Post on 13-Jan-2016
description
計量行動分析 第 7 回
離散選択モデルDiscrete (Qualitative)
Choice
ロジットモデルLogit Model
連続的選択と離散的選択Continuous choice and discrete choice
標準的消費者行動モデル ( 連続的選択 ) Continuous(Quantitative) Choice
一定期間に購入する 財・サービスの量を説明 （お金・時間をいくらずつ割り振るか？）
離散的（質的）選択モデル Discrete(Quanlitative)Choice
どの種類の財を選択するか？ （どこへ行くか？何を買うか？） 離散的 (discrete) な 選択肢 (alternative) からの選択
5 万円
離散的選択のモデル化Modeling of discrete choice 個人は，採りうる選択肢 alternative を列挙する Individual enumerates alternatives 各選択肢の特徴と費用を考え、評価点をつける Assign evaluation points for each alternative. 評価点が高いものを選ぶ Chose the alternative having the highest point
China 60 点 France 40 点 USA 50 点
確定的選択モデルDeterministic Choice Models
個人は「評価得点が少しでも高いほうの選択肢を必ず選ぶ」と考えるモデル
Individual always chose the alternative of the highest points. 同じ状況に直面する人は，全員同じ選択肢を選択
異なる条件の人の選択結果から効用関数を推定 判別関数モデル，数量化理論 II 類モデル
同じ状況に対する個人間での考え方の違いを考慮 例：高齢者は着席可能性を，若者は低運賃を重視 犠牲量 ( 最小化）モデル
比較における「あいまいさ」を認める ファジィ選択モデル Fuzzy Choice Model
確率的選択：評価点の差と選択率Probablistic Choice
実際には In many cases, 評価点に差が小さければ、どちらの選択肢も選ばれ
る可能性がある Both alternatives are possibly chosen, if the difference of utility was not large.
評価点の差が大きいときは，片方しか選ばれない．
選択肢 A が選ばれる可能性 Probability of choice of A
1
0 選択肢 A の得点－選択肢 B の得点Utility of A – Utility of B
2 つは同じ魅力50 ％ずつ
A が圧倒的に良い ほとんど A だけが選ばれるUtility of A is very superior to that of B, A is almost chosen.
A が圧倒的に劣るA が選ばれることはほとんどないUtility of A is very inferior to that of B, A is rarely chosen.
ロジットモデルLogit Model S 字型の曲線として , 次のような数式を使
うと
いろいろな計算が簡単にできる 3 つ以上の選択肢からの選択も同じ形になる
2000 年ノーベル経済学賞 McFadden （ 1937- ）が提案
各自の評価点が安定している部分と確率的に変動する部分の和である場合の選択から理論的に導いた。（ランダム効用モデル）
2 項ロジットモデルBinary （ Binomial) Logit Model
選択肢が 2 つの場合 2 項ロジットモデル When they have two alternatives
選択肢が多数（ｎ個）ある場合 Many alternatives 多項ロジットモデル Multinomial Logit
N
k k
jj
V
VP
1)exp(
)exp(
121
22
21
11 1
)exp()exp(
)exp(,
)exp()exp(
)exp(P
VV
VP
VV
VP
モデルの使用手順選択モデルの定式化（得点から S カーブで選択率を計算）Formulation of choice shares
パラメータ η ， β の推定 estimation of parameter values（実際の選択結果から、どういう得点付けだったかを推測）
将来の選択率の予測 prospect for choice share in the future（将来の選択肢の状況から得点を計算し、選択率を計算）
実際の観測結果から、もとの事象の発生確率を推定する 手品師がイカサマかも知れないコインを 6
回振ったら、表が 5 回、裏が 1回出た。 このコインは正しいコインだろうか？表が出や
すいようなイカサマのコインだろうか？ ( 検定 )
表 5 回、裏1回という観測結果から、このコインを 1回振ったときに表が出る確率 pの値を推測したい。 比率を用いてそのままｐ＝ 5/6 と推測する方法 さいゆうほう（最尤法） Maximum Likelihood
比率によるロジットモデルの推定Estimation of parameters in Logit model based on the aggregated shares
集計的方法 ( 集団の選択率にあわせる ) ある選択肢の状況下で観測された集団の選択確率ｐを用
いる Observed share ロジットモデルから、その時の 2 つの選択肢の魅力度
VA と VB の差が逆算できる
Utility differences are reversely calculated to meet with the obsereved share of two alternatives.
魅力度の差がうまく一致するように、魅力度の関数の形を調整する
Adjust the parameters (functions) such that the proper utility difference are obtained.
回帰分析 Linear Regression
1x 2x
y
2 つ ( 以上 ) の変数を用いて、目的の変数 y の値をうまく予測できるような平面を決める方法
]/ln[ BUSij
CARij PPy
)(1BUSij
CARij ttx
)(2BUSij
CARij ccx
21 xxy平面
をうまく決める
最尤法推定 Maximum Likelihood( 個々の事象の組み合わせを考える）
コインを続けて 6回振ったところ、そのうち 5 回が表であった．このコインの表の出る確率ｑはいくらか？
母数（ここでは表の出る確率ｑ )を変えたときに，観察された事象が実現する確率を求める ( 尤度関数 L(ｑ ) ）
観察された事象の発生確率が最大になるｑの値を求める
尤度関数を求めてみよう 確率ｑの事象が 5 回，（１－ｑ）の事象が 1回観察された
のだから、何回目が裏かが 6 通りあることを踏まえると、
)1(6)1(!1!5
!6)1()( 5515
56 qqqqqqCqL
最尤法による母数の推定の例Maximum likelihood: Example
L(θ )
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.00 0.20 0.40 0.60 0.80 1.00
)1(6)( 5 qqqL 尤度関数を最大化する
あるいは対数をとったものをｑについて最大化してもよい
333.86/5
0)1(
65
)1(
)1(5
)1(
15*
)1ln(ln56ln)(ln)(*
q
q
qqdq
dL
qqqLqL
これは， 6回のうち 5 回という割合に一致
0)65(6
)55(6)1(56)(
4
454
qqqqqqdq
qdL
最尤法によるロジットモデルの推定Disaggregated estimation by ML
非集計的方法 ( 各個人の選択を考える ) 魅力度の関数形を仮定する 各個人が直面した選択肢の状況をもとに、
ロジットモデルで各自の選択確率を求める。 それらを掛け合わせて、観測された事象の
発生確率 ( 尤度関数 ) を求める
尤度関数の値が最大になるように、魅力度の関数の形を調整する
njN
n nj
J
jPL
1 1
交通手段選択モデルTravel mode choice model ある OD を移動する消費者が ,複数の交通
手段の所要時間や費用を考えて手段を選択Utility for each mode are function of time
and cost
2 項ロジットモデル
CARij
CARij
CARij
BUSij
BUSij
BUSij
ctV
ctV
2,1
)exp(
)exp(
k k
i
V
V
jii VU
