Mercator: A Connection with Torsion

Introduction

In most presentations of Riemannian geometry, e.g. O’Neill (1983) and Wikipedia, the fundamental theorem of Riemannian geometry (“the miracle of Riemannian geometry”) is given: that for any semi-Riemannian manifold there is a unique torsion-free metric connection. I assume partly because of this and partly because the major application of Riemannian geometry is General Relativity, connections with torsion are given little if any attention.

It turns out we are all very familiar with a connection with torsion: the Mercator projection. Some mathematical physics texts, e.g. Nakahara (2003), allude to this but leave the details to the reader. Moreover, this connection respects the metric induced from Euclidean space.

We use SageManifolds to assist with the calculations. We hint at how this might be done more slickly in Haskell.

A Cartographic Aside

%matplotlib inline
/Applications/SageMath/local/lib/python2.7/site-packages/traitlets/traitlets.py:770: DeprecationWarning: A parent of InlineBackend._config_changed has adopted the new @observe(change) API
  clsname, change_or_name), DeprecationWarning)
import matplotlib
import numpy as np
import matplotlib.pyplot as plt
import cartopy
import cartopy.crs as ccrs
from cartopy.mpl.ticker import LongitudeFormatter, LatitudeFormatter
plt.figure(figsize=(8, 8))

ax = plt.axes(projection=cartopy.crs.Mercator())

ax.gridlines()

ax.add_feature(cartopy.feature.LAND)
ax.add_feature(cartopy.feature.COASTLINE)

plt.show()
png

png

We can see Greenland looks much broader at the North than in the middle. But if we use a polar projection (below) then we see this is not the case. Why then is the Mercator projection used in preference to e.g. the polar projection or the once controversial Gall-Peters – see here for more on map projections.

plt.figure(figsize=(8, 8))

bx = plt.axes(projection=cartopy.crs.NorthPolarStereo())

bx.set_extent([-180, 180, 90, 50], ccrs.PlateCarree())

bx.gridlines()

bx.add_feature(cartopy.feature.LAND)
bx.add_feature(cartopy.feature.COASTLINE)

plt.show()
png

png

Colophon

This is written as an Jupyter notebook. In theory, it should be possible to run it assuming you have installed at least sage and Haskell. To publish it, I used

jupyter-nbconvert --to markdown Mercator.ipynb
pandoc -s Mercator.md -t markdown+lhs -o Mercator.lhs \
       --filter pandoc-citeproc --bibliography DiffGeom.bib
BlogLiteratelyD --wplatex Mercator.lhs > Mercator.html

Not brilliant but good enough.

Some commands to jupyter to display things nicely.

%display latex
viewer3D = 'tachyon'

Warming Up With SageManifolds

Let us try a simple exercise: finding the connection coefficients of the Levi-Civita connection for the Euclidean metric on \mathbb{R}^2 in polar co-ordinates.

Define the manifold.

N = Manifold(2, 'N',r'\mathcal{N}', start_index=1)

Define a chart and frame with Cartesian co-ordinates.

ChartCartesianN.<x,y> = N.chart()
FrameCartesianN = ChartCartesianN.frame()

Define a chart and frame with polar co-ordinates.

ChartPolarN.<r,theta> = N.chart()
FramePolarN = ChartPolarN.frame()

The standard transformation from Cartesian to polar co-ordinates.

cartesianToPolar = ChartCartesianN.transition_map(ChartPolarN, (sqrt(x^2 + y^2), arctan(y/x)))
print(cartesianToPolar)
Change of coordinates from Chart (N, (x, y)) to Chart (N, (r, theta))
print(latex(cartesianToPolar.display()))

\displaystyle       \left\{\begin{array}{lcl} r & = & \sqrt{x^{2} + y^{2}} \\ \theta & = & \arctan\left(\frac{y}{x}\right) \end{array}\right.

cartesianToPolar.set_inverse(r * cos(theta), r * sin(theta))
Check of the inverse coordinate transformation:
   x == x
   y == y
   r == abs(r)
   theta == arctan(sin(theta)/cos(theta))

Now we define the metric to make the manifold Euclidean.

g_e = N.metric('g_e')
g_e[1,1], g_e[2,2] = 1, 1

We can display this in Cartesian co-ordinates.

print(latex(g_e.display(FrameCartesianN)))

\displaystyle       g_e = \mathrm{d} x\otimes \mathrm{d} x+\mathrm{d} y\otimes \mathrm{d} y

And we can display it in polar co-ordinates

print(latex(g_e.display(FramePolarN)))

\displaystyle       g_e = \mathrm{d} r\otimes \mathrm{d} r + \left( x^{2} + y^{2} \right) \mathrm{d} \theta\otimes \mathrm{d} \theta

Next let us compute the Levi-Civita connection from this metric.

nab_e = g_e.connection()
print(latex(nab_e))

\displaystyle       \nabla_{g_e}

If we use Cartesian co-ordinates, we expect that \Gamma^k_{ij} = 0, \forall i,j,k. Only non-zero entries get printed.

print(latex(nab_e.display(FrameCartesianN)))

Just to be sure, we can print out all the entries.

print(latex(nab_e[:]))

\displaystyle       \left[\left[\left[0, 0\right], \left[0, 0\right]\right], \left[\left[0, 0\right], \left[0, 0\right]\right]\right]

In polar co-ordinates, we get

print(latex(nab_e.display(FramePolarN)))

\displaystyle       \begin{array}{lcl} \Gamma_{ \phantom{\, r } \, \theta \, \theta }^{ \, r \phantom{\, \theta } \phantom{\, \theta } } & = & -\sqrt{x^{2} + y^{2}} \\ \Gamma_{ \phantom{\, \theta } \, r \, \theta }^{ \, \theta \phantom{\, r } \phantom{\, \theta } } & = & \frac{1}{\sqrt{x^{2} + y^{2}}} \\ \Gamma_{ \phantom{\, \theta } \, \theta \, r }^{ \, \theta \phantom{\, \theta } \phantom{\, r } } & = & \frac{1}{\sqrt{x^{2} + y^{2}}} \end{array}

Which we can rew-rewrite as

\displaystyle   \begin{aligned}  \Gamma^r_{\theta,\theta} &= -r \\  \Gamma^\theta_{r,\theta} &= 1/r \\  \Gamma^\theta_{\theta,r} &= 1/r  \end{aligned}

with all other entries being 0.

The Sphere

We define a 2 dimensional manifold. We call it the 2-dimensional (unit) sphere but it we are going to remove a meridian to allow us to define the desired connection with torsion on it.

S2 = Manifold(2, 'S^2', latex_name=r'\mathbb{S}^2', start_index=1)
print(latex(S2))

\displaystyle       \mathbb{S}^2

To start off with we cover the manifold with two charts.

polar.<th,ph> = S2.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi'); print(latex(polar))

\displaystyle       \left(\mathbb{S}^2,({\theta}, {\phi})\right)

mercator.<xi,ze> = S2.chart(r'xi:(-oo,oo):\xi ze:(0,2*pi):\zeta'); print(latex(mercator))

\displaystyle       \left(\mathbb{S}^2,({\xi}, {\zeta})\right)

We can now check that we have two charts.

print(latex(S2.atlas()))

\displaystyle       \left[\left(\mathbb{S}^2,({\theta}, {\phi})\right), \left(\mathbb{S}^2,({\xi}, {\zeta})\right)\right]

We can then define co-ordinate frames.

epolar = polar.frame(); print(latex(epolar))

\displaystyle       \left(\mathbb{S}^2 ,\left(\frac{\partial}{\partial {\theta} },\frac{\partial}{\partial {\phi} }\right)\right)

emercator = mercator.frame(); print(latex(emercator))

\displaystyle       \left(\mathbb{S}^2 ,\left(\frac{\partial}{\partial {\xi} },\frac{\partial}{\partial {\zeta} }\right)\right)

And define a transition map and its inverse from one frame to the other checking that they really are inverses.

xy_to_uv = polar.transition_map(mercator, (log(tan(th/2)), ph))
xy_to_uv.set_inverse(2*arctan(exp(xi)), ze)
Check of the inverse coordinate transformation:
   th == 2*arctan(sin(1/2*th)/cos(1/2*th))
   ph == ph
   xi == xi
   ze == ze

We can define the metric which is the pullback of the Euclidean metric on \mathbb{R}^3.

g = S2.metric('g')
g[1,1], g[2,2] = 1, (sin(th))^2

And then calculate the Levi-Civita connection defined by it.

nab_g = g.connection()
print(latex(nab_g.display()))

\displaystyle       \begin{array}{lcl} \Gamma_{ \phantom{\, {\theta} } \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi} } \phantom{\, {\phi} } } & = & -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\phi} } \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta} } \phantom{\, {\phi} } } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \\ \Gamma_{ \phantom{\, {\phi} } \, {\phi} \, {\theta} }^{ \, {\phi} \phantom{\, {\phi} } \phantom{\, {\theta} } } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}

We know the geodesics defined by this connection are the great circles.

We can check that this connection respects the metric.

print(latex(nab_g(g).display()))

\displaystyle       \nabla_{g} g = 0

And that it has no torsion.

print(latex(nab_g.torsion().display()))
0

A New Connection

Let us now define an orthonormal frame.

ch_basis = S2.automorphism_field()
ch_basis[1,1], ch_basis[2,2] = 1, 1/sin(th)
e = S2.default_frame().new_frame(ch_basis, 'e')
print(latex(e))

\displaystyle       \left(\mathbb{S}^2, \left(e_1,e_2\right)\right)

We can calculate the dual 1-forms.

dX = S2.coframes()[2] ; print(latex(dX))

\displaystyle       \left(\mathbb{S}^2, \left(e^1,e^2\right)\right)

print(latex((dX[1], dX[2])))

\displaystyle       \left(e^1, e^2\right)

print(latex((dX[1][:], dX[2][:])))

\displaystyle       \left(\left[1, 0\right], \left[0, \sin\left({\theta}\right)\right]\right)

In this case it is trivial to check that the frame and coframe really are orthonormal but we let sage do it anyway.

print(latex(((dX[1](e[1]).expr(), dX[1](e[2]).expr()), (dX[2](e[1]).expr(), dX[2](e[2]).expr()))))

\displaystyle       \left(\left(1, 0\right), \left(0, 1\right)\right)

Let us define two vectors to be parallel if their angles to a given meridian are the same. For this to be true we must have a connection \nabla with \nabla e_1 = \nabla e_2 = 0.

nab = S2.affine_connection('nabla', latex_name=r'\nabla')
nab.add_coef(e)

Displaying the connection only gives the non-zero components.

print(latex(nab.display(e)))

For safety, let us check all the components explicitly.

print(latex(nab[e,:]))

\displaystyle       \left[\left[\left[0, 0\right], \left[0, 0\right]\right], \left[\left[0, 0\right], \left[0, 0\right]\right]\right]

Of course the components are not non-zero in other frames.

print(latex(nab.display(epolar)))

\displaystyle       \begin{array}{lcl} \Gamma_{ \phantom{\, {\phi} } \, {\phi} \, {\theta} }^{ \, {\phi} \phantom{\, {\phi} } \phantom{\, {\theta} } } & = & \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}

print(latex(nab.display(emercator)))

\displaystyle       \begin{array}{lcl} \Gamma_{ \phantom{\, {\xi} } \, {\xi} \, {\xi} }^{ \, {\xi} \phantom{\, {\xi} } \phantom{\, {\xi} } } & = & 2 \, \cos\left(\frac{1}{2} \, {\theta}\right)^{2} - 1 \\ \Gamma_{ \phantom{\, {\zeta} } \, {\zeta} \, {\xi} }^{ \, {\zeta} \phantom{\, {\zeta} } \phantom{\, {\xi} } } & = & \frac{2 \, \cos\left(\frac{1}{2} \, {\theta}\right) \cos\left({\theta}\right) \sin\left(\frac{1}{2} \, {\theta}\right)}{\sin\left({\theta}\right)} \end{array}

This connection also respects the metric g.

print(latex(nab(g).display()))

\displaystyle       \nabla g = 0

Thus, since the Levi-Civita connection is unique, it must have torsion.

print(latex(nab.torsion().display(e)))

\displaystyle       \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} e_2\otimes e^1\otimes e^2 -\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} e_2\otimes e^2\otimes e^1

The equations for geodesics are

\displaystyle   \ddot{\gamma}^k + \Gamma_{ \phantom{\, {k} } \, {i} \, {j} }^{ \, {k} \phantom{\, {i} } \phantom{\, {j} } }\dot{\gamma}^i\dot{\gamma}^j = 0

Explicitly for both variables in the polar co-ordinates chart.

\displaystyle   \begin{aligned}  \ddot{\gamma}^\phi & + \frac{\cos\theta}{\sin\theta}\dot{\gamma}^\phi\dot{\gamma}^\theta &= 0 \\  \ddot{\gamma}^\theta & &= 0  \end{aligned}

We can check that \gamma^\phi(t) = \alpha\log\tan t/2 and \gamma^\theta(t) = t are solutions although sage needs a bit of prompting to help it.

t = var('t'); a = var('a')
print(latex(diff(a * log(tan(t/2)),t).simplify_full()))

\displaystyle       \frac{a}{2 \, \cos\left(\frac{1}{2} \, t\right) \sin\left(\frac{1}{2} \, t\right)}

We can simplify this further by recalling the trignometric identity.

print(latex(sin(2 * t).trig_expand()))

\displaystyle       2 \, \cos\left(t\right) \sin\left(t\right)

print(latex(diff (a / sin(t), t)))

\displaystyle       -\frac{a \cos\left(t\right)}{\sin\left(t\right)^{2}}

In the mercator co-ordinates chart this is

\displaystyle   \begin{aligned}  \gamma^\xi(t) &= \alpha\log\tan t/2 \\   \gamma^\zeta(t) &= \log\tan t/2  \end{aligned}

In other words: straight lines.

Reparametersing with s = \alpha\log\tan t/2 we obtain

\displaystyle   \begin{aligned}  \gamma^\phi(s) &= s \\  \gamma^\theta(s) &= 2\arctan e^\frac{s}{\alpha}  \end{aligned}

Let us draw such a curve.

R.<t> = RealLine() ; print(R)
Real number line R
print(dim(R))
1
c = S2.curve({polar: [2*atan(exp(-t/10)), t]}, (t, -oo, +oo), name='c')
print(latex(c.display()))

\displaystyle       \begin{array}{llcl} c:& \mathbb{R} & \longrightarrow & \mathbb{S}^2 \\ & t & \longmapsto & \left({\theta}, {\phi}\right) = \left(2 \, \arctan\left(e^{\left(-\frac{1}{10} \, t\right)}\right), t\right) \\ & t & \longmapsto & \left({\xi}, {\zeta}\right) = \left(-\frac{1}{10} \, t, t\right) \end{array}

c.parent()

\displaystyle       \mathrm{Hom}\left(\mathbb{R},\mathbb{S}^2\right)

c.plot(chart=polar, aspect_ratio=0.1)
png

png

It’s not totally clear this is curved so let’s try with another example.

d = S2.curve({polar: [2*atan(exp(-t)), t]}, (t, -oo, +oo), name='d')
print(latex(d.display()))

\displaystyle       \begin{array}{llcl} d:& \mathbb{R} & \longrightarrow & \mathbb{S}^2 \\ & t & \longmapsto & \left({\theta}, {\phi}\right) = \left(2 \, \arctan\left(e^{\left(-t\right)}\right), t\right) \\ & t & \longmapsto & \left({\xi}, {\zeta}\right) = \left(-t, t\right) \end{array}

d.plot(chart=polar, aspect_ratio=0.2)
png

png

Now it’s clear that a straight line is curved in polar co-ordinates.

But of course in Mercator co-ordinates, it is a straight line. This explains its popularity with mariners: if you draw a straight line on your chart and follow that bearing or rhumb line using a compass you will arrive at the end of the straight line. Of course, it is not the shortest path; great circles are but is much easier to navigate.

c.plot(chart=mercator, aspect_ratio=0.1)
png

png

d.plot(chart=mercator, aspect_ratio=1.0)
png

png

We can draw these curves on the sphere itself not just on its charts.

