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ercbk committed Dec 15, 2024
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<li>Normality among predictors isn’t necessary, but I think it improves fit or prediction somewhat</li>
</ul></li>
<li>log transformation may help some if the skew isn’t too extreme</li>
<li>See <a href="../qmd/mixed-effects-general.html#sec-me-gen-misc" style="color: green">Mixed Effects, General &gt;&gt; Misc</a> &gt;&gt; <span style="color: #990000">{skewlmm}</span></li>
</ul></li>
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<li><p>Density</p>
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<li class="sidebar-item">
<div class="sidebar-item-container">
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Expand Down Expand Up @@ -2684,15 +2684,15 @@ <h4 class="unnumbered anchored" data-anchor-id="sec-fcast-stat-alg-kalf-misc">Mi
</ul></li>
<li>If a dynamic system is linear and with Gaussian noise (inaccurate measurements, etc.), the optimal estimator of the hidden states is the Kalman Filter
<ul>
<li>For nonlinear systems, we use the extended Kalman filter, which works by simply linearizing the predictions and measurements about their mean. (I may do a second write-up on the EKF in the future)</li>
<li>For nonlinear systems, we use the extended Kalman filter, which works by simply linearizing the predictions and measurements about their mean.</li>
<li>Good for predictions where the measurements of the outcome variable over time can be noisy</li>
</ul></li>
<li>Assumptions
<ul>
<li>Gaussian noise</li>
<li>Markov property
<ul>
<li>If you know <span class="math inline">\(x_{t−1}\)</span>, then knowledge of <span class="math inline">\(x_{t−2},\ldots , x_0\)</span> doesn’t give any more information about xt (i.e.&nbsp;not much autocorrelation if at all)</li>
<li>If you know <span class="math inline">\(x_{t−1}\)</span>, then knowledge of <span class="math inline">\(x_{t−2},\ldots , x_0\)</span> doesn’t give any more information about <span class="math inline">\(x_t\)</span> (i.e.&nbsp;not much autocorrelation if at all)</li>
</ul></li>
</ul></li>
<li><strong>tl;dr</strong>
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<li>Light on memory (they don’t need to keep any history other than the previous state)</li>
<li>Very fast, making them well suited for real time problems and embedded systems</li>
</ul></li>
<li>Limitations
<ul>
<li>If model states have extremely high dimensions, we have to store a large covariance matrix and conduct intense matrix multiplications.</li>
<li>If some operator (that I didn’t understand) is nonlinear, then the covariance matrices can’t be calculated
<ul>
<li>An ensemble kalman filter gets around this by using Monte-Carlo simulation</li>
</ul></li>
</ul></li>
<li>Use cases
<ul>
<li>Engineering: common for reducing noise from sensor signals (i.e.&nbsp;smoothing out measurements)</li>
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