カルマンフィルタ

時間ごとに測定した位置から、次の位置の推測とかができる。

ガウシアン：

$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\{-\frac{1}{2} \cdot \frac{(x - \mu)^2}{\sigma^2}\}$

ガウシアンの合成：

$\begin{eqnarray*} \mu^{'} &=& \frac{1}{\gamma^2 + \sigma^2} (\gamma^2 \mu + \sigma^2 \nu)\\ \sigma^{2'} &=& \frac{1}{\frac{1}{\gamma^2} + \frac{1}{\sigma^2}} \end{eqnarray*}$

共分散 $\sigma^{2'}$ は、 $\sigma^2$ 、 $\gamma^2$ のどちらよりも小さくなる。
Kalman filterのアップデートで、事前のガウシアンと観測によるガウシアンから推測する新しいガウシアンのピークがどちらよりも高くなるのって直感には反するね。並が重なりあって高め合う場合にはわかるけど、そうじゃなくても高くなるってのは不思議。
予測はもっと簡単で、単に平均と共分散を足すだけ。
カルマンフィルタ予測
速度の予測ができる
平均はベクトル、共分散はマトリクス
1次元を位置（直接観測できる）、1次元を速度（観測できない）に割り振る
速度と位置の間には強い関係があるので(x’ = x + v)、速度を直接観測できなくても、観測された位置から推測できる
多次元にした場合の数式、むちゃくちゃ複雑だな

# Fill in the matrices P, F, H, R and I at the bottom
#
# This question requires NO CODING, just fill in the
# matrices where indicated. Please do not delete or modify
# any provided code OR comments. Good luck!

from math import *

class matrix:

    # implements basic operations of a matrix class

    def __init__(self, value):
        self.value = value
        self.dimx = len(value)
        self.dimy = len(value[0])
        if value == [[]]:
            self.dimx = 0

    def zero(self, dimx, dimy):
        # check if valid dimensions
        if dimx < 1 or dimy < 1:
            raise ValueError, "Invalid size of matrix"
        else:
            self.dimx = dimx
            self.dimy = dimy
            self.value = [[0 for row in range(dimy)] for col in range(dimx)]

    def identity(self, dim):
        # check if valid dimension
        if dim < 1:
            raise ValueError, "Invalid size of matrix"
        else:
            self.dimx = dim
            self.dimy = dim
            self.value = [[0 for row in range(dim)] for col in range(dim)]
            for i in range(dim):
                self.value[i][i] = 1

    def show(self):
        for i in range(self.dimx):
            print self.value[i]
        print ' '

    def __add__(self, other):
        # check if correct dimensions
        if self.dimx != other.dimx or self.dimy != other.dimy:
            raise ValueError, "Matrices must be of equal dimensions to add"
        else:
            # add if correct dimensions
            res = matrix([[]])
            res.zero(self.dimx, self.dimy)
            for i in range(self.dimx):
                for j in range(self.dimy):
                    res.value[i][j] = self.value[i][j] + other.value[i][j]
            return res

    def __sub__(self, other):
        # check if correct dimensions
        if self.dimx != other.dimx or self.dimy != other.dimy:
            raise ValueError, "Matrices must be of equal dimensions to subtract"
        else:
            # subtract if correct dimensions
            res = matrix([[]])
            res.zero(self.dimx, self.dimy)
            for i in range(self.dimx):
                for j in range(self.dimy):
                    res.value[i][j] = self.value[i][j] - other.value[i][j]
            return res

    def __mul__(self, other):
        # check if correct dimensions
        if self.dimy != other.dimx:
            raise ValueError, "Matrices must be m*n and n*p to multiply"
        else:
            # subtract if correct dimensions
            res = matrix([[]])
            res.zero(self.dimx, other.dimy)
            for i in range(self.dimx):
                for j in range(other.dimy):
                    for k in range(self.dimy):
                        res.value[i][j] += self.value[i][k] * other.value[k][j]
            return res

    def transpose(self):
        # compute transpose
        res = matrix([[]])
        res.zero(self.dimy, self.dimx)
        for i in range(self.dimx):
            for j in range(self.dimy):
                res.value[j][i] = self.value[i][j]
        return res

    # Thanks to Ernesto P. Adorio for use of Cholesky and CholeskyInverse functions

    def Cholesky(self, ztol=1.0e-5):
        # Computes the upper triangular Cholesky factorization of
        # a positive definite matrix.
        res = matrix([[]])
        res.zero(self.dimx, self.dimx)

        for i in range(self.dimx):
            S = sum([(res.value[k][i])**2 for k in range(i)])
            d = self.value[i][i] - S
            if abs(d) < ztol:
                res.value[i][i] = 0.0
            else:
                if d < 0.0:
                    raise ValueError, "Matrix not positive-definite"
                res.value[i][i] = sqrt(d)
            for j in range(i+1, self.dimx):
                S = sum([res.value[k][i] * res.value[k][j] for k in range(self.dimx)])
                if abs(S) < ztol:
                    S = 0.0
                res.value[i][j] = (self.value[i][j] - S)/res.value[i][i]
        return res

    def CholeskyInverse(self):
        # Computes inverse of matrix given its Cholesky upper Triangular
        # decomposition of matrix.
        res = matrix([[]])
        res.zero(self.dimx, self.dimx)

        # Backward step for inverse.
        for j in reversed(range(self.dimx)):
            tjj = self.value[j][j]
            S = sum([self.value[j][k]*res.value[j][k] for k in range(j+1, self.dimx)])
            res.value[j][j] = 1.0/tjj**2 - S/tjj
            for i in reversed(range(j)):
                res.value[j][i] = res.value[i][j] = -sum([self.value[i][k]*res.value[k][j] for k in range(i+1, self.dimx)])/self.value[i][i]
        return res

    def inverse(self):
        aux = self.Cholesky()
        res = aux.CholeskyInverse()
        return res

    def __repr__(self):
        return repr(self.value)


########################################

def filter(x, P):
    for n in range(len(measurements)):

        # prediction
        x = (F * x) + u
        P = F * P * F.transpose()

        # measurement update
        Z = matrix([measurements[n]])
        y = Z.transpose() - (H * x)
        S = H * P * H.transpose() + R
        K = P * H.transpose() * S.inverse()
        x = x + (K * y)
        P = (I - (K * H)) * P

    print 'x= '
    x.show()
    print 'P= '
    P.show()

########################################

print "### 4-dimensional example ###"

measurements = [[5., 10.], [6., 8.], [7., 6.], [8., 4.], [9., 2.], [10., 0.]]
initial_xy = [4., 12.]

# measurements = [[1., 4.], [6., 0.], [11., -4.], [16., -8.]]
# initial_xy = [-4., 8.]

# measurements = [[1., 17.], [1., 15.], [1., 13.], [1., 11.]]
# initial_xy = [1., 19.]

dt = 0.1

x = matrix([[initial_xy[0]], [initial_xy[1]], [0.], [0.]]) # initial state (location and velocity)
u = matrix([[0.], [0.], [0.], [0.]]) # external motion

#### DO NOT MODIFY ANYTHING ABOVE HERE ####
#### fill this in, remember to use the matrix() function!: ####

P = matrix([[0., 0, 0, 0],
            [0, 0., 0, 0],
            [0, 0, 1000., 0],
            [0, 0, 0, 1000.]])  # initial uncertainty
F = matrix([[1., 0, dt, 0],
            [0, 1., 0, dt],
            [0, 0, 1., 0],
            [0, 0, 0, 1.]])  # next state function
H = matrix([[1., 0, 0, 0],
            [0, 1., 0, 0]])  # measurement function
R = matrix([[0.1, 0],
            [0, 0.1]]) # measurement uncertainty
I = matrix([[1., 0, 0, 0],
            [0, 1., 0, 0],
            [0, 0, 1., 0],
            [0, 0, 0, 1.]]) # identity matrix

###### DO NOT MODIFY ANYTHING HERE #######

filter(x, P)

$x$ が位置と速度を表す、ガウシアンの平均 $\mu$
$P$ が、ガウシアンの共分散マトリクス $\Sigma$

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