R3 = Manifold(3, 'R^3', r'\mathbb{R}^3', start_index=1)
cart.<X,Y,Z> = R3.chart(); print(latex(cart))

\displaystyle       \left(\mathbb{R}^3,(X, Y, Z)\right)

Phi = S2.diff_map(R3, {
    (polar, cart): [sin(th) * cos(ph), sin(th) * sin(ph), cos(th)],
    (mercator, cart): [cos(ze) / cosh(xi), sin(ze) / cosh(xi),
                       sinh(xi) / cosh(xi)]
},
    name='Phi', latex_name=r'\Phi')

We can either plot using polar co-ordinates.

graph_polar = polar.plot(chart=cart, mapping=Phi, nb_values=25, color='blue')
show(graph_polar, viewer=viewer3D)
png

png

Or using Mercator co-ordinates. In either case we get the sphere (minus the prime meridian).

graph_mercator = mercator.plot(chart=cart, mapping=Phi, nb_values=25, color='red')
show(graph_mercator, viewer=viewer3D)
png

png

We can plot the curve with an angle to the meridian of \pi/2 - \arctan 1/10

graph_c = c.plot(mapping=Phi, max_range=40, plot_points=200, thickness=2)
show(graph_polar + graph_c, viewer=viewer3D)
png

png

And we can plot the curve at angle of \pi/4 to the meridian.

graph_d = d.plot(mapping=Phi, max_range=40, plot_points=200, thickness=2, color="green")
show(graph_polar + graph_c + graph_d, viewer=viewer3D)
png

png

Haskell

With automatic differentiation and symbolic numbers, symbolic differentiation is straigtforward in Haskell.

> import Data.Number.Symbolic
> import Numeric.AD
> 
> x = var "x"
> y = var "y"
> 
> test xs = jacobian ((\x -> [x]) . f) xs
>   where
>     f [x, y] = sqrt $ x^2 + y^2
ghci> test [1, 1]
  [[0.7071067811865475,0.7071067811865475]]

ghci> test [x, y]
  [[x/(2.0*sqrt (x*x+y*y))+x/(2.0*sqrt (x*x+y*y)),y/(2.0*sqrt (x*x+y*y))+y/(2.0*sqrt (x*x+y*y))]]

Anyone wishing to take on the task of producing a Haskell version of sagemanifolds is advised to look here before embarking on the task.

Appendix A: Conformal Equivalence

Agricola and Thier (2004) shows that the geodesics of the Levi-Civita connection of a conformally equivalent metric are the geodesics of a connection with vectorial torsion. Let’s put some but not all the flesh on the bones.

The Koszul formula (see e.g. (O’Neill 1983)) characterizes the Levi-Civita connection \nabla

\displaystyle   \begin{aligned}  2  \langle \nabla_X Y, Z\rangle & = X  \langle Y,Z\rangle + Y  \langle Z,X\rangle - Z  \langle X,Y\rangle \\  &-  \langle X,[Y,Z]\rangle +   \langle Y,[Z,X]\rangle +  \langle Z,[X,Y]\rangle  \end{aligned}

Being more explicit about the metric, this can be re-written as

\displaystyle   \begin{aligned}  2 g(\nabla^g_X Y, Z) & = X g(Y,Z) + Y g(Z,X) - Z g(X,Y) \\  &- g(X,[Y,Z]) +  g(Y,[Z,X]) + g(Z,[X,Y])  \end{aligned}

Let \nabla^h be the Levi-Civita connection for the metric h = e^{2\sigma}g where \sigma \in C^\infty M. Following [Gadea2010] and substituting into the Koszul formula and then applying the product rule

\displaystyle   \begin{aligned}  2 e^{2 \sigma} g(\nabla^h_X Y, Z) & = X  e^{2 \sigma} g(Y,Z) + Y e^{2 \sigma} g(Z,X) - Z e^{2 \sigma} g(X,Y) \\  & + e^{2 \sigma} g([X,Y],Z]) - e^{2 \sigma} g([Y,Z],X) + e^{2 \sigma} g([Z,X],Y) \\  & = 2 e^{2\sigma}[g(\nabla^{g}_X Y, Z) + X\sigma g(Y,Z) + Y\sigma g(Z,X) - Z\sigma g(X,Y)] \\  & = 2 e^{2\sigma}[g(\nabla^{g}_X Y + X\sigma Y + Y\sigma X - g(X,Y) \mathrm{grad}\sigma, Z)]  \end{aligned}

Where as usual the vector field, \mathrm{grad}\phi for \phi \in C^\infty M, is defined via g(\mathrm{grad}\phi, X) = \mathrm{d}\phi(X) = X\phi.

Let’s try an example.

nab_tilde = S2.affine_connection('nabla_t', r'\tilde_{\nabla}')
f = S2.scalar_field(-ln(sin(th)), name='f')
for i in S2.irange():
    for j in S2.irange():
        for k in S2.irange():
            nab_tilde.add_coef()[k,i,j] = \
                nab_g(polar.frame()[i])(polar.frame()[j])(polar.coframe()[k]) + \
                polar.frame()[i](f) * polar.frame()[j](polar.coframe()[k]) + \
                polar.frame()[j](f) * polar.frame()[i](polar.coframe()[k]) + \
                g(polar.frame()[i], polar.frame()[j]) * \
                polar.frame()[1](polar.coframe()[k]) * cos(th) / sin(th)
print(latex(nab_tilde.display()))

\displaystyle       \begin{array}{lcl} \Gamma_{ \phantom{\, {\theta} } \, {\theta} \, {\theta} }^{ \, {\theta} \phantom{\, {\theta} } \phantom{\, {\theta} } } & = & -\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}

print(latex(nab_tilde.torsion().display()))
0
g_tilde = exp(2 * f) * g
print(latex(g_tilde.parent()))

\displaystyle       \mathcal{T}^{(0,2)}\left(\mathbb{S}^2\right)

print(latex(g_tilde[:]))

\displaystyle       \left(\begin{array}{rr}      \frac{1}{\sin\left({\theta}\right)^{2}} & 0 \\      0 & 1      \end{array}\right)

nab_g_tilde = g_tilde.connection()
print(latex(nab_g_tilde.display()))

\displaystyle       \begin{array}{lcl} \Gamma_{ \phantom{\, {\theta} } \, {\theta} \, {\theta} }^{ \, {\theta} \phantom{\, {\theta} } \phantom{\, {\theta} } } & = & -\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}

It’s not clear (to me at any rate) what the solutions are to the geodesic equations despite the guarantees of Agricola and Thier (2004). But let’s try a different chart.

print(latex(nab_g_tilde[emercator,:]))

\displaystyle       \left[\left[\left[0, 0\right], \left[0, 0\right]\right], \left[\left[0, 0\right], \left[0, 0\right]\right]\right]

In this chart, the geodesics are clearly straight lines as we would hope.

References

Agricola, Ilka, and Christian Thier. 2004. “The geodesics of metric connections with vectorial torsion.” Annals of Global Analysis and Geometry 26 (4): 321–32. doi:10.1023/B:AGAG.0000047509.63818.4f.

Nakahara, M. 2003. “Geometry, Topology and Physics.” Text 822: 173–204. doi:10.1007/978-3-642-14700-5.

O’Neill, B. 1983. Semi-Riemannian Geometry with Applications to Relativity, 103. Pure and Applied Mathematics. Elsevier Science. https://books.google.com.au/books?id=CGk1eRSjFIIC.

Modelling an Ecosystem via Hamiltonian Monte Carlo

Introduction

Recall from the previous post that the Hare growth parameter undergoes Brownian motion so that the further into the future we go, the less certain we are about it. In order to ensure that this parameter remains positive, let’s model the log of it to be Brownian motion.

\displaystyle   \begin{aligned}  \frac{\mathrm{d}N_1}{\mathrm{d}t} & = & \rho_1 N_1 \bigg(1 - \frac{N_1}{K_1}\bigg) - c_1 N_1 N_2 \\  \frac{\mathrm{d}N_2}{\mathrm{d}t} & = & -\rho_2 N_2 \bigg(1 + \frac{N_2}{K_2}\bigg) + c_2 N_1 N_2 \\  \mathrm{d} \rho_1 & = & \rho_1 \sigma_{\rho_1} \mathrm{d}W_t  \end{aligned}

where the final equation is a stochastic differential equation with W_t being a Wiener process.

By Itô we have

\displaystyle   \mathrm{d} (\log{\rho_1}) = - \frac{\sigma_{\rho_1}^2}{2} \mathrm{d} t + \sigma_{\rho_1} \mathrm{d}W_t

Again, we see that the populations become noisier the further into the future we go.

Inference

Now let us infer the growth rate using Hamiltonian Monte Carlo and the domain specific probabilistic language Stan (Stan Development Team (2015b), Stan Development Team (2015a), Hoffman and Gelman (2014), Carpenter (2015)). Here’s the model expressed in Stan.

functions {
  real f1 (real a, real k1, real b, real p, real z) {
    real q;

    q = a * p * (1 - p / k1) - b * p * z;
    return q;
  }

  real f2 (real d, real k2, real c, real p, real z) {
    real q;

    q = -d * z * (1 + z / k2) + c * p * z;
    return q;
  }
}

data {
  int<lower=1> T;   // Number of observations
  real y[T];        // Observed hares
  real k1;          // Hare carrying capacity
  real b;           // Hare death rate per lynx
  real d;           // Lynx death rate
  real k2;          // Lynx carrying capacity
  real c;           // Lynx birth rate per hare
  real deltaT;      // Time step
}

parameters {
  real<lower=0> mu;
  real<lower=0> sigma;
  real<lower=0> p0;
  real<lower=0> z0;
  real<lower=0> rho0;
  real w[T];
}

transformed parameters {
  real<lower=0> p[T];
  real<lower=0> z[T];
  real<lower=0> rho[T];

  p[1] = p0;
  z[1] = z0;
  rho[1] = rho0;

  for (t in 1:(T-1)){
    p[t+1] = p[t] + deltaT * f1 (exp(rho[t]), k1, b, p[t], z[t]);
    z[t+1] = z[t] + deltaT * f2 (d, k2, c, p[t], z[t]);

    rho[t+1] = rho[t] * exp(sigma * sqrt(deltaT) * w[t] - 0.5 * sigma * sigma * deltaT);
  }
}

model {
  mu    ~ uniform(0.0,1.0);
  sigma ~ uniform(0.0, 0.5);
  p0    ~ lognormal(log(100.0), 0.2);
  z0    ~ lognormal(log(50.0), 0.1);
  rho0  ~ normal(log(mu), sigma);
  w     ~ normal(0.0,1.0);

  for (t in 1:T) {
    y[t] ~ lognormal(log(p[t]),0.1);
  }
}

Let’s look at the posteriors of the hyper-parameters for the Hare growth parameter.

Again, the estimate for \mu is pretty decent. For our generated data, \sigma =0 and given our observations are quite noisy maybe the estimate for this is not too bad also.

Appendix: The R Driving Code

All code including the R below can be downloaded from github.

install.packages("devtools")
library(devtools)
install_github("libbi/RBi",ref="master")
install_github("sbfnk/RBi.helpers",ref="master")

rm(list = ls(all.names=TRUE))
unlink(".RData")

library('RBi')
try(detach(package:RBi, unload = TRUE), silent = TRUE)
library(RBi, quietly = TRUE)

library('RBi.helpers')

library('ggplot2', quietly = TRUE)
library('gridExtra', quietly = TRUE)

endTime <- 50

PP <- bi_model("PP.bi")
synthetic_dataset_PP <- bi_generate_dataset(endtime=endTime,
                                            model=PP,
                                            seed="42",
                                            verbose=TRUE,
                                            add_options = list(
                                                noutputs=500))

rdata_PP <- bi_read(synthetic_dataset_PP)

df <- data.frame(rdata_PP$P$nr,
                 rdata_PP$P$value,
                 rdata_PP$Z$value,
                 rdata_PP$P_obs$value)

ggplot(df, aes(rdata_PP$P$nr, y = Population, color = variable), size = 0.1) +
    geom_line(aes(y = rdata_PP$P$value, col = "Hare"), size = 0.1) +
    geom_line(aes(y = rdata_PP$Z$value, col = "Lynx"), size = 0.1) +
    geom_point(aes(y = rdata_PP$P_obs$value, col = "Observations"), size = 0.1) +
    theme(legend.position="none") +
    ggtitle("Example Data") +
    xlab("Days") +
    theme(axis.text=element_text(size=4),
          axis.title=element_text(size=6,face="bold")) +
    theme(plot.title = element_text(size=10))
ggsave(filename="diagrams/LVdata.png",width=4,height=3)

library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

lvStanModel <- stan_model(file = "SHO.stan",verbose=TRUE)

lvFit <- sampling(lvStanModel,
                  seed=42,
                  data=list(T = length(rdata_PP$P_obs$value),
                            y = rdata_PP$P_obs$value,
                            k1 = 2.0e2,
                            b  = 2.0e-2,
                            d  = 4.0e-1,
                            k2 = 2.0e1,
                            c  = 4.0e-3,
                            deltaT = rdata_PP$P_obs$time[2] - rdata_PP$P_obs$time[1]
                            ),
                   chains=1)

samples <- extract(lvFit)

gs1 <- qplot(x = samples$mu, y = ..density.., geom = "histogram") + xlab(expression(\mu))
gs2 <- qplot(x = samples$sigma, y = ..density.., geom = "histogram") + xlab(expression(samples$sigma))
gs3 <- grid.arrange(gs1, gs2)
ggsave(plot=gs3,filename="diagrams/LvPosteriorStan.png",width=4,height=3)

synthetic_dataset_PP1 <- bi_generate_dataset(endtime=endTime,
                                             model=PP,
                                             init = list(P = 100, Z=50),
                                             seed="42",
                                             verbose=TRUE,
                                             add_options = list(
                                                 noutputs=500))

rdata_PP1 <- bi_read(synthetic_dataset_PP1)

synthetic_dataset_PP2 <- bi_generate_dataset(endtime=endTime,
                                             model=PP,
                                             init = list(P = 150, Z=25),
                                             seed="42",
                                             verbose=TRUE,
                                             add_options = list(
                                                 noutputs=500))

rdata_PP2 <- bi_read(synthetic_dataset_PP2)

df1 <- data.frame(hare = rdata_PP$P$value,
                  lynx = rdata_PP$Z$value,
                  hare1 = rdata_PP1$P$value,
                  lynx1 = rdata_PP1$Z$value,
                  hare2 = rdata_PP2$P$value,
                  lynx2 = rdata_PP2$Z$value)

ggplot(df1) +
    geom_path(aes(x=df1$hare,  y=df1$lynx, col = "0"), size = 0.1) +
    geom_path(aes(x=df1$hare1, y=df1$lynx1, col = "1"), size = 0.1) +
    geom_path(aes(x=df1$hare2, y=df1$lynx2, col = "2"), size = 0.1) +
    theme(legend.position="none") +
    ggtitle("Phase Space") +
    xlab("Hare") +
    ylab("Lynx") +
    theme(axis.text=element_text(size=4),
          axis.title=element_text(size=6,face="bold")) +
    theme(plot.title = element_text(size=10))
ggsave(filename="diagrams/PPviaLibBi.png",width=4,height=3)

PPInfer <- bi_model("PPInfer.bi")

bi_object_PP <- libbi(client="sample", model=PPInfer, obs = synthetic_dataset_PP)

bi_object_PP$run(add_options = list(
                     "end-time" = endTime,
                     noutputs = endTime,
                     nsamples = 2000,
                     nparticles = 128,
                     seed=42,
                     nthreads = 1),
                 verbose = TRUE,
                 stdoutput_file_name = tempfile(pattern="pmmhoutput", fileext=".txt"))

bi_file_summary(bi_object_PP$result$output_file_name)

mu <- bi_read(bi_object_PP, "mu")$value
g1 <- qplot(x = mu[2001:4000], y = ..density.., geom = "histogram") + xlab(expression(mu))
sigma <- bi_read(bi_object_PP, "sigma")$value
g2 <- qplot(x = sigma[2001:4000], y = ..density.., geom = "histogram") + xlab(expression(sigma))
g3 <- grid.arrange(g1, g2)
ggsave(plot=g3,filename="diagrams/LvPosterior.png",width=4,height=3)


df2 <- data.frame(hareActs = rdata_PP$P$value,
                  hareObs  = rdata_PP$P_obs$value)

ggplot(df, aes(rdata_PP$P$nr, y = value, color = variable)) +
    geom_line(aes(y = rdata_PP$P$value, col = "Phyto")) +
    geom_line(aes(y = rdata_PP$Z$value, col = "Zoo")) +
    geom_point(aes(y = rdata_PP$P_obs$value, col = "Phyto Obs"))

ln_alpha <- bi_read(bi_object_PP, "ln_alpha")$value

P <- matrix(bi_read(bi_object_PP, "P")$value,nrow=51,byrow=TRUE)
Z <- matrix(bi_read(bi_object_PP, "Z")$value,nrow=51,byrow=TRUE)

data50 <- bi_generate_dataset(endtime=endTime,
                              model=PP,
                              seed="42",
                              verbose=TRUE,
                              add_options = list(
                                  noutputs=50))

rdata50 <- bi_read(data50)

df3 <- data.frame(days = c(1:51), hares = rowMeans(P), lynxes = rowMeans(Z),
                                  actHs = rdata50$P$value, actLs = rdata50$Z$value)


ggplot(df3) +
    geom_line(aes(x = days, y = hares, col = "Est Phyto")) +
    geom_line(aes(x = days, y = lynxes, col = "Est Zoo")) +
    geom_line(aes(x = days, y = actHs, col = "Act Phyto")) +
    geom_line(aes(x = days, y = actLs, col = "Act Zoo"))

Bibliography

Carpenter, Bob. 2015. “Stan: A Probabilistic Programming Language.” Journal of Statistical Software.

Hoffman, Matthew D., and Andrew Gelman. 2014. “The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo.” Journal of Machine Learning Research.

Stan Development Team. 2015a. Stan Modeling Language User’s Guide and Reference Manual, Version 2.10.0. http://mc-stan.org/.

———. 2015b. “Stan: A C++ Library for Probability and Sampling, Version 2.10.0.” http://mc-stan.org/.

Ecology, Dynamical Systems and Inference via PMMH

Introduction

In the 1920s, Lotka (1909) and Volterra (1926) developed a model of a very simple predator-prey ecosystem.

\displaystyle   \begin{aligned}  \frac{\mathrm{d}N_1}{\mathrm{d}t} & = & \rho_1 N_1  - c_1 N_1 N_2 \label{eq2a} \\  \frac{\mathrm{d}N_2}{\mathrm{d}t} & = & c_2 N_1 N_2 - \rho_2 N2 \label{eq2b}  \end{aligned}

Although simple, it turns out that the Canadian lynx and showshoe hare are well represented by such a model. Furthermore, the Hudson Bay Company kept records of how many pelts of each species were trapped for almost a century, giving a good proxy of the population of each species.

We can capture the fact that we do not have a complete model by describing our state of ignorance about the parameters. In order to keep this as simple as possible let us assume that log parameters undergo Brownian motion. That is, we know the parameters will jiggle around and the further into the future we look the less certain we are about what values they will have taken. By making the log parameters undergo Brownian motion, we can also capture our modelling assumption that birth, death and predation rates are always positive. A similar approach is taken in Dureau, Kalogeropoulos, and Baguelin (2013) where the (log) parameters of an epidemiological model are taken to be Ornstein-Uhlenbeck processes (which is biologically more plausible although adds to the complexity of the model, something we wish to avoid in an example such as this).

Andrieu, Doucet, and Holenstein (2010) propose a method to estimate the parameters of such models (Particle Marginal Metropolis Hastings aka PMMH) and the domain specific probabilistic language LibBi (Murray (n.d.)) can be used to apply this (and other inference methods).

For the sake of simplicity, in this blog post, we only model one parameter as being unknown and undergoing Brownian motion. A future blog post will consider more sophisticated scenarios.

A Dynamical System Aside

The above dynamical system is structurally unstable (more on this in a future post), a possible indication that it should not be considered as a good model of predator–prey interaction. Let us modify this to include carrying capacities for the populations of both species.

\displaystyle   \begin{aligned}  \frac{\mathrm{d}N_1}{\mathrm{d}t} & = & \rho_1 N_1 \bigg(1 - \frac{N_1}{K_1}\bigg) - c_1 N_1 N_2 \\  \frac{\mathrm{d}N_2}{\mathrm{d}t} & = & -\rho_2 N_2 \bigg(1 + \frac{N_2}{K_2}\bigg) + c_2 N_1 N_2  \end{aligned}

Data Generation with LibBi

Let’s generate some data using LibBi.

// Generate data assuming a fixed growth rate for hares rather than
// e.g. a growth rate that undergoes Brownian motion.

model PP {
  const h         = 0.1;    // time step
  const delta_abs = 1.0e-3; // absolute error tolerance
  const delta_rel = 1.0e-6; // relative error tolerance

  const a  = 5.0e-1 // Hare growth rate
  const k1 = 2.0e2  // Hare carrying capacity
  const b  = 2.0e-2 // Hare death rate per lynx
  const d  = 4.0e-1 // Lynx death rate
  const k2 = 2.0e1  // Lynx carrying capacity
  const c  = 4.0e-3 // Lynx birth rate per hare

  state P, Z       // Hares and lynxes
  state ln_alpha   // Hare growth rate - we express it in log form for
                   // consistency with the inference model
  obs P_obs        // Observations of hares

  sub initial {
    P ~ log_normal(log(100.0), 0.2)
    Z ~ log_normal(log(50.0), 0.1)
  }

  sub transition(delta = h) {
    ode(h = h, atoler = delta_abs, rtoler = delta_rel, alg = 'RK4(3)') {
      dP/dt =  a * P * (1 - P / k1) - b * P * Z
      dZ/dt = -d * Z * (1 + Z / k2) + c * P * Z
    }
  }

  sub observation {
    P_obs ~ log_normal(log(P), 0.1)
  }
}

We can look at phase space starting with different populations and see they all converge to the same fixed point.

Data Generation with Haskell

Since at some point in the future, I plan to produce Haskell versions of the methods given in Andrieu, Doucet, and Holenstein (2010), let’s generate the data using Haskell.

> {-# OPTIONS_GHC -Wall                     #-}
> {-# OPTIONS_GHC -fno-warn-name-shadowing  #-}
> module LotkaVolterra (
>     solLv
>   , solPp
>   , h0
>   , l0
>   , baz
>   , logBM
>   , eulerEx
>   )where
> import Numeric.GSL.ODE
> import Numeric.LinearAlgebra
> import Data.Random.Source.PureMT
> import Data.Random hiding ( gamma )
> import Control.Monad.State

Here’s the unstable model.

> lvOde :: Double ->
>          Double ->
>          Double ->
>          Double ->
>          Double ->
>          [Double] ->
>          [Double]
> lvOde rho1 c1 rho2 c2 _t [h, l] =
>   [
>     rho1 * h - c1 * h * l
>   , c2 * h * l - rho2 * l
>   ]
> lvOde _rho1 _c1 _rho2 _c2 _t vars =
>   error $ "lvOde called with: " ++ show (length vars) ++ " variable"
> rho1, c1, rho2, c2 :: Double
> rho1 = 0.5
> c1 = 0.02
> rho2 = 0.4
> c2 = 0.004
> deltaT :: Double
> deltaT = 0.1
> solLv :: Matrix Double
> solLv = odeSolve (lvOde rho1 c1 rho2 c2)
>                  [50.0, 50.0]
>                  (fromList [0.0, deltaT .. 50])

And here’s the stable model.

> ppOde :: Double ->
>          Double ->
>          Double ->
>          Double ->
>          Double ->
>          Double ->
>          Double ->
>          [Double] ->
>          [Double]
> ppOde a k1 b d k2 c _t [p, z] =
>   [
>     a * p * (1 - p / k1) - b * p * z
>   , -d * z * (1 + z / k2) + c * p * z
>   ]
> ppOde _a _k1 _b _d _k2 _c _t vars =
>   error $ "ppOde called with: " ++ show (length vars) ++ " variable"
> a, k1, b, d, k2, c :: Double
> a = 0.5
> k1 = 200.0
> b = 0.02
> d = 0.4
> k2 = 50.0
> c = 0.004
> solPp :: Double -> Double -> Matrix Double
> solPp x y = odeSolve (ppOde a k1 b d k2 c)
>                  [x, y]
>                  (fromList [0.0, deltaT .. 50])
> gamma, alpha, beta :: Double
> gamma = d / a
> alpha = a / (c * k1)
> beta  = d / (a * k2)
> fp :: (Double, Double)
> fp = ((gamma + beta) / (1 + alpha * beta), (1 - gamma * alpha) / (1 + alpha * beta))
> h0, l0 :: Double
> h0 = a * fst fp / c
> l0 = a * snd fp / b
> foo, bar :: Matrix R
> foo = matrix 2 [a / k1, b, c, negate d / k2]
> bar = matrix 1 [a, d]
> baz :: Maybe (Matrix R)
> baz = linearSolve foo bar

This gives a stable fixed point of

ghci> baz
  Just (2><1)
   [ 120.00000000000001
   ,               10.0 ]

Here’s an example of convergence to that fixed point in phase space.

The Stochastic Model

Let us now assume that the Hare growth parameter undergoes Brownian motion so that the further into the future we go, the less certain we are about it. In order to ensure that this parameter remains positive, let’s model the log of it to be Brownian motion.

\displaystyle   \begin{aligned}  \frac{\mathrm{d}N_1}{\mathrm{d}t} & = & \rho_1 N_1 \bigg(1 - \frac{N_1}{K_1}\bigg) - c_1 N_1 N_2 \\  \frac{\mathrm{d}N_2}{\mathrm{d}t} & = & -\rho_2 N_2 \bigg(1 + \frac{N_2}{K_2}\bigg) + c_2 N_1 N_2 \\  \mathrm{d} \rho_1 & = & \rho_1 \sigma_{\rho_1} \mathrm{d}W_t  \end{aligned}

where the final equation is a stochastic differential equation with W_t being a Wiener process.

By Itô we have

\displaystyle   \mathrm{d} (\log{\rho_1}) = - \frac{\sigma_{\rho_1}^2}{2} \mathrm{d} t + \sigma_{\rho_1} \mathrm{d}W_t

We can use this to generate paths for \rho_1.

\displaystyle   \rho_1(t + \Delta t) = \rho_1(t)\exp{\bigg(- \frac{\sigma_{\rho_1}^2}{2} \Delta t + \sigma_{\rho_1} \sqrt{\Delta t} Z\bigg)}

where Z \sim {\mathcal{N}}(0,1).

> oneStepLogBM :: MonadRandom m => Double -> Double -> Double -> m Double
> oneStepLogBM deltaT sigma rhoPrev = do
>   x <- sample $ rvar StdNormal
>   return $ rhoPrev * exp(sigma * (sqrt deltaT) * x - 0.5 * sigma * sigma * deltaT)
> iterateM :: Monad m => (a -> m a) -> m a -> Int -> m [a]
> iterateM f mx n = sequence . take n . iterate (>>= f) $ mx
> logBMM :: MonadRandom m => Double -> Double -> Int -> Int -> m [Double]
> logBMM initRho sigma n m =
>   iterateM (oneStepLogBM (recip $ fromIntegral n) sigma) (return initRho) (n * m)
> logBM :: Double -> Double -> Int -> Int -> Int -> [Double]
> logBM initRho sigma n m seed =
>   evalState (logBMM initRho sigma n m) (pureMT $ fromIntegral seed)

We can see the further we go into the future the less certain we are about the value of the parameter.

Using this we can simulate the whole dynamical system which is now a stochastic process.

> f1, f2 :: Double -> Double -> Double ->
>           Double -> Double ->
>           Double
> f1 a k1 b p z = a * p * (1 - p / k1) - b * p * z
> f2 d k2 c p z = -d * z * (1 + z / k2) + c * p * z
> oneStepEuler :: MonadRandom m =>
>                 Double ->
>                 Double ->
>                 Double -> Double ->
>                 Double -> Double -> Double ->
>                 (Double, Double, Double) ->
>                 m (Double, Double, Double)
> oneStepEuler deltaT sigma k1 b d k2 c (rho1Prev, pPrev, zPrev) = do
>     let pNew = pPrev + deltaT * f1 (exp rho1Prev) k1 b pPrev zPrev
>     let zNew = zPrev + deltaT * f2 d k2 c pPrev zPrev
>     rho1New <- oneStepLogBM deltaT sigma rho1Prev
>     return (rho1New, pNew, zNew)
> euler :: MonadRandom m =>
>          (Double, Double, Double) ->
>          Double ->
>          Double -> Double ->
>          Double -> Double -> Double ->
>          Int -> Int ->
>          m [(Double, Double, Double)]
> euler stateInit sigma k1 b d k2 c n m =
>   iterateM (oneStepEuler (recip $ fromIntegral n) sigma k1 b d k2 c)
>            (return stateInit)
>            (n * m)
> eulerEx :: (Double, Double, Double) ->
>            Double -> Int -> Int -> Int ->
>            [(Double, Double, Double)]
> eulerEx stateInit sigma n m seed =
>   evalState (euler stateInit sigma k1 b d k2 c n m) (pureMT $ fromIntegral seed)

We see that the populations become noisier the further into the future we go.

Notice that the second order effects of the system are now to some extent captured by the fact that the growth rate of Hares can drift. In our simulation, this is demonstrated by our decreasing lack of knowledge the further we look into the future.

Inference

Now let us infer the growth rate using PMMH. Here’s the model expressed in LibBi.

// Infer growth rate for hares

model PP {
  const h         = 0.1;    // time step
  const delta_abs = 1.0e-3; // absolute error tolerance
  const delta_rel = 1.0e-6; // relative error tolerance

  const a  = 5.0e-1 // Hare growth rate - superfluous for inference
                    // but a reminder of what we should expect
  const k1 = 2.0e2  // Hare carrying capacity
  const b  = 2.0e-2 // Hare death rate per lynx
  const d  = 4.0e-1 // Lynx death rate
  const k2 = 2.0e1  // Lynx carrying capacity
  const c  = 4.0e-3 // Lynx birth rate per hare

  state P, Z       // Hares and lynxes
  state ln_alpha   // Hare growth rate - we express it in log form for
                   // consistency with the inference model
  obs P_obs        // Observations of hares
  param mu, sigma  // Mean and standard deviation of hare growth rate
  noise w          // Noise

  sub parameter {
    mu ~ uniform(0.0, 1.0)
    sigma ~ uniform(0.0, 0.5)
  }

  sub proposal_parameter {
     mu ~ truncated_gaussian(mu, 0.02, 0.0, 1.0);
     sigma ~ truncated_gaussian(sigma, 0.01, 0.0, 0.5);
   }

  sub initial {
    P ~ log_normal(log(100.0), 0.2)
    Z ~ log_normal(log(50.0), 0.1)
    ln_alpha ~ gaussian(log(mu), sigma)
  }

  sub transition(delta = h) {
    w ~ normal(0.0, sqrt(h));
    ode(h = h, atoler = delta_abs, rtoler = delta_rel, alg = 'RK4(3)') {
      dP/dt =  exp(ln_alpha) * P * (1 - P / k1) - b * P * Z
      dZ/dt = -d * Z * (1 + Z / k2) + c * P * Z
      dln_alpha/dt = -sigma * sigma / 2 - sigma * w / h
    }
  }

  sub observation {
    P_obs ~ log_normal(log(P), 0.1)
  }
}

Let’s look at the posteriors of the hyper-parameters for the Hare growth parameter.

The estimate for \mu is pretty decent. For our generated data, \sigma =0 and given our observations are quite noisy maybe the estimate for this is not too bad also.

