COECOE 出張報告　２００６年１２月出張報告　２００６年１２月１４日１４日

Tutorials – December 4, 2006Conference Sessions – December 4-7, 2006Hyatt Regency, Vancouver, B.C., Canada

Workshops – December 8-9, 2006Westin and Hilton, Whistler, B.C., Canada

Neural Information Processing Systems 2006

物理学第一教室　非線形動力学研究室　島崎 秀昭

参加者数参加者数 約１０００人約１０００人発表件数発表件数 ２０４件２０４件

論文投稿　８３３件

Peer Review 　 (Double blind)2~4 人の レビューアーによる採点　１０点満点４人による採点

合否　２０４件が受理　ポスター発表

６３件が口頭発表（講演 20 分 / スポットライト２分）

NIPS*0NIPS*066

HistogramHistogram

The duration for eruptions of the Old Faithful geyser in Yellowstone National Park (in minutes)

Bin widthBin width

2ˆMISE t t dt Histogram

Underlying Rate

2

2( )n

k vC

Optimizing a time-Optimizing a time-histogramhistogram

Unknown

MethodsMethods

1

1,

N

ii

k kN

2

1

1( )

N

ii

v k kN

2

2( )

( )n

k vC

n

Calculate the mean and variance of the number of spikes.

Compute the cost function

Repeat i through iii while changing the bin size Δ. Find Δ* that minimize the cost function.

Divide spike sequences with length T [s] into N bins of width Δ.

Method: Selection of the Bin SizeMethod: Selection of the Bin Size(i)

(ii)

(iii)

(iv)

Data : Britten et al. (2004) neural signal archive

Rate modulation of an MT Rate modulation of an MT neuronneuron

2

1 1( )m n

kC n C

m n n

Extrapolation

Too few to make

a Histogram !

Estimation: At least 12 trials are

required.

Optimized Histogram

How many trials are required to make a How many trials are required to make a Histogram?Histogram?

Required # of sequences (Estimation)

Extrapolated:

Finite optimal bin size

Original: Optimal bin size diverges

Optimal bin size v.s. m

# of sequences used

2

1 1( )m n

kC n C

m n n

( )nC

Required # of trials

2

1 2 1 22 0 0

1

nC n

t t dt dtn

2 31 1( ) 0 0 0 ( )

3 12nC On

(i) Expansion of the cost function by :

1 3

6~

0 n

2

2

1 1~ | |

1 1 1 1

n

c

C t dt t t dtn

un n

1cn t dt

(ii) Expansion of the cost function by 1/

Critical number of trials:

The second order phase transition.

Scaling of the optimal bin size:

Theoretical cost function:Theoretical cost function:

The mean and correlation function of the underlying rate is known.

See also Koyama, S. and Shinomoto, S. J. Phys. A, 37(29):7255–7265. 2004

ReferenceReference

A Method for Selecting the Bin Size of a Time Histogram

Hideaki Shimazaki and Shigeru ShinomotoNeural Computation in Press

島崎秀昭　学位論文　＠　４階図書室

•http://www.ton.scphys.kyoto-u.ac.jp/~hideaki/

Time-Varying Rate

Spike Sequences

Time Histogram

The spike count in the bin obeys the Poisson distribution*:

A histogram bar-height is an estimator of :

The mean underlying rate in an interval [0, ]:

0

1t dt

( )

k

nnp k n e

k

ˆ k

n

*When the spikes are obtained by repeating an independent trial, the accumulated data obeys the Poisson point process due to a general limit theorem.

TheoryTheory

22

0

1ˆMISE ( ) .tE dt

2 2

0 0

1 1ˆ ˆMISE ( ) ( )T

t t n tE dt dtE

T

Expectation by the Poisson statistics, given the rate.

2 22

0 0

1 1t tdt dt

Variance of the rate Variance of

an ideal histogram

Decomposition of the Systematic Error

Systematic ErrorSampling Error

Average over segmented bins.

Independent of

Method I. Selection of the Bin SizeMethod I. Selection of the Bin Size

2

0

22

1( ) MISE

ˆ( )

T

n t

n

C dtT

E

22ˆ ˆ ˆ2n n n nC E E E

22 ˆ ˆ ˆn n n nC E E E

n

Variance of a Histogram

2 22ˆ ˆ ˆ( ) .n n nE E E

2 1ˆ ˆ( )n nE En

Introduction of the cost function:

The variance decomposition:

Mean of a Histogram

The Poisson statistics obeys:

Unknown: Variance of ideal histogramSampling error

Variance of a histogram Sampling error