])(: ob[Pr KikUUP kii 選択肢 jの魅力度が他の選択肢よりも高い確率
2 項ロジットモデルの図解Scheme of binary logit model
X2
X1
V
V*1
0
集計データ ( 比率 ) を用いたロジットモデルの推定
表 3.15 バスの所要時間
ｔBij
1 2 3
1 5 11 13
2 10 12 12
3 14 16 7
単位：分
表 3.16 自動車の所要時間
tCij 1 2 3
1 3 8 10
2 8 7 11
3 10 11 3
単位：分
表 3.17 バスの所要費用
cBij 1 2 3
1 130 140 180
2 140 130 220
3 180 220 130
単位：円
表 3.18 自動車の所要費用
cCij 1 2 3
1 21 45 58
2 45 42 60
3 58 60 19
単位：円
表 3.19 バスの分担率
PBij 1 2 3
10.27
30.26
50.25
3
20.28
20.24
80.25
5
30.23
90.19
20.24
4
表 3.20 自動車の分担率
PCij 1 2 3
10.72
70.73
50.74
7
20.71
80.75
20.74
5
30.76
10.80
80.75
6
集計データ ( 比率 ) を用いた線形回帰分析による推定
)()(]/ln[ BUSij
CARij
BUSij
CARij
BUSij
CARij ccttPP
Bijｔ Cijｔ Bijｃ cCij PBij PCij ln(PC/ PB) C- tBｔ C- cBｃ5 3 130 21 0.273 0.727 0.979 - 2 - 109
10 8 140 45 0.282 0.718 0.935 - 2 - 9514 10 180 58 0.239 0.761 1.158 - 4 - 12211 8 140 45 0.265 0.735 1.020 - 3 - 9512 7 130 42 0.248 0.752 1.109 - 5 - 8816 11 220 60 0.192 0.808 1.437 - 5 - 16013 10 180 58 0.253 0.747 1.083 - 3 - 12212 11 220 60 0.255 0.745 1.072 - 1 - 1607 3 130 19 0.244 0.756 1.131 - 4 - 111
ロジットモデル式から、 2 つの選択率の比は以下のようになる
集計データ ( 比率 ) を用いた線形回帰分析による推定
390.000387.00796.0
00387.00796.0
CARij
CARij
CARij
BUSij
BUSij
BUSij
ctV
ctV
係数 標準誤差 t P-値 95%下限 95%上限 95.0%下限 95.0%上限切片 0.3898 0.045 8.731 1E-04 0.281 0.499055 0.281 0.499X 1値 -0.0796 0.006 -13.2 1E-05 -0.09 -0.0648 -0.09 -0.06X 2値 -0.0039 3E-04 -12.2 2E-05 -0 -0.00309 -0 -0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2 -1 0 1 2
回帰統計 R重相関0.9899 R2重決定0.9799
R2補正 0.9732標準誤差0.0237観測数 9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
- 2 - 1 0 1 2
VBUS-VCAR
バスの分担率
R による線形回帰Linear regression by R
tb <- c(5,10,14,11,12,16,13,12,7)tc <- c(3,8,10,8,7,11,10,11,3)cb <- c(130,140,180,140,130,220,180,220,130)cc <- c(21,45,58,45,42,60,58,60,19)pb <-
c(0.273,0.282,0.239,0.265,0.248,0.192,0.253,0.255,0.244)pc <- rep(1,9)-pblnpbpc <- log(pc/pb)tsa <- tc-tbcsa <- cc-cbans <- lm(lnpbpc ~ tsa+csa)summary(ans)
R による線形回帰の推定結果Result of linear regression
Call: lm(formula = lnpbpc ~ tsa + csa)Residuals: Min 1Q Median 3Q Max -0.021913 -0.017819 -0.006659 0.018098 0.030403 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.3898049 0.0446483 8.731 0.000125 ***tsa -0.0795878 0.0060416 -13.173 1.18e-05 ***csa -0.0038682 0.0003171 -12.200 1.85e-05 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.0237 on 6 degrees of freedomMultiple R-squared: 0.9799, Adjusted R-squared: 0.9732 F-statistic: 146 on 2 and 6 DF, p-value: 8.165e-06
最尤法によるロジットモデルの推定Maximum Likelihood Estimation
表 3.15 バスの所要時間
ｔBij
1 2 3
1 5 11 13
2 10 12 12
3 14 16 7
単位：分
表 3.16 自動車の所要時間
tCij 1 2 3
1 3 8 10
2 8 7 11
3 10 11 3
単位：分
表 3.17 バスの所要費用
cBij 1 2 3
1 130 140 180
2 140 130 220
3 180 220 130
単位：円
表 3.18 自動車の所要費用
cCij 1 2 3
1 21 45 58
2 45 42 60
3 58 60 19
単位：円
表 バスの利用者数NBij 1 2 3
1 39 22 212 11 31 253 16 15 50
表 自動車の利用者数NCij 1 2 3
1 104 61 622 28 94 733 51 63 155
最尤法によるロジットモデルの推定Maximum Likelihood Estimation
尤度関数
最大になるように値を調整
時間 費用 選択人数 ( )魅力度 仮定値 個人の選択率 事象発生確率Bijｔ Cijｔ Bijｃ cCij NBij NCij Vbij Vcij Pbij Pcij Pb̂ Nb Pc N̂c5 3 130 21 39 104 -0.8878 0.0831 0.27 0.72529 1.3E-22 3.1E-15
10 8 140 45 11 28 -1.3163 -0.399 0.29 0.71449 1E-06 8.2E-0514 10 180 58 16 51 -1.7816 -0.605 0.24 0.76436 9E-11 1.1E-0611 8 140 45 22 61 -1.3944 -0.399 0.27 0.73014 3.1E-13 4.7E-0912 7 130 42 31 94 -1.4341 -0.31 0.25 0.75484 1.2E-19 3.3E-1216 11 220 60 15 63 -2.0908 -0.691 0.2 0.80222 2.8E-11 9.3E-0713 10 180 58 21 62 -1.7035 -0.605 0.25 0.75001 2.3E-13 1.8E-0812 11 220 60 25 73 -1.7786 -0.691 0.25 0.74801 1.1E-15 6.2E-107 3 130 19 50 155 -1.0439 0.0907 0.24 0.75669 2E-31 1.7E-19
魅力度関数の係数の仮定値α -0.078045 6E-139 7.8E-87β -0.003828 尤度関数 5E-225γ 0.397569
最尤法によるロジットモデルの推定Maximum Likelihood Estimation
ソルバー機能： Solver in MS Excel表計算ソフト： Excel の中では、いくつかの数値が別のセルの数値に影響を与える場合、目的のセルの数値を最大にするように、元の数値を決定する計算ができる。実際には、この尤度関数は、あまりに値が小さいため、数値計算誤差の影響でうまく計算できない。尤度関数の対数 (log) を取ったものを考え、それを最大化する。 Logarithm of likelihood function nj
