A modern approach to AI encompasses advanced techniques such as machine learning and deep learning, utilizing neural networks to process complex data types like images, text, and sequential information. Natural language processing, reinforcement learning, and generative adversarial networks are key methodologies, with an increasing emphasis on ethical considerations and interdisciplinary collaboration. AI for Good initiatives aim to address societal challenges, while continual learning enables AI systems to adapt and evolve over time. Overall, modern AI strives for ethical, transparent, and impactful deployment, leveraging cutting-edge technologies to drive innovation and address real-world problem
(See Exercise 15.12.)
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4.3 The general case The preceding derivation illustrates the key property of Gaussian distributions that allows Kalman ltering to work: the fact that the exponent is a quadratic form. This is true not just for the univariate case; the full multivariate Gaussian distribution has the form N (, )(x) = e 1 2 (x)(cid:4)1 (x) . 587 (15.20) 588 KALMAN GAIN MATRIX Chapter 15.
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Probabilistic Reasoning over Time Multiplying out the terms in the exponent makes it clear that the exponent is also a quadratic function of the values xi in x. As in the univariate case, the ltering update preserves the Gaussian nature of the state distribution. Let us rst dene the general temporal model used with Kalman ltering. Both the transition model and the sensor model allow for a linear transformation with additive Gaussian noise. Thus, we have P (xt+1 | xt) = N (Fxt, x)(xt+1) P (zt | xt) = N (Hxt, z)(zt) , (15.21) where F and x are matrices describing the linear transition model and transition noise covariance, and H and z are the corresponding matrices for the sensor model. Now the update equations for the mean and covariance, in their full, hairy horribleness, are t+1 = (I Kt+1H)(FtF(cid:12) + x) , t+1 = F t + Kt+1(zt+1 HF
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22) where Kt+1 = (FtF(cid:12) + x)H(cid:12)(H(FtF(cid:12) + x)H(cid:12) + z)1 is called the Kalman gain matrix. Believe it or not, these equations make some intuitive sense. For example, consider the update for the mean state estimate . The term Ft is the predicted state at t + 1, so HFt is the predicted observation. Therefore, the term zt+1 HFt represents the error in the predicted observation. This is multiplied by Kt+1 to correct the predicted state; hence, Kt+1 is a measure of how seriously to take the new observation relative to the prediction. As in Equation (15.20), we also have the property that the variance update is independent of the observations. The sequence of values for t and Kt can therefore be computed ofine, and the actual calculations required during online tracking are quite modest.
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