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Course: FDA 2008, Fall 2009School: Cornell
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BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Constrained Functions Positive Smoothing We know that angular acceleration must be positive: Text: Chapter 6...
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BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Constrained Functions Positive Smoothing We know that angular acceleration must be positive: Text: Chapter 6 There are some situations in which we want to include known restrictions about x(t). x(t) is always positive x(t) is always increasing (or decreasing) x(t) is a density Idea: Enforce these conditions by transforming x(t). a(t) = [D 2 x(t)]2 + [D 2 y (t)]2 But negative values can occur because of smoothing/basis bias. BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Positive Smoothing We want to ensure that x(t) > 0. Observation: e w : ( , ) (0, ) So try the transformation x(t) = e W (t) with W (t) = (t)c and penalize the roughness of W (t) Estimating a Positive Smooth We now want to minimize n PENSSE (W ) = i=1 yi e W (ti ) 2 + [LW (t)]2 dt This does not have an explicit formula. But it is convex there is only one minimum. Requires numerical optimization, but this is generally fast. BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Positive Smoothing Vancouver Precipitation Constraints imply smoothness (of a certain type) tend to need less smoothing on W . Monotone Smoothing Berkeley growth study heights aged 1 - 18 BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Monotone Smoothing Estimating a Monotone Smooth We now want to minimize We need x(t) always increasing: n Dx(t) > 0 suggests Dx(t) = e W (t) x(t) = + again, W (t) = (t)c t t0 PENSSE (W ) = i=1 yi ti e t0 W (s) 2 ds + [LW (t)]2 dt e W (s) ds No explicit formula No good formula for the integral Still a convex problem; numerics work fairly quickly Note, LW (t) = D 2 W (t) suggests that any x(t) = + e t is smooth. BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Density Estimation Position of Beetles in Angles Density Estimation x(t) a density positive, integrates to 1 x(t) = e W (t) / But we observe only t1 , . . . , tn . Need to nd an objective to minimize. e W (t) dt BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Penalized Likelihood Likelihood of W (t) is probability of seeing t1 , . . . , tn if W is true. Easier to work with log likelihood n Thinking about Smoothness What is an appropriate measure of smoothness for densities? x(t) = Ce W (t) dt Compare to Normal density f (t) = [LW (t)] dt 2 l (W |t1 , . . . , tn ) = i=1 W (ti ) log e W (t) Minimize the penalized negative log likelihood: n 1 2 2 e (t ) / 2 PENLOGLIK (W ) = i=1 W (ti )+n log e W (t) then W (t) = t 2 should be smooth LW (t) = D 3 W (t). Alternatively, LW (t) = D 2 W (t) exponential distribution is smooth useful for positive data. dt+ Usual comments about numerics apply. BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing Rough to Smooth Densities Every 10th beetle More Penalized Likelihood Sometimes squared error is not appropriate Binary data: yi {0, 1}; want to measure p(ti ) = P(yi = 1|t). Count data; want a Poisson intensity at each ti . Measurements of event times t1 , . . . , tn ; Poisson-process intensity can change over time. In each case we can write down a log likelihood of observed data given (some transformation of) W (t) and apply a smoothing penalty. These are discussed in the text, but not implemented in software. Mathematical theory justi es process. BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a> : Constrained Smoothing BTRY 6150: Applied Functional <a href="/keyword/data-analysis/" >data analysis</a&...
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