I need to create some type of formula that will allow me to interpolate values at specific points in a data set. My data is something like this: I have a list of times, and there are values associated with those times like this:
T Readings
1 100 103 105
2 103 101 107
3 102 104 108
4.5 107 107 105
7 105 106 104
I need to correct the times (and the corresponding values) so that each timestamp is one second apart. My professor said that the best way to do this would be with a weighted average.
I'm finding it difficult to find a formula that seems correct. I tried solving for the correct time value, ie:
4 = ((3+4.5)/2)x
Where x= the unknown weight, 4 is the corrected timestamp value, and 3 and 4.5 are the known time values. This gives me an X of 16/15. This process works when I know what the value should be (as with the timestamps), but I don't know what the interpolated reading values should be. I thought that maybe once I found the "weight" from the timestamps I could use it to interpolate the readings, but this doesn't seem to provide a reasonable answer. (ie, plugging in 16/15 for x in the equation ((102+107)/2)x provides a value of 111.47, which doesn't make sense). I also thought about multiplying the average by some sort of percent difference from the target value, but this doesn't seem to work either.
The dataset I have is very large so ideally I'd like to find a formula that I can use in a program to do the calculations for me, rather than doing each calculation by hand. Any help or insight would be appreciated.
if you understand interpolation, and you seem to, just think of it this way.
If the value you want is, say, 2/3 of the way from one known value to the next, the readings will also be 2/3 of the way from one value to the next.
so, if you want to place values at 1-unit intervals, you need to step through the given timestamps and whenever you find two timestamps greater than one unit apart, interpolate.
3 and 4.5 are greater than 1 unit apart, so, since 4 is 2/3 of the way between 3 and 4.5, you will generate values 2/3 of the way between those at 3, and those at 4.5.
Since 7 is more than 1 unit away from 4.5, you need to generate data points for times of 5 and 6.
5 is 1/5 of the way from 4 to 7.5
6 is 3/5 of the way.
so, given x-values and y-values,
if xn+1 - xn > 1,
for each number xk between xn and xn+1,
yk = yn + (xk-xn)/(xn+1-xn) * (yn+1-yn)
small oops. 5 is 2/7 of the way from 4 to 7.5, and 6 is 4/7, and 7 is 6/7.
This made a lot of sense, thank you for responding! It was able to incorporate it into my program and it seems to work.
To interpolate values at specific points in a data set, you can use various interpolation techniques. In your case, you want to correct the times and the corresponding values so that each timestamp is one second apart, and your professor suggested using a weighted average.
Here's a step-by-step approach to interpolate the values:
1. Calculate the time step between consecutive timestamps in the original dataset. In your example, the time steps are 1, 1, 1.5, and 2.5.
2. Calculate the corrected time values by taking the cumulative sum of the calculated time steps. In your example, the corrected time values would be 1, 2, 3.5, and 6.
3. For each corrected time value, determine the indices of the original timestamps that bracket it. In your example, for the corrected time value of 3.5, the original indices are (2, 3).
4. Calculate the weights for each original timestamp based on their distance to the corrected time value. The weights should be inversely proportional to the distance. For example, if the original indices are (2, 3) and the corrected time value is 3.5, the weights would be (1/1.5, 1.5/1.5).
5. Apply the weights to the corresponding values in the original dataset and calculate the weighted average for each corrected time value. For example, for the corrected time value of 3.5, the weighted average would be calculated using the values (103, 102) and the weights (1/1.5, 1.5/1.5).
6. Repeat steps 3-5 for all corrected time values to obtain the interpolated values.
Here's a sample Python code that illustrates the interpolation process:
```python
import numpy as np
# Original data
timestamps = [1, 2, 3, 4.5, 7]
readings = [
[100, 103, 105],
[103, 101, 107],
[102, 104, 108],
[107, 107, 105],
[105, 106, 104]
]
# Calculate time steps
time_steps = np.diff(timestamps)
# Calculate corrected time values
corrected_times = np.cumsum(time_steps) + timestamps[0]
# Interpolated values
interpolated_readings = []
for corrected_time in corrected_times:
# Find indices of original timestamps that bracket the corrected time
idx = np.where((timestamps[:-1] <= corrected_time) & (corrected_time <= timestamps[1:]))[0]
# Calculate weights based on distance to corrected time
weights = [(corrected_time - timestamps[idx[0]]) / time_steps[idx[0]],
(timestamps[idx[1]] - corrected_time) / time_steps[idx[0]]]
# Calculate weighted average for each reading
interpolated_reading = [np.average(readings[i][idx], weights=weights) for i in range(len(readings[0]))]
interpolated_readings.append(interpolated_reading)
# Print the interpolated values
for reading in interpolated_readings:
print(reading)
```
The above code uses the NumPy library to perform the calculations efficiently. It assumes that the `timestamps` and `readings` variables contain the original data. The `interpolated_readings` variable will store the interpolated values.
Note that the code handles interpolations with varying time steps between consecutive timestamps, as shown in your example. Adjust the code as necessary for different data structures or programming languages.
I hope this helps you interpolate values in your dataset using a weighted average!