Question

我必须遗漏一些东西，但我想要做的就是运行一个带有多个变量的基本线性回归。问题在于变量具有自定义边界（一些是0 - > 1，其他可能不同）。我希望看到解决方案的系数与statsmodels.api一样完成 - 就像t和P值的输出一样。

我可以使用statsmodels.api.OLS运行summary()，但我似乎无法将变量的范围限制为非负。

我可以运行scipy.optimize.nnls，但这并不能给我任何关于每个变量置信度的输出。

我还尝试使用scipy.optimize.lsq_linear参数bounds，但这似乎并不像我期望的那样工作。

如何结合这些功能来获得我想要的东西？一个例子可能是：

Ys = [1,2,3,4]
Xs = [[4,2,6,4], [6,2,1,4], [1,2,4,9]]
bounds = [[0,1], [0,1], [0,5], [0.2,0.4]]

期望的输出：

                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          1.3292      0.585     -2.274      0.024      -2.483      -0.176
x1             0.0184      0.010      1.859      0.065      -0.001       0.038
x2             0.0253      0.006      4.462      0.000       0.014       0.036
x3             0.0309      0.192      0.057      0.955      -0.368       0.390

所有系数都与边界匹配。

Answer 1

scipy对此案例有一个特殊的优化器，nnls。

问题在于标准错误和推理不是标准的，并且对于一般情况并不容易实现。（我还没有弄清楚如何获得标准错误。）

https://github.com/statsmodels/statsmodels/issues/1211

在我不需要标准错误并且工作正常的情况下我使用了nnls。

在优化期间需要不等式或边界约束的第二个用例，然而，结果通常在内部。这些情况可以通过重新参数化来处理，并且在几种情况下使用，例如，用于离散或广义线性模型或用于方差函数估计的log或logit链接函数。如果最优在内部，则适用标准推断。

修改

获得“近似”标准误差的一种方法是使用适当的优化器找到不等式约束问题的参数，然后在估计标准模型（如OLS）时强加约束。对于非负性约束，可以丢弃在零边界处具有估计参数的变量。也就是说，我们将约束的不等式约束视为等式约束。

然而，在这种情况下计算出的标准误差在假设之下，我们知道哪些约束具有约束力，并且不考虑可能会或可能不会考虑约束的不平等约束的不确定性。 / p>

Answer 2

如果使用R可以接受，我想您也可以使用bbmle的{{1}}函数来优化最小二乘似然函数，并计算非负nnls系数的95％置信区间。此外，您可以考虑通过优化系数的对数来确定系数不会变为负值，以便在逆变换范围内它们永远不会变为负值。

这里是一个数值示例，说明了这种方法，此处是对高斯形色谱峰与高斯噪声的叠加进行反卷积的背景：（欢迎任何评论）

首先让我们模拟一些数据：

mle2

现在让我们用一个带状矩阵对卷积的噪声信号require(Matrix) n = 200 x = 1:n npeaks = 20 set.seed(123) u = sample(x, npeaks, replace=FALSE) # peak locations which later need to be estimated peakhrange = c(10,1E3) # peak height range h = 10^runif(npeaks, min=log10(min(peakhrange)), max=log10(max(peakhrange))) # simulated peak heights, to be estimated a = rep(0, n) # locations of spikes of simulated spike train, need to be estimated a[u] = h gauspeak = function(x, u, w, h=1) h*exp(((x-u)^2)/(-2*(w^2))) # shape of single peak, assumed to be known bM = do.call(cbind, lapply(1:n, function (u) gauspeak(x, u=u, w=5, h=1) )) # banded matrix with theoretical peak shape function used y_nonoise = as.vector(bM %*% a) # noiseless simulated signal = linear convolution of spike train with peak shape function y = y_nonoise + rnorm(n, mean=0, sd=100) # simulated signal with gaussian noise on it y = pmax(y,0) par(mfrow=c(1,1)) plot(y, type="l", ylab="Signal", xlab="x", main="Simulated spike train (red) to be estimated given known blur kernel & with Gaussian noise") lines(a, type="h", col="red")去卷积，该带状矩阵包含已知高斯形状模糊核y的移位副本（这是我们的协变量/设计矩阵）。

