如何在分布不均的数据集中拟合平滑滞后？
How to fit a smooth hysteresis in a poorly distributed data set?
这是 的后续问题。 smooth.spline 的直接应用很不适合我的实际情况，尽管它被证明对这类问题很有用，并且在我的模拟玩具数据集上运行良好。
我上传了我的数据集示例 here。
下图由最后的代码创建。
磁滞没有完全关闭 - 但这不是问题。
应用 smooth.spline 会产生过于混乱的输出（红色）。在 smooth.spline 中使用参数 spar 我能够获得足够平滑的近似值（蓝色）——尽管它只是很差地遵循右上角的数据点曲线。
然而真正让我不安的是左下角点周围的 "knot" - 即使是平滑曲线（蓝色）它也在那里。当它从左下角移动到右上角时，它也有一个人为的 "loop"，尽管那里附近没有数据点。
library(dplyr)
library(ggplot2)
library(tidyr)
exdata <- readRDS("data/example.Rdata")
exdata %<>%
mutate(y_smooth = smofun(y),
x_smooth = smofun(x),
y_smooth2 = smofun(y, spar = 0.7),
x_smooth2= smofun(x, spar = 0.7)
)
ggplot(exdata, mapping = aes(x=x, y=y)) +
geom_point(alpha = 0.3) +
geom_path(mapping = aes(x=x_smooth, y=y_smooth), colour = "red") +
geom_path(mapping = aes(x=x_smooth2, y=y_smooth2), colour = "blue") +
theme_light()
正如我在之前的帖子中所说：，我们希望分别对每个坐标进行平滑处理。
## you have 713 data
plot(1:713, exdata$x); points(1:713, exdata$y)
通常我会为每个使用 smooth.spline。它在您的原始玩具数据集上运行良好，但在您上面勾画的真实数据集上效果不佳。直接使用 smooth.spline 将导致我们陷入尴尬的境地：。显然，您的数据中有两个变化点，并且您需要不同段的不同平滑度/自由度。在 linked 案例研究中，我能够想出一个转换来克服这个问题，但在这里我不知道我可以做什么合适的转换。相反，我建议我们手动放置样条的结，以便直接控制平滑度。
smooth.spline 不允许我们手动放置结。你可以告诉它你想要多少结（通过使用参数 nknots），它在内部按分位数放置结。事实上，我们不必依赖smooth.spline。这只是一个回归问题。我们可以用任何一种回归方法来做，只要它让你满意。在下文中，我使用 splines 软件包提供的回归 B 样条曲线（其中一个带有 R，因此无需安装）并使用普通最小二乘法拟合模型（通过 lm）。
library(splines)
test <- function (b1, b2, n1, n2, n3, degree = 1, exdata) {
x <- exdata$x; y <- exdata$y; t <- 1:length(x)
## place knots:
## n1 knots on [1, b1)
## n2 knots on [b1, b2]
## n3 knots on (b3, 713]
t1 <- seq(1, b1, length = n1 + 1)
t2 <- seq(b1, b2, length = n2)
t3 <- seq(b2, 713, length = n3 + 1)
kt <- c(t1[1:n1], t2, t3[-1])
kt_inner <- kt[2:(length(kt) - 1)]
fitx <- lm( x ~ bs(t, degree = degree, knots = kt_inner) )
fity <- lm( y ~ bs(t, degree = degree, knots = kt_inner) )
xh <- fitted(fitx); yh <- fitted(fity)
par(mfrow = c(1, 2))
plot(t, x, col = 8); points(t, y, col = 8)
abline(v = kt, lty = 2, col = 8)
lines(t, xh, lwd = 2); lines(t, yh, lwd = 2)
plot(x, y, col = 8)
lines(xh, yh, lwd = 2)
z <- list(x = xh, y = yh)
}
测试功能允许您指定两个断点b1、b2。然后，在 [1, b1) 上你想要 n1 节，在 [b1, b2] 上你想要 n2 节，在 (b3, 713] 上你想要 n3 节。设置 degree 允许我们尝试线性样条、二次样条和三次样条。测试函数将产生拟合样条曲线，并且 return 不可见地平滑 x、y 值。
经过几次尝试，我发现以下可能是最好的：
