Curve Fitting in .NET framework

Creator QR Code in .NET framework Curve Fitting

Temperature of cooling block 300 250 Temp(F) 200 150 100 0 1 2 3 Time(h) 4 5 6 7

Figure 10-5

Let s create data for the time axis:

>> t = linspace(0,7);

Next define the fit function:

>> y = a*t^3 + b*t^2 + c*t + d;

We will also need an array of data evaluated at the exact time points where the temperature data was collected:

>> w = a*time^3 + b*time^2 + c*time + d;

Let s plot the collected data and the fit curve on the same graph:

>> plot(time,temp,'o',t,y),xlabel('time(h)'),ylabel ('temp(F)'),title('Third Order fit for cooling metal block')

The result is shown in Figure 10-6 It looks like the third order polynomial has generated a good fit Let s compute r2 First we find the mean of the collected data and calculate S:

Third order fit for cooling metal block 400 Temp(F) 300 200 100

3 Time(h)

Figure 10-6

Now let s find A:

We find r2 to be:

It looks like we have a perfect fit to the data However this is misleading and in fact the real function describing this data is: T = 100 + 20259e 023t 259e 180t Let s have MATLAB print the r-squared value in long format:

The RMS is less than one, so we can take it to be a good approximation Now that we have a function y(t), we can estimate the temperature at various times for which we have no data For instance, the data is collected out to a time of 7 hours Let s build the function out to longer times First we extend the time line:

>> t = [0:01:15];

Regenerate the fit polynomial:

>> y = a*t^3 + b*t^2 + c*t + d;

We can use the find command to ask questions about the data For instance, when is the temperature less than 80 degrees, so maybe we can risk picking up the metal block

The command find has returned the array indices for temperatures that satisfy this requirement We can reference these locations in the array t that contains the times and the array y that contains the temperatures First we extract the times, adding a quote mark at the end to transpose the data into a column vector:

Now let s get the corresponding temperatures:

We can arrange the data into a two column table, with the left column containing the times and the right column the temperatures:

So for instance, we see that at 149 hours the block is estimated to be at about 75 degrees Now let s generate a plot of the data from 10 hours to 15 hours First we find the array index for the time at 10 hours We had generated the time in increments of 01, so we can find it by searching for t > 99:

>> find(t >99) ans = Columns 1 through 18 101 111 112 102 113 103 114 104 115 105 116 106 117 107 118 108 109 110

120 Temp(F) 100 80 60 10

125 13 Time(h)

Figure 10-7 A plot of estimated temperatures in a time range well beyond that where the data was collected

Next we create a new array containing these time values:

>> T1 = t(101:151);

Now let s grab the temperatures over that range and store them:

>> TEMP2 = y(101:151);

Then we can plot them:

>> plot(T1,TEMP2),xlabel('time(h)'),ylabel('Temp(F)'), axis([10 15 60 120])

The result is shown in Figure 10-7 The model predicts that over this 5 hour range the temperature of the metal block will drop more than 40 degrees

Other types of fits besides fitting to a polynomial are possible, we briefly mention one In some cases it may be necessary to fit the data to an exponential function The fit in this case is: y = b(10)mx where x is the independent variable and y is the dependent variable Now we define the relations: w = log10 y z=x and then fit the data to a line of the form: w = p1z + p2