首先，让我们用非负最小二乘法对信号进行去卷积：

bM

现在，让我们优化高斯损失目标的负对数似然性，并优化系数的对数，以便在逆变换范围内它们永远不会为负：

library(nnls)
library(microbenchmark)
microbenchmark(a_nnls <- nnls(A=bM,b=y)$x) # 5.5 ms
plot(x, y, type="l", main="Ground truth (red), nnls estimate (blue)", ylab="Signal (black) & peaks (red & blue)", xlab="Time", ylim=c(-max(y),max(y)))
lines(x,-y)
lines(a, type="h", col="red", lwd=2)
lines(-a_nnls, type="h", col="blue", lwd=2)
yhat = as.vector(bM %*% a_nnls) # predicted values
residuals = (y-yhat)
nonzero = (a_nnls!=0) # nonzero coefficients
n = nrow(X)
p = sum(nonzero)+1 # nr of estimated parameters = nr of nonzero coefficients+estimated variance
variance = sum(residuals^2)/(n-p) # estimated variance = 8114.505

我还没有尝试比较这种方法相对于非参数引导或非参数引导的性能，但是肯定更快。

我也倾向于认为，我应该能够基于信息矩阵为非负nnls系数计算Wald置信区间，并以对数变换的比例尺计算以实施非负约束，并以nnls估计值进行评估。我认为就像这样：