Appendix: The R Driving Code

All code including the R below can be downloaded from github but make sure you use the straight-libbi branch and not master.

install.packages("devtools")
library(devtools)
install_github("sbfnk/RBi",ref="master")
install_github("sbfnk/RBi.helpers",ref="master")

rm(list = ls(all.names=TRUE))
unlink(".RData")

library('RBi')
try(detach(package:RBi, unload = TRUE), silent = TRUE)
library(RBi, quietly = TRUE)

library('RBi.helpers')

library('ggplot2', quietly = TRUE)
library('gridExtra', quietly = TRUE)

endTime <- 50

PP <- bi_model("PP.bi")
synthetic_dataset_PP <- bi_generate_dataset(endtime=endTime,
                                            model=PP,
                                            seed="42",
                                            verbose=TRUE,
                                            add_options = list(
                                                noutputs=500))

rdata_PP <- bi_read(synthetic_dataset_PP)

df <- data.frame(rdata_PP$P$nr,
                 rdata_PP$P$value,
                 rdata_PP$Z$value,
                 rdata_PP$P_obs$value)

ggplot(df, aes(rdata_PP$P$nr, y = Population, color = variable), size = 0.1) +
    geom_line(aes(y = rdata_PP$P$value, col = "Hare"), size = 0.1) +
    geom_line(aes(y = rdata_PP$Z$value, col = "Lynx"), size = 0.1) +
    geom_point(aes(y = rdata_PP$P_obs$value, col = "Observations"), size = 0.1) +
    theme(legend.position="none") +
    ggtitle("Example Data") +
    xlab("Days") +
    theme(axis.text=element_text(size=4),
          axis.title=element_text(size=6,face="bold")) +
    theme(plot.title = element_text(size=10))
ggsave(filename="diagrams/LVdata.png",width=4,height=3)

synthetic_dataset_PP1 <- bi_generate_dataset(endtime=endTime,
                                             model=PP,
                                             init = list(P = 100, Z=50),
                                             seed="42",
                                             verbose=TRUE,
                                             add_options = list(
                                                 noutputs=500))

rdata_PP1 <- bi_read(synthetic_dataset_PP1)

synthetic_dataset_PP2 <- bi_generate_dataset(endtime=endTime,
                                             model=PP,
                                             init = list(P = 150, Z=25),
                                             seed="42",
                                             verbose=TRUE,
                                             add_options = list(
                                                 noutputs=500))

rdata_PP2 <- bi_read(synthetic_dataset_PP2)

df1 <- data.frame(hare = rdata_PP$P$value,
                  lynx = rdata_PP$Z$value,
                  hare1 = rdata_PP1$P$value,
                  lynx1 = rdata_PP1$Z$value,
                  hare2 = rdata_PP2$P$value,
                  lynx2 = rdata_PP2$Z$value)

ggplot(df1) +
    geom_path(aes(x=df1$hare,  y=df1$lynx, col = "0"), size = 0.1) +
    geom_path(aes(x=df1$hare1, y=df1$lynx1, col = "1"), size = 0.1) +
    geom_path(aes(x=df1$hare2, y=df1$lynx2, col = "2"), size = 0.1) +
    theme(legend.position="none") +
    ggtitle("Phase Space") +
    xlab("Hare") +
    ylab("Lynx") +
    theme(axis.text=element_text(size=4),
          axis.title=element_text(size=6,face="bold")) +
    theme(plot.title = element_text(size=10))
ggsave(filename="diagrams/PPviaLibBi.png",width=4,height=3)

PPInfer <- bi_model("PPInfer.bi")

bi_object_PP <- libbi(client="sample", model=PPInfer, obs = synthetic_dataset_PP)

bi_object_PP$run(add_options = list(
                     "end-time" = endTime,
                     noutputs = endTime,
                     nsamples = 4000,
                     nparticles = 128,
                     seed=42,
                     nthreads = 1),
                 ## verbose = TRUE,
                 stdoutput_file_name = tempfile(pattern="pmmhoutput", fileext=".txt"))

bi_file_summary(bi_object_PP$result$output_file_name)

mu <- bi_read(bi_object_PP, "mu")$value
g1 <- qplot(x = mu[2001:4000], y = ..density.., geom = "histogram") + xlab(expression(mu))
sigma <- bi_read(bi_object_PP, "sigma")$value
g2 <- qplot(x = sigma[2001:4000], y = ..density.., geom = "histogram") + xlab(expression(sigma))
g3 <- grid.arrange(g1, g2)
ggsave(plot=g3,filename="diagrams/LvPosterior.png",width=4,height=3)


df2 <- data.frame(hareActs = rdata_PP$P$value,
                  hareObs  = rdata_PP$P_obs$value)

ggplot(df, aes(rdata_PP$P$nr, y = value, color = variable)) +
    geom_line(aes(y = rdata_PP$P$value, col = "Phyto")) +
    geom_line(aes(y = rdata_PP$Z$value, col = "Zoo")) +
    geom_point(aes(y = rdata_PP$P_obs$value, col = "Phyto Obs"))

ln_alpha <- bi_read(bi_object_PP, "ln_alpha")$value

P <- matrix(bi_read(bi_object_PP, "P")$value,nrow=51,byrow=TRUE)
Z <- matrix(bi_read(bi_object_PP, "Z")$value,nrow=51,byrow=TRUE)

data50 <- bi_generate_dataset(endtime=endTime,
                              model=PP,
                              seed="42",
                              verbose=TRUE,
                              add_options = list(
                                  noutputs=50))

rdata50 <- bi_read(data50)

df3 <- data.frame(days = c(1:51), hares = rowMeans(P), lynxes = rowMeans(Z),
                                  actHs = rdata50$P$value, actLs = rdata50$Z$value)


ggplot(df3) +
    geom_line(aes(x = days, y = hares, col = "Est Phyto")) +
    geom_line(aes(x = days, y = lynxes, col = "Est Zoo")) +
    geom_line(aes(x = days, y = actHs, col = "Act Phyto")) +
    geom_line(aes(x = days, y = actLs, col = "Act Zoo"))

Bibliography

Andrieu, Christophe, Arnaud Doucet, and Roman Holenstein. 2010. “Particle Markov chain Monte Carlo methods.” Journal of the Royal Statistical Society. Series B: Statistical Methodology 72 (3): 269–342. doi:10.1111/j.1467-9868.2009.00736.x.

Dureau, Joseph, Konstantinos Kalogeropoulos, and Marc Baguelin. 2013. “Capturing the time-varying drivers of an epidemic using stochastic dynamical systems.” Biostatistics (Oxford, England) 14 (3): 541–55. doi:10.1093/biostatistics/kxs052.

Lotka, Alfred J. 1909. “Contribution to the Theory of Periodic Reactions.” The Journal of Physical Chemistry 14 (3): 271–74. doi:10.1021/j150111a004.

Murray, Lawrence M. n.d. “Bayesian State-Space Modelling on High-Performance Hardware Using LibBi.”

Volterra, Vito. 1926. “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi.” Memorie Della R. Accademia Dei Lincei 6 (2): 31–113. http://www.liberliber.it/biblioteca/v/volterra/variazioni{\_}e{\_}fluttuazioni/pdf/volterra{\_}variazioni{\_}e{\_}fluttuazioni.pdf.

Fun with LibBi and Influenza

Introduction

This is a bit different from my usual posts (well apart from my write up of hacking at Odessa) in that it is a log of how I managed to get LibBi (Library for Bayesian Inference) to run on my MacBook and then not totally satisfactorily (as you will see if you read on).

The intention is to try a few more approaches to the same problem, for example, Stan, monad-bayes and hand-crafted.

Kermack and McKendrick (1927) give a simple model of the spread of an infectious disease. Individuals move from being susceptible (S) to infected (I) to recovered (R).

\displaystyle   \begin{aligned}  \frac{dS}{dt} & = & - \delta S(t) I(t) & \\  \frac{dI}{dt} & = & \delta S(t) I(t) - & \gamma I(t) \\  \frac{dR}{dt} & = &                    & \gamma I(t)  \end{aligned}

In 1978, anonymous authors sent a note to the British Medical Journal reporting an influenza outbreak in a boarding school in the north of England (“Influenza in a boarding school” 1978). The chart below shows the solution of the SIR (Susceptible, Infected, Record) model with parameters which give roughly the results observed in the school.

LibBi

Step 1

~/LibBi-stable/SIR-master $ ./init.sh
error: 'ncread' undefined near line 6 column 7

The README says this is optional so we can skip over it. Still it would be nice to fit the bridge weight function as described in Moral and Murray (2015).

The README does say that GPML is required but since we don’t (yet) need to do this step, let’s move on.

~/LibBi-stable/SIR-master $ ./run.sh
./run.sh

Error: ./configure failed with return code 77. See
.SIR/build_openmp_cuda_single/configure.log and
.SIR/build_openmp_cuda_single/config.log for details

It seems the example is configured to run on CUDA and it is highly likely that my installation of LibBI was not set up to allow this. We can change config.conf from

--disable-assert
--enable-single
--enable-cuda
--nthreads 2

to

--nthreads 4
--enable-sse
--disable-assert

On to the next issue.

~/LibBi-stable/SIR-master $ ./run.sh
./run.sh
Error: ./configure failed with return code 1. required QRUpdate
library not found. See .SIR/build_sse/configure.log and
.SIR/build_sse/config.log for details

But QRUpdate is installed!

~/LibBi-stable/SIR-master $ brew info QRUpdate
brew info QRUpdate
homebrew/science/qrupdate: stable 1.1.2 (bottled)
http://sourceforge.net/projects/qrupdate/
/usr/local/Cellar/qrupdate/1.1.2 (3 files, 302.6K)
/usr/local/Cellar/qrupdate/1.1.2_2 (6 files, 336.3K)
  Poured from bottle
/usr/local/Cellar/qrupdate/1.1.2_3 (6 files, 337.3K) *
  Poured from bottle
From: https://github.com/Homebrew/homebrew-science/blob/master/qrupdate.rb
==> Dependencies
Required: veclibfort ✔
Optional: openblas ✔
==> Options
--with-openblas
	Build with openblas support
--without-check
	Skip build-time tests (not recommended)

Let’s look in the log as advised. So it seems that a certain symbol cannot be found.

checking for dch1dn_ in -lqrupdate

Let’s try ourselves.

nm -g /usr/local/Cellar/qrupdate/1.1.2_3/lib/libqrupdate.a | grep dch1dn_
0000000000000000 T _dch1dn_

So the symbol is there! What gives? Let’s try setting one of the environment variables.

export LDFLAGS='-L/usr/local/lib'

Now we get further.

./run.sh
Error: ./configure failed with return code 1. required NetCDF header
not found. See .SIR/build_sse/configure.log and
.SIR/build_sse/config.log for details

So we just need to set another environment variable.

export CPPFLAGS='-I/usr/local/include/'

This is more mysterious.

./run.sh
Error: ./configure failed with return code 1. required Boost header
not found. See .SIR/build_sse/configure.log and
.SIR/build_sse/config.log for details ~/LibBi-stable/SIR-master

Let’s see what we have.

brew list | grep -i boost

Nothing! I recall having some problems with boost when trying to use a completely different package. So let’s install boost.

brew install boost

Now we get a different error.

./run.sh
Error: make failed with return code 2, see .SIR/build_sse/make.log for details

Fortunately at some time in the past sbfnk took pity on me and advised me here to use boost155, a step that should not be lightly undertaken.

/usr/local/Cellar/boost155/1.55.0_1: 10,036 files, 451.6M, built in 15 minutes 9 seconds

Even then I had to say

brew link --force boost155

Finally it runs.

./run.sh 2> out.txt

And produces a lot of output

wc -l out.txt
   49999 out.txt

ls -ltrh results/posterior.nc
   1.7G Apr 30 19:57 results/posterior.nc

Rather worringly, out.txt has all lines of the form

1: -51.9191 -23.2045 nan beats -inf -inf -inf accept=0.5

nan beating -inf does not sound good.

Now we are in a position to analyse the results.

octave --path oct/ --eval "plot_and_print"
error: 'bi_plot_quantiles' undefined near line 23 column 5

I previously found an Octave package(?) called OctBi so let’s create an .octaverc file which adds this to the path. We’ll also need to load the netcdf package which we previously installed.

addpath ("../OctBi-stable/inst")
pkg load netcdf
~/LibBi-stable/SIR-master $ octave --path oct/ --eval "plot_and_print"
octave --path oct/ --eval "plot_and_print"
warning: division by zero
warning: called from
    mean at line 117 column 7
    read_hist_simulator at line 47 column 11
    bi_read_hist at line 85 column 12
    bi_hist at line 63 column 12
    plot_and_print at line 56 column 5
warning: division by zero
warning: division by zero
warning: division by zero
warning: division by zero
warning: division by zero
warning: print.m: fig2dev binary is not available.
Some output formats are not available.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
warning: opengl_renderer: x/y/zdata should have the same dimensions. Not rendering.
sh: pdfcrop: command not found

I actually get a chart from this so some kind of success.

This does not look like the chart in the Moral and Murray (2015), the fitted number of infected patients looks a lot smoother and the “rates” parameters also vary in a much smoother manner. For reasons I haven’t yet investigated, it looks like over-fitting. Here’s the charts in the paper.

Bibliography

“Influenza in a boarding school.” 1978. British Medical Journal, March, 587.

Kermack, W. O., and A. G. McKendrick. 1927. “A Contribution to the Mathematical Theory of Epidemics.” Proceedings of the Royal Society of London Series A 115 (August): 700–721. doi:10.1098/rspa.1927.0118.

Moral, Pierre Del, and Lawrence M Murray. 2015. “Sequential Monte Carlo with Highly Informative Observations.”

Every Manifold is Paracompact

Introduction

In their paper Betancourt et al. (2014), the authors give a corollary which starts with the phrase “Because the manifold is paracompact”. It wasn’t immediately clear why the manifold was paracompact or indeed what paracompactness meant although it was clearly something like compactness which means that every cover has a finite sub-cover.

It turns out that every manifold is paracompact and that this is intimately related to partitions of unity.

Most of what I have written below is taken from some hand-written anonymous lecture notes I found by chance in the DPMMS library in Cambridge University. To whomever wrote them: thank you very much.

Limbering Up

Let \{U_i : i \in {\mathcal{I}}\} be an open cover of a smooth manifold M. A partition of unity on M, subordinate to the cover \{U_i : i \in {\mathcal{I}}\} is a finite collection of smooth functions

\displaystyle   X_j : M^n \longrightarrow \mathbb{R}_+

where j = 1, 2, \ldots N for some N such that

\displaystyle   \sum_{j = 0}^N X_j(x) = 1 \,\mathrm{for all}\, x \in M

and for each j there exists i(j) \in {\mathcal{I}} such that

\displaystyle   {\mathrm{supp}}{X_j} \subset U_{i(j)}

We don’t yet know partitions of unity exist.

First define

\displaystyle   f(t) \triangleq  \begin{cases}  0            & \text{if } t \leq 0 \\  \exp{(-1/t)} & \text{if } t > 0 \\  \end{cases}

Techniques of classical analysis easily show that f is smooth (t=0 is the only point that might be in doubt and it can be checked from first principles that f^{(n)}(0) = 0 for all n).

Next define

\displaystyle   \begin{aligned}  g(t) &\triangleq \frac{f(t)}{f(t) + f(1 - t)} \\  h(t) &\triangleq g(t + 2)g(2 - t)  \end{aligned}

Finally we can define F: \mathbb{R}^n \rightarrow \mathbb{R} by F(x) = h(\|x\|). This has the properties

  • F(x) = 1 if \|x\| \leq 1
  • 0 \leq F(x) \leq 1
  • F(x) = 0 if \|x\| > 2

Now take a point p \in M centred in a chart (U_p, \phi_p) so that, without loss of generality, B(0,3) \subseteq \phi_p(U_p) (we can always choose r_p so that the open ball B(0,3r_p) \subseteq \phi'_p(U_p) and then define another chart (U_p, \phi_p) with \phi_p(x) = \phi'_p(x)/\|x\|).

Define the images of the open and closed balls of radius 1 and 2 respectively

\displaystyle   \begin{aligned}  V_p &= \phi_p^{-1}(B(0,1)) \\  W_p &= \phi_p^{-1}\big(\overline{B(0,2)}\big) \\  \end{aligned}

and further define bump functions

\displaystyle   \psi_p(y) \triangleq  \begin{cases}  F(\phi_p(y)) & \text{if } y \in U_p\\  0            & \text{otherwise} \\  \end{cases}

Then \psi_p is smooth and its support lies in W_p \subset U_p.

By compactness, the open cover \{V_p : p \in M\} has a finite subcover \{V_{p_1},\ldots,V_{p_K}\}. Now define

\displaystyle   X_j : M^n \longrightarrow \mathbb{R}_+

by

\displaystyle   X_j(y) = \frac{\psi_{p_j}(y)}{\sum_{i=1}^K \psi_{p_i}(y)}

Then X_j is smooth, {\mathrm{supp}}{X_j} = {\mathrm{supp}}{\psi_{p_j}} \subset U_{p_j} and \sum_{j=1}^K X_j(y) = 1. Thus \{X_j\} is the required partition of unity.

Paracompactness

Because M is a manifold, it has a countable basis \{A_i\}_{i \in \mathbb{N}} and for any point p, there must exist A_i \subset V_p with p \in A_i. Choose one of these and call it V_{p_i}. This gives a countable cover of M by such sets.

Now define

\displaystyle   L_1 = W_{p_1} \subset V_{p_1} \cup V_{p_2} \cup \ldots \cup V_{p_{i(2)}}

where, since L_1 is compact, V_{p_2}, \ldots, V_{p_{i(2)}} is a finite subcover.

And further define

\displaystyle   L_n = W_{p_1} \cup W_{p_2} \cup \ldots \cup W_{p_{i(n)}}        \subset        V_{p_1} \cup V_{p_2} \cup \ldots \cup V_{p_{i(n+1)}}

where again, since L_n is compact, V_{p_{i(n)+1}}, \ldots, V_{p_{i(n+1)}} is a finite subcover.

Now define

\displaystyle   \begin{aligned}  K_n &= L_n \setminus {\mathrm{int}}{L_{n-1}} \\  U_n &= {\mathrm{int}}(L_{n+1}) \setminus L_n  \end{aligned}

Then K_n is compact, U_n is open and K_n \subset U_n. Furthermore, \bigcup_{n \in \mathbb{N}} K_n = M and U_n only intersects with U_{n-1} and U_{n+1}.

Given any open cover {\mathcal{O}} of M, each K_n can be covered by a finite number of open sets in U_n contained in some member of {\mathcal{O}}. Thus every point in K_n can be covered by at most a finite number of sets from U_{n-1}, U_n and U_{n+1} and which are contained in some member of {\mathcal{O}}. This is a locally finite refinement of {\mathcal{O}} and which is precisely the definition of paracompactness.

To produce a partition of unity we define bump functions \psi_j as above on this locally finite cover and note that locally finite implies that \sum_j \psi_j is well defined. Again, as above, define

\displaystyle   X_j(y) = \frac{\psi_{j}(y)}{\sum_{i=1} \psi_{i}(y)}

to get the required result.

