset to 70°f and warm air at 120 f begins to flov in at 200ft'/min. the air is well mixed within the room, and an equal mass flow of air at room temperature (changing with time) is withdrawn through a return duct. the air pressure is nearly 1 atm everywhere. ignoring heat transfer with the surroundings and kinetic and potential energy effects, estimate how long it takes for the room temperature to reach 70°f. plot the room temperature as a function of time.