z <- test(b1 = 90, b2 = 130, n1 = 4, n2 = 5, n3 = 10, degree = 3, exdata)
这里没有处罚。但是通过使用 mgcv::gam 代替 lm + splines::bs，惩罚回归是可能的。包 mgcv 允许我们指定节点（参见：）。
如果我们不想费心指定节点，我们可以在 mgcv 中使用 "adaptive spline" 平滑器，它允许平滑参数在局部变化（它实质上会产生一组平滑参数）。这确实克服了 smooth.spline 中只有一个平滑参数控制全局惩罚的限制。
请注意，在样条方法中，选择断点隐藏在结的选择中，因为断点在结的集合中。
还有 none 基于样条的方法。包 segmented 通过自动选择变化/断点来进行分段线性回归（但你必须使用参数 psi 提前告诉它有多少断点）。
对不同的方法进行比较和讨论的空间很大。作为未来可能重新调查的备份，我在这里附上了你的数据集（以防保管箱 link 损坏）。
exdata <-
structure(list(x = c(129.5695, 131.7681, 111.0764, 104.072, 117.4685,
124.6366, 123.3217, 127.4646, 128.0695, 110.6105, 105.3012, 98.3239,
94.7057, 103.349, 101.5618, 97.0452, 99.2581, 96.869, 87.8457,
89.5727, 101.3341, 103.7006, 100.9025, 101.2322, 107.0066, 99.4793,
101.446, 110.8227, 115.8369, 113.8645, 119.1815, 115.382, 106.6097,
100.0037, 99.4583, 93.7219, 93.5242, 94.6776, 98.66, 92.5156,
99.0487, 99.2332, 97.9979, 98.5008, 99.4881, 96.2035, 99.3862,
108.1677, 107.2952, 107.5039, 109.8523, 107.0338, 98.331, 106.3463,
105.3598, 100.5509, 108.4587, 111.785, 102.911, 104.0221, 103.0966,
96.0376, 95.0978, 92.9253, 94.8009, 96.7117, 101.732, 107.1491,
107.0559, 107.6543, 108.8192, 101.6353, 94.3223, 94.3889, 93.4375,
99.0691, 101.1942, 102.7105, 105.8936, 104.8027, 93.6381, 99.6715,
98.8952, 100.2534, 104.2578, 105.9263, 106.0875, 118.8101, 121.278,
117.5301, 121.667, 428.0926, 731.1513, 775.6144, 795.4499, 817.3595,
535.1731, 276.7803, 330.6044, 439.8171, 541.2278, 648.0471, 794.4545,
865.3579, 909.2367, 978.2347, 1067.6434, 1088.8355, 1153.1032,
1230.6436, 1262.7867, 1284.1465, 1291.3056, 1282.8526, 1277.1739,
1301.0525, 1300.5524, 1319.1291, 1332.0036, 1339.5618, 1342.3142,
1351.3678, 1355.4577, 1364.5274, 1386.3259, 1388.0149, 1393.6825,
1407.7905, 1410.5657, 1389.9132, 1386.5312, 1376.0732, 1371.785,
1375.1907, 1387.2856, 1386.5125, 1387.6686, 1379.9445, 1378.8248,
1363.9721, 1367.6901, 1361.4242, 1355.2429, 1347.776, 1348.4306,
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1335.5, 1343.194, 1331.6572, 1323.6614, 1314.4852, 1322.7351,
1320.4661, 1329.9164, 1326.9096, 1336.5571, 1333.9171, 1330.2649,
1325.3137, 1337.4239, 1333.6401, 1326.0902, 1320.4635, 1309.8197,
1293.1904, 1292.3817, 1281.3893, 1285.0823, 1292.0862, 1294.0099,
1286.4846, 1297.9813, 1289.5944, 1286.8668, 1282.2538, 1275.3989,
