가우스과정회귀모델gydF4y2Ba
가우스과정회귀(gpr)모델은비모수커널기반의확률적모델입니다。gydF4y2BafitrgpgydF4y2Ba
함수를사용하여探地雷达모델을훈련시킬수있습니다。gydF4y2Ba
훈련 세트gydF4y2Ba 을살펴보겠습니다。여기서gydF4y2Ba 및gydF4y2Ba 이며,알수없는분포에서추출되었습니다。探地雷达모델은새입력벡터gydF4y2Ba 및훈련데이터가주어진경우응답변수gydF4y2Ba 의값을예측하는문제를해결합니다。선형회귀모델의형식은다음과같습니다。gydF4y2Ba
여기서gydF4y2Ba 입니다。오차분산σgydF4y2Ba2gydF4y2Ba및계수β는데이터에서추정됩니다。GPR모델은가우스과정(gp)에서잠재변수gydF4y2Ba 및명시적기저함수h를추가하여응답변수를설명합니다。잠재변수의공분산함수는응답변수의매끄러움정도를포착하고기저함수는입력값gydF4y2Ba 를p차원특징공간에합니다。gydF4y2Ba
GP는확률변수들의집합으로,이집합에속한임의의유한개의확률변수들은결합가우스분포를가집니다。gydF4y2Ba 가GP이고n개의관측값gydF4y2Ba 이주어진경우확률변수gydF4y2Ba 의결합분포는가우스분포입니다。Gp는평균함수gydF4y2Ba 및공분산함수gydF4y2Ba 에의해정의됩니다。즉,gydF4y2Ba 가가우스과정이면gydF4y2Ba 및gydF4y2Ba 입니다。gydF4y2Ba
이제다음모델을살펴보겠습니다。gydF4y2Ba
여기서gydF4y2Ba 이며,즉f(x)는공분산함수gydF4y2Ba 을갖는평균이0 tmp에서도출됩니다。h(x)는RgydF4y2BadgydF4y2Ba의원래특징벡터x를RgydF4y2BapgydF4y2Ba의새특징벡터h(x)로변환하는기저함수집합입니다。β는기저함수계수로구성된p×1벡터입니다。이모델은探地雷达모델을나타냅니다。응답변수y의일개사례는다음과같이모델링할수있습니다。gydF4y2Ba
따라서,GPR모델은확률적모델입니다。여기에는각관측값gydF4y2Ba 마다探地雷达모델을비모수적으로만드는잠재변수f(xgydF4y2Ba我gydF4y2Ba)가추가되어있습니다。벡터형식에서이모델은다음과동일합니다。gydF4y2Ba
여기서는다음이성립합니다。gydF4y2Ba
探地雷达모델에서잠재변수gydF4y2Ba 의결합분포는다음과같습니다。gydF4y2Ba
이는선형회귀모델에가까우며,여기서gydF4y2Ba 는다음과같이@ @시됩니다。gydF4y2Ba
공분산함수gydF4y2Ba 은일반적으로커널모수또는하이퍼파라미터집합gydF4y2Ba 로모수화됩니다。많은경우,gydF4y2Ba 은gydF4y2Ba 에대한종속성을명시적으로나타내기위해gydF4y2Ba 로@ @기됩니다。gydF4y2Ba
fitrgpgydF4y2Ba
는探地雷达모델을훈련시키는동데이터에서커널함수의기저함수계수gydF4y2Ba
,잡음분산gydF4y2Ba
,하이퍼파라미터gydF4y2Ba
를추정합니다。기저함수,커널(공분산)함수,모수의초기값을지정할수있습니다。gydF4y2Ba
探地雷达모델이확률적이므로훈련된모델을사용하여예측구간을계산할수있습니다(gydF4y2Ba预测gydF4y2Ba
및gydF4y2BaresubPredictgydF4y2Ba
참조)。gydF4y2Ba
또한훈련된探地雷达모델을사용하여회귀오차를계산할수도있습니다(gydF4y2Ba损失gydF4y2Ba
및gydF4y2BaresubLossgydF4y2Ba
참조)。gydF4y2Ba
探地雷达모델의예측구간비교하기gydF4y2Ba
이예제에서는잡음이없는데이터세트와잡음이있는데이터세트에GPR모델을피팅합니다。그런다음두개의피팅된探地雷达모델의예측된응답변수와예측구간을비교합니다。gydF4y2Ba
함수gydF4y2Ba 에서두개의관측값데이터세트를생성합니다。gydF4y2Ba
rng (gydF4y2Ba“默认”gydF4y2Ba)gydF4y2Ba%用于再现性gydF4y2BaX_observed = linspace(0,10,21)';Y_observed1 = x_observed.*sin(x_observed);Y_observed2 = y_observed1 + 0.5*randn(size(x_observed));gydF4y2Ba
y_observed1gydF4y2Ba
의값에는잡음이없고gydF4y2Bay_observed2gydF4y2Ba
의값에는얼마간의랜덤잡음이있습니다。gydF4y2Ba
探地雷达모델을관측된데이터세트에피팅합니다。gydF4y2Ba
