Cobb-Douglas production function in TI Nspire

The Cobb-Douglas function in Economics is a function to represent relationship between labour and capital input with respect to technology. The basic form of the function is

Q=KL^αC^1-α

A more common form is

y_t = β₀ · x_t1^β1 · x_t2^β2 · e^μt

The calculation can be simplified by taking log on the function as below:

ln y_t = ln β₀ + β₁ ln x_t1 + β₂ ln x_t2 + μ_t

Not only Multiple Regression is built-in in the TI Nspire so the calculation for this function is trivial, the 3-D graphing capability can also be utilized to visualise the function.

The lower plane is the plot of the Cobb-Douglas function using a set of sample data, with the upper plane plotting the return of scale, i.e. α + β. In this case it is > 1.

Implementing parallel GPU function in CUDA for R

There are existing R packages for CUDA. But if there is a need to customize your own parallel code on NVIDIA GPU to be called from R, it is possible to do so with the CUDA Toolkit. This post demonstrates a sample function to approximate the value of Pi using Monte Carlo method which is accelerated by GPU. The sample is built using Visual Studio 2010 but the Toolkit is supported on linux platforms as well. It is assumed that the Visual Studio is integrated with the CUDA Toolkit.

The first thing to do is to create a New Project using the Win32 Console Application template, and specify DLL with Empty project option.

And then, some standard project environment customization including:

CUDA Build Customization:

CUDA Runtime, select Shared/dynamic CUDA runtime library:

Project Dependencies setting. Since the CUDA code in this example utilize curand for Monte Carlo, the corresponding library must be included or else the linking will fail.

Finally the time to code. Only a cu file is needed which resembles the standard directives. It is important to include the extern declaration as below for R to call.

After a successful compile, the DLL will be created with the CUDA code. This DLL will be registered in R for calling.

Finally, start R and issue the dyn.load command to load the DLL into the running environment. Shown below is a “wrapper” R function to make calling the CUDA code easier. Notice at the heart of this wrapper is the .C function.

Last but not least, the CUDA Toolkit comes with a visual profiler which is capable to be launched for profiling the performance of the NVIDIA GPU. It can be launched from the GUI, or using a command line like the example below. It should be noted that the command line profiler must be started before R or it might not be able to profile properly.

The GUI profiler is equipped with a nice interface to show performance statistics.

Data input for ANOVA in TI nspire and R

In TI nspire CX, the application Lists & Spreadsheet provided a convenient Excel list interface for data input.

The data can also be named by columns and recalled from the Calculator application. Statistical functions can then be applied. Using a sample from the classical TI-89 statistics guide book on determining the interaction between two factors using 2-way ANOVA, the same output is obtained from the TI nspire CX.

In R, data are usually imported from CSV file using read.csv() command. There are also other supported formats including SPSS and Excel. For more casual data entry that command line input is suffice, raw data are usually stored into list variable using c() command. Working with ANOVA for data entry in this way is not as straightforward because dimension is required for the analysis on data stored in the list variable.

To accomplish the ANOVA, factor data types are used in conjunction with list variable. The below is the same TI example completed in R. Firstly we define the list variable in a fashion of the order by club (c1 = driver, c2 = five iron) then brand (b1-, b2-, b3-, with the last digit as the sample number), i.e.
{c1,b1-1}; {c1,b1-2}; {c1,b1-3}; {c1,b1-4};
{c1,b2-1}; {c1,b2-2};…
{c2,b1-1}; {c2,b1-2};…

Two Factor variables are then created, one for club (with twelve 1’s followed by twelve 2’s), and another for brand (1 to 3 each repeating four times for each sample, and then completed by another identical sequence).

These two Factor variables essentially represent the position (or index in array’s term) of the nth data value in respect of the factor it belongs to, and can be better visualized in the following table.

Finally, the 2-way ANOVA can be performed using the following commands.

Interaction plot in R.

White test in TI Nspire and R

The White test is a statistical test to determine whether homoskedasticity exists in a data set. This test is based on the variance from the residual values. The TI Npsire is capable of computing this test even though it is not part of built-in functions, as the residual values can be recalled from regression tests. An example including multiple regression is shown below.

A scatter plot for visual inspection of heteroskedasticity.

In spreadsheet mode the calculation of the data set.

And in R.