N
n
J
j nj
N
n
J
j nj PP nj
1 11 1lnln
最尤法によるロジットモデルの推定
時間 費用 選択人数 ( )魅力度 仮定値 個人の選択率選択率の対数 人数分を合計Bijｔ Cijｔ Bijｃ cCij NBij NCij Vbij Vcij Pbij Pcij lnPbij lnPcij NblnPb NclnPc5 3 130 21 39 104 -0.8878 0.0831 0.27 0.725 -1.292 -0.321 -50.39 -33.4
10 8 140 45 11 28 -1.3163 -0.399 0.29 0.714 -1.253 -0.336 -13.79 -9.41314 10 180 58 16 51 -1.7816 -0.605 0.24 0.764 -1.445 -0.269 -23.13 -13.711 8 140 45 22 61 -1.3944 -0.399 0.27 0.73 -1.31 -0.315 -28.82 -19.1912 7 130 42 31 94 -1.4341 -0.31 0.25 0.755 -1.406 -0.281 -43.58 -26.4416 11 220 60 15 63 -2.0908 -0.691 0.2 0.802 -1.621 -0.22 -24.31 -13.8813 10 180 58 21 62 -1.7035 -0.605 0.25 0.75 -1.386 -0.288 -29.11 -17.8412 11 220 60 25 73 -1.7786 -0.691 0.25 0.748 -1.378 -0.29 -34.46 -21.197 3 130 19 50 155 -1.0439 0.0907 0.24 0.757 -1.413 -0.279 -70.67 -43.21
魅力度関数の係数の仮定値α -0.078045 -318.3 -198.3β -0.003828 対数尤度関数 -516.5γ 0.397569
最大になるように値を調整
作成されたロジットモデルEstimated Logit model
3975.00038.0078.0
0038.0078.0
CARCARCAR
BUSBUSBUS
ctV
ctV
BUSCAR
CARBUS
BUSBUS
PP
VV
VP
1
)exp()exp(
)exp(
R における尤度関数の定義と最大化#対数尤度関数の定義 係数ベクトルｘの関数と見なす．
LL <- function (x){ vbus <- x[1] * tb + x[2] * cb vcar <- x[1] * tc + x[2] * cc + x[3] ppb <- 1/(1+exp(vcar - vbus)) ppc <- 1- ppb return(sum(pb*log(ppb)+pc*log(ppc))) }#Optim 関数で最大化を行い，結果を res に代入する．初期値は (0,0,0)b0=c(0,0,0)res<-optim(b0,LL, method = "BFGS", hessian = TRUE,
control=list(fnscale=-1))
R による推定結果の表示> res$par[1] -0.081037584 -0.004007811
0.369193890$value[1] -5.047779$countsfunction gradient 62 18 $convergence[1] 0$messageNULL$hessian [,1] [,2] [,3][1,] -19.628306 -613.3939 5.313007[2,] -613.393875 -23980.3173 196.530577[3,] 5.313007 196.5306 -1.681972
R による非集計最尤法の推定Maximum Likelihood by R
完全非集計データ (1 サンプル 1行）の時 Pure disaggregate data (one line for one sample)
glm （従属変数～独立変数１＋・・・＋独立変数 n,family=binomial(link=“logit”))
従属変数が比率の場合 Share data glm （従属変数～独立変数１＋・・・＋独立
変数 n,family=quasibinomial(link=“logit”))
R による最尤法での推定比率を説明する場合 (share data)
ans2 <- glm(pc ~ tsa+csa, family=quasibinomial(link="logit"))summary(ans2)
Call:glm(formula = pc ~ tsa + csa, family = quasibinomial(link = "logit"))Deviance Residuals: Min 1Q Median 3Q Max -0.008856 -0.007615 -0.002659 0.007188 0.013667 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.3991916 0.0446219 8.946 0.000109 ***tsa -0.0787542 0.0060030 -13.119 1.21e-05 ***csa -0.0038095 0.0003174 -12.001 2.03e-05 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for quasibinomial family taken to be 0.0001007740)
Null deviance: 0.02932541 on 8 degrees of freedomResidual deviance: 0.00060564 on 6 degrees of freedomAIC: NA
Number of Fisher Scoring iterations: 4
各個人の選択を説明する場合R により，個人ごとのデータを作成
nb <- c(39,11,16,22,31,15,21,25,50)nc <- c(104,28,51,61,94,63,62,73,155)nn <- nb + nc nnf <-numeric(length=10)for (i in 1:9){
nnf[i+1]=nnf[i]+nn[i] }
chc <- numeric(length=nnf[10]-1)tim <- numeric(length=nnf[10]-1) cst <- numeric(length=nnf[10]-1) for (i in 1:9){
chc[nnf[i]:(nnf[i]+nn[i]-1)] <- c(rep(1,nb[i]), rep(0,nc[i]))tim[nnf[i]:(nnf[i]+nn[i]-1)] <- rep((tb[i]-tc[i]),nn[i])cst[nnf[i]:(nnf[i]+nn[i]-1)] <- rep((cb[i]-cc[i]),nn[i])}
各個人の選択を説明する場合pure disaggregate data
ans3 <- glm(chc ~ tim+cst, family=binomial(link="logit"))summary(ans3)
Call:glm(formula = chc ~ tim + cst, family = binomial(link = "logit"))Deviance Residuals: Min 1Q Median 3Q Max -0.8215 -0.7630 -0.7470 -0.1019 1.8016 Coefficients: Estimate Std. Error z value Pr(>|z|)(Intercept) -0.385889 0.507795 -0.760 0.447tim -0.079514 0.061803 -1.287 0.198cst -0.003873 0.003465 -1.118 0.264(Dispersion parameter for binomial family taken to be 1) Null deviance: 1034.7 on 919 degrees of freedomResidual deviance: 1032.4 on 917 degrees of freedomAIC: 1038.4Number of Fisher Scoring iterations: 4
理論的補足ランダム項がある場合の選択
ランダム項の確率分布の仮定 集団全体における各自の効用の誤差 εの頻度
分布を連続的な確率密度関数で表現
選択率の導出
選択率の導出 ( 図解 )
ε1
ε2 選択肢１が選ばれる確率Prob(U1>U2)=Prob(V1+ε1>V2＋ ε
２）＝ Prob(ε2<V1-V2+ε1)
V1-V2+ε1V1-V2
（＞０）
