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DSPA2: Data Science and Predictive Analytics (UMich HS650)
Appendix 11: Foundational Generative Artificial Intelligence Models (GAIMs)
SOCR/MIDAS (Ivo Dinov)
June 14, 2024
Expanding on DSPA Chapter 14 (Deep Learning) and the GAIM GitHub website, this DSPA2 appendix demonstrates the utilization (application) of pre-trained large language models, such as foundational generative artificial intelligence models (GAIMs). Specifically, we show various SOCR GAIMs and use LM Studio, an integrated development environment for GAIM applications, that parallels what to RStudio does for computational data modeling and analytics but for generative AI.
This DSPA GAIM tutorial demonstrates AI-augmented human intelligence and provides specific examples of four complementary applications of AI decision-support including:
- Theoretical research and abstract scientific discovery,
- Applied and exploratory scientific inquiry,
- Interdisciplinary studies, and
- Health science scholarship and clinical decision-support.
1 SOCR GAIM Applications
The SOCR HTML5 Apps website includes a number of interactive data science, computational modeling, statistical analysis, and ML/AI applications. Several of these apps demonstrate GAIM implementations and foudational AI services, e.g.,
- The SOCR
AI Bot provides access to SOCR data analytics, synthetic text
generation, image and code generation, and
R-training) - The SOCR GAIM 2D/3D Synthetic Brain Image/Volume Generation App is used for simulating realistic normal and pathological brain scans/images (or hypervolumes) based on user specified participant profiles
- The SOCR Dimensionality Reduction app includes functionality fo r linear and nonlinear projections of high dimensional data into 3D and 3D spaces
- The Interactive Virtual Hospital/Simulated Patient Data (VH/SimPat) App is used for simulation of realistic patient profiles as digital twins of real electronic medical records
- The Interactive Pressure Injury Prediction Model (PIPM) App demonstrated the use of ML/AI for on-demand prediction and quantification of risks for developing a hospital-acquired condition - bed-ulcers (skin pressure injuries)
- The SOCR Analytical Toolkit (SOCRAT) is a scalable web platform for in-browser applications for interactive data analysis and visualization
- The SOCR 2D/3D Hierarchical Clustering and Gaussian Mixture app Modeling and the General Clustering App offer mechanisms for unsupervised segmentation, clustering, and tessellation.
2 LM Studio
LM Studio is designed to provide the following functionality:
- On-demand downloading of compatible GAIM models from HuggingFace repositories.
- Offline running, applying, and testing open AI pretrained-models shared on huggingface, i.e., large language models (LLMs), just like RStudio enables the programming design, execution, packaging and reporting of end-to-end computational and data science protocols in self-contained electronic markdown notebooks.
LM Studio supports any ggml Llama, MPT, and StarCoder models available on HuggingFace (e.g., Llama2, Orca, Vicuna, NousHermes, WizardCoder, MPT, etc.). LM Studio is available for all major computer systems, Mac, Windows/PC, and Linux.
The power of LMStudio is in the ability to
immersively experiment, interactively create, and
dynamically innovate in a wide range of settings:
- any theoretical research area,
- any applied science domain,
- any interdisciplinary studies,
- any health science scholarship and clinical decision-support.
In the following examples, some of these three basic activities are demonstrated using free-text prompts.
Clearly, the automated GAIM responses to human prompts are
probabilistic, not guaranteed to be correct, and may
be incomplete, incomprehensible, or
regressive. Some of the potential issues are clearly
identifiable, e.g., erroneous LaTeX math typesetting,
whereas other GAIM response problems may be a lot more subtle.
Hence, constant, vigorous, and holistic human-expert review,
interpretation, and evaluation of GAIM-generated responses is
always warranted (required) prior to communicating, disseminating, or
decision-making based on synthetic outputs from GAIM
decision-support systems.
Yet, the level of synthetic understanding of the tasks and the depth of the GAIM responses suggest that the potential use under direct and appropriate human-supervision may be beneficial in many circumstances.
3 Theoretical Research Prompts
Modern science is predicated on four core scientific pillars - experimental, theoretical, computational and data sciences. Next, we will demonstrate synthetic GAIM responses in each of these pillars, starting with some theoretical (model-based) scientific examples.
3.1 Prompt: Derive a solution to the wave equation in 3D use English language
Using this specific GAIM, mistral-portuguese-luana-7b-mathematics.Q8_0.gguf, we obtain the following result (which is expected to change each time we run the GAIM model).
3.1.1 Response
To derive a solution to the wave equation in 3D, we need to use the generalized form of the wave equation in 3D, which is given by:
\[\frac{\partial\psi}{\partial t} = v^2\nabla^2 \psi,\]
where \(\psi(x, y, z, t)\) is the displacement function, or wavefunction, \(t\) is time, \(v\) is the velocity of the waves, and \(\nabla^2\) is the Laplacian operator, which operates on the spatial coordinates \(x\), \(y\), and \(z\).
We can find a solution to this equation by assuming a form for the displacement function that separates the spatial and temporal parts:
\[\psi(x, y, z, t) = X(x)Y(y)Z(z)T(t)\ .\]
Now let’s substitute this form into the wave equation and simplify:
\[\frac{\partial (XT)}{\partial t} = v^2\nabla^2(XYT)\ .\]
Recall that the Laplacian operator can be written as the sum of the second partial derivatives:
\[\nabla^2 = \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} +\frac{\partial ^2}{\partial z^2} .\]
Now we have:
\[v²(x''t + y''zt + z''xt) = v²(xy''t + xyt'' + yz''x + zx''t)\]
By equating the coefficients of XYT on both sides, we can see that:
\[x''/v² + y''/v² + z''/v² = xyt''/v² + yzt''/v² + zxt''/v²\]
Now, let’s divide both sides by \(v^2\) to simplify further:
\[(1/v²)(x'' + y'' + z''t) = (xy'' + yzt'' + zxt'') / v²\]
This shows that \(x'' + y'' + z'' = xyt'' + yzt'' + zxt''\).
Since \(x\), \(y\), and \(z\) are functions of spatial coordinates only, and \(t\) is a function of time only, their second partial derivatives will be zero unless they depend on the other coordinate. However, since we assumed that \(X(x)\), \(Y(y)\), and \(Z(z)\) do not depend on \(t\), their second partial derivatives will be zero. Therefore, we have:
\[x'' + y'' + z'' = 0.\]
This implies that \(x\), \(y\), and \(z\) must be functions of the form:
\[x(x) = a \ \sin(kx - wt)\] \[y(y) = b\ \sin(ly - wt)\] \[z(z) = c\ \sin(mz - wt)\ .\]
where \(a\), \(b\), and \(c\) are constants, \(k\), \(l\), and \(m\) are wavenumbers, and \(w\) is the angular frequency.
Now let’s substitute these forms of \(x\), \(y\), and \(z\) into the displacement function \(\psi\):
\[\psi(x, y, z, t) = \sin(kx - wt)\sin(ly - wt)\sin(mz - wt).\]
This is a solution to the wave equation in 3D, where \(\psi(x, y, z, t)\) is a function of all four coordinates and depends on both time and spatial position. The constants \(a, b, c, k, l, m, w\) determine the specific properties of the wave, including its shape, frequency, velocity, and spatial variation.
Note that this solution is valid only for waves traveling in free space (i.e., without any boundaries or obstacles), and it assumes that the initial conditions are such that the waves are sinusoidal at \(t = 0\). If the initial conditions are different, the form of the solution will change accordingly.
In summary, we derived a general solution to the wave equation in 3D using a function that separates the spatial and temporal parts, where each spatial coordinate depends on its own position and time through sinusoidal functions. The constants and wave numbers determine the specific properties of the wave.
3.2 Prompt: Derive a solution to the ultrahyperbolic wave equation with 2 time dimensions in 3D use English language
3.2.1 Response
Let’s begin by writing down the wave equation with two time dimensions in three dimensions. The wave equation in 3D is given by:
\[\nabla^2 \phi - (1/c_1)^2 (\partial_t \phi) - (1/c_2)^2(\partial_t \phi) - (1/c_3)^2 (\partial_t \phi) = 0\] Where \(\phi\) is the wave potential, \(c_1\), \(c_2\), and \(c_3\) are the velocities of wave propagation in each spatial dimension, and \(\nabla^2\) is the Laplacian operator. The first term in the equation represents the second spatial derivative, which describes how the wave changes in space. The remaining terms represent the second temporal derivatives, which describe how the wave changes over time.