Bibliography

Betancourt, M. J., Simon Byrne, Samuel Livingstone, and Mark Girolami. 2014. “The Geometric Foundations of Hamiltonian Monte Carlo,” October, 45. http://arxiv.org/abs/1410.5110.

The Lie Derivative

Introduction

In proposition 58 Chapter 1 in the excellent book O’Neill (1983), the author demonstrates that the Lie derivative of one vector field with respect to another is the same as the Lie bracket (of the two vector fields) although he calls the Lie bracket just bracket and does not define the Lie derivative preferring just to use its definition with giving it a name. The proof relies on a prior result where he shows a co-ordinate system at a point p can be given to a vector field X for which X_p \neq 0 so that X = \frac{\partial}{\partial x_1}.

Here’s a proof seems clearer (to me at any rate) and avoids having to distinguish the case wehere the vector field is zero or non-zero. These notes give a similar proof but, strangely for undergraduate level, elide some of the details.

A Few Definitions

Let \phi: M \longrightarrow N be a smooth mapping and let A be a 0,s tensor with s \geq 0 then define the pullback of A by \phi to be

\displaystyle   \phi^*A(v_1,\ldots,v_s) = A(\mathrm{d}\phi v_1,\ldots,\mathrm{d}\phi v_s)

For a (0,0) tensor f \in {\mathscr{F}}(N) the pullback is defined to be \phi^*(f) = f \circ \phi \in {\mathscr{F}}(M).

Standard manipulations show that \phi^*A is a smooth (covariant) tensor field and that \phi^* is \mathbb{R}-linear and that \phi^*(A\otimes B) = \phi^*A \otimes \phi^*B.

Let F : M \longrightarrow N be a diffeomorphism and Y a vector field on N we define the pullback of this field to be

\displaystyle   (F^*{Y})_x = D(F^{-1})_{F(x)}(Y_{F(x)})

Note that the pullback of a vector field only exists in the case where F is a diffeomorphism; in contradistinction, in the case of pullbacks of purely covariant tensors, the pullback always exists.

For the proof below, we only need the pullback of functions and vector fields; the pullback for (0,s) tensors with s \geq 1 is purely to give a bit of context.

From O’Neill (1983) Chapter 1 Definition 20, let \phi: M \rightarrow N be a smooth mapping. Vector fields X on M and Y on N are Frelated written X \underset{F}{\sim} Y if and only if dF({X}_p) = Y_{Fp}.

The Alternative Proof

By Lemma 21 Chapter 1 of O’Neill (1983), X and Y are F-related if and only if X(f \circ F) = Yf \circ F.

Recalling that dF(X_p)(f) = X_p(F \circ f) and since

\displaystyle   dF_x d(F^{-1})_{Fx}(X_{Fx}) = X_{Fx}

we see that the fields F^*{Y} and Y are F-related: F^*{Y}_x \underset{F}{\sim} Y_{Fx}. Thus we can apply the Lemma.

\displaystyle   (F^*{Y})(f \circ F) = (F^*{Y})(F^*{f}) =  Yf \circ F = F^*(Yf)

Although we don’t need this, we can express the immediately above equivalence in a way similar to the rule for covariant tensors

\displaystyle   (F^*{Y})(f \otimes F) = (F^*{Y})\otimes(F^*{f})

First let’s calculate the Lie derivative of a function f with respect to a vector field X where \phi_t is its flow

\displaystyle   \begin{aligned}  L_X f &\triangleq \lim_{t \rightarrow 0} \frac{\phi_t^*(f) - f}{t} \\        &=          \lim_{t \rightarrow 0} \frac{f \circ \phi_t - f \circ \phi_0}{t} \\        &=          \lim_{t \rightarrow 0} \frac{f \circ \phi (t,x) - f \circ \phi (0, x)}{t} \\        &=          (\phi_x)'_0(f) \\        &=          X_x(f) \\        &=          (Xf)_x  \end{aligned}

Analogously defining the Lie derivative of Y with respect to X

\displaystyle   (L_X Y) \triangleq \frac{(\phi_t^*{Y}) - Y}{t}f

we have

\displaystyle   \begin{aligned}  L_X(Yf) &= \lim_{t \rightarrow 0} \frac{\phi_t^*(Yf) - Yf}{t} \\          &= \lim_{t \rightarrow 0} \frac{(\phi_t^*{Y})(\phi_t^*{f}) - Yf}{t} \\          &= \lim_{t \rightarrow 0}             \frac{(\phi_t^*{Y})(\phi_t^*{f}) - (\phi_t^*{Y})f + (\phi_t^*{Y})f - Yf}{t} \\          &= \lim_{t \rightarrow 0}             \Bigg(             (\phi_t^*{Y})\frac{\phi_t^*{f} - f}{t} +             \frac{(\phi_t^*{Y}) - Y}{t}f             \Bigg) \\          &= Y(L_X f) + (L_X Y)f  \end{aligned}

Since L_X f = Xf we have

\displaystyle   X(Yf) = Y(Xf) + (L_X Y)f

Thus

\displaystyle   (L_X Y)f  = Y(Xf) - X(Yf) = [X,Y]f

as required.

Bibliography

O’Neill, B. 1983. Semi-Riemannian Geometry with Applications to Relativity, 103. Pure and Applied Mathematics. Elsevier Science. https://books.google.com.au/books?id=CGk1eRSjFIIC.

Particle Smoothing

Introduction

The equation of motion for a pendulum of unit length subject to Gaussian white noise is

\displaystyle   \frac{\mathrm{d}^2\alpha}{\mathrm{d}t^2} = -g\sin\alpha + w(t)

We can discretize this via the usual Euler method

\displaystyle   \begin{bmatrix}  x_{1,i} \\  x_{2,i}  \end{bmatrix}  =  \begin{bmatrix}  x_{1,i-1} + x_{2,i-1}\Delta t \\  x_{2,i-1} - g\sin x_{1,i-1}\Delta t  \end{bmatrix}  +  \mathbf{q}_{i-1}

where q_i \sim {\mathcal{N}}(0,Q) and

\displaystyle   Q  =  \begin{bmatrix}  \frac{q^c \Delta t^3}{3} & \frac{q^c \Delta t^2}{2} \\  \frac{q^c \Delta t^2}{2} & {q^c \Delta t}  \end{bmatrix}

The explanation of the precise form of the covariance matrix will be the subject of another blog post; for the purpose of exposition of forward filtering / backward smoothing, this detail is not important.

Assume that we can only measure the horizontal position of the pendulum and further that this measurement is subject to error so that

\displaystyle   y_i = \sin x_i + r_k

where r_i \sim {\mathcal{N}}(0,R).

Particle Filtering can give us an estimate of where the pendulum is and its velocity using all the observations up to that point in time. But now suppose we have observed the pendulum for a fixed period of time then at times earlier than the time at which we stop our observations we now have observations in the future as well as in the past. If we can somehow take account of these future observations then we should be able to improve our estimate of where the pendulum was at any given point in time (as well as its velocity). Forward Filtering / Backward Smoothing is a technique for doing this.

Haskell Preamble

> {-# OPTIONS_GHC -Wall                     #-}
> {-# OPTIONS_GHC -fno-warn-name-shadowing  #-}
> {-# OPTIONS_GHC -fno-warn-type-defaults   #-}
> {-# OPTIONS_GHC -fno-warn-unused-do-bind  #-}
> {-# OPTIONS_GHC -fno-warn-missing-methods #-}
> {-# OPTIONS_GHC -fno-warn-orphans         #-}
> {-# LANGUAGE MultiParamTypeClasses        #-}
> {-# LANGUAGE TypeFamilies                 #-}
> {-# LANGUAGE ScopedTypeVariables          #-}
> {-# LANGUAGE ExplicitForAll               #-}
> {-# LANGUAGE DataKinds                    #-}
> {-# LANGUAGE FlexibleInstances            #-}
> {-# LANGUAGE MultiParamTypeClasses        #-}
> {-# LANGUAGE FlexibleContexts             #-}
> {-# LANGUAGE TypeFamilies                 #-}
> {-# LANGUAGE BangPatterns                 #-}
> {-# LANGUAGE GeneralizedNewtypeDeriving   #-}
> {-# LANGUAGE TemplateHaskell              #-}
> {-# LANGUAGE DataKinds                    #-}
> {-# LANGUAGE DeriveGeneric                #-}
> module PendulumSamples ( pendulumSamples
>                        , pendulumSamples'
>                        , testFiltering
>                        , testSmoothing
>                        , testFilteringG
>                        , testSmoothingG
>                        ) where
> import           Data.Random hiding ( StdNormal, Normal )
> import           Data.Random.Source.PureMT ( pureMT )
> import           Control.Monad.State ( evalState, replicateM )
> import qualified Control.Monad.Loops as ML
> import           Control.Monad.Writer ( tell, WriterT, lift,
>                                         runWriterT
>                                       )
> import           Numeric.LinearAlgebra.Static
>                  ( R, vector, Sym,
>                    headTail, matrix, sym,
>                    diag
>                  )
> import           GHC.TypeLits ( KnownNat )
> import           MultivariateNormal ( MultivariateNormal(..) )
> import qualified Data.Vector as V
> import           Data.Bits ( shiftR )
> import           Data.List ( transpose )
> import           Control.Parallel.Strategies
> import           GHC.Generics (Generic)

Simulation

Let’s first plot some typical trajectories of the pendulum.

> deltaT, g :: Double
> deltaT = 0.01
> g  = 9.81
> type PendulumState = R 2
> type PendulumObs = R 1
> pendulumSample :: MonadRandom m =>
>                   Sym 2 ->
>                   Sym 1 ->
>                   PendulumState ->
>                   m (Maybe ((PendulumState, PendulumObs), PendulumState))
> pendulumSample bigQ bigR xPrev = do
>   let x1Prev = fst $ headTail xPrev
>       x2Prev = fst $ headTail $ snd $ headTail xPrev
>   eta <- sample $ rvar (MultivariateNormal 0.0 bigQ)
>   let x1= x1Prev + x2Prev * deltaT
>       x2 = x2Prev - g * (sin x1Prev) * deltaT
>       xNew = vector [x1, x2] + eta
>       x1New = fst $ headTail xNew
>   epsilon <-  sample $ rvar (MultivariateNormal 0.0 bigR)
>   let yNew = vector [sin x1New] + epsilon
>   return $ Just ((xNew, yNew), xNew)

Let’s try plotting some samples when we are in the linear region with which we are familiar from school \sin\alpha \approx \alpha.

\displaystyle   \frac{\mathrm{d}^2\alpha}{\mathrm{d}t^2} = -g\alpha + w(t)

In this case we expect the horizontal displacement to be approximately equal to the angle of displacement and thus the observations to be symmetric about the actuals.

> bigQ :: Sym 2
> bigQ = sym $ matrix bigQl
> qc1 :: Double
> qc1 = 0.0001
> bigQl :: [Double]
> bigQl = [ qc1 * deltaT^3 / 3, qc1 * deltaT^2 / 2,
>           qc1 * deltaT^2 / 2,       qc1 * deltaT
>         ]
> bigR :: Sym 1
> bigR  = sym $ matrix [0.0001]
> m0 :: PendulumState
> m0 = vector [0.01, 0]
> pendulumSamples :: [(PendulumState, PendulumObs)]
> pendulumSamples = evalState (ML.unfoldrM (pendulumSample bigQ bigR) m0) (pureMT 17)

But if we work in a region in which linearity breaks down then the observations are no longer symmetrical about the actuals.

> bigQ' :: Sym 2
> bigQ' = sym $ matrix bigQl'
> qc1' :: Double
> qc1' = 0.01
> bigQl' :: [Double]
> bigQl' = [ qc1' * deltaT^3 / 3, qc1' * deltaT^2 / 2,
>            qc1' * deltaT^2 / 2,       qc1' * deltaT
>          ]
> bigR' :: Sym 1
> bigR'  = sym $ matrix [0.1]
> m0' :: PendulumState
> m0' = vector [1.6, 0]
> pendulumSamples' :: [(PendulumState, PendulumObs)]
> pendulumSamples' = evalState (ML.unfoldrM (pendulumSample bigQ' bigR') m0') (pureMT 17)

Filtering

We do not give the theory behind particle filtering. The interested reader can either consult Särkkä (2013) or wait for a future blog post on the subject.

> nParticles :: Int
> nParticles = 500

The usual Bayesian update step.

> type Particles a = V.Vector a
> oneFilteringStep ::
>   MonadRandom m =>
>   (Particles a -> m (Particles a)) ->
>   (Particles a -> Particles b) ->
>   (b -> b -> Double) ->
>   Particles a ->
>   b ->
>   WriterT [Particles a] m (Particles a)
> oneFilteringStep stateUpdate obsUpdate weight statePrevs obs = do
>   tell [statePrevs]
>   stateNews <- lift $ stateUpdate statePrevs
>   let obsNews = obsUpdate stateNews
>   let weights       = V.map (weight obs) obsNews
>       cumSumWeights = V.tail $ V.scanl (+) 0 weights
>       totWeight     = V.last cumSumWeights
>   vs <- lift $ V.replicateM nParticles (sample $ uniform 0.0 totWeight)
>   let js = indices cumSumWeights vs
>       stateTildes = V.map (stateNews V.!) js
>   return stateTildes

The system state and observable.

> data SystemState a = SystemState { x1  :: a, x2  :: a }
>   deriving (Show, Generic)
> instance NFData a => NFData (SystemState a)
> newtype SystemObs a = SystemObs { y1  :: a }
>   deriving Show

To make the system state update a bit more readable, let’s introduce some lifted arithmetic operators.

> (.+), (.*), (.-) :: (Num a) => V.Vector a -> V.Vector a -> V.Vector a
> (.+) = V.zipWith (+)
> (.*) = V.zipWith (*)
> (.-) = V.zipWith (-)

The state update itself

> stateUpdate :: Particles (SystemState Double) ->
>                 Particles (SystemState Double)
> stateUpdate xPrevs = V.zipWith SystemState x1s x2s
>   where
>     ix = V.length xPrevs
> 
>     x1Prevs = V.map x1 xPrevs
>     x2Prevs = V.map x2 xPrevs
> 
>     deltaTs = V.replicate ix deltaT
>     gs = V.replicate ix g
>     x1s = x1Prevs .+ (x2Prevs .* deltaTs)
>     x2s = x2Prevs .- (gs .* (V.map sin x1Prevs) .* deltaTs)

and its noisy version.

> stateUpdateNoisy :: MonadRandom m =>
>                     Sym 2 ->
>                     Particles (SystemState Double) ->
>                     m (Particles (SystemState Double))
> stateUpdateNoisy bigQ xPrevs = do
>   let xs = stateUpdate xPrevs
> 
>       x1s = V.map x1 xs
>       x2s = V.map x2 xs
> 
>   let ix = V.length xPrevs
>   etas <- replicateM ix $ sample $ rvar (MultivariateNormal 0.0 bigQ)
> 
>   let eta1s, eta2s :: V.Vector Double
>       eta1s = V.fromList $ map (fst . headTail) etas
>       eta2s = V.fromList $ map (fst . headTail . snd . headTail) etas
> 
>   return (V.zipWith SystemState (x1s .+ eta1s) (x2s .+ eta2s))

The function which maps the state to the observable.

> obsUpdate :: Particles (SystemState Double) ->
>              Particles (SystemObs Double)
> obsUpdate xs = V.map (SystemObs . sin . x1) xs

And finally a function to calculate the weight of each particle given an observation.

> weight :: forall a n . KnownNat n =>
>           (a -> R n) ->
>           Sym n ->
>           a -> a -> Double
> weight f bigR obs obsNew = pdf (MultivariateNormal (f obsNew) bigR) (f obs)

The variance of the prior.

> bigP :: Sym 2
> bigP = sym $ diag 0.1

Generate our ensemble of particles chosen from the prior,

> initParticles :: MonadRandom m =>
>                  m (Particles (SystemState Double))
> initParticles = V.replicateM nParticles $ do
>   r <- sample $ rvar (MultivariateNormal m0' bigP)
>   let x1 = fst $ headTail r
>       x2 = fst $ headTail $ snd $ headTail r
>   return $ SystemState { x1 = x1, x2 = x2}

run the particle filter,

> runFilter :: Int -> [Particles (SystemState Double)]
> runFilter nTimeSteps = snd $ evalState action (pureMT 19)
>   where
>     action = runWriterT $ do
>       xs <- lift $ initParticles
>       V.foldM
>         (oneFilteringStep (stateUpdateNoisy bigQ') obsUpdate (weight f bigR'))
>         xs
>         (V.fromList $ map (SystemObs . fst . headTail . snd)
>                           (take nTimeSteps pendulumSamples'))

and extract the estimated position from the filter.

