Category Archives: TI Nspire CX CAS

Logistic Regression – from Nspire to R to Theano

Logistic regression is a very powerful tool for classification and prediction. It works very well with linearly separable problem. This installment will attempt to recap on its practical implementation, from traditional perspective by maximum likelihood, to more machine learning approach by neural network, as well as from handheld calculator to GPU cores.

The heart of the logistic regression model is the logistic function. It takes in any real value and return value in the range from 0 to 1. This is ideal for binary classifier system. The following is a graph of this function.
theanologistic1

TI Nspire

In the TI Nspire calculator, logistic regression is provided as a built-in function but is limited to single variable. For multi-valued problems, custom programming is required to apply optimization techniques to determine the coefficients of the regression model. One such application as shown below is the Nelder-Mead method in TI Nspire calculator.

Suppose in a data set from university admission records, there are four attributes (independent variables: SAT score, GPA, Interview score, Aptitude score) and one outcome (“Admission“) as the dependent variable.
theano-new1

Through the use of a Nelder-Mead program, the logistic function is first defined as l. It takes all regression coefficients (a1, a2, a3, a4, b), dependent variable (s), independent variables (x1, x2, x3, x4), and then simply return the logistic probability. Next, the function to optimize in the Nelder-Mead program is defined as nmfunc. This is the likelihood function on the logistic function. Since Nelder-Mead is a minimization algorithm the negative of this function is taken. On completion of the program run, the regression coefficients in the result matrix are available for prediction, as in the following case of a sample data with [GPA=1500, SAT=3, Interview=8, Aptitude=60].

theanologistic2(nspire1)

R

In R, as a sophisticated statistical package, the calculation is much simpler. Consider the sample case above, it is just a few lines of commands to invoke its built-in logistic model.

theano-new2

Theano

Apart from the traditional methods, modern advances in computing paradigms made possible neural network coupled with specialized hardware, for example GPU, for solving these problem in a manner much more efficiently, especially on huge volume of data. The Python library Theano is a complex library supporting and enriching these calculations through optimization and symbolic expression evaluation. It also features compiler capabilities for CUDA and integrates Computer Algebra System into Python.

One of the examples come with the Theano documentation depicted the application of logistic regression to showcase various Theano features. It first initializes a random set of data as the sample input and outcome using numpy.random. And then the regression model is created by defining expressions required for the logistic model, including the logistic function and likelihood function. Lastly by using the theano.function method, the symbolic expression graph coded for the regression model is finally compiled into callable objects for the training of neural network and subsequent prediction application.

theanologistic5(theano1)

A nice feature from Theano is the pretty printing of the expression model in a tree like text format. This is such a feel-like-home reminiscence of my days reading SQL query plans for tuning database queries.

theanologistic5(theano2).PNG

 

Advertisements

Unit conversion quirks in TI Nspire

The TI Nspire calculator provided a rich set of common units from area, length, mass, etc. Units start with an understore in Nspire, for example, kg is represented as _kg. User are free to create their own units. On the desktop version of the Nspire software, a short cut for the conversion symbol (► ) is “@>”.

Recently over a conversation with a friend living overseas we are curious of the lowest we can get for a cut of meat at our own places. I am getting 12 per 500 gm on discount a few days ago. He gets 7.8 per 1 lb at best.

Some mental calculations for we have different units, but I decided to fire up the Nspire for this inequality to see what will happen:
units1

Alas, doesn’t work. Obviously I was expecting a boolean. The more verbose inequality with some pre-calculation didn’t work either.
units2

That’s where I realize Nspire might not be handling unit in equations the way we expected. An easy fix of course is to times a common unit (e.g. _kg) on both sides, but that pretty much defeat the whole purpose of simplicity of calculations of this kind.

Building blocks of genetic algorithm in TI Nspire

Genetic algorithm is one of the more popular evolutionary algorithm with wide range of usage including optimization. While there are a lot of implementation of this technique, including the one as an option in the Excel solver, building one is a very good choice to understand the underlying process.

The TI Nspire provided a rich set of matrix operations that can be utilized to model the data structure required genetic algorithm. For example, creating an initial population with arbitrary size of binary, integer, or real numbers.

ga1

The cross over operation can be modeled as extracting part of a matrix using the augment function.

ga2

Fitness function can be dynamically defined using the expr function available in the Nspire environment.ga3

 

Exploring Lorenz system in TI Nspire

The Lorenz system is a non-linear system involving three parameters. It is three dimensional and can be plotted for visualization. Although the Nspire is capable to plot 3D graphs, sequence functions is supported in 2D plot only. Even so, it is still good to explore this chaotic system.

The three axis x,y,z are represented using the sequence functions u1, u2, and u3 respectively.lorenz1
Clicking CTRL-T will open the Data page alongside the plot.
lorenz3

Resulting pattern in scatter plot resembles the famous 3D plot even if it is 2D.
lorenz2

Experimenting with convergence time in neural network models

After setting up Keras and Theano and have some basic benchmark on the Nvidia GPU, the next thing to get a taste of neural network through these deep learning models are to compare these with one to solve the same problem (an XOR classification) that run on a modern calculator, the TI Nspire, using the Nelder-Mead algorithm for convergence of neural network weights.

A sample of SGD settings in Keras Theano with 30000 iterations converged in around 84 seconds. While the TI Nspire  completed with comparable results in 19 seconds. This is not a fair game of course, as there are lots of parameters that can be tuned in the model.

keras-nspire4

 

Training neural network using Nelder-Mead algorithm on TI Nspire

In this installment the Nelder-Mead method is used to train a simple neural network for the XOR problem. The network consisted of 2-input, 1-output, and 2 hidden layers, and is fully connected. In mainstream practical neural network, back propagation and other evolutionary algorithms are much more popular for training neural network for real world problem. Nelder-Mead is used here just out of curiosity to see how this general optimization routine performed under neural network settings on TI Nspire.
neural-network-xor1

The sigmoid function is declared in an TI Nspire function.
neural-network-xor2

For the XOR problem, the inputs are defined as two lists, and the expected output in another.
neural-network-xor3

The activation functions for each neuron are declared.
neural-network-xor4

To train the network, the sum of squared error function is used to feed into the Nelder-Mead algorithm for minimization. Random numbers are used for initial parameters.
neural-network-xor6
neural-network-xor7

Finally the resulting weights and bias are obtained from running the Nelder-Mead program.
neural-network-xor8

The comparison graph of the performance of the Nelder-Mead trained XOR neural network against expected values.
neural-network-xor9

 

Fast Fourier transform in TI Nspire

The FFT is not available as built in function in the TI Nspire, but it is trivial to write a program for doing this calculation. Instead of using the standard TI Basic program, the Lua scripting is attempted this time. Unlike the commonly used TI Basic program, variables are not shared directly. Things get complex when working with lists and matrices. However, there are some utility functions from the Lua scripting in Nspire that make it possible to exchange data with the Calculator page. The example below shown the FFT results from the Lua script in the Table page.

fftlua1

fftlua2

The HP Prime provided built in function for FFT.

fftlua3