To find a solution to this equation, we’ll assume that the potential \(\phi\) is a product of functions of each spatial and temporal coordinate: \[\phi(x_1, x_2, x_3, t_1, t_2) = \psi(x_1, t_1)\varphi(x_2, t_2)g(x_3, t_1, t_2)\]
Now let’s substitute this assumed form of the solution into the wave equation and separate the variables by setting each term equal to zero:
\[\frac{1}{\psi} \nabla^2 \psi - \frac{1}{c_1^2}(\partial_t\psi) - \frac{1}{c_2^2}(\partial_{t_1}\varphi) - \frac{1}{c_3^2}(\partial_{t_1}g) = 0\] \[\frac{1}{\varphi} (\nabla^2 \varphi) - \frac{1}{c_1^2}(\partial_t\psi) - \frac{1}{c_2^2}(\partial_{t_2}\varphi) - \frac{1}{c_3^2}(\partial_{t_2}g) = 0\] \[\frac{1}{g} (\nabla^2 g) - \frac{1}{c_1^2}(\partial_{t_1}\psi) - \frac{1}{c_2^2}(\partial_{t_1}\varphi) - \frac{1}{c_3^2}(\partial_{t_2}g) = 0\]
Each of these equations must be satisfied independently, so we have three separate wave equations:
\[\nabla^2 \psi - (1/c_1)^2(\partial_t\psi) = 0\] \[\nabla^2 \varphi - (1/c_2)^2(\partial_{t_2}\varphi) = 0\] \[\nabla^2 g - (1/c_3)^2(\partial_{t_2}g) = 0\]
These are simply the one-dimensional wave equation in each spatial dimension, with an additional temporal dependence. To find a solution to each of these equations, we’ll assume that the function only depends on one variable (either \(x\) or \(t\)), and we’ll find solutions for each case:
- For \(\psi\): We can assume a solution of the form \(\psi(x_1, t_1) = f(x_1)g(t_1)\) where \(f\) and \(g\) are functions. Plugging this into the wave equation, we get: \[\frac{1}{f}(\partial^2_x f) - \frac{1}{c_1^2}g' = 0\]
Separating variables again gives:
\[\partial^2_x f/f - (1/c_1)^2 g'/g = 0\]
This can be solved by assuming \(f\) is a harmonic function, which satisfies \(\partial^2_x f = -\omega^2 f\). Plugging in \(\omega^2 = (1/c_1)^2 g'\), we have:
\[\partial^2_x f = -\omega^2 f\]
This is the one-dimensional wave equation, and its general solution is a linear combination of sine and cosine functions:
\[f(x_1) = A \cos(\omega x_1) + B \sin(\omega x_1)\] where \(A\) and \(B\) are constants.
- For \(\varphi\): Assuming a solution of the form \(\varphi(x_2, t_2) = h(x_2)k(t_2)\), we get:
\[\partial^2_{x_2} h/h - (1/c_2)^2 k'/k = 0\]
This can be solved similarly to the previous case, and its general solution is a linear combination of sine and cosine functions:
\[h(x_2) = C \cos(\Omega x_2) + D \sin(\Omega x_2)\]
where \(C\) and \(D\) are constants.
- For \(g\): Assuming a solution of the form \(g(x_3, t_1, t_2) = u(x_3)v(t_1)w(t_2)\), we get:
\[\partial^2_{x_3} u/u - (1/c_3)^2 w'/w = 0\]
This is the one-dimensional wave equation, and its general solution is a linear combination of sine and cosine functions:
\[u(x_3) = E \cos(\Pi x_3) + F \sin(\Pi x_3)\]
where \(E\) and \(F\) are constants.
Now we have three separate solutions for the potential \(\phi\):
\[\phi(x_1, x_2, x_3, t_1, t_2) = (A \cos(\omega x_1) + B \sin(\omega x_1))(C \cos(\Omega x_2) + D \sin(\Omega x_2))(E \cos(\Pi x_3) + F \sin(\Pi x_3))\]
This is a solution to the 3D ultrahyperbolic wave equation with two time dimensions. It’s a linear combination of harmonic functions that describe waves propagating in each spatial dimension, with different temporal dependencies. The specific form of the solution depends on the values of the constants \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\).
3.3 Prompt: Derive a solution to the ultrahyperbolic wave equation with 2 time dimensions and Cauchy boundary conditions in 3D use English language and LaTeX typesetting
3.3.1 Response
\[\begin{align*} & \nabla^2 p - c^2 \frac {\partial^2 p} {\partial t_1^2} - b^2 \frac {\partial^2 p} {\partial t_2^2} = 0 \\ & \text {subject to the Cauchy boundary conditions:} \\ & p(x,y,z,0) = f(x,y,z) \\ & \frac {\partial p} {\partial t_1}(x,y,z,0) = g(x,y,z) \\ & \frac {\partial p} {\partial t_2}(x,y,z,0) = h(x,y,z) \end{align*}\] where \(p\) is the wave function in 3D, \((x, y, z)\) are the spatial coordinates, \(t_1\) and \(t_2\) are time dimensions, \(c\) and \(b\) are constant velocity values for each direction, and \(f\), \(g\), and \(h\) are given functions of \((x,y,z)\).
To find a solution to this wave equation, we use the method of separation of variables. We assume that the solution \(p\) can be written as a product of a spatial part and a temporal part:
\[\begin{align*} p(x, y, z, t_1, t_2) = X(x,y,z) Y(t_1,t_2) \end{align*}\]
We now substitute this assumed form of the solution into the wave equation and separate the variables: \[\begin{align*} & \nabla^2 p - c^2 \frac {\partial^2 p} {\partial t_1^2} - b^2 \frac {\partial^2 p} {\partial t_2^2} = 0 \\ &\Rightarrow \nabla^2 X Y - c^2 \frac {\partial^2 X Y} {\partial t_1^2} - b^2 \frac {\partial^2 X Y} {\partial t_2^2} = 0 \\ &\Rightarrow (\nabla^2 X - c^2 \frac {\partial^2 x} {\partial t_1^2})y + x (\nabla^2 y - b^2 \frac {\partial^2 y} {\partial t_2^2}) = 0 \end{align*}\]
Now, we separate the variables by setting both parts equal to zero: \[\begin{align*} & (\nabla^2 X - c^2 \frac {\partial^2 x} {\partial t_1^2})y = 0 \\ & x (\nabla^2 y - b^2 \frac {\partial^2 y} {\partial t_2^2}) = 0 \end{align*}\]
Since \(x\) and \(y\) are not equal to zero, we have: \[\begin{align*} & \nabla^2 x - c^2 \frac {\partial^2 x} {\partial t_1^2} = 0 \\ & \nabla^2 y - b^2 \frac {\partial^2 y} {\partial t_2^2} = 0 \end{align*}\]
Now we have two separate partial differential equations, one for \(x\) and one for \(y\). We can assume that each part depends only on its respective spatial and temporal variables:
\[\begin{align*} & x(x,y,z,t_1) = X(x,y,z) T_1(t_1) \\ & y(x,y,z,t_2) = Y(x,y,z) T_2(t_2) \end{align*}\]
Substituting these into the original PDEs and separating variables again gives us: \[\begin{align*} & \nabla^2 X - c^2 x \frac {\partial^2 t_1} {\partial t_1^2} = 0 \\ & \nabla^2 y - b^2 y \frac {\partial^2 t_2} {\partial t_2^2} = 0 \end{align*}\]
We now have two Laplacian PDEs with time-dependent coefficients. To find solutions to these, we use the method of eigenfunctions and eigenvalues. We assume that each part can be written as a product of a spatial part and a temporal part: \[\begin{align*} & x(x,y,z,t_1) = X(k_1,k_2,k_3) e^{i \omega t_1} \\ & y(x,y,z,t_2) = Y(K_1,K_2,K_3) e^{i \Omega t_2} \end{align*}\]
where \(k_1\), \(k_2\), and \(k_3\) are spatial frequencies, \(\omega\) and \(\Omega\) are temporal frequencies, and the eigenfunctions \(X\) and \(Y\) depend only on the spatial frequencies.
Substituting these assumed forms into the PDEs for \(x\) and \(y\) gives us: \[\begin{align*} & \nabla^2 X - c^2 k_1^2 x = 0 \\ & \nabla^2 Y - b^2 K_1^2 y = 0 \end{align*}\]
These are now ordinary differential equations in the spatial frequencies. The solutions will depend on the boundary conditions and the specific shapes of \(X\) and \(Y\).
To find a general solution, we need to solve for \(x\) and \(y\) separately, subject to their respective Laplacian PDEs and Cauchy boundary conditions. This may involve complex analysis or numerical methods, depending on the complexity of the problem. Once we have solutions for \(x\) and \(y\), we can multiply them together to get a solution \(p\):
\[\begin{align*} p(x,y,z,t_1,t_2) = x(k_1,k_2,k_3) e^{i \omega t_1} y(K_1,K_2,K_3) e^{i \Omega t_2} \end{align*}\]
This is a general solution to the ultrahyperbolic wave equation with Cauchy boundary conditions in 3D. The specific form of \(p\) will depend on the solutions for \(x\) and \(y\).
It’s important to note that this is a very general solution, and there may be specific cases where additional constraints or simplifications apply. For example, if the problem has symmetry properties, these could be used to simplify the solutions. Additionally, numerical methods may be required to find an accurate solution for complex boundary conditions or when analytical solutions are difficult to obtain.
In summary, we have derived a general solution to the ultrahyperbolic wave equation with Cauchy boundary conditions in 3D by separating variables and assuming a product form of the solution. The specific form of the solution will depend on the solutions for \(x\) and \(y\), which may involve complex analysis or numerical methods. Additionally, symmetry properties or simplifying assumptions could be used to further simplify the solution.
\[\begin{align*} & p(x, y, z, t_1, t_2) = X(k_1,k_2,k_3) e^{i \omega t_1} Y(K_1,K_2,K_3) e^{i \Omega t_2} \end{align*}\]
where \(X\) and \(Y\) are the solutions for the spatial parts of the wave function, subject to their respective Laplacian PDEs and Cauchy boundary conditions.
\[\begin{align*} & X(x,y,z) = \sum_{n=1}^{\infty} A_n \sin (k_n x + \phi_n) \sin (k_n y + \psi_n) \sin (k_n z + \theta_n) \\ & Y(K_1,K_2,K_3) = \sum_{m=1}^{\infty} B_M e^{i K_M T_1} e^{i K_M T_2} \end{align*}\]
where \(A_n\) and \(B_M\) are constants, \(\phi_n\), \(\psi_n\), and \(\theta_n\) are phases, and \(k_n\), \(K_M\), \(\omega\), and \(\Omega\) are spatial and temporal frequencies. These solutions may be modified depending on the specific boundary conditions and symmetry properties of the problem.
Finally, it’s worth noting that this solution assumes a homogeneous medium with constant velocity values in each direction. If the medium is anisotropic or the velocity values vary with position or time, the solution will need to be adjusted accordingly. Additionally, if the problem involves nonlinear effects or interactions between different wave modes, the solution may become more complex and require additional assumptions or modeling techniques.
In conclusion, the ultrahyperbolic wave equation in 3D with Cauchy boundary conditions can be solved by separating variables and assuming a product form of the solution. The specific form of the solution will depend on the solutions for \(x\) and \(y\), which may involve complex analysis or numerical methods. Symmetry properties or simplifying assumptions could also be used to further simplify the solution. The resulting solution will depend on the boundary conditions, the shape of the medium, and any nonlinear effects or interactions present in the problem.
(Answer: \(\boxed{p(x, y, z, t_1, t_2) = x(k_1,k_2,k_3) e^{i \omega t_1} y(K_1,K_2,K_3) e^{i \Omega t_2}}\))
The general solution to the ultrahyperbolic wave equation with Cauchy boundary conditions in 3D is given by:
\[p(x, y, z, t_1, t_2) = x(k_1,k_2,k_3) e^{i\omega t_1} y(K_1,K_2,K_3) e^{i\Omega t_2}\]
Where \(x(k_1,k_2,k_3)\) and \(y(k_1,k_2,k_3)\) are the solutions for the spatial parts of the wave function, subject to their respective Laplacian PDEs and Cauchy boundary conditions. The temporal part of the solution is given by \(e^{i\omega t_1}\) and \(e^{i\Omega t_2}\), where omega and Omega are the temporal frequencies.
The specific form of \(x(k_1,k_2,k_3)\) and \(y(k_1,k_2,k_3)\) will depend on the boundary conditions and symmetry properties of the problem. For example, if the problem has cylindrical symmetry, we may assume that the wave function depends only on the radial and z directions:
\[x(r, z) = \sum_{n=1}^{\infty} A_n \sin (k_n r) \sin (k_n z + \theta_n)\] \[y(R, Z) = \sum_{m=1}^{\infty} B_M e^{i K_M T} \cos (K_M R).\]
Here, \(A_n\) and \(B_M\) are constants, \(k_n\) and \(K_M\) are spatial frequencies, \(r\) and \(R\) are radial coordinates, \(z\) and \(Z\) are vertical coordinates, and \(t\) and \(t_2\) are temporal coordinates. The phases \(\theta_n\) and \(\psi_m\) depend on the boundary conditions and symmetry properties of the problem.
If the medium is homogeneous and isotropic, with constant velocity values in each direction, we can assume that the spatial frequencies satisfy:
\[k_n^2 + K_M^2 = k^2\]
where \(k\) is a constant related to the dispersion relation for the wave equation. The temporal frequencies \(\omega\) and Omega are related to the dispersion relation through:
\[\omega = \sqrt{k^2 - k_n^2}\] \[\Omega = \sqrt{k^2 - K_M^2}\]
If the problem involves nonlinear effects or interactions between different wave modes, the solution may become more complex and require additional assumptions or modeling techniques. For example, if we have a wave equation with nonlinearity of the form:
\[(1/c^2) \frac{\partial^2 p}{\partial t_1^2} - (1/c^2) \frac{\partial^2 p}{\partial t_2^2} + c^2 |\nabla p|^2 = 0\]
The solution may involve multiple spatial and temporal frequencies, as well as nonlinear terms related to the wave function and its derivatives. In this case, numerical methods or simplifying assumptions (such as assuming small-amplitude waves) may be required to find an accurate solution.
Overall, the general form of the solution for the ultrahyperbolic wave equation in 3D with Cauchy boundary conditions is given by:
\[p(x, y, z, t_1, t_2) = x(k_1,k_2,k_3) e^{i\omega t_1} y(K_1,K_2,K_3) e^{i\Omega t_2}\]
Where \(x(k_1,k_2,k_3)\) and \(y(k_1,k_2,k_3)\) are the solutions for the spatial parts of the wave function, subject to their respective Laplacian PDEs and Cauchy boundary conditions. The temporal part of the solution is given by \(e^{i\omega t_1}\) and \(e^{i\Omega t_2}\), where omega and \(\Omega\) are the temporal frequencies. The specific form of these functions will depend on the problem at hand, including the boundary conditions, symmetry properties, and presence of nonlinear effects or interactions.
The general solution to a homogeneous differential equation is given by the method of superposition, which involves adding solutions of individual homogeneous equations with different initial and boundary conditions. In this case, since we are considering the wave equation in 3D with Cauchy boundary conditions, we can assume that the solution can be represented as a linear combination of different spatial harmonics (sinusoidal functions) multiplied by temporal exponentials.
Let’s assume that the wave function \(p(x, y, z, t_1, t_2)\) can be represented as:
\[p(x, y, z, t_1, t_2) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} \sin(k_n x) \sin(k_n y) \sin(k_nz) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
Where \(k_n, k_m\) are the spatial frequencies, \(\omega_n\), \(\Omega_m\) are the temporal frequencies, and \(A_{nm}\) are constants that depend on the boundary conditions. The sums run over all possible values of \(n\) and \(m\). This representation assumes that the wave function can be expressed as a Fourier series in both \(x\) and \(z\) directions, with coefficients determined by the boundary conditions in those directions.
Now, let’s derive the partial differential equation (PDE) for \(p(x, y, z, t_1, t_2)\):
\[(1/c^2) \frac{\partial^2 p}{\partial t_1^2} - (1/c^2) \frac{\partial^2 p}{\partial t_2^2} + c^2 |\nabla p|^2 = 0\]
Let’s take the second partial derivatives with respect to \(t_1\) and \(t_2\):
\[\frac{\partial^2 p}{\partial t_1^2} = -c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} k_n^2 \sin(k_nx) \sin(k_ny) \sin(k_nz) e^{i\omega_n t_1} e^{i\Omega_m t_2} \frac{\partial^2 p}{\partial t_2^2} =\\ -c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} \Omega_M^2 \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
The cross-derivative term is:
\[\frac{\partial p}{\partial t_1} \frac{\partial p}{\partial t_2} = -c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} k_n \Omega_M \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
Now, let’s take the Laplacian of \(p\) with respect to \(x, y\), and \(z\):
\[\nabla^2 p = -\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (k_n^2 + k_m^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
Now, we can substitute these expressions into the PDE and set it equal to zero:
\[\frac{1}{c^2} \left (-c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} k_n^2 \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} -\\ c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} \Omega_M^2 \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} + \\ c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (k_n^2 + k_m^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} \right ) = 0\]
Simplifying the expression inside the braces:
\[c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (k_n^2 + \Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
\[-c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
Dividing both sides by \(-c^2\):
\[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
Since the sum of sine functions is nonzero unless all coefficients are zero, we have:
\[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) = 0\]
This implies that either \(\Omega_M^2 = k_n^2\) for all \(n\) and \(m\), or \(A_{nm} = 0\) for all \(n\) and \(m\). Since we are considering Cauchy boundary conditions, the latter case is not possible as it would mean that the solution is identically zero everywhere. Therefore, we must have:
\[\Omega_M^2 = k_n^2 , \ \ \forall n, m\]
This implies that the temporal frequencies depend on the spatial frequencies in a specific way. The solution can be written as:
\[p(x, y, z, t_1, t_2) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
Where \(\Omega_n = \sqrt(k_n^2 + \Omega_M^2)\), and \(\Omega_m\) is a constant determined by the boundary conditions. This solution represents a superposition of different spatial harmonics (sinusoidal functions) with temporal frequencies that depend on the spatial frequencies and boundary conditions.
The general solution to a homogeneous differential equation is given by the method of superposition, which involves adding solutions of individual homogeneous equations with different initial and boundary conditions. In this case, since we are considering the wave equation in 3D with Cauchy boundary conditions, we can assume that the solution can be represented as a linear combination of different spatial harmonics (sinusoidal functions) multiplied by temporal exponentials.
Let’s assume that the wave function \(p(x, y, z, t_1, t_2)\) can be represented as:
\[p(x, y, z, t_1, t_2) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
Where \(k_n\), \(k_m\) are the spatial frequencies, omega_n, \(\Omega_M\) are the temporal frequencies, and \(A_{nm}\) is a constant determined by the boundary conditions. The temporal frequencies depend on the spatial frequencies and boundary conditions.
The wave equation in 3D with 2 time dimensions can be written as:
\[\frac{\partial^2 p}{\partial t_1^2} - \frac{c^2}{a^2} \frac{\partial^2 p}{\partial t_2^2} + \frac{c^2}{a^2} \nabla^2 p = 0\]
where c and a are constants, and \(\nabla^2\) is the Laplacian operator in 3D.
Substituting the assumed solution into the wave equation, we get:
\[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (k_n^2 + \Omega_M^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} - \\ \frac{c^2}{a^2} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} + \\ \frac{c^2}{a^2} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} k_n^2 \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
Simplifying the expression inside the braces:
\[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (k_n^2 + \Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
\[-c^2 \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
Dividing both sides by \(-c^2\):
\[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2} = 0\]
Since the sum of sine functions is nonzero unless all coefficients are zero, we have:
\[\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} (\Omega_M^2 - k_n^2) = 0\]
This implies that either \(\Omega_M^2 = k_n^2\) for all \(n\) and \(m\), or \(A_{nm} = 0\) for all \(n\) and \(m\). Since we are considering Cauchy boundary conditions, the latter case is not possible as it would mean that the solution is identically zero everywhere. Therefore, we must have:
\[\Omega_M^2 = k_n^2\ , \ \forall\ n, m\]
This implies that the temporal frequencies depend on the spatial frequencies in a specific way. The solution can be written as:
\[p(x, y, z, t_1, t_2) = \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} A_{nm} \sin(k_n x) \sin(k_n y) \sin(k_n z) e^{i\omega_n t_1} e^{i\Omega_m t_2}\]
Where \(\omega_n = \sqrt{k_n^2 + \Omega_M^2}\), and \(\Omega_m\) is a constant determined by the boundary conditions. This solution represents a superposition of different spatial harmonics (sinusoidal functions) with temporal frequencies that depend on the spatial frequencies and boundary conditions.
Note that the exact form of \(\Omega_M\) depends on the boundary conditions, which are not specified in the given problem. Different boundary conditions would result in different forms of \(\Omega_M\). In general, Cauchy boundary conditions specify the values of the solution and its normal derivative on a hypersurface at a particular time, but do not provide enough information to uniquely determine the temporal frequencies \(\Omega_M\). Additional information, such as initial values or other boundary conditions, would be required to fully determine the solution in terms of both spatial and temporal frequencies.
(The ultrahyperbolic wave equation refers to a hyperbolic PDE with multiple time dimensions. In this case, we have 2 time dimensions, denoted by \(t_1\) and \(t_2\).)
4 Applied Science Domains
Next, we show synthetic GAIM responses in experimental and applied scientific domains.
4.1 Prompt: Explain how complex-time (kime) representation can be used to support spacekime analytics and brain mapping research using fMRI imaging data
Using this specific GAIM, ruslanmv/Medical-Llama3-8B-GGUF/medical-llama3-8b.Q5_K_M.gguf, we obtain the following result (which is expected to change each time we run the GAIM model).
4.1.1 Response
Kime is an abbreviation of kinetics time, which refers to the temporal duration of neural processes. Kime is a measure that accounts for the relative speed differences among various neural events within specific temporal intervals (e.g., seconds). Thus, kime is a unitless and dimensionless measure that describes the rate or pace at which neural activities are occurring in a given time interval. For example, an event with kime of 1 means that its duration is the same as the reference time interval, while an event with kime of 0.5 means that it takes half the time to occur. Kime has been used successfully to measure and represent brain activations (or deactivations) in functional magnetic resonance imaging (fMRI). The brain map data collected by fMRI is usually presented as a spatial-temporal matrix, with the x- and y-axis of the brain map representing the cortical regions, while the z-axis represents time. However, time is often represented as number of scans or TRs (the repetition time between each scan) instead of seconds in the matrix. Thus, kime can be used to represent a 3-D space-time structure with spatial-temporal dimensions on the x- and y-axis and kime dimension on z-axis. By transforming brain map data from temporal units such as scans or TRs into kime, kime can facilitate data integration across various time scales. It also allows for the extraction of the spatiotemporal patterns of neural activities within a given time interval (i.e., seconds) in an objective manner. In addition, by integrating brain map data collected from different fMRI experiments using kime representation, it is possible to study the functional connectivity and interactions between various parts of the brain at different levels of temporal resolution.
(Human-Expert) Note: Although in this example, the GAIM understands the clinical aspects of examining brain function using fMRI, clearly this GAIM response is inaccurate, this may be explained by the fact that the model may not have been trained using the most recent Time Complexity, Inferential Uncertainty, and Spacekime Analytics materials or the latest scientific publications.
4.2 Prompt: Try again to more accurately explain and provide specificity of how complex-time (kime) representation can be used to support spacekime analytics and brain mapping research using fMRI imaging data
Using this specific GAIM, ruslanmv/Medical-Llama3-8B-GGUF/medical-llama3-8b.Q5_K_M.gguf, we obtain the following result (which is expected to change each time we run the GAIM model).
4.2.1 Response
The complexity-time (kime) representation can be used in the following way for brain mapping with fMRI:
kime representation is a higher dimensional embedding of temporal dynamics than the usual 3D MRI volumes of time-series voxels or the time-frequency representations which use Fourier or wavelet transforms and other frequency-domain embeddings.
kime representation can be used to find regions in space-time that exhibit complex behaviors with respect to a specific task, such as language tasks involving speech generation.
kime representations may also be useful for detecting pathological conditions in the brain, e.g., tumors or neurodegenerative diseases, which appear to alter the temporal dynamics of different brain regions. For example, it may be possible to find regions that exhibit a loss of kime complexity due to such changes.
One advantage of using space-time embeddings is that they preserve spatial and topological relationships between voxels and can be used for clustering or classification tasks, which can then be related back to specific brain regions. For example, one could use kime representation to find clusters of voxels in the fMRI data that behave similarly to each other, but differently from the rest of the voxels.
Another advantage is that it may be possible to apply machine learning algorithms such as neural networks or deep learning methods directly on these embeddings without having to flatten them into a 1D vector, which can reduce the dimensionality and increase accuracy. This would be similar to the idea behind using convolutions in convolutional neural networks for image recognition.
One challenge of using kime representation is that it requires much more data than traditional methods, such as MRI or fMRI imaging. However, with recent advances in data collection techniques and machine learning algorithms, this may not be a major issue anymore. Also, it can provide additional insights into brain dynamics that are not captured by conventional approaches.
4.3 Prompt: Formulate a Kime-phpase Operator
Use the complex-time (kime) representation and spacekime analytics, see content in [https://www.socr.umich.edu/TCIU/HTMLs/](https://www.socr.umich.edu/TCIU/HTMLs/) and [https://www.degruyter.com/document/doi/10.1515/9783110697827/pdf](https://www.degruyter.com/document/doi/10.1515/9783110697827/pdf), to mathematically formulate a kime-phase operator whose eigenvalues are observable complex-time kime-phases.
4.4 Response
To solve the problem via programming, we first need to understand that a “kime-phase operator” is a theoretical concept from space-time physics, specifically related to time and space representation known as Kime (Complex Time). In the given references, it’s mentioned that such an operator might not have a straightforward mathematical formulation.
However, we can try to represent this concept mathematically in a simplified form:
Assuming that the kime-phase operator is a linear operator \(A\) acting on a complex Hilbert space \(H\), we may write \(A : H \to H.\)
Since it’s an observable property, it must be Hermitian (its eigenvalues are real). Also, since it corresponds to phase, its eigenvectors should form an orthonormal basis for the Hilbert space. Hence, \(A\) can be written as a diagonal matrix with the phases as its diagonal elements.
We can implement this using Python’s NumPy library, which provides functions to perform linear algebra operations efficiently. The following code snippet will generate a Hermitian operator (phase operator) of size \(n\times n\).
import numpy as np
def create_kime_phase_operator(n):
# Generating random phases between -pi and pi
phases = np.exp(1j * np.random.uniform(-np.pi, np.pi, n))
# Creating a diagonal matrix with the phases as its elements
kime_phase_operator = np.diag(phases)
return kime_phase_operatorYou can call this function to create a phase operator of any size. For example, for a \(3\times 3\) operator:
5 Interdisciplinary Studies
The third type of synthetic GAIM responses involves interdisciplinary scientific examples.
5.1 Prompt 1: Provide arguments for and against human geo-engineering
Using this specific GAIM, lmstudio-community/Meta-Llama-3-8B-Instruct-GGUF/Meta-Llama-3-8B-Instruct-Q4_K_M.gguf, we obtain the following result (which is expected to change each time we run the GAIM model).
5.1.1 Response
Human Geo-Engineering, also known as Climate Engineering or Weather Modification, refers to the deliberate large-scale intervention in the Earth’s climate system with the goal of mitigating the effects of global warming. Here are some arguments for and against human geo-engineering:
5.1.1.1 Arguments For Human Geo-Engineering
Rapid Climate Change Mitigation: Human geo-engineering could potentially slow down or reverse the effects of climate change, giving us more time to transition to renewable energy sources and reduce greenhouse gas emissions.
Immediate Impact: Geo-engineering methods like stratospheric sulfate injection or ocean fertilization could have a rapid impact on global temperatures, providing immediate relief from extreme weather events and heatwaves.
Cost-Effective: Some geo-engineering methods may be more cost-effective than traditional mitigation strategies, such as reducing carbon emissions through energy transitions.
Complementarity with Existing Efforts: Human geo-engineering could complement existing efforts to reduce greenhouse gas emissions, providing a supplementary approach to mitigate climate change.
5.1.1.2 Arguments Against Human Geo-Engineering
Uncertainty and Unpredictability: The potential consequences of large-scale geo-engineering are still unknown, making it difficult to predict the outcomes, and potentially creating unintended side effects.
Risk of Malfunction or Failure: Geo-engineering technologies are complex and vulnerable to malfunctions or failures, which could have catastrophic consequences for the environment and human societies.
Distributive Justice: Some geo-engineering methods may disproportionately benefit certain regions or groups, exacerbating existing social and economic inequalities.
Lack of International Governance: There is currently no international framework or regulation governing geo-engineering activities, making it difficult to ensure responsible and transparent use.
Alternative Solutions Exist: Reducing greenhouse gas emissions through energy transitions and sustainable development could be more effective and sustainable in the long run.
Ecological Consequences: Large-scale geo-engineering may have unintended ecological consequences, such as disrupting natural nutrient cycles or altering global weather patterns.
5.1.1.3 Additional Considerations
Scientific Research Priorities: Before deploying large-scale geo-engineering methods, it is essential to prioritize scientific research and monitoring to better understand the potential risks and benefits.
International Cooperation and Governance: Establishing international frameworks and regulations for geo-engineering activities would be crucial for ensuring responsible and transparent use.
Alternative Solutions and Adaptation: Focusing on reducing greenhouse gas emissions through energy transitions and sustainable development, as well as adapting to climate change impacts, may be a more effective and sustainable approach.
In conclusion, while human geo-engineering has been proposed as a potential solution to mitigate climate change, it is essential to weigh the potential benefits against the risks and uncertainties. Further scientific research and international cooperation are necessary to develop responsible and effective strategies for addressing global warming.
5.2 Prompt 2: Write an R or Rmd script to load this CSV dataset https://umich.instructure.com/files/1789624/download?download_frd=1. Then run the script to generate several exploratory data analytics, graphs
Note that we are providing a URL link to a data import file (ALS_TrainingData_2223.csv).
Using this specific GAIM, andrewcanis/c4ai-command-r-v01-GGUF/c4ai-command-r-v01-Q3_K_S.gguf, which only uses 3-bits to quantize the model coefficients (lower-fidelity trading with speed), to obtain the following result (that is expected to change each time we run the GAIM model).
5.2.1 Response
Sure! Here’s a solution using Rmarkdown. First, install and load the following packages:
##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionHere is the exploratory data analysis and visualization …
# Set file path
file_path <- "https://umich.instructure.com/files/1789624/download?download_frd=1"
row.names <- ""
# Load dataset
data <- read.csv(file_path) # %>%
# dplyr::select(-c(row.names))
# Explore the structure of the dataset
str(data)## 'data.frame': 2223 obs. of 101 variables:
## $ ID : int 1 2 3 4 5 6 7 8 9 11 ...
## $ Age_mean : int 65 48 38 63 63 36 55 55 37 72 ...
## $ Albumin_max : num 57 45 50 47 47 51 46 45 48 44 ...
## $ Albumin_median : num 40.5 41 47 44 45.5 47 44 42 46 42 ...
## $ Albumin_min : num 38 39 45 41 42 46 40 38 41 38 ...
## $ Albumin_range : num 0.0662 0.01045 0.00893 0.01211 0.00829 ...
## $ ALSFRS_slope : num -0.966 -0.922 -0.915 -0.598 -0.444 ...
## $ ALSFRS_Total_max : int 30 37 24 30 32 37 34 30 35 28 ...
## $ ALSFRS_Total_median : num 28 33 14 29 27.5 34.5 24 27.5 28.5 25.5 ...
## $ ALSFRS_Total_min : int 22 21 10 24 20 27 10 20 24 23 ...
## $ ALSFRS_Total_range : num 0.0212 0.0287 0.025 0.015 0.0204 ...
## $ ALT.SGPT._max : num 24 25 25 62 38 34 80 38 47 39 ...
## $ ALT.SGPT._median : num 22 13 20 60 26.5 23 46 27 42 20 ...
## $ ALT.SGPT._min : num 18 8 14 41 22 18 19 15 25 11 ...
## $ ALT.SGPT._range : num 0.0209 0.0296 0.0196 0.0524 0.0265 ...
## $ AST.SGOT._max : int 31 31 24 46 35 31 57 26 43 49 ...
## $ AST.SGOT._median : num 27.5 17 19 40 26.5 26 37 25 30 24 ...
## $ AST.SGOT._min : num 23 14 18 33 20 21 22 16 24 17 ...
## $ AST.SGOT._range : num 0.0279 0.0296 0.0107 0.0324 0.0249 ...
## $ Bicarbonate_max : num 30 32 35 23 32 29 32 29 36 32 ...
## $ Bicarbonate_median : num 28 28 29 20 28 26 27.5 28 29 29.5 ...
## $ Bicarbonate_min : num 25 25 24 20 23 22 23 25 20 27 ...
## $ Bicarbonate_range : num 0.01742 0.0122 0.01964 0.00748 0.01493 ...
## $ Blood.Urea.Nitrogen..BUN._max : num 8.03 8.4 5.48 8.03 5.11 ...
## $ Blood.Urea.Nitrogen..BUN._median: num 7.12 4.75 4.38 8.03 4.2 ...
## $ Blood.Urea.Nitrogen..BUN._min : num 6.57 4.02 3.65 6.57 3.65 ...
## $ Blood.Urea.Nitrogen..BUN._range : num 0.00509 0.00763 0.00326 0.00364 0.00242 ...
## $ bp_diastolic_max : int 90 80 86 90 100 84 98 80 90 90 ...
## $ bp_diastolic_median : num 83 78 76 80 80 80 86 74 80 80 ...
## $ bp_diastolic_min : int 69 64 58 70 68 60 80 64 80 70 ...
## $ bp_diastolic_range : num 0.0556 0.0287 0.05 0.0499 0.0531 ...
## $ bp_systolic_max : int 160 140 120 150 160 140 134 134 135 140 ...
## $ bp_systolic_median : num 139 132 110 130 130 ...
## $ bp_systolic_min : int 129 104 90 120 104 100 110 104 115 120 ...
## $ bp_systolic_range : num 0.082 0.0646 0.0536 0.0748 0.0929 ...
## $ Calcium_max : num 2.5 2.32 2.47 2.47 2.42 ...
## $ Calcium_median : num 2.22 2.17 2.3 2.35 2.26 ...
## $ Calcium_min : num 2.22 2.02 2.2 2.23 2.17 ...
## $ Calcium_range : num 0.000956 0.000522 0.00049 0.000474 0.000414 ...
## $ Chloride_max : num 109 108 108 109 107 110 108 107 110 103 ...
## $ Chloride_median : num 108 102 106 107 104 105 104 106 105 99 ...
## $ Chloride_min : num 103 100 104 106 100 101 100 101 101 95 ...
## $ Chloride_range : num 0.02091 0.01394 0.00714 0.00748 0.01161 ...
## $ Creatinine_max : num 79.6 61.9 88.4 70.7 61.9 ...
## $ Creatinine_median : num 79.6 53 79.6 61.9 48.6 ...
## $ Creatinine_min : num 70.7 44.2 70.7 53 26.5 ...
## $ Creatinine_range : num 0.0308 0.0308 0.0316 0.0441 0.0586 ...
## $ Gender_mean : int 1 1 2 2 1 2 2 1 2 1 ...
## $ Glucose_max : num 7.44 6.77 5.66 5.11 7.49 ...
## $ Glucose_median : num 4.5 5 5.11 4.77 5.72 ...
## $ Glucose_min : num 4.22 4.05 4.22 4.66 5.05 ...
## $ Glucose_range : num 0.01122 0.00474 0.00258 0.00111 0.00405 ...
## $ hands_max : int 8 8 4 6 8 8 6 8 6 8 ...
## $ hands_median : num 7.5 6 1 5.5 6.5 7 4 8 1.5 7 ...
## $ hands_min : int 6 6 0 4 3 5 1 5 0 6 ...
## $ hands_range : num 0.00529 0.00359 0.00714 0.00499 0.00849 ...
## $ Hematocrit_max : num 44.6 41.9 49.1 46.3 44 46.8 50.5 45.5 48 42 ...
## $ Hematocrit_median : num 43.1 39.6 46.2 43 42.9 ...
## $ Hematocrit_min : num 40.7 37.7 44 41.7 39.5 41.9 44.1 37.1 45 38 ...
## $ Hematocrit_range : num 0.01359 0.00732 0.00911 0.01147 0.00746 ...
## $ Hemoglobin_max : num 156 138 161 154 152 157 165 152 156 139 ...
## $ Hemoglobin_median : num 146 132 154 145 146 ...
## $ Hemoglobin_min : num 143 128 151 144 138 142 151 122 149 125 ...
## $ Hemoglobin_range : num 0.0453 0.0174 0.0179 0.0249 0.0232 ...
## $ leg_max : int 8 8 4 4 2 8 8 1 8 1 ...
## $ leg_median : num 6.5 7.5 3 3.5 2 8 6 0.5 8 0 ...
## $ leg_min : int 4 3 2 2 0 4 4 0 5 0 ...
## $ leg_range : num 0.01058 0.00898 0.00357 0.00499 0.0034 ...
## $ mouth_max : int 5 9 10 12 12 9 10 12 12 12 ...
## $ mouth_median : num 3.5 8 7 12 12 8 6 12 12 12 ...
## $ mouth_min : int 0 4 4 12 12 7 0 10 12 11 ...
## $ mouth_range : num 0.01323 0.00898 0.01071 0 0 ...
## $ onset_delta_mean : int -1023 -341 -1181 -365 -1768 -334 -268 -763 -440 -1324 ...
## $ onset_site_mean : int 1 1 1 2 2 1 2 2 2 2 ...
## $ Platelets_max : int 172 286 233 275 313 220 245 487 149 378 ...
## $ Platelets_median : num 169 264 213 233 284 ...
## $ Platelets_min : num 152 230 167 204 268 178 191 212 109 281 ...
## $ Potassium_max : num 4.5 5 4.1 4.3 4.6 4.5 4.5 4.7 4.6 4.4 ...
## $ Potassium_median : num 4.25 4.3 4 4.2 3.75 4.3 4.1 4.5 4.3 3.7 ...
## $ Potassium_min : num 4 3.9 3.9 4 3.5 4.2 3.6 4.2 4 3.2 ...
## $ Potassium_range : num 0.001742 0.001916 0.000357 0.000748 0.001824 ...
## $ pulse_max : int 79 90 82 84 101 88 96 100 84 100 ...
## $ pulse_median : num 68 76 73 72 96 66 80 80 68 100 ...
## $ pulse_min : int 61 64 60 68 74 60 66 64 59 80 ...
## $ pulse_range : num 0.0476 0.0467 0.0393 0.0399 0.0448 ...
## $ respiratory_max : int 4 4 4 3 4 4 4 4 4 4 ...
## $ respiratory_median : num 3 4 4 3 4 4 3 4 4 4 ...
## $ respiratory_min : int 3 3 4 3 3 3 2 1 4 4 ...
## $ respiratory_range : num 0.00265 0.0018 0 0 0.0017 ...
## $ Sodium_max : num 148 142 145 143 143 145 145 145 146 147 ...
## $ Sodium_median : num 146 138 143 139 140 ...
## $ Sodium_min : num 143 136 140 138 138 137 136 139 138 132 ...
## $ Sodium_range : num 0.01742 0.01045 0.00893 0.01247 0.00829 ...
## $ SubjectID : int 533 649 1234 2492 2956 3085 3551 3971 4390 4772 ...
## $ trunk_max : int 8 8 5 5 6 8 7 5 6 3 ...
## $ trunk_median : num 7 7 0 5 4 8 5 3 3 3 ...
## $ trunk_min : int 7 5 0 3 1 7 2 2 2 1 ...
## $ trunk_range : num 0.00265 0.00539 0.00893 0.00499 0.00849 ...
## $ Urine.Ph_max : num 6 7 6 7 6 8 9 6 7 7 ...
## [list output truncated]# Count the number of missing values per column
sum_missings <- data %>%
sapply(function(x) sum(is.na(x)))
# Drop columns with more than 10 missing values - this AI Code DOESN'T work
# clean_data <- data %>%
# dplyr::select(-if_all(~sum(.x. %in% NA) > 10))
threshold <- 0.10
clean_data <- data %>% select(where(~mean(is.na(.)) < threshold))
# Basic statistics of numerical variables
summary(clean_data)## ID Age_mean Albumin_max Albumin_median
## Min. : 1.0 Min. :18.00 Min. :37.00 Min. :34.50
## 1st Qu.: 614.5 1st Qu.:47.00 1st Qu.:45.00 1st Qu.:42.00
## Median :1213.0 Median :55.00 Median :47.00 Median :44.00
## Mean :1214.9 Mean :54.55 Mean :47.01 Mean :43.95
## 3rd Qu.:1815.5 3rd Qu.:63.00 3rd Qu.:49.00 3rd Qu.:46.00
## Max. :2424.0 Max. :81.00 Max. :70.30 Max. :51.10
## Albumin_min Albumin_range ALSFRS_slope ALSFRS_Total_max
## Min. :24.00 Min. :0.000000 Min. :-4.3452 Min. :11.00
## 1st Qu.:39.00 1st Qu.:0.009042 1st Qu.:-1.0863 1st Qu.:29.00
## Median :41.00 Median :0.012111 Median :-0.6207 Median :33.00
## Mean :40.77 Mean :0.013779 Mean :-0.7283 Mean :31.69
## 3rd Qu.:43.00 3rd Qu.:0.015873 3rd Qu.:-0.2838 3rd Qu.:36.00
## Max. :49.00 Max. :0.243902 Max. : 1.2070 Max. :40.00
## ALSFRS_Total_median ALSFRS_Total_min ALSFRS_Total_range ALT.SGPT._max
## Min. : 2.5 Min. : 0.00 Min. :0.00000 Min. : 10.00
## 1st Qu.:23.0 1st Qu.:14.00 1st Qu.:0.01404 1st Qu.: 32.00
## Median :28.0 Median :20.00 Median :0.02330 Median : 45.00
## Mean :27.1 Mean :19.88 Mean :0.02604 Mean : 54.44
## 3rd Qu.:32.0 3rd Qu.:27.00 3rd Qu.:0.03480 3rd Qu.: 65.00
## Max. :40.0 Max. :40.00 Max. :0.11765 Max. :944.00
## ALT.SGPT._median ALT.SGPT._min ALT.SGPT._range AST.SGOT._max
## Min. : 8.00 Min. : 1.60 Min. :0.002747 Min. : 11.00
## 1st Qu.: 22.00 1st Qu.: 15.00 1st Qu.:0.030303 1st Qu.: 30.00
## Median : 30.00 Median : 21.00 Median :0.047619 Median : 38.00
## Mean : 32.99 Mean : 23.01 Mean :0.071137 Mean : 43.13
## 3rd Qu.: 40.00 3rd Qu.: 28.00 3rd Qu.:0.077539 3rd Qu.: 48.00
## Max. :193.00 Max. :109.00 Max. :2.383117 Max. :911.00
## AST.SGOT._median AST.SGOT._min AST.SGOT._range Bicarbonate_max
## Min. : 9.00 Min. : 1.00 Min. :0.00000 Min. :20.0
## 1st Qu.: 22.00 1st Qu.:17.00 1st Qu.:0.02352 1st Qu.:29.0
## Median : 27.00 Median :20.00 Median :0.03502 Median :31.0
## Mean : 29.08 Mean :21.54 Mean :0.04919 Mean :30.9
## 3rd Qu.: 34.00 3rd Qu.:25.00 3rd Qu.:0.05243 3rd Qu.:32.0
## Max. :100.00 Max. :86.00 Max. :1.91667 Max. :52.0
## Bicarbonate_median Bicarbonate_min Bicarbonate_range
## Min. :19.50 Min. : 2.50 Min. :0.00000
## 1st Qu.:26.00 1st Qu.:22.00 1st Qu.:0.01266
## Median :27.00 Median :23.00 Median :0.01493
## Mean :26.96 Mean :23.16 Mean :0.01687
## 3rd Qu.:28.00 3rd Qu.:24.45 3rd Qu.:0.01815
## Max. :39.50 Max. :34.00 Max. :0.21429
## Blood.Urea.Nitrogen..BUN._max Blood.Urea.Nitrogen..BUN._median
## Min. : 2.921 Min. : 2.191
## 1st Qu.: 5.842 1st Qu.: 4.640
## Median : 6.937 Median : 5.423
## Mean : 7.353 Mean : 5.558
## 3rd Qu.: 8.210 3rd Qu.: 6.353
## Max. :25.192 Max. :11.866
## Blood.Urea.Nitrogen..BUN._min Blood.Urea.Nitrogen..BUN._range bp_diastolic_max
## Min. : 0.5842 Min. :0.000000 Min. : 70.00
## 1st Qu.: 3.2859 1st Qu.:0.004109 1st Qu.: 88.00
## Median : 4.0700 Median :0.005817 Median : 90.00
## Mean : 4.1609 Mean :0.007133 Mean : 92.03
## 3rd Qu.: 5.0000 3rd Qu.:0.008353 3rd Qu.: 98.00
## Max. :10.2228 Max. :0.069543 Max. :140.00
## bp_diastolic_median bp_diastolic_min bp_diastolic_range bp_systolic_max
## Min. : 56.00 Min. : 20.00 Min. :0.00000 Min. :100.0
## 1st Qu.: 78.00 1st Qu.: 65.00 1st Qu.:0.03527 1st Qu.:138.0
## Median : 80.00 Median : 70.00 Median :0.04337 Median :145.0
## Mean : 81.11 Mean : 69.89 Mean :0.04766 Mean :147.1
## 3rd Qu.: 85.00 3rd Qu.: 75.00 3rd Qu.:0.05435 3rd Qu.:157.0
## Max. :110.00 Max. :100.00 Max. :0.71429 Max. :220.0
## bp_systolic_median bp_systolic_min bp_systolic_range Calcium_max
## Min. : 90.0 Min. : 72.0 Min. :0.00000 Min. :2.171
## 1st Qu.:120.0 1st Qu.:108.0 1st Qu.:0.05272 1st Qu.:2.400
## Median :130.0 Median :110.0 Median :0.06494 Median :2.470
## Mean :129.6 Mean :113.4 Mean :0.07118 Mean :2.475
## 3rd Qu.:136.0 3rd Qu.:120.0 3rd Qu.:0.08190 3rd Qu.:2.530
## Max. :190.0 Max. :165.0 Max. :0.40462 Max. :9.460
## Calcium_median Calcium_min Calcium_range Chloride_max
## Min. :2.046 Min. :0.2438 Min. :0.0000000 Min. : 96.0
## 1st Qu.:2.283 1st Qu.:2.1707 1st Qu.:0.0003741 1st Qu.:106.0
## Median :2.345 Median :2.2300 Median :0.0004739 Median :107.0
## Mean :2.346 Mean :2.2229 Mean :0.0005407 Mean :107.2
## 3rd Qu.:2.400 3rd Qu.:2.2977 3rd Qu.:0.0005893 3rd Qu.:109.0
## Max. :2.800 Max. :2.6500 Max. :0.0129009 Max. :119.0
## Chloride_median Chloride_min Chloride_range Creatinine_max
## Min. : 90.0 Min. : 76.00 Min. :0.00000 Min. : 22.00
## 1st Qu.:102.0 1st Qu.: 98.00 1st Qu.:0.01250 1st Qu.: 65.00
## Median :104.0 Median :100.00 Median :0.01587 Median : 79.56
## Mean :103.5 Mean : 99.26 Mean :0.01787 Mean : 78.78
## 3rd Qu.:105.0 3rd Qu.:101.00 3rd Qu.:0.01990 3rd Qu.: 88.40
## Max. :111.0 Max. :109.00 Max. :0.21429 Max. :248.00
## Creatinine_median Creatinine_min Creatinine_range Gender_mean
## Min. : 18.00 Min. : 0.00 Min. :0.00000 Min. :1.000
## 1st Qu.: 53.04 1st Qu.: 39.00 1st Qu.:0.03824 1st Qu.:1.000
## Median : 62.00 Median : 53.00 Median :0.04865 Median :2.000
## Mean : 65.19 Mean : 51.98 Mean :0.05842 Mean :1.637
## 3rd Qu.: 78.85 3rd Qu.: 61.88 3rd Qu.:0.07026 3rd Qu.:2.000
## Max. :176.80 Max. :167.96 Max. :0.42095 Max. :2.000
## Glucose_max Glucose_median Glucose_min Glucose_range
## Min. : 4.160 Min. : 3.497 Min. : 0.000 Min. :0.000000
## 1st Qu.: 5.827 1st Qu.: 4.911 1st Qu.: 4.051 1st Qu.:0.003051
## Median : 6.500 Median : 5.300 Median : 4.440 Median :0.004695
## Mean : 7.160 Mean : 5.487 Mean : 4.265 Mean :0.006319
## 3rd Qu.: 7.600 3rd Qu.: 5.695 3rd Qu.: 4.800 3rd Qu.:0.007373
## Max. :33.688 Max. :26.196 Max. :12.200 Max. :0.097463
## hands_max hands_median hands_min hands_range
## Min. :0.000 Min. :0.000 Min. :0.000 Min. :0.000000
## 1st Qu.:5.000 1st Qu.:3.000 1st Qu.:0.000 1st Qu.:0.003610
## Median :7.000 Median :5.500 Median :3.000 Median :0.006652
## Mean :6.181 Mean :4.905 Mean :3.047 Mean :0.006883
## 3rd Qu.:8.000 3rd Qu.:7.000 3rd Qu.:5.000 3rd Qu.:0.009513
## Max. :8.000 Max. :8.000 Max. :8.000 Max. :0.042857
## Hematocrit_max Hematocrit_median Hematocrit_min Hematocrit_range
## Min. : 0.373 Min. : 0.362 Min. : 0.311 Min. :0.000000
## 1st Qu.:42.300 1st Qu.:40.000 1st Qu.:37.000 1st Qu.:0.007164
## Median :45.200 Median :42.600 Median :40.000 Median :0.009701
## Mean :41.939 Mean :39.467 Mean :36.962 Mean :0.011431
## 3rd Qu.:47.700 3rd Qu.:45.000 3rd Qu.:42.700 3rd Qu.:0.013579
## Max. :81.000 Max. :56.000 Max. :52.900 Max. :0.185714
## Hemoglobin_max Hemoglobin_median Hemoglobin_min Hemoglobin_range
## Min. :116.0 Min. :106.0 Min. : 6.204 Min. :0.00000
## 1st Qu.:144.0 1st Qu.:136.0 1st Qu.:128.000 1st Qu.:0.02321
## Median :152.0 Median :145.0 Median :136.000 Median :0.03106
## Mean :152.1 Mean :144.3 Mean :135.461 Mean :0.03824
## 3rd Qu.:160.0 3rd Qu.:152.0 3rd Qu.:145.000 3rd Qu.:0.04205
## Max. :280.0 Max. :182.0 Max. :180.000 Max. :0.56180
## leg_max leg_median leg_min leg_range
## Min. :0.00 Min. :0.00 Min. :0.000 Min. :0.000000
## 1st Qu.:3.00 1st Qu.:2.50 1st Qu.:1.000 1st Qu.:0.003378
## Median :5.00 Median :3.00 Median :2.000 Median :0.005435
## Mean :5.31 Mean :4.05 Mean :2.493 Mean :0.006163
## 3rd Qu.:8.00 3rd Qu.:6.00 3rd Qu.:3.000 3rd Qu.:0.008718
## Max. :8.00 Max. :8.00 Max. :8.000 Max. :0.042017
## mouth_max mouth_median mouth_min mouth_range
## Min. : 1.00 Min. : 0.000 Min. : 0.000 Min. :0.000000
## 1st Qu.:10.00 1st Qu.: 8.000 1st Qu.: 5.000 1st Qu.:0.001815
## Median :12.00 Median :11.000 Median : 9.000 Median :0.005329
## Mean :10.74 Mean : 9.703 Mean : 7.778 Mean :0.006595
## 3rd Qu.:12.00 3rd Qu.:12.000 3rd Qu.:11.000 3rd Qu.:0.010251
## Max. :12.00 Max. :12.000 Max. :12.000 Max. :0.036765
## onset_delta_mean onset_site_mean Platelets_max Platelets_median
## Min. :-3119 Min. :1.000 Min. : 84.0 Min. : 73.0
## 1st Qu.: -887 1st Qu.:2.000 1st Qu.:239.0 1st Qu.:204.0
## Median : -572 Median :2.000 Median :275.0 Median :233.0
## Mean : -683 Mean :1.801 Mean :285.3 Mean :238.8
## 3rd Qu.: -374 3rd Qu.:2.000 3rd Qu.:320.0 3rd Qu.:270.0
## Max. : -16 Max. :3.000 Max. :866.0 Max. :526.0
## Platelets_min Potassium_max Potassium_median Potassium_min
## Min. : 0.197 Min. : 3.400 Min. :3.000 Min. :2.400
## 1st Qu.:175.000 1st Qu.: 4.400 1st Qu.:4.000 1st Qu.:3.700
## Median :204.000 Median : 4.500 Median :4.200 Median :3.900
## Mean :208.382 Mean : 4.628 Mean :4.189 Mean :3.857
## 3rd Qu.:236.000 3rd Qu.: 4.800 3rd Qu.:4.300 3rd Qu.:4.000
## Max. :476.000 Max. :43.000 Max. :5.100 Max. :5.100
## Potassium_range pulse_max pulse_median pulse_min
## Min. :0.000000 Min. : 53.00 Min. : 50.00 Min. : 18.00
## 1st Qu.:0.001058 1st Qu.: 84.00 1st Qu.: 72.00 1st Qu.: 60.00
## Median :0.001425 Median : 90.00 Median : 77.00 Median : 64.00
## Mean :0.001744 Mean : 90.64 Mean : 76.97 Mean : 65.37
## 3rd Qu.:0.001913 3rd Qu.: 96.00 3rd Qu.: 81.00 3rd Qu.: 70.00
## Max. :0.098674 Max. :144.00 Max. :115.00 Max. :102.00
## pulse_range respiratory_max respiratory_median respiratory_min
## Min. :0.005425 Min. :2.00 Min. :0.000 Min. :0.000
## 1st Qu.:0.036755 1st Qu.:4.00 1st Qu.:3.000 1st Qu.:2.000
## Median :0.048821 Median :4.00 Median :4.000 Median :3.000
## Mean :0.053587 Mean :3.91 Mean :3.593 Mean :2.791
## 3rd Qu.:0.062365 3rd Qu.:4.00 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :0.500000 Max. :4.00 Max. :4.000 Max. :4.000
## respiratory_range Sodium_max Sodium_median Sodium_min
## Min. :0.000000 Min. :134.0 Min. :128.0 Min. :112.0
## 1st Qu.:0.000000 1st Qu.:142.0 1st Qu.:139.0 1st Qu.:135.0
## Median :0.001828 Median :143.0 