Human Growth: From data to functions

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Human Growth From data to functions
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Challenges to measuring growth

• We need repeated and regular access to subjects
  for up to 20 years.
• Height changes over the day, and must be measured
  at a fixed time.
• Height is measured in supine position in infancy,
  followed by standing height. The change involves
  an adjustment of about 1 cm.
• Measurement error is about 0.5 cm in later years,
  but is rather larger in infancy.

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Challenges to functional modeling

• We want smooth curves that fit the data as well
  as is reasonable.
• We will want to look at velocity and
  acceleration, so we want to differentiate twice
  and still be smooth.
• In principle the curves should be monotone i.
  e., have a positive derivative.

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The monotonicity problem

• The tibia of a newborn measured daily shows us
  that over the short term growth takes places in
  spurts.
• This babys tibia grows as fast as 2 mm/day!
• How can we fit a smooth monotone function?

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Weighted sums of basis functions

• We need a flexible method for constructing curves
  to fit the data.
• We begin with a set of basic functional building
  blocks fk(t), called basis functions.
• Our fitting function x(t) is a weighted sum of
  these

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What are the main choices for basis functions?

• Fourier series
• a constant term,
• a sine/cosine pair of fixed frequency, and
• followed by a series of sine/cosine pairs with
  integer multiples of the base frequency.
• Fourier series are best for periodic data.

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Five Fourier basis functions
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B-splines

• These basis functions are piecewise polynomials
  defined by a set of discrete values called knots.
  
• The order of the polynomials (degree 1)
  controls their smoothness.
• Each basis function is nonzero only over a number
  of contiguous inter-knot intervals equal to the
  order.
• Polynomials are a special type of B-spline, and
  are thus included within the system.

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When should I use B-splines?

• B-splines are the basis of choice for most
  non-periodic.
• They give complete control over flexibility,
  allowing more flexibility where needed and less
  where not needed.
• Computing with B-splines is extremely efficient.

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Five order 2 B-spline basis functions A basis
for polygonal lines
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Eight order 4 B-spline basis functions A basis
for twice-differentiable functions
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B-splines for growth data

• We use order 6 B-splines because we want to
  differentiate the result at least twice. Order 4
  splines look smooth, but their second derivatives
  are rough.
• We place a knot at each of the 31 ages.
• The total number of basis functions order
  number of interior knots. 35 in this case.

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Isnt using 35 basis functions to fit 31
observations a problem?

• Yes. We will fit each observation exactly.
• This will ignore the fact that the measurement
  error is typically about 0.5 cm.
• But well fix this up later, when we look at
  roughness penalties.

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Okay, lets see what happens

• These two Matlab commands define the basis and
  fit the data
• hgtbasis
• create_bspline_basis(1,18, 35, 6, age)
• hgtfd
• data2fd(hgtfmat, age, hgtbasis)

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Why we need to smooth

• Noise in the data has a huge impact on derivative
  estimates.

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Please let me smooth the data!

• This command sets up 12 B-spline basis functions
  defined by equally spaced knots. This gives us
  about the right amount of fitting power given the
  error level.
• hgtbasis
• create_bspline_basis(1,18, 12, 6)

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• These are velocities are much better.
• They go negative on the right, though.

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Lets see some accelerations

• These acceleration curves are too unstable at the
  ends.
• We need something better.

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A measure of roughness

• What do we mean by smooth?
• A function that is smooth has limited curvature.
• Curvature depends on the second derivative. A
  straight line is completely smooth.

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Total curvature

• We can measure the roughness of a function x(t)
  by integrating its squared second derivative.
• The second derivative notation is D2x(t).

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Total curvature of acceleration

• Since we want acceleration to be smooth, we
  measure roughness at the level of acceleration

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The penalized least squares criterion

• We strike a compromise between fitting the data
  and keeping the fit smooth.

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How does this control roughness?

• Smoothing parameter ? controls roughness.
• When ? 0, only fitting the data matters.
• But as ? increases, we place more and more
  emphasis on penalizing roughness.
• As ? ? 8, only roughness matters, and functions
  having zero roughness are used.

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• We can either smooth at the data fitting step, or
  smooth a rough function.
• This Matlab command smooths the fit to the data
  obtained using knots at ages. The roughness of
  the fourth derivative is controlled.
• lambda 0.01
• hgtfd smooth_fd(hgtfd, lambda, 4)

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Accelerations using a roughness penalty

• These accelerations are much less variable at the
  extremes.

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The corresponding velocities
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How did you choose ??

• We smooth just enough to obtain tolerable
  roughness in the estimated curves (accelerations
  in this case), but not so much as to lose
  interesting variation.
• There are data-driven methods for choosing ?, but
  they offer only a reasonable place to begin
  exploring.
• But smoothing inevitably involves judgment.

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What about monotonicity?

• The growth curves should be monotonic.
• The velocities should be non-negative.
• Its hard to prevent linear combinations of
  anything from breaking the rules.
• We need an indirect approach to constructing a
  monotonic model

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A differential equation for monotonicity

• Any strictly monotonic function x(t) must satisfy
  a simple linear differential equation

The reason is simple because of strict
monotonicity, the first derivative Dx(t) will
never be 0, and function w(t) is therefore
simply D2x(t)/Dx(t).
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The solution of the differential equation

• Consequently, any strictly monotonic function
  x(t) must be expressible in the form

This suggests that we transform the monotone
smoothing problem into one of estimating function
w(t), and constants ß0 and ß1.
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What we have learned

• B-spline bases are a good choice for fitting
  non-periodic functions Fourier series are right
  for periodic situations.
• We can control smoothness by either using a
  restricted number of basis functions, or by
  imposing a roughness penalty.
• Roughness penalty methods generally work better.
• Differential equations can play a useful role
  when fitting constrained functions to data.

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More information

• Ramsay Silverman (1997, 2004), Chs. 3, 4, 13
• Ramsay Silverman (2002), Ch. 6.
• The long-term growth data are from the Berkeley
  growth study.
• The infant growth data were collected by Michael
  Hermanussen.

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Where do we go from here?

• We need to look more systematically at how to
  smooth data.
• This involves deciding what basis function system
  to use.
• Splines are so important that we have to look at
  them in more detail.
• Heres a serious problem

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Whats wrong with the mean?

• The cross-sectional mean is the heavy blue line.
• It has less amplitude variation than any single
  curve.
• The pubertal growth spurt for the mean lasts
  longer than does any single curve.
• The problem is that we are averaging over curves
  in quite different stages of growth.

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Whats wrong with the mean?

• The cross-sectional mean is the heavy blue line.
• It has less amplitude variation than any single
  curve.
• The pubertal growth spurt for the mean lasts
  longer than does any single curve.
• The problem is that we are averaging over curves
  in quite different stages of growth.

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Phase and Amplitude Variation

• Functional data like growth curves often show
  variation in the timing of events, like the
  pubertal growth spurt.
• This is called phase variation.
• We have to find out how to separate phase from
  amplitude variation before we can do even simple
  things like compute mean curves.