library(bbmle)
XM=as.matrix(bM)[,nonzero,drop=FALSE] # design matrix, keeping only covariates with nonnegative nnls coefs
colnames(XM)=paste0("v",as.character(1:n))[nonzero]
yv=as.vector(y) # response
# negative log likelihood function for gaussian loss
NEGLL_gaus_logbetas <- function(logbetas, X=XM, y=yv, sd=sqrt(variance)) {
  -sum(stats::dnorm(x = y, mean = X %*% exp(logbetas), sd = sd, log = TRUE))
}  
parnames(NEGLL_gaus_logbetas) <- colnames(XM)
system.time(fit <- mle2(
  minuslogl = NEGLL_gaus_logbetas, 
  start = setNames(log(a_nnls[nonzero]+1E-10), colnames(XM)), # we initialise with nnls estimates
  vecpar = TRUE,
  optimizer = "nlminb"
)) # takes 0.86s
AIC(fit) # 2394.857
summary(fit) # now gives log(coefficients) (note that p values are 2 sided)
# Coefficients:
#       Estimate Std. Error z value     Pr(z)    
# v10    4.57339    2.28665  2.0000 0.0454962 *  
# v11    5.30521    1.10127  4.8173 1.455e-06 ***
# v27    3.36162    1.37185  2.4504 0.0142689 *  
# v38    3.08328   23.98324  0.1286 0.8977059    
# v39    3.88101   12.01675  0.3230 0.7467206    
# v48    5.63771    3.33932  1.6883 0.0913571 .  
# v49    4.07475   16.21209  0.2513 0.8015511    
# v58    3.77749   19.78448  0.1909 0.8485789    
# v59    6.28745    1.53541  4.0950 4.222e-05 ***
# v70    1.23613  222.34992  0.0056 0.9955643    
# v71    2.67320   54.28789  0.0492 0.9607271    
# v80    5.54908    1.12656  4.9257 8.407e-07 ***
# v86    5.96813    9.31872  0.6404 0.5218830    
# v87    4.27829   84.86010  0.0504 0.9597911    
# v88    4.83853   21.42043  0.2259 0.8212918    
# v107   6.11318    0.64794  9.4348 < 2.2e-16 ***
# v108   4.13673    4.85345  0.8523 0.3940316    
# v117   3.27223    1.86578  1.7538 0.0794627 .  
# v129   4.48811    2.82435  1.5891 0.1120434    
# v130   4.79551    2.04481  2.3452 0.0190165 *  
# v145   3.97314    0.60547  6.5620 5.308e-11 ***
# v157   5.49003    0.13670 40.1608 < 2.2e-16 ***
# v172   5.88622    1.65908  3.5479 0.0003884 ***
# v173   6.49017    1.08156  6.0008 1.964e-09 ***
# v181   6.79913    1.81802  3.7399 0.0001841 ***
# v182   5.43450    7.66955  0.7086 0.4785848    
# v188   1.51878  233.81977  0.0065 0.9948174    
# v189   5.06634    5.20058  0.9742 0.3299632    
# ---
#   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# 
# -2 log L: 2338.857 
exp(confint(fit, method="quad"))  # backtransformed confidence intervals calculated via quadratic approximation (=Wald confidence intervals)
#              2.5 %        97.5 %
# v10   1.095964e+00  8.562480e+03
# v11   2.326040e+01  1.743531e+03
# v27   1.959787e+00  4.242829e+02
# v38   8.403942e-20  5.670507e+21
# v39   2.863032e-09  8.206810e+11
# v48   4.036402e-01  1.953696e+05
# v49   9.330044e-13  3.710221e+15
# v58   6.309090e-16  3.027742e+18
# v59   2.652533e+01  1.090313e+04
# v70  1.871739e-189 6.330566e+189
# v71   8.933534e-46  2.349031e+47
# v80   2.824905e+01  2.338118e+03
# v86   4.568985e-06  3.342200e+10
# v87   4.216892e-71  1.233336e+74
# v88   7.383119e-17  2.159994e+20
# v107  1.268806e+02  1.608602e+03
# v108  4.626990e-03  8.468795e+05
# v117  6.806996e-01  1.021572e+03
# v129  3.508065e-01  2.255556e+04
# v130  2.198449e+00  6.655952e+03
# v145  1.622306e+01  1.741383e+02
# v157  1.853224e+02  3.167003e+02
# v172  1.393601e+01  9.301732e+03
# v173  7.907170e+01  5.486191e+03
# v181  2.542890e+01  3.164652e+04
# v182  6.789470e-05  7.735850e+08
# v188 4.284006e-199 4.867958e+199
# v189  5.936664e-03  4.236704e+06
library(broom)
signlevels = tidy(fit)$p.value/2 # 1-sided p values for peak to be sign higher than 1
adjsignlevels = p.adjust(signlevels, method="fdr") # FDR corrected p values
a_nnlsbbmle = exp(coef(fit)) # exp to backtransform
max(a_nnls[nonzero]-a_nnlsbbmle) # -9.981704e-11, coefficients as expected almost the same
plot(x, y, type="l", main="Ground truth (red), nnls bbmle logcoeff estimate (blue & green, green=FDR p value<0.05)", ylab="Signal (black) & peaks (red & blue)", xlab="Time", ylim=c(-max(y),max(y)))
lines(x,-y)
lines(a, type="h", col="red", lwd=2)
lines(x[nonzero], -a_nnlsbbmle, type="h", col="blue", lwd=2)
lines(x[nonzero][(adjsignlevels<0.05)&(a_nnlsbbmle>1)], -a_nnlsbbmle[(adjsignlevels<0.05)&(a_nnlsbbmle>1)], 
      type="h", col="green", lwd=2)
sum((signlevels<0.05)&(a_nnlsbbmle>1)) # 14 peaks significantly higher than 1 before FDR correction
sum((adjsignlevels<0.05)&(a_nnlsbbmle>1)) # 11 peaks significant after FDR correction