> testFiltering :: Int -> [Double]
> testFiltering nTimeSteps = map ((/ (fromIntegral nParticles)). sum . V.map x1)
>                                (runFilter nTimeSteps)

Smoothing

If we could calculate the marginal smoothing distributions \{p(x_t \,|\, y_{1:T})\}_{i=1}^T then we might be able to sample from them. Using the Markov assumption of our model that x_i is independent of y_{i+1:N} given x_{i+1}, we have

\displaystyle   \begin{aligned}  \overbrace{p(x_i \,|\, y_{1:N})}^{\mathrm{smoother}\,\mathrm{at}\, i} &=  \int p(x_i, x_{i+1} \,|\, y_{1:N}) \,\mathrm{d}x_{i+1} & \text{marginal distribution} \\  &=  \int p(x_{i+1} \,|\, y_{1:N}) \,p(x_{i} \,|\, y_{1:N}, x_{i+1}) \,\mathrm{d}x_{i+1} & \text{conditional density} \\  &=  \int p(x_{i+1} \,|\, y_{1:N}) \,p(x_{i} \,|\, y_{1:i}, x_{i+1}) \,\mathrm{d}x_{i+1} & \text{Markov model} \\  &=  \int p(x_{i+1} \,|\, y_{1:N}) \,  \frac{p(x_{i}, x_{i+1} \,|\, y_{1:i})}{p(x_{i+1} \,|\, y_{1:i})}  \,\mathrm{d}x_{i+1}  & \text{conditional density} \\  &=  \int p(x_{i+1} \,|\, y_{1:N}) \,  \frac{p(x_{i+1} \,|\, x_{i}, y_{1:i})  \,p(x_{i} \,|\, y_{1:i})}{p(x_{i+1} \,|\, y_{1:i})}  \,\mathrm{d}x_{i+1}  & \text{conditional density} \\  &=  \int \overbrace{p(x_{i+1} \,|\, y_{1:N})}^{\text{smoother at }i+1} \,  \underbrace{  \overbrace{p(x_{i} \,|\, y_{1:i})}^{\text{filter at }i}  \frac{p(x_{i+1} \,|\, x_{i})}       {p(x_{i+1} \,|\, y_{1:i})}  }  _{\text{backward transition }p(x_{i} \,|\, y_{1:i},\,x_{i+1})}  \,\mathrm{d}x_{i+1}  & \text{Markov model}  \end{aligned}

We observe that this is a (continuous state space) Markov process with a non-homogeneous transition function albeit one which goes backwards in time. Apparently for conditionally Gaussian linear state-space models, this is known as RTS, or Rauch-Tung-Striebel smoothing (Rauch, Striebel, and Tung (1965)).

According to Cappé (2008),

  • It appears to be efficient and stable in the long term (although no proof was available at the time the slides were presented).

  • It is not sequential (in particular, one needs to store all particle positions and weights).

  • It has numerical complexity proportional O(n^2) where N is the number of particles.

We can use this to sample paths starting at time i = N and working backwards. From above we have

\displaystyle   p(x_i \,|\, X_{i+1}, Y_{1:N}) =  \frac{p(X_{i+1} \,|\, x_{i})  \,p(x_{i} \,|\, Y_{1:i})}{p(X_{i+1} \,|\, Y_{1:i})}  =  Z  \,p(X_{i+1} \,|\, x_{i})  \,p(x_{i} \,|\, Y_{1:i})

where Z is some normalisation constant (Z for “Zustandssumme” – sum over states).

From particle filtering we know that

\displaystyle   {p}(x_i \,|\, y_{1:i}) \approx \hat{p}(x_i \,|\, y_{1:i}) \triangleq \sum_{j=1}^M w_i^{(j)}\delta(x_i - x_i^{(j)})

Thus

\displaystyle   \hat{p}(x_i \,|\, X_{i+1}, Y_{1:i})  =  Z  \,p(X_{i+1} \,|\, x_{i})  \,\hat{p}(x_{i} \,|\, Y_{1:i})  =  \sum_{j=1}^M w_i^{(j)}\delta(x_i - x_i^{(j)})  \,p(X_{i+1} \,|\, x_{i})

and we can sample x_i from \{x_i^{(j)}\}_{1 \leq j \leq M} with probability

\displaystyle   \frac{w_k^{(i)}  \,p(X_{i+1} \,|\, x_{i})}  {\sum_{i=1}^N w_k^{(i)}  \,p(X_{i+1} \,|\, x_{i})}

Recalling that we have re-sampled the particles in the filtering algorithm so that their weights are all 1/M and abstracting the state update and state density function, we can encode this update step in Haskell as

> oneSmoothingStep :: MonadRandom m =>
>          (Particles a -> V.Vector a) ->
>          (a -> a -> Double) ->
>          a ->
>          Particles a ->
>          WriterT (Particles a) m a
> oneSmoothingStep stateUpdate
>                  stateDensity
>                  smoothingSampleAtiPlus1
>                  filterSamplesAti = do it
>   where
>     it = do
>       let mus = stateUpdate filterSamplesAti
>           weights =  V.map (stateDensity smoothingSampleAtiPlus1) mus
>           cumSumWeights = V.tail $ V.scanl (+) 0 weights
>           totWeight     = V.last cumSumWeights
>       v <- lift $ sample $ uniform 0.0 totWeight
>       let ix = binarySearch cumSumWeights v
>           xnNew = filterSamplesAti V.! ix
>       tell $ V.singleton xnNew
>       return $ xnNew

To sample a complete path we start with a sample from the filtering distribution at at time i = N (which is also the smoothing distribution)

> oneSmoothingPath :: MonadRandom m =>
>              (Int -> V.Vector (Particles a)) ->
>              (a -> Particles a -> WriterT (Particles a) m a) ->
>              Int -> m (a, V.Vector a)
> oneSmoothingPath filterEstss oneSmoothingStep nTimeSteps = do
>   let ys = filterEstss nTimeSteps
>   ix <- sample $ uniform 0 (nParticles - 1)
>   let xn = (V.head ys) V.! ix
>   runWriterT $ V.foldM oneSmoothingStep xn (V.tail ys)
> oneSmoothingPath' :: (MonadRandom m, Show a) =>
>              (Int -> V.Vector (Particles a)) ->
>              (a -> Particles a -> WriterT (Particles a) m a) ->
>              Int -> WriterT (Particles a) m a
> oneSmoothingPath' filterEstss oneSmoothingStep nTimeSteps = do
>   let ys = filterEstss nTimeSteps
>   ix <- lift $ sample $ uniform 0 (nParticles - 1)
>   let xn = (V.head ys) V.! ix
>   V.foldM oneSmoothingStep xn (V.tail ys)

Of course we need to run through the filtering distributions starting at i = N

> filterEstss :: Int -> V.Vector (Particles (SystemState Double))
> filterEstss n = V.reverse $ V.fromList $ runFilter n
> testSmoothing :: Int -> Int -> [Double]
> testSmoothing m n = V.toList $ evalState action (pureMT 23)
>   where
>     action = do
>       xss <- V.replicateM n $
>              snd <$> (oneSmoothingPath filterEstss (oneSmoothingStep stateUpdate (weight h bigQ')) m)
>       let yss = V.fromList $ map V.fromList $
>                 transpose $
>                 V.toList $ V.map (V.toList) $
>                 xss
>       return $ V.map (/ (fromIntegral n)) $ V.map V.sum $ V.map (V.map x1) yss

By eye we can see we get a better fit

and calculating the mean square error for filtering gives 1.87\times10^{-2} against the mean square error for smoothing of 9.52\times10^{-3}; this confirms the fit really is better as one would hope.

Unknown Gravity

Let us continue with the same example but now assume that g is unknown and that we wish to estimate it. Let us also assume that our apparatus is not subject to noise.

Again we have

\displaystyle   \frac{\mathrm{d}^2\alpha}{\mathrm{d}t^2} = -g\sin\alpha + w(t)

But now when we discretize it we include a third variable

\displaystyle   \begin{bmatrix}  x_{1,i} \\  x_{2,i} \\  x_{3,i}  \end{bmatrix}  =  \begin{bmatrix}  x_{1,i-1} + x_{2,i-1}\Delta t \\  x_{2,i-1} - x_{3,i-1}\sin x_{1,i-1}\Delta t \\  x_{3,i-1}  \end{bmatrix}  +  \mathbf{q}_{i-1}

where q_i \sim {\mathcal{N}}(0,Q)

\displaystyle   Q  =  \begin{bmatrix}  0 & 0 & 0 \\  0 & 0 & 0 \\  0 & 0 & \sigma^2_g  \end{bmatrix}

Again we assume that we can only measure the horizontal position of the pendulum so that

\displaystyle   y_i = \sin x_i + r_k

where r_i \sim {\mathcal{N}}(0,R).

> type PendulumStateG = R 3
> pendulumSampleG :: MonadRandom m =>
>                   Sym 3 ->
>                   Sym 1 ->
>                   PendulumStateG ->
>                   m (Maybe ((PendulumStateG, PendulumObs), PendulumStateG))
> pendulumSampleG bigQ bigR xPrev = do
>   let x1Prev = fst $ headTail xPrev
>       x2Prev = fst $ headTail $ snd $ headTail xPrev
>       x3Prev = fst $ headTail $ snd $ headTail $ snd $ headTail xPrev
>   eta <- sample $ rvar (MultivariateNormal 0.0 bigQ)
>   let x1= x1Prev + x2Prev * deltaT
>       x2 = x2Prev - g * (sin x1Prev) * deltaT
>       x3 = x3Prev
>       xNew = vector [x1, x2, x3] + eta
>       x1New = fst $ headTail xNew
>   epsilon <-  sample $ rvar (MultivariateNormal 0.0 bigR)
>   let yNew = vector [sin x1New] + epsilon
>   return $ Just ((xNew, yNew), xNew)
> pendulumSampleGs :: [(PendulumStateG, PendulumObs)]
> pendulumSampleGs = evalState (ML.unfoldrM (pendulumSampleG bigQg bigRg) mG) (pureMT 29)
> data SystemStateG a = SystemStateG { gx1  :: a, gx2  :: a, gx3 :: a }
>   deriving Show

The state update itself

> stateUpdateG :: Particles (SystemStateG Double) ->
>                 Particles (SystemStateG Double)
> stateUpdateG xPrevs = V.zipWith3 SystemStateG x1s x2s x3s
>   where
>     ix = V.length xPrevs
> 
>     x1Prevs = V.map gx1 xPrevs
>     x2Prevs = V.map gx2 xPrevs
>     x3Prevs = V.map gx3 xPrevs
> 
>     deltaTs = V.replicate ix deltaT
>     x1s = x1Prevs .+ (x2Prevs .* deltaTs)
>     x2s = x2Prevs .- (x3Prevs .* (V.map sin x1Prevs) .* deltaTs)
>     x3s = x3Prevs

and its noisy version.

> stateUpdateNoisyG :: MonadRandom m =>
>                      Sym 3 ->
>                      Particles (SystemStateG Double) ->
>                      m (Particles (SystemStateG Double))
> stateUpdateNoisyG bigQ xPrevs = do
>   let ix = V.length xPrevs
> 
>   let xs = stateUpdateG xPrevs
> 
>       x1s = V.map gx1 xs
>       x2s = V.map gx2 xs
>       x3s = V.map gx3 xs
> 
>   etas <- replicateM ix $ sample $ rvar (MultivariateNormal 0.0 bigQ)
>   let eta1s, eta2s, eta3s :: V.Vector Double
>       eta1s = V.fromList $ map (fst . headTail) etas
>       eta2s = V.fromList $ map (fst . headTail . snd . headTail) etas
>       eta3s = V.fromList $ map (fst . headTail . snd . headTail . snd . headTail) etas
> 
>   return (V.zipWith3 SystemStateG (x1s .+ eta1s) (x2s .+ eta2s) (x3s .+ eta3s))

The function which maps the state to the observable.

> obsUpdateG :: Particles (SystemStateG Double) ->
>              Particles (SystemObs Double)
> obsUpdateG xs = V.map (SystemObs . sin . gx1) xs

The mean and variance of the prior.

> mG :: R 3
> mG = vector [1.6, 0.0, 8.00]
> bigPg :: Sym 3
> bigPg = sym $ matrix [
>     1e-6, 0.0, 0.0
>   , 0.0, 1e-6, 0.0
>   , 0.0, 0.0, 1e-2
>   ]

Parameters for the state update; note that the variance is not exactly the same as in the formulation above.

> bigQg :: Sym 3
> bigQg = sym $ matrix bigQgl
> qc1G :: Double
> qc1G = 0.0001
> sigmaG :: Double
> sigmaG = 1.0e-2
> bigQgl :: [Double]
> bigQgl = [ qc1G * deltaT^3 / 3, qc1G * deltaT^2 / 2, 0.0,
>            qc1G * deltaT^2 / 2,       qc1G * deltaT, 0.0,
>                            0.0,                 0.0, sigmaG
>          ]

The noise of the measurement.

> bigRg :: Sym 1
> bigRg  = sym $ matrix [0.1]

Generate the ensemble of particles from the prior,

> initParticlesG :: MonadRandom m =>
>                  m (Particles (SystemStateG Double))
> initParticlesG = V.replicateM nParticles $ do
>   r <- sample $ rvar (MultivariateNormal mG bigPg)
>   let x1 = fst $ headTail r
>       x2 = fst $ headTail $ snd $ headTail r
>       x3 = fst $ headTail $ snd $ headTail $ snd $ headTail r
>   return $ SystemStateG { gx1 = x1, gx2 = x2, gx3 = x3}

run the particle filter,

> runFilterG :: Int -> [Particles (SystemStateG Double)]
> runFilterG n = snd $ evalState action (pureMT 19)
>   where
>     action = runWriterT $ do
>       xs <- lift $ initParticlesG
>       V.foldM
>         (oneFilteringStep (stateUpdateNoisyG bigQg) obsUpdateG (weight f bigRg))
>         xs
>         (V.fromList $ map (SystemObs . fst . headTail . snd) (take n pendulumSampleGs))

and extract the estimated parameter from the filter.

> testFilteringG :: Int -> [Double]
> testFilteringG n = map ((/ (fromIntegral nParticles)). sum . V.map gx3) (runFilterG n)

Again we need to run through the filtering distributions starting at i = N

> filterGEstss :: Int -> V.Vector (Particles (SystemStateG Double))
> filterGEstss n = V.reverse $ V.fromList $ runFilterG n
> testSmoothingG :: Int -> Int -> ([Double], [Double], [Double])
> testSmoothingG m n = (\(x, y, z) -> (V.toList x, V.toList y, V.toList z))  $
>                      mkMeans $
>                      chunks
>   where
> 
>     chunks =
>       V.fromList $ map V.fromList $
>       transpose $
>       V.toList $ V.map (V.toList) $
>       chunksOf m $
>       snd $ evalState (runWriterT action) (pureMT 23)
> 
>     mkMeans yss = (
>       V.map (/ (fromIntegral n)) $ V.map V.sum $ V.map (V.map gx1) yss,
>       V.map (/ (fromIntegral n)) $ V.map V.sum $ V.map (V.map gx2) yss,
>       V.map (/ (fromIntegral n)) $ V.map V.sum $ V.map (V.map gx3) yss
>       )
> 
>     action =
>       V.replicateM n $
>       oneSmoothingPath' filterGEstss
>                         (oneSmoothingStep stateUpdateG (weight hG bigQg))
>                         m

Again by eye we get a better fit but note that we are using the samples in which the state update is noisy as well as the observation so we don’t expect to get a very good fit.

Notes

Helpers for Converting Types

> f :: SystemObs Double -> R 1
> f = vector . pure . y1
> h :: SystemState Double -> R 2
> h u = vector [x1 u , x2 u]
> hG :: SystemStateG Double -> R 3
> hG u = vector [gx1 u , gx2 u, gx3 u]

Helpers for the Inverse CDF

That these are helpers for the inverse CDF is delayed to another blog post.

> indices :: V.Vector Double -> V.Vector Double -> V.Vector Int
> indices bs xs = V.map (binarySearch bs) xs
> binarySearch :: (Ord a) =>
>                 V.Vector a -> a -> Int
> binarySearch vec x = loop 0 (V.length vec - 1)
>   where
>     loop !l !u
>       | u <= l    = l
>       | otherwise = let e = vec V.! k in if x <= e then loop l k else loop (k+1) u
>       where k = l + (u - l) `shiftR` 1

Vector Helpers

> chunksOf :: Int -> V.Vector a -> V.Vector (V.Vector a)
> chunksOf n xs = ys
>   where
>     l = V.length xs
>     m  = 1 + (l - 1) `div` n
>     ys = V.unfoldrN m (\us -> Just (V.take n us, V.drop n us)) xs

Bibliography

Cappé, Olivier. 2008. “An Introduction to Sequential Monte Carlo for Filtering and Smoothing.” http://www-irma.u-strasbg.fr/~guillou/meeting/cappe.pdf.