V1-V2 （＜０）の場合
ε1
ε2
釣鐘（つりがね）まんじゅうを，中央からずれた位置で包丁で切る右下の方の切れ端の体積が選択率を表わす．
誤差項をどのような分布で考えるか？( まんじゅうのふくらみ方 )
具体的なランダム効用モデル ロジット（ LOGIT) モデル
各選択肢の誤差項が独立で同一の Gumbel 分布に従う と仮定したモデル 第 k 選択肢の選択確率は，
プロビット（ PROBIT) モデル 誤差項が多変量正規分布に従うと仮定したモデル 選択確率を解析的に表示することができない モンテカルロシミュレーションなどを用い近似値を
計算
)exp(exp)(Prob)( xxxF )()exp()(')( xFxxFxf
k
kii VVP )exp(/)exp( η ：誤差項の分散に 反比例するパラメータ
非集計データによるロジットモデルのパラメータ推定
尤度関数
尤度関数の値を最大にするように β を定める
パラメータ推定（２）
ニュートン・ラプソン法Newton Raphson Method
多項ロジットモデルMultinomial Logit Model
k
kii VVP )exp(/)exp(
モデル作成の手順
多項ロジットモデルMultinomial Logit Model
個人によって全ての選択肢が利用できない場合もありうる
k
kii VVP )exp(/)exp(
推定結果の評価
的中率
モデル上の選択結果と，実際の選択結果との当てはまりを示す指標
多項ロジットモデル用のデータ ( 愛媛大学作成： )ehime.csv
SEQ,選択,年齢,保有台数,鉄道可能性,時間,費用,バス可能性,時間,費用,４輪可能性,時間,費用3869,1,7,5,1,37.2 ,250,1,129.0 ,850,1,25.61,120.617447,1,7,5,1,37.2 ,250,1,121.3 ,1174.33,1,25.61,120.617924,1,7,5,1,42.1 ,230,1,148.3 ,960,1,29.85,142.342460,1,5,5,1,39.0 ,230,1,106.6 ,970,1,37.06,177.843800,1,5,5,1,39.0 ,230,1,91.3 ,850,1,37.06,177.843347,1,3,5,1,22.0 ,140,1,77.9 ,300,1,9.41,34.044143,1,3,5,1,22.0 ,140,1,77.9 ,300,1,9.41,34.042529,1,1,5,1,45.0 ,320,1,94.8 ,910,1,34.42,165.44985,1,1,5,1,45.0 ,320,1,128.5 ,1050,1,34.42,165.396424,1,6,4,1,83.3 ,570,1,179.8 ,1604.61,1,62.26,511.517791,1,6,4,1,83.3 ,570,1,172.0 ,1720,1,62.26,511.511498,1,5,4,1,43.6 ,650,1,169.3 ,1932.33,1,77.59,400.491962,1,5,4,1,54.1 ,609,1,136.0 ,1182.28,1,61.78,313.432367,1,5,4,1,120.3 ,1100,1,112.2 ,1449.06,1,60.44,306.362369,1,5,4,1,120.3 ,1100,1,112.2 ,1449.06,1,60.44,306.363985,1,5,4,1,54.1 ,609,1,134.2 ,1182.28,1,61.78,313.434307,1,5,4,1,43.6 ,650,1,183.8 ,2082.33,1,77.59,400.494931,1,5,4,1,155.0 ,1100,1,143.1 ,1449.06,1,60.44,306.364932,1,5,4,1,155.0 ,1100,1,143.1 ,1449.06,1,60.44,306.364701,1,1,4,1,54.7 ,320,1,105.9 ,1020,1,31.65,160.455774,1,1,4,1,29.7 ,180,1,67.4 ,450,1,14.63,61.376840,1,1,4,1,29.7 ,180,1,97.8 ,450,1,14.63,61.377242,1,1,4,1,54.7 ,320,1,98.9 ,882.17,1,31.65,160.452368,1,7,3,1,120.3 ,1100,1,112.2 ,1449.06,1,60.44,306.363309,1,7,3,1,75.7 ,570,1,167.2 ,1460,1,58.17,497.654662,1,7,3,1,18.0 ,140,1,70.4 ,160,1,6.81,20.014830,1,7,3,1,35.3 ,296,1,95.3 ,440,1,20.37,92.234933,1,7,3,1,155.0 ,1100,1,143.1 ,1449.06,1,60.44,306.364980,1,7,3,1,33.7 ,190,1,125.1 ,800,1,22.24,102.385011,1,7,3,1,16.9 ,140,1,86.1 ,780,1,7.19,21.755648,1,7,3,1,18.0 ,140,1,71.0 ,160,1,6.81,20.015656,1,7,3,1,16.9 ,140,1,71.4 ,180,1,7.19,21.756138,1,7,3,1,32.7 ,190,1,80.3 ,350,1,15.26,64.696343,1,7,3,1,43.0 ,180,1,46.9 ,280,1,12.08,48.077198,1,7,3,1,32.7 ,190,1,49.0 ,350,1,15.26,64.697199,1,7,3,1,43.0 ,180,1,45.5 ,280,1,12.08,48.077630,1,7,3,1,35.3 ,296,1,67.8 ,440,1,20.37,92.239099,1,7,3,1,33.7 ,190,1,108.1 ,670,1,22.24,102.38133,1,6,3,1,92.3 ,820,1,170.0 ,2265.89,1,95.32,495.61147,1,6,3,1,99.0 ,950,1,212.5 ,2374,1,101.79,534.35280,1,6,3,1,100.7 ,950,1,226.1 ,2515.28,1,99.95,524.3820,1,6,3,1,50.8 ,650,1,140.6 ,1877.17,1,78.09,403.982538,1,6,3,1,49.8 ,455,1,106.1 ,1010,1,31.09,351.853312,1,6,3,1,92.3 ,820,1,276.8 ,3055.28,1,95.32,495.613564,1,6,3,1,50.8 ,650,1,170.3 ,1873.11,1,78.09,403.983875,1,6,3,1,40.8 ,332,1,85.1 ,590,1,25.4,124.424462,1,6,3,1,59.2 ,230,1,99.1 ,750,1,25.66,128.914575,1,6,3,1,99.0 ,950,1,214.6 ,2473.11,1,101.79,534.354669,1,6,3,1,52.6 ,400,1,110.8 ,1040,1,31.09,351.854691,1,6,3,1,25.3 ,190,1,68.0 ,450,1,15.07,67.514909,1,6,3,1,36.0 ,312,1,56.3 ,360,1,8.85,31.544964,1,6,3,1,33.1 ,190,1,105.4 ,660,1,23.88,111.024965,1,6,3,1,33.1 ,190,1,105.4 ,660,1,23.88,111.025235,1,6,3,1,36.0 ,312,1,41.8 ,210,1,8.85,31.545301,1,6,3,1,24.5 ,140,1,44.3 ,290,1,12.45,49.265775,1,6,3,1,60.0 ,320,1,79.9 ,840,1,29.17,147.46149,1,6,3,1,25.3 ,190,1,68.2 ,460,1,15.07,67.516970,1,6,3,1,100.7 ,950,1,231.2 ,2526.61,1,99.95,524.37194,1,6,3,1,59.2 ,230,1,93.9 ,830,1,25.66,128.917196,1,6,3,1,60.0 ,320,1,78.3 ,720,1,29.17,147.47601,1,6,3,1,47.2 ,355,1,84.9 ,590,1,25.4,124.427800,1,6,3,1,50.7 ,190,1,100.0 ,170,1,19.08,73.997954,1,6,3,1,50.7 ,190,1,74.5 ,170,1,19.08,73.998040,1,6,3,1,33.1 ,190,1,78.2 ,660,1,23.88,111.028042,1,6,3,1,33.1 ,190,1,78.2 ,660,1,23.88,111.029016,1,6,3,1,24.5 ,140,1,43.9 ,290,1,12.45,49.26446,1,5,3,1,50.5 ,650,1,153.9 ,1927.17,1,81.34,419.241424,1,5,3,1,47.4 ,650,1,139.9 ,1857.17,1,77.44,399.042464,1,5,3,1,37.4 ,230,1,134.5 ,1020,1,37.65,181.162989,1,5,3,1,79.8 ,480,1,124.5 ,1180.56,1,47.5,247.483567,1,5,3,1,47.4 ,650,1,168.6 ,1853.11,1,77.44,399.043797,1,5,3,1,50.5 ,650,1,196.7 ,2033.11,1,81.34,419.244269,1,5,3,1,48.3 ,190,1,68.4 ,520,1,22.24,101.954311,1,5,3,1,37.4 ,230,1,89.5 ,910.56,1,37.65,181.174785,1,5,3,1,68.5 ,570,1,140.5 ,1240,1,49.62,450.684957,1,5,3,1,21.2 ,180,1,43.9 ,240,1,9.69,35.295914,1,5,3,1,21.2 ,180,1,43.9 ,240,1,9.69,35.297236,1,5,3,1,48.3 ,190,1,64.6 ,520,1,22.24,101.957940,1,5,3,1,68.5 ,570,1,141.5 ,1230,1,49.62,450.683839,1,4,3,1,29.1 ,190,1,81.1 ,660,1,19.39,90.514667,1,4,3,1,29.6 ,190,1,88.1 ,520,1,21.22,96.835637,1,4,3,1,29.1 ,190,1,114.9 ,620,1,19.39,90.518601,1,4,3,1,29.6 ,190,1,88.7 ,520,1,21.22,96.834361,1,3,3,1,46.4 ,320,1,90.8 ,734.06,1,32.71,162.736575,1,3,3,1,49.3 ,320,1,165.7 ,1194.61,1,34.25,166.358087,1,3,3,1,46.4 ,320,1,89.4 ,734.06,1,32.71,162.738231,1,3,3,1,49.3 ,320,1,132.5 ,1310,1,34.25,166.353965,1,2,3,1,31.0 ,190,1,105.6 ,560,1,18.76,88.974355,1,2,3,1,58.3 ,230,1,119.7 ,730,1,28.65,135.935207,1,2,3,1,34.8 ,230,1,70.2 ,540,1,22.22,106.565611,1,2,3,1,31.0 ,190,1,107.1 ,560,1,18.76,88.976347,1,2,3,1,55.6 ,230,1,67.8 ,520,1,24.32,112.618572,1,2,3,1,34.8 ,230,1,69.3 ,540,1,22.22,106.564700,1,1,3,1,54.7 ,320,1,105.9 ,1020,1,31.65,160.454979,1,1,3,1,33.7 ,190,1,125.1 ,800,1,22.24,102.385016,1,1,3,1,56.3 ,320,1,114.4 ,758.39,1,31.68,162.167241,1,1,3,1,54.7 ,320,1,98.9 ,882.17,1,31.65,160.457248,1,1,3,1,56.3 ,320,1,96.2 ,882.17,1,31.68,162.169098,1,1,3,1,33.7 ,190,1,108.1 ,670,1,22.24,102.3832,1,7,2,1,40.5 ,230,1,113.5 ,570,1,36.29,175.66695,1,7,2,1,43.6 ,320,1,66.5 ,580,1,36.29,175.661580,1,7,2,1,72.7 ,740,1,167.0 ,2067.17,1,82.98,431.552361,1,7,2,1,95.3 ,1020,1,129.9 ,1316.11,1,53.15,267.42371,1,7,2,1,126.0 ,1100,1,114.2 ,1479.06,1,61.4,312.512531,1,7,2,1,49.5 ,230,1,109.7 ,1040,1,33.06,158.282548,1,7,2,1,48.1 ,230,1,104.6 ,910,1,28.43,133.892988,1,7,2,1,79.8 ,480,1,124.5 ,1180.56,1,47.5,247.484310,1,7,2,1,47.8 ,480,1,128.8 ,1316.11,1,53.15,267.44342,1,7,2,1,25.4 ,140,1,102.4 ,380,1,9.74,36.444370,1,7,2,1,36.2 ,230,1,83.4 ,690,1,27.29,134.14928,1,7,2,1,45.9 ,320,1,126.7 ,890,1,28.08,132.365106,1,7,2,1,25.2 ,180,1,52.3 ,330,1,12.54,53.635447,1,7,2,1,63.6 ,480,1,120.3 ,1130,1,44.71,424.285448,1,7,2,1,64.4 ,480,1,124.3 ,1130,1,47.6,439.775452,1,7,2,1,160.8 ,1100,1,144.5 ,1479.06,1,61.4,312.515759,1,7,2,1,25.2 ,180,1,42.6 ,310,1,12.54,53.636052,1,7,2,1,112.7 ,740,1,177.9 ,1780,1,69.54,565.296609,1,7,2,1,74.1 ,609,1,180.4 ,1923.11,1,61.17,312.156825,1,7,2,1,56.0 ,650,1,210.4 ,2163.11,1,82.98,431.556831,1,7,2,1,25.4 ,140,1,72.1 ,380,1,9.74,36.446972,1,7,2,1,49.5 ,230,1,121.8 ,1060,1,33.06,158.286973,1,7,2,1,48.1 ,230,1,127.3 ,930,1,28.43,133.897917,1,7,2,1,63.6 ,480,1,123.0 ,1120,1,44.71,424.287943,1,7,2,1,64.4 ,480,1,127.0 ,1120,1,47.6,439.777949,1,7,2,1,142.0 ,980,1,180.4 ,1850,1,69.54,565.298474,1,7,2,1,36.2 ,230,1,82.5 ,690,1,27.29,134.18622,1,7,2,1,45.9 ,320,1,130.0 ,890,1,28.08,132.362473,1,6,2,1,135.1 ,962,1,199.8 ,1548.11,1,88.33,448.432971,1,6,2,1,21.7 ,140,1,114.1 ,520,1,9.9,36.253868,1,6,2,1,50.5 ,320,1,126.6 ,890.28,1,35.98,178.054210,1,6,2,1,44.5 ,320,1,144.4 ,1022.17,1,34.62,170.285742,1,6,2,1,64.9 ,480,1,118.2 ,1090,1,44.89,425.545743,1,6,2,1,64.9 ,480,1,118.2 ,1090,1,44.89,425.545881,1,6,2,1,21.7 ,140,1,102.7 ,990,1,9.9,36.257477,1,6,2,1,135.1 ,962,1,177.1 ,1851.11,1,88.33,448.437918,1,6,2,1,64.9 ,480,1,120.7 ,1080,1,44.89,425.547919,1,6,2,1,64.9 ,480,1,120.7 ,1080,1,44.89,425.549144,1,6,2,1,98.0 ,540,1,126.5 ,890.28,1,34.62,170.2895,1,5,2,1,31.2 ,190,1,88.2 ,430,1,22.12,101.06108,1,5,2,1,31.9 ,190,1,78.6 ,600,1,22.7,103.821392,1,5,2,1,31.2 ,190,1,42.7 ,450,1,22.12,101.061499,1,5,2,1,43.6 ,650,1,169.3 ,1932.33,1,77.59,400.491642,1,5,2,1,31.9 ,190,1,78.4 ,600,1,22.7,103.821937,1,5,2,1,37.9 ,320,1,85.9 ,890,1,43.71,213.192474,1,5,2,1,37.9 ,320,1,114.5 ,880,1,43.71,213.192685,1,5,2,1,104.0 ,530,1,96.8 ,840,1,29.55,147.63899,1,5,2,1,25.7 ,190,1,73.7 ,550,1,16.32,72.833980,1,5,2,1,39.4 ,230,1,81.6 ,650,1,29.47,145.194002,1,5,2,1,20.9 ,140,1,49.1 ,300,1,8.79,31.894175,1,5,2,1,32.1 ,190,1,58.4 ,450,1,18.37,86.754308,1,5,2,1,43.6 ,650,1,183.8 ,2082.33,1,77.59,400.494564,1,5,2,1,55.7 ,230,1,106.7 ,640,1,26.01,128.954679,1,5,2,1,25.7 ,190,1,62.6 ,550,1,16.32,72.834705,1,5,2,1,27.3 ,190,1,90.3 ,500,1,22.54,103.994967,1,5,2,1,35.0 ,190,1,112.6 ,660,1,23.74,109.675068,1,5,2,1,20.9 ,140,1,50.5 ,300,1,8.79,31.895685,1,5,2,1,32.1 ,190,1,45.5 ,440,1,18.37,86.756608,1,5,2,1,74.1 ,609,1,180.4 ,1923.11,1,61.17,312.156855,1,5,2,1,59.2 ,190,1,90.5 ,610,1,20.78,99.246862,1,5,2,1,41.6 ,230,1,93.6 ,820,1,29.55,141.316863,1,5,2,1,41.6 ,230,1,93.6 ,820,1,29.55,141.317205,1,5,2,1,106.1 ,740,1,187.8 ,1816.22,1,75.46,597.027208,1,5,2,1,40.5 ,190,1,89.7 ,620,1,20.78,99.247218,1,5,2,1,57.5 ,230,1,92.1 ,820,1,29.55,147.67240,1,5,2,1,55.7 ,230,1,73.1 ,620,1,26.01,128.957825,1,5,2,1,45.6 ,190,1,145.9 ,890,1,34.12,153.817879,1,5,2,1,106.1 ,740,1,190.6 ,1920,1,75.46,597.028135,1,5,2,1,27.3 ,190,1,63.1 ,500,1,22.54,103.998155,1,5,2,1,41.6 ,230,1,94.9 ,840,1,29.55,141.318156,1,5,2,1,41.6 ,230,1,94.9 ,840,1,29.55,141.318194,1,5,2,1,35.0 ,190,1,86.3 ,670,1,23.74,109.678321,1,5,2,1,39.4 ,230,1,78.9 ,760,1,29.47,145.198626,1,5,2,1,45.6 ,190,1,120.5 ,890,1,34.12,153.813358,1,4,2,1,20.2 ,180,1,92.7 ,400,1,10.66,40.423827,1,4,2,1,28.3 ,190,1,59.6 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,460,1,17.01,82.846842,2,6,2,1,29.2 ,140,1,48.6 ,190,1,8.18,27.166904,2,6,2,1,27.5 ,180,1,50.3 ,330,1,12.06,48.387139,2,6,2,1,53.5 ,230,1,80.3 ,660,1,23.6,109.47340,2,5,2,1,33.3 ,311,1,61.3 ,450,1,22.31,100.921483,2,5,2,1,33.3 ,311,1,91.3 ,430,1,22.31,100.923370,2,5,2,1,25.5 ,140,1,104.8 ,520,1,13.01,58.693689,2,5,2,1,28.1 ,140,1,108.1 ,430,1,11.24,44.554164,2,5,2,1,21.7 ,140,1,39.1 ,200,1,8.92,32.914165,2,5,2,1,21.7 ,140,1,39.1 ,200,1,8.92,32.914953,2,5,2,1,20.9 ,180,1,46.2 ,250,1,10.03,39.044955,2,5,2,1,17.9 ,140,1,70.9 ,170,1,6.7,19.515077,2,5,2,1,21.7 ,140,1,39.2 ,200,1,8.92,32.915078,2,5,2,1,21.7 ,140,1,39.2 ,200,1,8.92,32.915092,2,5,2,1,20.9 ,180,1,44.6 ,250,1,10.03,39.045654,2,5,2,1,17.9 ,140,1,85.0 ,280,1,6.7,19.516467,2,5,2,1,25.5 ,140,1,104.6 ,530,1,13.01,58.696830,2,5,2,1,28.1 ,140,1,108.2 ,430,1,11.24,44.55832,2,4,2,1,22.1 ,180,1,43.3 ,250,1,10.2,38.831802,2,4,2,1,22.1 ,180,1,43.2 ,250,1,10.2,38.833852,2,4,2,1,37.3 ,190,1,117.2 ,640,1,18.39,87.194694,2,4,2,1,22.9 ,180,1,53.9 ,380,1,12.75,51.494776,2,4,2,1,108.2 ,640,1,89.3 ,550,1,21.69,99.475034,2,4,2,1,27.6 ,180,1,50.6 ,310,1,14,58.25263,2,4,2,1,32.0 ,190,1,50.3 ,310,1,14,58.25919,2,4,2,1,23.3 ,180,1,54.1 ,300,1,8.44,28.496018,2,4,2,1,23.3 ,180,1,85.3 ,300,1,8.44,28.496025,2,4,2,1,112.9 ,640,1,90.3 ,550,1,22.5,103.616835,2,4,2,1,22.9 ,180,1,55.1 ,380,1,12.75,51.496977,2,4,2,1,37.3 ,190,1,84.6 ,640,1,18.39,87.197515,2,4,2,1,72.1 ,640,1,64.2 ,550,1,21.69,99.477517,2,4,2,1,71.7 ,640,1,110.2 ,690,1,22.5,103.614259,2,3,2,1,27.6 ,140,1,59.0 ,290,1,10.52,40.647096,2,3,2,1,27.6 ,140,1,59.1 ,290,1,10.52,40.641178,2,7,1,1,26.6 ,180,1,52.7 ,370,1,15.22,62.111364,2,7,1,1,26.6 ,180,1,52.7 ,370,1,15.22,62.111372,2,7,1,1,22.7 ,180,1,43.1 ,320,1,12.66,53.031379,2,7,1,1,28.6 ,298,1,55.8 ,460,1,20.54,91.61494,2,7,1,1,106.9 ,640,1,74.7 ,740,1,28.25,136.421721,2,7,1,1,22.7 ,180,1,43.1 ,320,1,12.66,53.031915,2,7,1,1,28.6 ,298,1,55.8 ,460,1,20.54,91.62121,2,7,1,1,39.7 ,320,1,122.3 ,1048.44,1,42.69,209.82129,2,7,1,1,47.8 ,320,1,114.7 ,1139.22,1,45.74,227.582273,2,7,1,1,34.2 ,230,1,120.3 ,890,1,28.25,136.422566,2,7,1,1,27.6 ,190,1,50.0 ,280,1,11.25,42.242583,2,7,1,1,51.2 ,459,1,79.3 ,794.89,1,50.1,246.742627,2,7,1,1,29.4 ,190,1,44.3 ,250,1,11.99,44.722651,2,7,1,1,32.1 ,190,1,65.7 ,450,1,12.56,50.193241,2,7,1,1,30.7 ,190,1,96.5 ,520,1,17.74,77.153292,2,7,1,1,26.1 ,140,1,74.8 ,260,1,9.95,36.763398,2,7,1,1,22.3 ,180,1,76.8 ,630,1,12.8,51.763738,2,7,1,1,40.9 ,190,1,111.9 ,620,1,20.27,96.763836,2,7,1,1,24.4 ,140,1,46.0 ,240,1,8.24,27.593919,2,7,1,1,