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939.6842, 942.3164, 928.0374, 919.9734, 911.6006), y = c(788.1826,
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1304.3444, 1300.2792, 1305.167, 1307.8903, 1306.3299, 1293.7967,
1278.7566, 1274.368, 1245.5256)), row.names = c(NA, -713L), .Names = c("x",
"y"), class = "data.frame")
这是 smooth.spline 的直接应用很不适合我的实际情况，尽管它被证明对这类问题很有用，并且在我的模拟玩具数据集上运行良好。
我上传了我的数据集示例 here。
下图由最后的代码创建。
磁滞没有完全关闭 - 但这不是问题。
应用 smooth.spline 会产生过于混乱的输出（红色）。在 smooth.spline 中使用参数 spar 我能够获得足够平滑的近似值（蓝色）——尽管它只是很差地遵循右上角的数据点曲线。
然而真正让我不安的是左下角点周围的 "knot" - 即使是平滑曲线（蓝色）它也在那里。当它从左下角移动到右上角时，它也有一个人为的 "loop"，尽管那里附近没有数据点。
library(dplyr)
library(ggplot2)
library(tidyr)
exdata <- readRDS("data/example.Rdata")
exdata %<>%
mutate(y_smooth = smofun(y),
x_smooth = smofun(x),
y_smooth2 = smofun(y, spar = 0.7),
x_smooth2= smofun(x, spar = 0.7)
)
ggplot(exdata, mapping = aes(x=x, y=y)) +
geom_point(alpha = 0.3) +
geom_path(mapping = aes(x=x_smooth, y=y_smooth), colour = "red") +
geom_path(mapping = aes(x=x_smooth2, y=y_smooth2), colour = "blue") +
theme_light()
正如我在之前的帖子中所说：
## you have 713 data
plot(1:713, exdata$x); points(1:713, exdata$y)
通常我会为每个使用 smooth.spline。它在您的原始玩具数据集上运行良好，但在您上面勾画的真实数据集上效果不佳。直接使用 smooth.spline 将导致我们陷入尴尬的境地：
smooth.spline 不允许我们手动放置结。你可以告诉它你想要多少结（通过使用参数 nknots），它在内部按分位数放置结。事实上，我们不必依赖smooth.spline。这只是一个回归问题。我们可以用任何一种回归方法来做，只要它让你满意。在下文中，我使用 splines 软件包提供的回归 B 样条曲线（其中一个带有 R，因此无需安装）并使用普通最小二乘法拟合模型（通过 lm）。
library(splines)
test <- function (b1, b2, n1, n2, n3, degree = 1, exdata) {
x <- exdata$x; y <- exdata$y; t <- 1:length(x)
## place knots:
## n1 knots on [1, b1)
## n2 knots on [b1, b2]
## n3 knots on (b3, 713]
t1 <- seq(1, b1, length = n1 + 1)
t2 <- seq(b1, b2, length = n2)
t3 <- seq(b2, 713, length = n3 + 1)
kt <- c(t1[1:n1], t2, t3[-1])
kt_inner <- kt[2:(length(kt) - 1)]
fitx <- lm( x ~ bs(t, degree = degree, knots = kt_inner) )
fity <- lm( y ~ bs(t, degree = degree, knots = kt_inner) )
xh <- fitted(fitx); yh <- fitted(fity)
par(mfrow = c(1, 2))
plot(t, x, col = 8); points(t, y, col = 8)
abline(v = kt, lty = 2, col = 8)
lines(t, xh, lwd = 2); lines(t, yh, lwd = 2)
plot(x, y, col = 8)
lines(xh, yh, lwd = 2)
z <- list(x = xh, y = yh)
}
测试功能允许您指定两个断点b1、b2。然后，在 [1, b1) 上你想要 n1 节，在 [b1, b2] 上你想要 n2 节，在 (b3, 713] 上你想要 n3 节。设置 degree 允许我们尝试线性样条、二次样条和三次样条。测试函数将产生拟合样条曲线，并且 return 不可见地平滑 x、y 值。