Rauch, H. E., C. T. Striebel, and F. Tung. 1965. “Maximum Likelihood Estimates of Linear Dynamic Systems.” Journal of the American Institute of Aeronautics and Astronautics 3 (8): 1445–50.

Särkkä, Simo. 2013. Bayesian Filtering and Smoothing. New York, NY, USA: Cambridge University Press.

Inferring Parameters for ODEs using Stan

Introduction

The equation of motion for a pendulum of unit length subject to Gaussian white noise is

\displaystyle  \frac{\mathrm{d}^2\alpha}{\mathrm{d}t^2} = -g\sin\alpha + w(t)

We can discretize this via the usual Euler method

\displaystyle  \begin{bmatrix} x_{1,i} \\ x_{2,i} \end{bmatrix} = \begin{bmatrix} x_{1,i-1} + x_{2,i-1}\Delta t \\ x_{2,i-1} - g\sin x_{1,i-1}\Delta t \end{bmatrix} + \mathbf{q}_{i-1}

where q_i \sim {\mathcal{N}}(0,Q) and

\displaystyle  Q = \begin{bmatrix} \frac{q^c \Delta t^3}{3} & \frac{q^c \Delta t^2}{2} \\ \frac{q^c \Delta t^2}{2} & {q^c \Delta t} \end{bmatrix}

The explanation of the precise form of the covariance matrix will be the subject of another blog post; for the purpose of exposition of using Stan and, in particular, Stan’s ability to handle ODEs, this detail is not important.

Instead of assuming that we know g let us take it to be unknown and that we wish to infer its value using the pendulum as our experimental apparatus.

Stan is a probabilistic programming language which should be welll suited to perform such an inference. We use its interface via the R package rstan.

A Stan and R Implementation

Let’s generate some samples using Stan but rather than generating exactly the model we have given above, instead let’s solve the differential equation and then add some noise. Of course this won’t quite give us samples from the model the parameters of which we wish to estimate but it will allow us to use Stan’s ODE solving capability.

Here’s the Stan

functions {
  real[] pendulum(real t,
                  real[] y,
                  real[] theta,
                  real[] x_r,
                  int[] x_i) {
    real dydt[2];
    dydt[1] <- y[2];
    dydt[2] <- - theta[1] * sin(y[1]);
    return dydt;
  }
}
data {
  int<lower=1> T;
  real y0[2];
  real t0;
  real ts[T];
  real theta[1];
  real sigma[2];
}
transformed data {
  real x_r[0];
  int x_i[0];
}
model {
}
generated quantities {
  real y_hat[T,2];
  y_hat <- integrate_ode(pendulum, y0, t0, ts, theta, x_r, x_i);
  for (t in 1:T) {
    y_hat[t,1] <- y_hat[t,1] + normal_rng(0,sigma[1]);
    y_hat[t,2] <- y_hat[t,2] + normal_rng(0,sigma[2]);
  }
}

And here’s the R to invoke it

library(rstan)
library(mvtnorm)

qc1 = 0.0001
deltaT = 0.01
nSamples = 100
m0 = c(1.6, 0)
g = 9.81
t0 = 0.0
ts = seq(deltaT,nSamples * deltaT,deltaT)

bigQ = matrix(c(qc1 * deltaT^3 / 3, qc1 * deltaT^2 / 2,
                qc1 * deltaT^2 / 2,       qc1 * deltaT
                ),
              nrow = 2,
              ncol = 2,
              byrow = TRUE
              )

samples <- stan(file = 'Pendulum.stan',
                data = list (
                    T  = nSamples,
                    y0 = m0,
                    t0 = t0,
                    ts = ts,
                    theta = array(g, dim = 1),
                    sigma = c(bigQ[1,1], bigQ[2,2]),
                    refresh = -1
                ),
                algorithm="Fixed_param",
                seed = 42,
                chains = 1,
                iter =1
                )

We can plot the angle the pendulum subtends to the vertical over time. Note that this is not very noisy.

s <- extract(samples,permuted=FALSE)
plot(s[1,1,1:100])

Now let us suppose that we don’t know the value of g and we can only observe the horizontal displacement.

zStan <- sin(s[1,1,1:nSamples])

Now we can use Stan to infer the value of g.

functions {
  real[] pendulum(real t,
                  real[] y,
                  real[] theta,
                  real[] x_r,
                  int[] x_i) {
    real dydt[2];
    dydt[1] <- y[2];
    dydt[2] <- - theta[1] * sin(y[1]);
    return dydt;
  }
}
data {
  int<lower=1> T;
  real y0[2];
  real z[T];
  real t0;
  real ts[T];
}
transformed data {
  real x_r[0];
  int x_i[0];
}
parameters {
  real theta[1];
  vector<lower=0>[1] sigma;
}
model {
  real y_hat[T,2];
  real z_hat[T];
  theta ~ normal(0,1);
  sigma ~ cauchy(0,2.5);
  y_hat <- integrate_ode(pendulum, y0, t0, ts, theta, x_r, x_i);
  for (t in 1:T) {
    z_hat[t] <- sin(y_hat[t,1]);
    z[t] ~ normal(z_hat[t], sigma);
  }
}

Here’s the R to invoke it.

estimates <- stan(file = 'PendulumInfer.stan',
                  data = list (
                      T  = nSamples,
                      y0 = m0,
                      z  = zStan,
                      t0 = t0,
                      ts = ts
                  ),
                  seed = 42,
                  chains = 1,
                  iter = 1000,
                  warmup = 500,
                  refresh = -1
                  )
e <- extract(estimates,pars=c("theta[1]","sigma[1]","lp__"),permuted=TRUE)

This gives an estiamted valeu for g of 9.809999 which is what we would hope.

Now let’s try adding some noise to our observations.

set.seed(42)
epsilons <- rmvnorm(n=nSamples,mean=c(0.0),sigma=bigR)

zStanNoisy <- sin(s[1,1,1:nSamples] + epsilons[,1])

estimatesNoisy <- stan(file = 'PendulumInfer.stan',
                       data = list (T  = nSamples,
                                    y0 = m0,
                                    z  = zStanNoisy,
                                    t0 = t0,
                                    ts = ts
                                    ),
                       seed = 42,
                       chains = 1,
                       iter = 1000,
                       warmup = 500,
                       refresh = -1
                       )
eNoisy <- extract(estimatesNoisy,pars=c("theta[1]","sigma[1]","lp__"),permuted=TRUE)

This gives an estiamted value for g of 8.5871024 which is ok but not great.

Postamble

To build this page, download the relevant files from github and run this in R:

library(knitr)
knit('Pendulum.Rmd')

And this from command line:

pandoc -s Pendulum.md --filter=./Include > PendulumExpanded.html

Naive Particle Smoothing is Degenerate

Introduction

Let \{X_t\}_{t \geq 1} be a (hidden) Markov process. By hidden, we mean that we are not able to observe it.

\displaystyle  X_1 \sim \mu(\centerdot) \quad X_t \,|\, (X_{t-1} = x) \sim f(\centerdot \,|\, x)

And let \{Y_t\}_{t \geq 1} be an observable Markov process such that

\displaystyle  Y_t \,|\, (X_{t} = x) \sim g(\centerdot \,|\, x)

That is the observations are conditionally independent given the state of the hidden process.

As an example let us take the one given in Särkkä (2013) where the movement of a car is given by Newton’s laws of motion and the acceleration is modelled as white noise.

\displaystyle  \begin{aligned} X_t &= AX_{t-1} + Q \epsilon_t \\ Y_t &= HX_t + R \eta_t \end{aligned}

Although we do not do so here, A, Q, H and R can be derived from the dynamics. For the purpose of this blog post, we note that they are given by

\displaystyle  A = \begin{bmatrix} 1 & 0 & \Delta t & 0        \\ 0 & 1 & 0        & \Delta t \\ 0 & 0 & 1        & 0        \\ 0 & 0 & 0        & 1 \end{bmatrix} , \quad Q = \begin{bmatrix} \frac{q_1^c \Delta t^3}{3} & 0                          & \frac{q_1^c \Delta t^2}{2} & 0                          \\ 0                          & \frac{q_2^c \Delta t^3}{3} & 0                          & \frac{q_2^c \Delta t^2}{2} \\ \frac{q_1^c \Delta t^2}{2} & 0                          & {q_1^c \Delta t}           & 0                          \\ 0                          & \frac{q_2^c \Delta t^2}{2} & 0                          & {q_2^c \Delta t} \end{bmatrix}

and

\displaystyle  H = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} , \quad R = \begin{bmatrix} \sigma_1^2 & 0 \\ 0 & \sigma_2^2 \end{bmatrix}

We wish to determine the position and velocity of the car given noisy observations of the position. In general we need the distribution of the hidden path given the observable path. We use the notation x_{m:n} to mean the path of x starting a m and finishing at n.

\displaystyle  p(x_{1:n} \,|\, y_{1:n}) = \frac{p(x_{1:n}, y_{1:n})}{p(y_{1:n})}

Haskell Preamble

> {-# OPTIONS_GHC -Wall                     #-}
> {-# OPTIONS_GHC -fno-warn-name-shadowing  #-}
> {-# OPTIONS_GHC -fno-warn-type-defaults   #-}
> {-# OPTIONS_GHC -fno-warn-unused-do-bind  #-}
> {-# OPTIONS_GHC -fno-warn-missing-methods #-}
> {-# OPTIONS_GHC -fno-warn-orphans         #-}
> {-# LANGUAGE FlexibleInstances            #-}
> {-# LANGUAGE MultiParamTypeClasses        #-}
> {-# LANGUAGE FlexibleContexts             #-}
> {-# LANGUAGE TypeFamilies                 #-}
> {-# LANGUAGE BangPatterns                 #-}
> {-# LANGUAGE GeneralizedNewtypeDeriving   #-}
> {-# LANGUAGE ScopedTypeVariables          #-}
> {-# LANGUAGE TemplateHaskell              #-}
> module ParticleSmoothing
>   ( simpleSamples
>   , carSamples
>   , testCar
>   , testSimple
>   ) where
> import Data.Random.Source.PureMT
> import Data.Random hiding ( StdNormal, Normal )
> import qualified Data.Random as R
> import Control.Monad.State
> import Control.Monad.Writer hiding ( Any, All )
> import qualified Numeric.LinearAlgebra.HMatrix as H
> import Foreign.Storable ( Storable )
> import Data.Maybe ( fromJust )
> import Data.Bits ( shiftR )
> import qualified Data.Vector as V
> import qualified Data.Vector.Unboxed as U
> import Control.Monad.ST
> import System.Random.MWC
> import Data.Array.Repa ( Z(..), (:.)(..), Any(..), computeP,
>                          extent, DIM1, DIM2, slice, All(..)
>                        )
> import qualified Data.Array.Repa as Repa
> import qualified Control.Monad.Loops as ML
> import PrettyPrint ()
> import Text.PrettyPrint.HughesPJClass ( Pretty, pPrint )
> import Data.Vector.Unboxed.Deriving

Some Theory

If we could sample X_{1:n}^{(i)} \sim p(x_{1:n} \,|\, y_{1:n}) then we could approximate the posterior as

\displaystyle  \hat{p}(x_{1:n} \,|\, y_{1:n}) = \frac{1}{N}\sum_{i=1}^N \delta_{X_{1:n}^{(i)}}(x_{1:n})

If we wish to, we can create marginal estimates

\displaystyle  \hat{p}(x_k \,|\, y_{1:n}) = \frac{1}{N}\sum_{i=1}^N \delta_{X_{k}^{(i)}}(x_{k})

When k = N, this is the filtering estimate.

Standard Bayesian Recursion

Prediction

\displaystyle  \begin{aligned} p(x_n \,|\, y_{1:n-1}) &= \int p(x_{n-1:n} \,|\, y_{1:n-1}) \,\mathrm{d}x_{n-1} \\  &= \int p(x_{n} \,|\, x_{n-1}, y_{1:n-1}) \, p(x_{n-1} \,|\, y_{1:n-1}) \,\mathrm{d}x_{n-1} \\  &= \int f(x_{n} \,|\, x_{n-1}) \, p(x_{n-1} \,|\, y_{1:n-1}) \,\mathrm{d}x_{n-1} \\ \end{aligned}

Update

\displaystyle  \begin{aligned} p(x_n \,|\, y_{1:n}) &= \frac{p(y_n \,|\, x_n, y_{1:n-1}) \, p(x_n \,|\, y_{1:n-1})}                              {p(y_n \,|\, y_{1:n-1})} \\                      &= \frac{g(y_n \,|\, x_n) \, p(x_n \,|\, y_{1:n-1})}                              {p(y_n \,|\, y_{1:n-1})} \end{aligned}

where by definition

\displaystyle  {p(y_n \,|\, y_{1:n-1})} = \int {g(y_n \,|\, x_n) \, p(x_n \,|\, y_{1:n-1})} \,\mathrm{d}x_n

Path Space Recursion

We have

\displaystyle  \begin{aligned} p(x_{1:n} \,|\, y_{1:n}) &= \frac{p(x_{1:n}, y_{1:n})}{p(y_{1:n})} \\ &= \frac{p(x_n, y_n \,|\, x_{1:n-1}, y_{1:n-1})}{p(y_{1:n})} \, p(x_{1:n-1}, y_{1:n-1}) \\ &= \frac{p(y_n \,|\, x_{1:n}, y_{1:n-1}) \, p(x_n \,|\, x_{1:n-1}, y_{1:n-1}) }{p(y_{1:n})} \, p(x_{1:n-1}, y_{1:n-1}) \\ &= \frac{g(y_n \,|\, x_{n}) \, f(x_n \,|\, x_{n-1})}{p(y_n \,|\, y_{1:n-1})} \, \frac{p(x_{1:n-1}, y_{1:n-1})}{ \, p(y_{1:n-1})} \\ &= \frac{g(y_n \,|\, x_{n}) \, f(x_n \,|\, x_{n-1})}{p(y_n \,|\, y_{1:n-1})} \, {p(x_{1:n-1} \,|\,y_{1:n-1})} \\ &= \frac{g(y_n \,|\, x_{n}) \, \overbrace{f(x_n \,|\, x_{n-1}) \, {p(x_{1:n-1} \,|\,y_{1:n-1})}}^{\mathrm{predictive}\,p(x_{1:n} \,|\, y_{1:n-1})}} {p(y_n \,|\, y_{1:n-1})} \\ \end{aligned}

where by definition

\displaystyle  p(y_n \,|\, y_{1:n-1}) = \int g(y_n \,|\, x_n) \, p(x_{1:n} \,|\, y_{1:n-1}) \,\mathrm{d}x_{1:n}

Prediction

\displaystyle  p(x_{1:n} \,|\, y_{1:n-1}) = f(x_n \,|\, x_{n-1}) \, {p(x_{1:n-1} \,|\,y_{1:n-1})}

Update

\displaystyle  p(x_{1:n} \,|\, y_{1:n}) = \frac{g(y_n \,|\, x_{n}) \, {p(x_{1:n} \,|\, y_{1:n-1})}} {p(y_n \,|\, y_{1:n-1})}

Algorithm

The idea is to simulate paths using the recursion we derived above.

At time n-1 we have an approximating distribution

\displaystyle  \hat{p}(x_{1:n-1} \,|\, y_{1:n-1}) = \frac{1}{N}\sum_{i=1}^N \delta_{X_{1:n-1}}^{(i)}(x_{1:n-1})

Sample \tilde{X}_n^{(i)} \sim f(\centerdot \,|\, X_{n-1}^{(i)}) and set \tilde{X}_{1:n}^{(i)} = (\tilde{X}_{1:n-1}^{(i)}, \tilde{X}_n^{(i)}). We then have an approximation of the prediction step

\displaystyle  \hat{p}(x_{1:n} \,|\, y_{1:n-1}) = \frac{1}{N}\sum_{i=1}^N \delta_{\tilde{X}_{1:n}}^{(i)}(x_{1:n})

Substituting

\displaystyle  \begin{aligned} {\hat{p}(y_n \,|\, y_{1:n-1})} &= \int {g(y_n \,|\, x_n) \, \hat{p}(x_{1:n} \,|\, y_{1:n-1})} \,\mathrm{d}x_n \\ &= \int {g(y_n \,|\, x_n)}\frac{1}{N}\sum_{i=1}^N \delta_{\tilde{X}_{1:n-1}}^{(i)}(x_{1:n}) \,\mathrm{d}x_n \\ &= \frac{1}{N}\sum_{i=1}^N {g(y_n \,|\, \tilde{X}_n^{(i)})} \end{aligned}

and again

\displaystyle  \begin{aligned} \tilde{p}(x_{1:n} \,|\, y_{1:n}) &= \frac{g(y_n \,|\, x_{n}) \, {\hat{p}(x_{1:n} \,|\, y_{1:n-1})}}      {\hat{p}(y_n \,|\, y_{1:n-1})} \\ &= \frac{g(y_n \,|\, x_{n}) \, \frac{1}{N}\sum_{i=1}^N \delta_{\tilde{X}_{1:n}}^{(i)}(x_{1:n})}      {\frac{1}{N}\sum_{i=1}^N {g(y_n \,|\, \tilde{X}_n^{(i)})}} \\ &= \frac{ \sum_{i=1}^N g(y_n \,|\, \tilde{X}_n^{(i)}) \, \delta_{\tilde{X}_{1:n}}^{(i)}(x_{1:n})}      {\sum_{i=1}^N {g(y_n \,|\, \tilde{X}_n^{(i)})}} \\ &= \sum_{i=1}^N W_n^{(i)} \delta_{\tilde{X}_{1:n}^{(i)}} (x_{1:n}) \end{aligned}

where W_n^{(i)} \propto g(y_n \,|\, \tilde{X}_n^{(i)}) and \sum_{i=1}^N W_n^{(i)} = 1.