这些计算的结果与XM=as.matrix(bM)[,nonzero,drop=FALSE] # design matrix posbetas = a_nnls[nonzero] # nonzero nnls coefficients dispersion=sum(residuals^2)/(n-p) # estimated dispersion (variance in case of gaussian noise) (1 if noise were poisson or binomial) information_matrix = t(XM) %*% XM # observed Fisher information matrix for nonzero coefs, ie negative of the 2nd derivative (Hessian) of the log likelihood at param estimates scaled_information_matrix = (t(XM) %*% XM)*(1/dispersion) # information matrix scaled by 1/dispersion # let's now calculate this scaled information matrix on a log transformed Y scale (cf. stat.psu.edu/~sesa/stat504/Lecture/lec2part2.pdf, slide 20 eqn 8 & Table 1) to take into account the nonnegativity constraints on the parameters scaled_information_matrix_logscale = scaled_information_matrix/((1/posbetas)^2) # scaled information_matrix on transformed log scale=scaled information matrix/(PHI'(betas)^2) if PHI(beta)=log(beta) vcov_logscale = solve(scaled_information_matrix_logscale) # scaled variance-covariance matrix of coefs on log scale ie of log(posbetas) # PS maybe figure out how to do this in better way using chol2inv & QR decomposition - in R unscaled covariance matrix is calculated as chol2inv(qr(XW_glm)$qr) SEs_logscale = sqrt(diag(vcov_logscale)) # SEs of coefs on log scale ie of log(posbetas) posbetas_LOWER95CL = exp(log(posbetas) - 1.96*SEs_logscale) posbetas_UPPER95CL = exp(log(posbetas) + 1.96*SEs_logscale) data.frame("2.5 %"=posbetas_LOWER95CL,"97.5 %"=posbetas_UPPER95CL,check.names=F) # 2.5 % 97.5 % # 1 1.095874e+00 8.563185e+03 # 2 2.325947e+01 1.743600e+03 # 3 1.959691e+00 4.243037e+02 # 4 8.397159e-20 5.675087e+21 # 5 2.861885e-09 8.210098e+11 # 6 4.036017e-01 1.953882e+05 # 7 9.325838e-13 3.711894e+15 # 8 6.306894e-16 3.028796e+18 # 9 2.652467e+01 1.090340e+04 # 10 1.870702e-189 6.334074e+189 # 11 8.932335e-46 2.349347e+47 # 12 2.824872e+01 2.338145e+03 # 13 4.568282e-06 3.342714e+10 # 14 4.210592e-71 1.235182e+74 # 15 7.380152e-17 2.160863e+20 # 16 1.268778e+02 1.608639e+03 # 17 4.626207e-03 8.470228e+05 # 18 6.806543e-01 1.021640e+03 # 19 3.507709e-01 2.255786e+04 # 20 2.198287e+00 6.656441e+03 # 21 1.622270e+01 1.741421e+02 # 22 1.853214e+02 3.167018e+02 # 23 1.393520e+01 9.302273e+03 # 24 7.906871e+01 5.486398e+03 # 25 2.542730e+01 3.164851e+04 # 26 6.787667e-05 7.737904e+08 # 27 4.249153e-199 4.907886e+199 # 28 5.935583e-03 4.237476e+06 z_logscale = log(posbetas)/SEs_logscale # z values for log(coefs) being greater than 0, ie coefs being > 1 (since log(1) = 0) pvals = pnorm(z_logscale, lower.tail=FALSE) # one-sided p values for log(coefs) being greater than 0, ie coefs being > 1 (since log(1) = 0) pvals.adj = p.adjust(pvals, method="fdr") # FDR corrected p values plot(x, y, type="l", main="Ground truth (red), nnls estimates (blue & green, green=FDR Wald p value<0.05)", ylab="Signal (black) & peaks (red & blue)", xlab="Time", ylim=c(-max(y),max(y))) lines(x,-y) lines(a, type="h", col="red", lwd=2) lines(-a_nnls, type="h", col="blue", lwd=2) lines(x[nonzero][pvals.adj<0.05], -a_nnls[nonzero][pvals.adj<0.05], type="h", col="green", lwd=2) sum((pvals<0.05)&(posbetas>1)) # 14 peaks significantly higher than 1 before FDR correction sum((pvals.adj<0.05)&(posbetas>1)) # 11 peaks significantly higher than 1 after FDR correction返回的结果几乎相同（但速度更快），所以我认为这是正确的，并且将与我们对mle2所做的隐含操作相对应。 ..

仅使用常规线性模型拟合在mle2拟合中用正系数重新拟合协变量是行不通的，因为这样的线性模型拟合不会考虑非负约束，因此会导致无意义的置信度间隔可能会变为负数。本文"Exact post model selection inference for marginal screening" by Jason Lee & Jonathan Taylor还提出了一种对非负nnls（或LASSO）系数进行模型后选择推断的方法，并为此使用了截断的高斯分布。我还没有看到针对nnls fits的这种方法的任何公开可用的实现-对于LASSO fits，有selectiveInference软件包做了类似的事情。如果有人碰巧实现了，请告诉我！

带有t值的Python非负线性回归

2 个答案: