Making the logistic formula more versatile, stage two—different step sizes, up/down shifts
Transitions that follow a logistic pattern need not involve only steps from zero to one. To complete our generalization of the logistic formula, we will add parameters that let us adjust the curve to take other step sizes, and to have other baselines.
Both new parameters are straightforward to implement mathematically. To make the amount of vertical change something different than 1, we will multiply the simple formula by a scale factor. We will call this the height parameter since the curve will now have y values ranging from zero to height.
Moving the baseline up or down is even simpler mathematically. We will add a floor parameter to the scaled formula. Positive values for floor will move the curve up; negative values for floor will move the curve down. (Note that we add floor at the very end. It is not multiplied by height.)
So the full logistic model we have constructed has four parameters, and lets the data (via the model-fitting process) determine what their values are:
- rate (cell G3) — The controls the slope of the graph at the midpoint, which reflects how fast (and in which direction) the transition occurs. The slope at the midpoint equals rate×height.
- center (cell G4) — The midpoint of a transition is not usually at an input value of zero, even though the midpoint of the graph will be when the exponent expression is equal to zero. The formula can be made to reflect this by subtracting a center parameter from the input variable x.
- height (cell G5) — The amount of change in output will not always be 1.0. Multiplying the basic logistic model by a height scaling parameter will make its values go from zero to height instead of from zero to one.
- floor (cell G6) — Not all transitions have zero as the lower limit for the output value. We can allow for this with a floor parameter that is always added to the basic logistic formula.
The resulting full logistic model formula is [latex]y=\frac{height}{1+{{0.018316}^{rate\cdot(x-center)}}}+floor[/latex]
Corresponding spreadsheet formula:=$G$5/(1+0.018316^($G$3*(A3-$G$4)))+$G$6
Example 3: Fit a full logistic model to the dataset to the right showing water temperature during the first 10 seconds after a hot-water faucet is opened, then answer these questions about the model:
[a] What real-world quantity corresponds to the floor parameter? What is its value for the best-fit model for this data?
[b] What real-world quantity corresponds to the center parameter? What is its value for the best-fit model for this data?
[c] How can you compute an estimate of the hot-water temperature from the best-fit parameters? What is its value in this case?
[d] How fast was the water warming up when the water temperature was halfway through the transition?
Water Temperature | |
Seconds | Degrees F |
1 | 68.8 |
2 | 68.7 |
3 | 70.9 |
4 | 86.9 |
5 | 115.5 |
6 | 124.5 |
7 | 125.4 |
8 | 125.5 |
9 | 125.7 |
10 | 125.7 |
[i] Use the General Model template to make a logistic worksheet, using the formula shown above.Solution approach:
[ii] Copy the data to the worksheet and make a graph of data and model.
[iii] Preset to approximate values to speed up Solver’s search: rate=1, floor=70, height=55, center=4
[iv] Use Solver to find the best-fit parameters.
Answers:
[a] The floor parameter corresponds to the temperature of the cooled-off water that has been standing in the hot-water pipes. In this case, that temperature was 68.5 °F.
[b] The center parameter corresponds to the time in seconds from when data-taking was started until the water temperature had risen halfway to the hot-water temperature. In this case, the midpoint was at 4.31 seconds.
[c] In this model, the hot-water temperature is the sum of the floor and height parameters, 125.6 °F.
[d] This is the product of the rate and height parameters, which is 31.9 °F/second in this case.
A | B | C | D | E | F | G | H | |
1 | X | y data | y model | Residual | Squared | |||
2 | Secs | Deg F | prediction | deviation | deviation | Logistic parameters | ||
3 | 1 | 68.5 | 68.56844 | -0.068438 | 0.004684 | 0.559017 | Rate | |
4 | 2 | 69.1 | 68.85719 | 0.242814 | 0.058959 | 4.310686 | Center | |
5 | 3 | 71.2 | 71.42354 | -0.223536 | 0.049968 | 57.05165 | Height | |
6 | 4 | 87.6 | 87.53105 | 0.068945 | 0.004753 | 68.53369 | Floor | |
7 | 5 | 115.5 | 115.5249 | -0.024861 | 0.000618 | |||
8 | 6 | 124.2 | 124.3091 | -0.109083 | 0.011899 | |||
9 | 7 | 125.5 | 125.4462 | 0.053849 | 0.0029 | |||
10 | 8 | 125.4 | 125.5704 | -0.170428 | 0.029046 | Model and data value counts | ||
11 | 9 | 125.8 | 125.5837 | 0.216258 | 0.046767 | 4 | Number of parameters | |
12 | 10 | 125.6 | 125.5852 | 0.014834 | 0.00022 | 10 | Number of dev. averaged | |
13 | ||||||||
14 | Goodness of fit of this model | |||||||
15 | 0.209814 | Sum of squared deviations | ||||||
16 | 0.187 | Standard deviation |