Now sample

\displaystyle  X_{1:n}^{(i)} \sim \tilde{p}(x_{1:n} \,|\, y_{1:n})

A Haskell Implementation

Let’s specify some values for the example of the car moving in two dimensions.

> deltaT, sigma1, sigma2, qc1, qc2 :: Double
> deltaT = 0.1
> sigma1 = 1/2
> sigma2 = 1/2
> qc1 = 1
> qc2 = 1
> bigA :: H.Matrix Double
> bigA = (4 H.>< 4) bigAl
> bigAl :: [Double]
> bigAl = [1, 0 , deltaT,      0,
>          0, 1,       0, deltaT,
>          0, 0,       1,      0,
>          0, 0,       0,      1]
> bigQ :: H.Herm Double
> bigQ = H.trustSym $ (4 H.>< 4) bigQl
> bigQl :: [Double]
> bigQl = [qc1 * deltaT^3 / 3,                  0, qc1 * deltaT^2 / 2,                  0,
>                           0, qc2 * deltaT^3 / 3,                  0, qc2 * deltaT^2 / 2,
>          qc1 * deltaT^2 / 2,                  0,       qc1 * deltaT,                  0,
>                           0, qc2 * deltaT^2 / 2,                  0,       qc2 * deltaT]
> bigH :: H.Matrix Double
> bigH = (2 H.>< 4) [1, 0, 0, 0,
>                    0, 1, 0, 0]
> bigR :: H.Herm Double
> bigR = H.trustSym $ (2 H.>< 2) [sigma1^2,        0,
>                                        0, sigma2^2]
> m0 :: H.Vector Double
> m0 = H.fromList [0, 0, 1, -1]
> bigP0 :: H.Herm Double
> bigP0 = H.trustSym $ H.ident 4
> n :: Int
> n = 23

With these we generate hidden and observable sample path.

> carSample :: MonadRandom m =>
>              H.Vector Double ->
>              m (Maybe ((H.Vector Double, H.Vector Double), H.Vector Double))
> carSample xPrev = do
>   xNew <- sample $ rvar (Normal (bigA H.#> xPrev) bigQ)
>   yNew <- sample $ rvar (Normal (bigH H.#> xNew) bigR)
>   return $ Just ((xNew, yNew), xNew)
> carSamples :: [(H.Vector Double, H.Vector Double)]
> carSamples = evalState (ML.unfoldrM carSample m0) (pureMT 17)

We can plot an example trajectory for the car and the noisy observations that are available to the smoother / filter.

Sadly there is no equivalent to numpy in Haskell. Random number packages generate vectors, for multi-rank arrays there is repa and for fast matrix manipulation there is hmtatrix. Thus for our single step path update function, we have to pass in functions to perform type conversion. Clearly with all the copying inherent in this approach, performance is not going to be great.

The type synonym ArraySmoothing is used to denote the cloud of particles at each time step.

> type ArraySmoothing = Repa.Array Repa.U DIM2
> singleStep :: forall a . U.Unbox a =>
>               (a -> H.Vector Double) ->
>               (H.Vector Double -> a) ->
>               H.Matrix Double ->
>               H.Herm Double ->
>               H.Matrix Double ->
>               H.Herm Double ->
>               ArraySmoothing a -> H.Vector Double ->
>               WriterT [ArraySmoothing a] (StateT PureMT IO) (ArraySmoothing a)
> singleStep f g bigA bigQ bigH bigR x y = do
>   tell[x]
>   let (Z :. ix :. jx) = extent x
> 
>   xHatR <- lift $ computeP $ Repa.slice x (Any :. jx - 1)
>   let xHatH = map f $ Repa.toList (xHatR  :: Repa.Array Repa.U DIM1 a)
>   xTildeNextH <- lift $ mapM (\x -> sample $ rvar (Normal (bigA H.#> x) bigQ)) xHatH
> 
>   let xTildeNextR = Repa.fromListUnboxed (Z :. ix :. (1 :: Int)) $
>                     map g xTildeNextH
>       xTilde = Repa.append x xTildeNextR
> 
>       weights = map (normalPdf y bigR) $
>                 map (bigH H.#>) xTildeNextH
>       vs = runST (create >>= (asGenST $ \gen -> uniformVector gen n))
>       cumSumWeights = V.scanl (+) 0 (V.fromList weights)
>       totWeight = sum weights
>       js = indices (V.map (/ totWeight) $ V.tail cumSumWeights) vs
>       xNewV = V.map (\j -> Repa.transpose $
>                            Repa.reshape (Z :. (1 :: Int) :. jx + 1) $
>                            slice xTilde (Any :. j :. All)) js
>       xNewR = Repa.transpose $ V.foldr Repa.append (xNewV V.! 0) (V.tail xNewV)
>   computeP xNewR

The state for the car is a 4-tuple.

> data SystemState a = SystemState { xPos  :: a
>                                  , yPos  :: a
>                                  , _xSpd :: a
>                                  , _ySpd :: a
>                                  }

We initialize the smoother from some prior distribution.

> initCar :: StateT PureMT IO (ArraySmoothing (SystemState Double))
> initCar = do
>   xTilde1 <- replicateM n $ sample $ rvar (Normal m0 bigP0)
>   let weights = map (normalPdf (snd $ head carSamples) bigR) $
>                 map (bigH H.#>) xTilde1
>       vs = runST (create >>= (asGenST $ \gen -> uniformVector gen n))
>       cumSumWeights = V.scanl (+) 0 (V.fromList weights)
>       js = indices (V.tail cumSumWeights) vs
>       xHat1 = Repa.fromListUnboxed (Z :. n :. (1 :: Int)) $
>               map ((\[a,b,c,d] -> SystemState a b c d) . H.toList) $
>               V.toList $
>               V.map ((V.fromList xTilde1) V.!) js
>   return xHat1

Now we can run the smoother.

> smootherCar :: StateT PureMT IO
>             (ArraySmoothing (SystemState Double)
>             , [ArraySmoothing (SystemState Double)])
> smootherCar = runWriterT $ do
>   xHat1 <- lift initCar
>   foldM (singleStep f g bigA bigQ bigH bigR) xHat1 (take 100 $ map snd $ tail carSamples)
> f :: SystemState Double -> H.Vector Double
> f (SystemState a b c d) = H.fromList [a, b, c, d]
> g :: H.Vector Double -> SystemState Double
> g = (\[a,b,c,d] -> (SystemState a b c d)) . H.toList

And create inferred positions for the car which we then plot.

> testCar :: IO ([Double], [Double])
> testCar = do
>   states <- snd <$> evalStateT smootherCar (pureMT 24)
>   let xs :: [Repa.Array Repa.D DIM2 Double]
>       xs = map (Repa.map xPos) states
>   sumXs :: [Repa.Array Repa.U DIM1 Double] <- mapM Repa.sumP (map Repa.transpose xs)
>   let ixs = map extent sumXs
>       sumLastXs = map (* (recip $ fromIntegral n)) $
>                   zipWith (Repa.!) sumXs (map (\(Z :. x) -> Z :. (x - 1)) ixs)
>   let ys :: [Repa.Array Repa.D DIM2 Double]
>       ys = map (Repa.map yPos) states
>   sumYs :: [Repa.Array Repa.U DIM1 Double] <- mapM Repa.sumP (map Repa.transpose ys)
>   let ixsY = map extent sumYs
>       sumLastYs = map (* (recip $ fromIntegral n)) $
>                   zipWith (Repa.!) sumYs (map (\(Z :. x) -> Z :. (x - 1)) ixsY)
>   return (sumLastXs, sumLastYs)

So it seems our smoother does quite well at inferring the state at the latest observation, that is, when it is working as a filter. But what about estimates for earlier times? We should do better as we have observations in the past and in the future. Let’s try with a simpler example and a smaller number of particles.

First we create some samples for our simple 1 dimensional linear Gaussian model.

> bigA1, bigQ1, bigR1, bigH1 :: Double
> bigA1 = 0.5
> bigQ1 = 0.1
> bigR1 = 0.1
> bigH1 = 1.0
> simpleSample :: MonadRandom m =>
>               Double ->
>               m (Maybe ((Double, Double), Double))
> simpleSample xPrev = do
>   xNew <- sample $ rvar (R.Normal (bigA1 * xPrev) bigQ1)
>   yNew <- sample $ rvar (R.Normal (bigH1 * xNew) bigR1)
>   return $ Just ((xNew, yNew), xNew)
> simpleSamples :: [(Double, Double)]
> simpleSamples = evalState (ML.unfoldrM simpleSample 0.0) (pureMT 17)

Again create a prior.

> initSimple :: MonadRandom m => m (ArraySmoothing Double)
> initSimple = do
>   let y = snd $ head simpleSamples
>   xTilde1 <- replicateM n $ sample $ rvar $ R.Normal y bigR1
>   let weights = map (pdf (R.Normal y bigR1)) $
>                 map (bigH1 *) xTilde1
>       totWeight = sum weights
>       vs = runST (create >>= (asGenST $ \gen -> uniformVector gen n))
>       cumSumWeights = V.scanl (+) 0 (V.fromList $ map (/ totWeight) weights)
>       js = indices (V.tail cumSumWeights) vs
>       xHat1 = Repa.fromListUnboxed (Z :. n :. (1 :: Int)) $
>               V.toList $
>               V.map ((V.fromList xTilde1) V.!) js
>   return xHat1

Now we can run the smoother.

> smootherSimple :: StateT PureMT IO (ArraySmoothing Double, [ArraySmoothing Double])
> smootherSimple = runWriterT $ do
>   xHat1 <- lift initSimple
>   foldM (singleStep f1 g1 ((1 H.>< 1) [bigA1]) (H.trustSym $ (1 H.>< 1) [bigQ1^2])
>                           ((1 H.>< 1) [bigH1]) (H.trustSym $ (1 H.>< 1) [bigR1^2]))
>         xHat1
>         (take 20 $ map H.fromList $ map return . map snd $ tail simpleSamples)
> f1 :: Double -> H.Vector Double
> f1 a = H.fromList [a]
> g1 :: H.Vector Double -> Double
> g1 = (\[a] -> a) . H.toList

And finally we can look at the paths not just the means of the marginal distributions at the latest observation time.

> testSimple :: IO [[Double]]
> testSimple = do
>   states <- snd <$> evalStateT smootherSimple (pureMT 24)
>   let path :: Int -> IO (Repa.Array Repa.U DIM1 Double)
>       path i = computeP $ Repa.slice (last states) (Any :. i :. All)
>   paths <- mapM path [0..n - 1]
>   return $ map Repa.toList paths

We can see that at some point in the past all the current particles have one ancestor. The marginals of the smoothing distribution (at some point in the past) have collapsed on to one particle.

Notes

Helpers for the Inverse CDF

That these are helpers for the inverse CDF is delayed to another blog post.

> indices :: V.Vector Double -> V.Vector Double -> V.Vector Int
> indices bs xs = V.map (binarySearch bs) xs
> binarySearch :: Ord a =>
>                 V.Vector a -> a -> Int
> binarySearch vec x = loop 0 (V.length vec - 1)
>   where
>     loop !l !u
>       | u <= l    = l
>       | otherwise = let e = vec V.! k in if x <= e then loop l k else loop (k+1) u
>       where k = l + (u - l) `shiftR` 1

Multivariate Normal

The random-fu package does not contain a sampler or pdf for a multivariate normal so we create our own. This should be added to random-fu-multivariate package or something similar.

> normalMultivariate :: H.Vector Double -> H.Herm Double -> RVarT m (H.Vector Double)
> normalMultivariate mu bigSigma = do
>   z <- replicateM (H.size mu) (rvarT R.StdNormal)
>   return $ mu + bigA H.#> (H.fromList z)
>   where
>     (vals, bigU) = H.eigSH bigSigma
>     lSqrt = H.diag $ H.cmap sqrt vals
>     bigA = bigU H.<> lSqrt
> data family Normal k :: *
> data instance Normal (H.Vector Double) = Normal (H.Vector Double) (H.Herm Double)
> instance Distribution Normal (H.Vector Double) where
>   rvar (Normal m s) = normalMultivariate m s
> normalPdf :: (H.Numeric a, H.Field a, H.Indexable (H.Vector a) a, Num (H.Vector a)) =>
>              H.Vector a -> H.Herm a -> H.Vector a -> a
> normalPdf mu sigma x = exp $ normalLogPdf mu sigma x
> normalLogPdf :: (H.Numeric a, H.Field a, H.Indexable (H.Vector a) a, Num (H.Vector a)) =>
>                  H.Vector a -> H.Herm a -> H.Vector a -> a
> normalLogPdf mu bigSigma x = - H.sumElements (H.cmap log (diagonals dec))
>                               - 0.5 * (fromIntegral (H.size mu)) * log (2 * pi)
>                               - 0.5 * s
>   where
>     dec = fromJust $ H.mbChol bigSigma
>     t = fromJust $ H.linearSolve (H.tr dec) (H.asColumn $ x - mu)
>     u = H.cmap (\x -> x * x) t
>     s = H.sumElements u
> diagonals :: (Storable a, H.Element t, H.Indexable (H.Vector t) a) =>
>              H.Matrix t -> H.Vector a
> diagonals m = H.fromList (map (\i -> m H.! i H.! i) [0..n-1])
>   where
>     n = max (H.rows m) (H.cols m)
> instance PDF Normal (H.Vector Double) where
>   pdf (Normal m s) = normalPdf m s
>   logPdf (Normal m s) = normalLogPdf m s

Misellaneous

> derivingUnbox "SystemState"
>     [t| forall a . (U.Unbox a) => SystemState a -> (a, a, a, a) |]
>     [| \ (SystemState x y xdot ydot) -> (x, y, xdot, ydot) |]
>     [| \ (x, y, xdot, ydot) -> SystemState x y xdot ydot |]
> instance Pretty a => Pretty (SystemState a) where
>   pPrint (SystemState x y xdot ydot ) = pPrint (x, y, xdot, ydot)

Bibliography

Särkkä, Simo. 2013. Bayesian Filtering and Smoothing. New York, NY, USA: Cambridge University Press.

Floating Point: A Faustian Bargain?

Every so often, someone bitten by floating point arithmetic behaving in an unexpected way is tempted to suggest that a calculation should be done be precisely and rounding done at the end. With floating point rounding is done at every step.

Here’s an example of why floating point might really be the best option for numerical calculations.

Suppose you wish to find the roots of a quintic equation.

> import Numeric.AD
> import Data.List
> import Data.Ratio
> p :: Num a => a -> a
> p x = x^5 - 2*x^4 - 3*x^3 + 3*x^2 - 2*x - 1

We can do so using Newton-Raphson using automatic differentiation to calculate the derivative (even though for polynomials this is trivial).

> nr :: Fractional a => [a]
> nr = unfoldr g 0
>   where
>     g z = let u = z - (p z) / (h z) in Just (u, u)
>     h z = let [y] = grad (\[x] -> p x) [z] in y

After 7 iterations we see the size of the denominator is quite large (33308 digits) and the calculation takes many seconds.

ghci> length $ show $ denominator (nr!!7)
  33308

On the other hand if we use floating point we get an answer accurate to 1 in 2^{53} after 7 iterations very quickly.

ghci> mapM_ putStrLn $ map show $ take 7 nr
  -0.5
  -0.3368421052631579
  -0.31572844839628944
  -0.31530116270327685
  -0.31530098645936266
  -0.3153009864593327
  -0.3153009864593327

The example is taken from here who refers the reader to Nick Higham’s book: Accuracy and Stability of Numerical Algorithms.

Of course we should check we found a right answer.

ghci> p $ nr!!6
  0.0