经过几次尝试，我发现以下可能是最好的：
z <- test(b1 = 90, b2 = 130, n1 = 4, n2 = 5, n3 = 10, degree = 3, exdata)
这里没有处罚。但是通过使用 mgcv::gam 代替 lm + splines::bs，惩罚回归是可能的。包 mgcv 允许我们指定节点（参见：
如果我们不想费心指定节点，我们可以在 mgcv 中使用 "adaptive spline" 平滑器，它允许平滑参数在局部变化（它实质上会产生一组平滑参数）。这确实克服了 smooth.spline 中只有一个平滑参数控制全局惩罚的限制。
请注意，在样条方法中，选择断点隐藏在结的选择中，因为断点在结的集合中。
还有 none 基于样条的方法。包 segmented 通过自动选择变化/断点来进行分段线性回归（但你必须使用参数 psi 提前告诉它有多少断点）。
对不同的方法进行比较和讨论的空间很大。作为未来可能重新调查的备份，我在这里附上了你的数据集（以防保管箱 link 损坏）。
exdata <- structure(list(x = c(129.5695, 131.7681, 111.0764, 104.072, 117.4685, 124.6366, 123.3217, 127.4646, 128.0695, 110.6105, 105.3012, 98.3239, 94.7057, 103.349, 101.5618, 97.0452, 99.2581, 96.869, 87.8457, 89.5727, 101.3341, 103.7006, 100.9025, 101.2322, 107.0066, 99.4793, 101.446, 110.8227, 115.8369, 113.8645, 119.1815, 115.382, 106.6097, 100.0037, 99.4583, 93.7219, 93.5242, 94.6776, 98.66, 92.5156, 99.0487, 99.2332, 97.9979, 98.5008, 99.4881, 96.2035, 99.3862, 108.1677, 107.2952, 107.5039, 109.8523, 107.0338, 98.331, 106.3463, 105.3598, 100.5509, 108.4587, 111.785, 102.911, 104.0221, 103.0966, 96.0376, 95.0978, 92.9253, 94.8009, 96.7117, 101.732, 107.1491, 107.0559, 107.6543, 108.8192, 101.6353, 94.3223, 94.3889, 93.4375, 99.0691, 101.1942, 102.7105, 105.8936, 104.8027, 93.6381, 99.6715, 98.8952, 100.2534, 104.2578, 105.9263, 106.0875, 118.8101, 121.278, 117.5301, 121.667, 428.0926, 731.1513, 775.6144, 795.4499, 817.3595, 535.1731, 276.7803, 330.6044, 439.8171, 541.2278, 648.0471, 794.4545, 865.3579, 909.2367, 978.2347, 1067.6434, 1088.8355, 1153.1032, 1230.6436, 1262.7867, 1284.1465, 1291.3056, 1282.8526, 1277.1739, 1301.0525, 1300.5524, 1319.1291, 1332.0036, 1339.5618, 1342.3142, 1351.3678, 1355.4577, 1364.5274, 1386.3259, 1388.0149, 1393.6825, 1407.7905, 1410.5657, 1389.9132, 1386.5312, 1376.0732, 1371.785, 1375.1907, 1387.2856, 1386.5125, 1387.6686, 1379.9445, 1378.8248, 1363.9721, 1367.6901, 1361.4242, 1355.2429, 1347.776, 1348.4306, 1346.6911, 1352.2614, 1347.0287, 1349.8785, 1349.8588, 1345.208, 1335.5, 1343.194, 1331.6572, 1323.6614, 1314.4852, 1322.7351, 1320.4661, 1329.9164, 1326.9096, 1336.5571, 1333.9171, 1330.2649, 1325.3137, 1337.4239, 1333.6401, 1326.0902, 1320.4635, 1309.8197, 1293.1904, 1292.3817, 1281.3893, 1285.0823, 1292.0862, 1294.0099, 1286.4846, 1297.9813, 1289.5944, 1286.8668, 1282.2538, 1275.3989, 1273.0393, 1275.9338, 1272.8439, 1270.2929, 1263.9435, 1253.0364, 1247.754, 1250.2937, 1259.3518, 1273.8355, 1286.2065, 1293.7979, 1286.6026, 1290.399, 1285.4735, 1273.8435, 1261.1018, 1264.7923, 1250.2116, 1245.0338, 1247.965, 1253.4547, 1248.2883, 1250.0202, 1251.6438, 1260.2338, 1255.7682, 1252.6586, 1245.6251, 1238.162, 1226.5238, 1233.5413, 1241.1029, 1246.857, 1255.0828, 1257.7526, 1254.8524, 1257.7671, 1263.4184, 1261.9403, 1260.0229, 1258.0052, 1246.7327, 1249.2547, 1249.9542, 1246.4968, 1242.6747, 1232.7555, 1232.7507, 1240.5921, 1235.1443, 1230.6716, 1254.277, 1244.1764, 1236.4669, 1253.6932, 1263.0974, 1248.3615, 1255.2735, 1257.1036, 1251.1783, 1247.3818, 1249.0185, 1238.181, 1232.3126, 1224.0232, 1224.9371, 1222.0632, 1224.4354, 1222.4038, 1226.4081, 1215.139, 1215.3999, 1207.859, 1207.6245, 1204.4897, 1212.6945, 1206.7439, 1211.7789, 1211.6164, 1221.6207, 1226.8173, 1224.421, 1223.8749, 1217.5781, 1204.7366, 1192.8738, 1196.071, 1195.3558, 1198.7697, 1196.0119, 1191.8406, 1181.6458, 1184.4447, 1180.3189, 1187.6651, 1189.3141, 1204.3338, 1203.0549, 1204.9328, 1204.4187, 1205.1438, 1193.5248, 1189.1359, 1185.1723, 1172.775, 1170.4184, 1175.5543, 1176.8246, 1167.7341, 1163.6372, 1161.8829, 1147.2916, 1131.1096, 1128.5597, 1131.7037, 1126.5585, 1128.371, 1141.5219, 1149.2273, 1136.943, 1143.498, 1148.9494, 1147.0205, 1151.1578, 1165.1582, 1157.0784, 1155.4953, 1149.6255, 1142.4007, 1128.9415, 1129.4853, 1129.0765, 1128.6933, 1123.9743, 1133.2886, 1131.7012, 1134.0293, 1136.5361, 1136.5032, 1138.8941, 1144.9796, 1146.6936, 1141.7335, 1146.273, 1139.3171, 1144.506, 1142.9864, 1144.5049, 1147.8466, 1152.4488, 1136.4797, 1134.9687, 1132.3612, 1131.8932, 1129.8746, 1135.4942, 1137.3314, 1135.515, 1123.3012, 1123.8599, 1127.5974, 1121.888, 1121.8971, 1122.7738, 1121.3582, 1101.4597, 1101.6568, 1106.9945, 1111.0663, 1109.69, 1123.6524, 1117.9505, 1113.164, 1109.8204, 1112.7482, 1105.8838, 1104.777, 1101.0453, 1102.3828, 1102.1349, 1106.6481, 1114.2183, 1117.4973, 1115.0706, 1105.0425, 1098.5148, 1095.9327, 1091.0444, 1094.8351, 1097.9752, 1102.8526, 1098.7115, 1104.0875, 1100.6438, 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