Non linear interpolation to find the missing values using R

I have a data.table test_dt, in which some of the values are missing for each group. Can someone suggest the fastest way to interpolate and fill the missing value?

In the graph form x-axis is column a and y-axis is column b, hench the interval distance of column a should be considered while predicting b.

test_dt = structure(list(group = c("B1", "B1", "B1", "B1", "B1", "B1", "B1", "B1", "C1", "C1", "C1", "C1", "C1", "C1", "C1", "C1"), a = c(165, 170, 185, 195, 200, 210, 220, 240, 1, 1.5, 2, 4.5, 5, 5.5, 7, 10), b = c(1.925, 0.575, 0.3, NA, NA, 2.825, 9.05, 27.9, 3.775, 3.225, 2.75, 0.255, 0.04, NA, NA, NA)), row.names = c(NA, -16L), class = c("data.table", "data.frame"), index = integer(0))
> test_dt 
    group     a      b
 1:    B1 165.0  1.925
 2:    B1 170.0  0.575
 3:    B1 185.0  0.300
 4:    B1 195.0     NA
 5:    B1 200.0     NA
 6:    B1 210.0  2.825
 7:    B1 220.0  9.050
 8:    B1 240.0 27.900
 9:    C1   1.0  3.775
10:    C1   1.5  3.225
11:    C1   2.0  2.750
12:    C1   4.5  0.255
13:    C1   5.0  0.040
14:    C1   5.5     NA
15:    C1   7.0     NA
16:    C1  10.0     NA

Following are some conditions -

Interpolated values should not be zero or negative value
Interpolated values should not be a copy of the last non-NA value

I tried to solve it using na.spline but the result is not correct, especially for group C1, where value of column a is 7 and 10

test_dt[, predicted := zoo::na.spline(zoo(.SD), x = a)[, 2], by = c("group")]
> test_dt
    group     a      b   predicted
 1:    B1 165.0  1.925  1.92500000
 2:    B1 170.0  0.575  0.57500000
 3:    B1 185.0  0.300  0.30000000
 4:    B1 195.0     NA  0.18981191
 5:    B1 200.0     NA  0.40167712
 6:    B1 210.0  2.825  2.82500000
 7:    B1 220.0  9.050  9.05000000
 8:    B1 240.0 27.900 27.90000000
 9:    C1   1.0  3.775  3.77500000
10:    C1   1.5  3.225  3.22500000
11:    C1   2.0  2.750  2.75000000
12:    C1   4.5  0.255  0.25500000
13:    C1   5.0  0.040  0.04000000
14:    C1   5.5     NA  0.03038963
15:    C1   7.0     NA  1.67389631
16:    C1  10.0     NA 16.44642968

Update - 1 Following is the second dataset. Using the solutions below, the predicted values are very high.

temp_dt = structure(list(group = c("K1", "K1", "K1", "K1", "K1", "K1", 
"K1", "K1", "K1", "K1", "K1", "K1", "K1", "F1", "F1", "F1", "F1", 
"F1", "F1"), a = c(185, 190, 195, 200, 202.5, 205, 210, 212.5, 
215, 217.5, 220, 222.5, 225, 22, 23, 24, 25, 26, 35), b = c(NA, 
NA, 0.45, 0.6, 1.05, 1.45, 2.7, 3.125, 4.3, NA, 7.05, 8.95, 10.35, 
0.5, 0.7, 0.925, 1.2, 1.75, NA), predicted = c(463741.812236, 
14.7721571, 0.45, 0.6, 1.05, 1.45, 2.7, 3.125, 4.3, 5.6346081, 
7.05, 8.95, 10.35, 0.5, 0.7, 0.925, 1.2, 1.75, 65226323253857.9
)), row.names = c(NA, -19L), class = c("data.table", "data.frame"
), .internal.selfref = <pointer: 0x563cccc504c0>, index = integer(0))

test_dt[, predicted := round(exp(na.spline(log(b), x = a, na.rm = FALSE)), 7), by = group]


    group     a      b    predicted
 1:    K1 185.0     NA 4.637418e+05
 2:    K1 190.0     NA 1.477216e+01
 3:    K1 195.0  0.450 4.500000e-01
 4:    K1 200.0  0.600 6.000000e-01
 5:    K1 202.5  1.050 1.050000e+00
 6:    K1 205.0  1.450 1.450000e+00
 7:    K1 210.0  2.700 2.700000e+00
 8:    K1 212.5  3.125 3.125000e+00
 9:    K1 215.0  4.300 4.300000e+00
10:    K1 217.5     NA 5.634608e+00
11:    K1 220.0  7.050 7.050000e+00
12:    K1 222.5  8.950 8.950000e+00
13:    K1 225.0 10.350 1.035000e+01
14:    F1  22.0  0.500 5.000000e-01
15:    F1  23.0  0.700 7.000000e-01
16:    F1  24.0  0.925 9.250000e-01
17:    F1  25.0  1.200 1.200000e+00
18:    F1  26.0  1.750 1.750000e+00
19:    F1  35.0     NA 6.522632e+13

Solution

Ultimately, you have to decide the interpolation procedure based on scientific background. However, in order to avoid producing negative values, the log-transformation is useful. In the following, I combine that with a spline interpolation.

library(data.table)
test_dt = data.table(group = c("B1", "B1", "B1", "B1", "B1", "B1", 
                               "B1", "B1", "C1", "C1", "C1", "C1", "C1", "C1", "C1", "C1"), 
                     a = c(165, 170, 185, 195, 200, 210, 220, 240, 1, 1.5, 2, 4.5, 5, 5.5, 7, 10), 
                     b = c(1.925, 0.575, 0.3, NA, NA, 2.825, 9.05, 27.9, 3.775, 3.225, 2.75, 0.255, 
                           0.04, NA, NA, NA))

library(zoo)
test_dt[, c := exp(na.spline(log(b), x = a, na.rm = FALSE)), by = group]

library(ggplot2)
ggplot(test_dt, aes(x = a, color = group)) +
  geom_line(aes(y = c)) +
  geom_point(aes(y = c, color = "interpolated")) +
  geom_point(aes(y = b))