gprMdl1 = fitrgp(x_observed, y_observved1);gprMdl2 = fitrgp(x_observed, y_observved2);gydF4y2Ba
피팅된모델을사용하여예측된응답변수와95%예측구간을계산합니다。gydF4y2Ba
X = linspace(0,10)';[ypred1,~,yint1] = predict(gprMdl1,x);[ypred2,~,yint2] = predict(gprMdl2,x);gydF4y2Ba
图의크기를조정하여하나의图에두개의플롯을gydF4y2Ba
图;fig.Position(3) = fig.Position(3)*2;gydF4y2Ba
1×2타일형식차트레이아웃을만듭니다。gydF4y2Ba
tiledlayout(1、2、gydF4y2Ba“TileSpacing”gydF4y2Ba,gydF4y2Ba“紧凑”gydF4y2Ba)gydF4y2Ba
각타일에관측된데이터점들의산점도플롯과gydF4y2Ba 의함수플롯을그립니다。그런다음gp예측응답변수의플롯과예측구간의패치를추가합니다。gydF4y2Ba
nexttile举行gydF4y2Ba在gydF4y2Ba散射(x_observed y_observed1,gydF4y2Ba“r”gydF4y2Ba)gydF4y2Ba观测数据点百分比gydF4y2Bafplot (@ x (x)。* sin (x) [0, 10],gydF4y2Ba“——r”gydF4y2Ba)gydF4y2Bax*sin(x)的函数图gydF4y2Ba情节(x, ypred1,gydF4y2Ba‘g’gydF4y2Ba)gydF4y2Ba%探地雷达预测gydF4y2Ba补丁([x; flipud (x)], [yint1 (: 1); flipud (yint1 (:, 2))),gydF4y2Ba“k”gydF4y2Ba,gydF4y2Ba“FaceAlpha”gydF4y2Ba, 0.1);gydF4y2Ba%预测区间gydF4y2Ba持有gydF4y2Ba从gydF4y2Ba标题(gydF4y2Ba“无噪声观测的GPR拟合”gydF4y2Ba)({传奇gydF4y2Ba“无噪声的观察”gydF4y2Ba,gydF4y2Ba'g(x) = x*sin(x)'gydF4y2Ba,gydF4y2Ba“探地雷达预测”gydF4y2Ba,gydF4y2Ba“95%预测区间”gydF4y2Ba},gydF4y2Ba“位置”gydF4y2Ba,gydF4y2Ba“最佳”gydF4y2Ba) nexttile holdgydF4y2Ba在gydF4y2Ba散射(x_observed y_observed2,gydF4y2Ba“xr”gydF4y2Ba)gydF4y2Ba观测数据点百分比gydF4y2Bafplot (@ x (x)。* sin (x) [0, 10],gydF4y2Ba“——r”gydF4y2Ba)gydF4y2Bax*sin(x)的函数图gydF4y2Ba情节(x, ypred2,gydF4y2Ba‘g’gydF4y2Ba)gydF4y2Ba%探地雷达预测gydF4y2Ba补丁([x; flipud (x)], [yint2 (: 1); flipud (yint2 (:, 2))),gydF4y2Ba“k”gydF4y2Ba,gydF4y2Ba“FaceAlpha”gydF4y2Ba, 0.1);gydF4y2Ba%预测区间gydF4y2Ba持有gydF4y2Ba从gydF4y2Ba标题(gydF4y2Ba“噪声观测的GPR拟合”gydF4y2Ba)({传奇gydF4y2Ba“嘈杂的观察”gydF4y2Ba,gydF4y2Ba'g(x) = x*sin(x)'gydF4y2Ba,gydF4y2Ba“探地雷达预测”gydF4y2Ba,gydF4y2Ba“95%预测区间”gydF4y2Ba},gydF4y2Ba“位置”gydF4y2Ba,gydF4y2Ba“最佳”gydF4y2Ba)gydF4y2Ba
관측값에잡음이없는경우探地雷达피팅의예측된응답변수가관측값을지나갑니다。예측된응답변수의준편차는거의0입니다。따라서예측구간이매우좁습니다。관측값에잡음이포함된경우에는예측된응답변수가관측값을지나가지않으며예측구간이넓어집니다。gydF4y2Ba
참고 문헌gydF4y2Ba
[1]拉斯穆森,c.e.和c.k.i.威廉姆斯。机器学习的高斯过程。麻省理工学院出版社。剑桥,马萨诸塞州,2006年。gydF4y2Ba
참고 항목gydF4y2Ba
fitrgpgydF4y2Ba
|gydF4y2BaRegressionGPgydF4y2Ba
|gydF4y2Ba预测gydF4y2Ba