多項ロジットモデルの最尤法プログラム ( 愛媛大学作成： )
### Multi Nomial Logit (MNL) estimation program (Original code by EHIME University)###データファイルの読み込みData<-read.csv("ehime.csv",header=T)hh<-nrow(Data) ##データ数 :Data の行数を数えるprint(hh)ch<- 3 ##今回用いる選択肢の数b0<-c(0, 0, 0, 0, 0, 0)Srail <- sum(Data[,14]== 1); Sbus <- sum(Data[,14]== 2); Scar <- sum(Data[,14]== 3)cat("rail:",Srail," bus:",Sbus," car:",Scar,"\n")## Logit model の対数尤度関数の定義fr <- function(x) {LL=0##効用の計算rail <- x[1]*Data[, 6]/100 + x[2]*Data[, 7]/100 + x[5]*matrix(1,nrow =hh,ncol=1)bus <- x[1]*Data[, 9]/100 + x[2]*Data[,10]/100 + x[3]*(Data[, 3]>=6) + x[6]*matrix(1,nrow=hh,ncol=1)car <- x[1]*Data[,12]/100 + x[2]*Data[,13]/100 + x[4]*(Data[, 4]>=2)##効用の指数化Erail <- exp(rail)*Data[, 5]Ebus <- exp(bus )*Data[, 8]Ecar <- exp(car )*Data[,11]PPrail <- Erail/(Erail+Ebus+Ecar)PPbus <- Ebus /(Erail+Ebus+Ecar)PPcar <- Ecar /(Erail+Ebus+Ecar)##選択結果の確率のみを有効化Prail <- (PPrail!=0)*PPrail + (PPrail==0)Pbus <- (PPbus !=0)*PPbus + (PPbus ==0)Pcar <- (PPcar !=0)*PPcar + (PPcar ==0)##選択結果Crail <- Data[,14]== 1Cbus <- Data[,14]== 2Ccar <- Data[,14]== 3##対数尤度の計算LL <- colSums( Crail*log(Prail) + Cbus*log(Pbus) + Ccar*log(Pcar) )return(LL)}## 対数尤度関数 fr の最大化res<-optim(b0,fr, method = "BFGS", hessian = TRUE, control=list(fnscale=-1))## estimated parameterb<-res$parhhh<-res$hessian## t 値の計算tval<-b/sqrt(-diag(solve(hhh)))##初期尤度L0 <- Srail*log(Srail/hh)+Sbus*log(Sbus/hh)+Scar*log(Scar/hh)##最終尤度LL <- res$value## 適合度の計算##結果の出力##ρ^2 値cat(" roh = ",(L0-LL)/L0,"\n")##修正済ρ^2 値cat(" rohbar= ",(L0-(LL-length(b)))/L0,"\n")print(res)print(tval)
多項ロジットモデルの最尤法による推定結果 rail: 493 bus: 432 car: 708roh = 0.2432912rohbar= 0.2398755$par[1] -1.7411817 -0.1757195 1.7302273 2.5890795 2.0644240 2.3457336$value[1] -1329.232$countsfunction gradient38 14$convergence[1] 0$messageNULL$hessian[,1] [,2] [,3] [,4] [,5] [,6][1,] -80.15815 -538.9584 -67.42988 65.52220 27.40143 -114.7780[2,] -538.95843 -4708.3230 -493.74059 503.37025 123.55640 -800.5406[3,] -67.42988 -493.7406 -127.96682 31.64655 79.67259 -127.9668[4,] 65.52220 503.3703 31.64655 -214.17601 136.44173 77.7343[5,] 27.40143 123.5564 79.67259 136.44173 -293.24442 123.6202[6,] -114.77804 -800.5406 -127.96682 77.73429 123.62021 -224.7471
[1] -5.857365 -5.520454 12.528884 17.136179 13.928912 10.259022
多項ロジットモデルの最尤法による推定結果 rail: 493 bus: 432 car: 708
roh = 0.2432912
rohbar= 0.2398755$par
時間 費用 高齢 B 台数 C 鉄道 バス
[1] -1.7411817 -0.1757195 1.7302273 2.5890795 2.0644240 2.3457336
[1] -5.857365 -5.520454 12.528884 17.136179 13.928912 10.259022
$value
[1] -1329.232
所要時間は -0.0174［／分］，費用は -0.00176［／円］でパラメータ比から得られる時間評価値は 10［円／分］程度と，若干低い
多項ロジットモデル （パッケージ） Multinomial Logit Model
install.packages("mlogit")library("mlogit")Dehime<-read.csv("ehime.csv",header=T)Ehime <-mlogit.data(Dehime,varying=c(5:13), shape="wide",choice="choice")#MNL without constant termsummary(mlogit(choice~time+cost-1,data=Ehime))#MNL with constant termsummary(mlogit(choice~time+cost,data=Ehime))
多項ロジットモデル ( 定数項なし ) Multinomial Logit Model
Call:mlogit(formula = Choice ~ time + cost - 1, data = Ehime)Frequencies of alternatives: bus car rail 0.26454 0.43356 0.30190 Newton-Raphson maximisation gradient close to zero. May be a solution 5 iterations, 0h:0m:0s g'(-H)^-1g = 7.47E-31 Coefficients : Estimate Std. Error t-value Pr(>|t|) time -0.00329405 0.00197725 -1.6660 0.0957185 . cost -0.00096882 0.00025323 -3.8259 0.0001303 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Log-Likelihood: -1715.7
多項ロジットモデル ( 定数項あり ) Multinomial Logit Model
Call:mlogit(formula = Choice ~ time + cost, data = Ehime)Frequencies of alternatives: bus car rail 0.26454 0.43356 0.30190 Newton-Raphson maximisation gradient close to zero. May be a solution 5 iterations, 0h:0m:0s g'(-H)^-1g = 9.17E-26 Coefficients : Estimate Std. Error t-value Pr(>|t|)
altcar -1.07985435 0.14856671 -7.2685 3.635e-13 ***altrail -0.95903599 0.11625041 -8.2497 2.220e-16 ***time -0.01607098 0.00261456 -6.1467 7.910e-10 ***cost -0.00119568 0.00027125 -4.4081 1.043e-05 ***
Log-Likelihood: -1680.5McFadden R^2: 0.68776 Likelihood ratio test : chisq = 152.13 (p.value=< 2.22e-16
MNL モデルの限界と注意点 青バス・赤バス問題（ I.I.A 特性 )例：バスと自動車の交通機関選択モデル バスも自動車も共に一般化交通費用が等しい →バス, 自動車の選択率は 1/2ずつになる バスの半数を赤く, 半数を青く塗って区別 →青バス, 赤バス , 自動車の選択率は 1/3ずつ 色を変えたらバスのシェアが 1/2から 2/3に上昇？
各選択肢の確率的効用の間に相関がある場合には，非現実的な選択率を与える
ネスティッド ( 入れ子 ) ロジットモデルNested Logit Model （ NL)
下位の選択肢のうち大きいほうを取ったときの期待値
ネスティッド ( 入れ子 ) ロジットモデルNested Logit Model （ NL)
下位の選択肢の間の相関のほうが ,上位の選択肢間の相関より大きい必要がある
ネスティッド ( 入れ子 ) ロジットモデルNested Logit Model （ NL)
•段階推定法：下の階層から順次推定して，ログサム変数値を計算して上位階層を推定する
•階層間でパラメータ値が大きく異なるなど ,解釈が困難
•同時推定法：全体の対数尤度関数を作り，全パラメータ値とスケールパラメータを一気に推定
ネスティッドロジットモデルの最尤法プログラム ( 愛媛大学作成： )
### Nested Logit estimation program (Original code by EHIME University)
###データファイルの読み込みData<-read.csv(“ehime.csv",header=T)hh<-nrow(Data) ##データ数:Data の行数を数えるprint(hh)
ch<- 3 ##今回用いる選択肢の数b0<-c(0, 0, 0, 0, 0, 0, 1) ##初期パラメータ値( 全て0)## サンプルにおける各選択肢のシェアSrail <- sum(Data[,2]==“rail”); Sbus <- sum(Data[,2]==“ bus”); Scar <- sum(Data[,2]==“car”)cat("rail:",Srail," bus:",Sbus," car:",Scar,"\n")
## Logit model の対数尤度関数の定義fr <- function(x) {LL=0##効用の計算rail <- x[1]*Data[, 6]/100 + x[2]*Data[, 7]/100 + x[5]*matrix(1,nrow =hh,ncol=1)bus <- x[1]*Data[, 9]/100 + x[2]*Data[,10]/100 + x[3]*(Data[, 3]>=6) + x[6]*matrix(1,nrow=hh,ncol=1)car <- x[1]*Data[,12]/100 + x[2]*Data[,13]/100 + x[4]*(Data[, 4]>=2)##効用の指数化 Erail <- exp(rail)*Data[, 5]Ebus <- exp(bus )*Data[, 8]Ecar <- exp(car )*Data[,11]
##ここから，採用したいツリーの部分を残す##（Rail+Bus)+Car# LSrb <- exp( x[7]*log( Erail + Ebus ) )# LSc <- exp( x[7]*log( Ecar ) )# PPrail <- Erail/(Erail+Ebus) * LSrb/(LSrb + LSc)# PPbus <- Ebus /(Erail+Ebus) * LSrb/(LSrb + LSc)# PPcar <- LSc/(LSrb + LSc)##（Rail+(Bus+Car）LSr <- exp( x[7]*log( Erail ) )LSbc <- exp( x[7]*log( Ebus + Ecar ) )PPrail <- LSr/(LSr + LSbc)PPbus <- Ebus /(Ebus+Ecar) * LSbc/(LSr + LSbc)PPcar <- Ecar /(Ebus+Ecar) * LSbc/(LSr + LSbc)##（Rail+Car)+Bus ##（実際には考えにくいツリー）# LSrc <- exp( x[7]*log( Erail+ Ecar ) )# LSb <- exp( x[7]*log( Ebus ) )# PPrail <- Erail/(Erail+Ecar) * LSrc/(LSrc + LSb)# PPbus <- LSb/(LSrc + LSb)# PPcar <- Ecar /(Erail+Ecar) * LSrc/(LSrc + LSb)
##選択結果の確率のみを有効化##（非選択の確率は1におけば,対数尤度の際に0として扱われるので効率的）Prail <- (PPrail!=0)*PPrail + (PPrail==0)Pbus <- (PPbus !=0)*PPbus + (PPbus ==0)Pcar <- (PPcar !=0)*PPcar + (PPcar ==0)
##選択結果Crail <- Data[,14] == 1Cbus <- Data[,14] == 2 Ccar <- Data[,14] == 3 ##対数尤度の計算LL <- colSums( Crail*log(Prail) + Cbus*log(Pbus) + Ccar*log(Pcar) )return(LL)}
## 対数尤度関数fr の最大化（最適化関数optim()による)res<-optim(b0,fr, method = "BFGS", hessian = TRUE, control=list(fnscale=-1))## estimated parameterb<-res$parhhh<-res$hessian## t 値の計算tval<-b/sqrt(-diag(solve(hhh)))##初期尤度L0 <- Srail*log(Srail/hh)+Sbus*log(Sbus/hh)+Scar*log(Scar/hh)##最終尤度LL <- res$value
## 適合度の計算##結果の出力##ρ^2 値cat(" roh = ",(L0-LL)/L0,"\n")##修正済ρ^2 値cat(" rohbar= ",(L0-(LL-length(b)))/L0,"\n")print(res)print(tval)
NL モデルの計算結果(ehime)
ケース A ： rohbar= 0.2652 (最も大きい )パラメータ： -0.8978 0.01277 1.178 0.5003 0.03899 -0.43394 4.873
t 値： -5.62 1.41 8.43 6.39 0.42 -3.40 7.04
時間 費用 高齢 B 台数 C 鉄道 バス スケール
ケース B ： rohbar= 0.2416パラメータ： -1.327 -0.1613 1.244 2.413 1.877 2.382 1.354
t 値： -4.86 -6.01 7.20 14.97 11.86 12.29 9.82
時間 費用 高齢 B 台数 C 鉄道 バス スケール
ケース C ： rohbar= 0.2470パラメータ： -2.357 -0.1976 2.297 3.008 1.992 2.814 0.5830
t 値： -5.69 -4.70 11.29 15.79 9.53 9.81 8.72
時間 費用 高齢 B 台数 C 鉄道 バス スケール
スケールパラメータが 1以
上