Median :140.0 Median :137.0
## Mean :0.002513 Mean :143.4 Mean :140.1 Mean :136.8
## 3rd Qu.:0.003653 3rd Qu.:145.0 3rd Qu.:141.0 3rd Qu.:138.0
## Max. :0.025424 Max. :169.0 Max. :146.5 Max. :145.0
## Sodium_range SubjectID trunk_max trunk_median
## Min. :0.00000 Min. : 533 Min. :0.000 Min. :0.000
## 1st Qu.:0.01058 1st Qu.:240826 1st Qu.:5.000 1st Qu.:3.000
## Median :0.01312 Median :496835 Median :7.000 Median :5.000
## Mean :0.01500 Mean :498880 Mean :6.204 Mean :4.893
## 3rd Qu.:0.01728 3rd Qu.:750301 3rd Qu.:8.000 3rd Qu.:6.500
## Max. :0.14286 Max. :999482 Max. :8.000 Max. :8.000
## trunk_min trunk_range Urine.Ph_max Urine.Ph_median
## Min. :0.000 Min. :0.000000 Min. :5.00 Min. :5.000
## 1st Qu.:1.000 1st Qu.:0.003643 1st Qu.:6.00 1st Qu.:5.000
## Median :3.000 Median :0.006920 Median :7.00 Median :6.000
## Mean :2.956 Mean :0.007136 Mean :6.82 Mean :5.711
## 3rd Qu.:5.000 3rd Qu.:0.009639 3rd Qu.:7.00 3rd Qu.:6.000
## Max. :8.000 Max. :0.042017 Max. :9.00 Max. :9.000
## Urine.Ph_min
## Min. :5.000
## 1st Qu.:5.000
## Median :5.000
## Mean :5.183
## 3rd Qu.:5.000
## Max. :8.000# Pair plot of the numeric variables - this AI Code DOESN'T work
# ggplot(clean_data, aes(y=..density.., x=..count..)) +
# geom_bin2d() +
# geom_rug(position=position_dodge()) +
# scale_x_continuous(limits=c(0, 450))+
# facet_wrap(.~variable, scales="free")
# Boxplots of numerical variables by category variable - this AI Code DOESN'T work
# ggplot(clean_data, aes(y=value, x=factor(category))) +
# geom_boxplot() +
# ylab("Value") + xlab("Category") + ggtitle("Distribution of Value by Category")
# Correlation heatmap - there is no package called ‘GGSciChart’
# corr <- clean_data %>% select_if(is.numeric) %>% cor()
# ggplot(corr, GGSciChart::geom_rcorheatmap())
# First, make sure you have the following packages installed:
# install.packages("readr")
# install.packages("dplyr")
# install.packages("ggplot2")
# install.packages("plotly")5.3 Prompt 3: Write an Rmd script to load this CSV dataset “https://umich.instructure.com/files/1789624/download?download_frd=1”. Then, generate several “plot_ly” graphs showing exploratory data analytics.
Note that we are providing a URL link to a data import file (ALS_TrainingData_2223.csv).
This time, we will use a much larger GAIM model of size \(35B\) parameters andrewcanis/c4ai-command-r-v01-GGUF/c4ai-command-r-v01-Q8_0.gguf, which uses 8-bit quantization and requires over \(30GB\) Virtual RAM to run an yields the following (better) result (that again may change each time we run the GAIM model).
5.3.1 Response
# Install packages
# install.packages("dplyr")
# install.packages("ggplot2")
library("dplyr")
library("ggplot2")Here is the exploratory data analysis and visualization, which runs much slower due to the large VRAM demands of the 8Bit-quantized LLM.
##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layout# Read the dataset from URL
dataset <- read_csv("https://umich.instructure.com/files/1789624/download?download_frd=1")## Rows: 2223 Columns: 101## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (101): ID, Age_mean, Albumin_max, Albumin_median, Albumin_min, Albumin_r...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Exploratory Data Analysis (EDA) with Plotly
# 1. Histogram of Age
plot_ly(data = dataset, x = ~Age_mean, type = "histogram") %>%
layout(title = "(ALS_TrainingData_2223) Histogram of (mean) Age")# 2. Box plot of Heart-rate by Gender
plot_ly(data = dataset, y = ~pulse_median, x = ~factor(Gender_mean),
color=~factor(Gender_mean), type = 'box') %>%
layout(title = "Box Plot of Heart-rate by Gender")## Warning in RColorBrewer::brewer.pal(N, "Set2"): minimal value for n is 3, returning requested palette with 3 different levels
## Warning in RColorBrewer::brewer.pal(N, "Set2"): minimal value for n is 3, returning requested palette with 3 different levels# 3. Bubble chart of Potassium vs. Blood Pressure (size=Platelets, color=Onset-site)
plot_ly(data=dataset, x=~bp_systolic_median, y=~Potassium_median, type="scatter",
size=~Platelets_median, color=~onset_site_mean, text=~SubjectID) %>%
layout(title = "Potassium vs. Blood Pressure (size=Platelets, color=Onset-site)",
xaxis = list(title = "Blood pressure (systolic)"),
yaxis = list(title = "Potassium level (median)"))## No scatter mode specifed:
## Setting the mode to markers
## Read more about this attribute -> https://plotly.com/r/reference/#scatter-mode## Warning: `line.width` does not currently support multiple values.5.4 Prompt 4: Using DSPA Content to generate Rmd scripts
Use the **DSPA Materials**, [https://socr.umich.edu/DSPA2/DSPA2_notes/](https://socr.umich.edu/DSPA2/DSPA2_notes/) and [https://link.springer.com/content/pdf/10.1007/978-3-319-72347-1.pdf](https://link.springer.com/content/pdf/10.1007/978-3-319-72347-1.pdf), to write an Rmd script that loads this CSV dataset "[https://umich.instructure.com/files/1789624/download?download_frd=1](https://umich.instructure.com/files/1789624/download?download_frd=1)". Then, generate several "plot_ly" graphs showing exploratory data analytics.
For this experiment, we are using a large Retrieval Augmented Generation (RAG) model (DOI: 10.48550/arXiv.2005.11401), TroyDoesAI/Codestral-22B-RAG-Q8-gguf, which combines pre-trained parametric and non-parametric memory for language or code generation. The parametric memory relies on a pre-trained seq2seq model and the non-parametric memory is a dense vector index from an openly accessible web resource, e.g., wikipedia, that is accessed with a pre-trained neural retriever.
5.5 Response
Sure, here’s a basic example of how you might structure your Rmd script. This script loads the dataset, performs some exploratory data analysis and generates several interactive plots using ‘plotly’.
Note: The complete RAG-GAIM response will now work (out-of-the-box) and may require some expert fine-tuning.
5.5.1 Load the dataset
5.5.3 Exploratory data analysis
5.5.4 Box plot
Next, let’s create a box plot of ‘var1’ grouped by ‘var3’.
6 Healthcare applications and clinical decision-support
Since 2020, there have been many published studies demonstrating the AI utility to provide clinical decision support, e.g., DOI: 10.1038/s41598-023-32032-6, DOI: 10.3390/bioengineering10050581, and DOI: 10.1007/978-3-031-17483-4.
To demonstrate the utility of an AI clinical decision support system, we will employ this pre-trained \(8B\)-parameter GAIM Medical-Llama3-8B-GGUF/medical-llama3-8b.Q5_K_M.gguf and medical test-bank questions from:
- United States Medical Licensing Examination program and their USMLE Test-bank, and
- National Council of State Boards of Nursing (NCSBN), National Council Licensure Examination (NCLEX), and quizzes from the NurseLabs.
6.1 Prompt 1: Type = Clinical Diagnosis
A 32-year-old woman with type 1 diabetes mellitus has had progressive renal failure during the past 2 years. She has not yet started dialysis. Examination shows no abnormalities. Her hemoglobin concentration is 9 g/dL, hematocrit is 28%, and mean corpuscular volume is 94 μm3. A blood smear shows normochromic, normocytic cells. Which of the following is the most likely cause?
6.1.1 AI Response 1
- Anemia from iron deficiency.
- Explanation: The patient has a microcytic anemia due to iron deficiency. The typical appearance of a red cell in microcytic anemia is the presence of small dark cells on a blood smear, with normal white cell morphology and no other evidence of hemolysis. The cause is usually chronic gastrointestinal bleeding from peptic ulcer disease, colonic cancer, or a nonmalignant condition such as angiodysplasia. The patient in this question has anemia caused by iron deficiency. She does not have hemolysis. The red cell morphology and reticulocyte count (not provided) will help confirm the diagnosis. A bone marrow aspirate may be required to identify a rare cause of microcytic anemia such as lead poisoning.
Correct answer: Erythropoietin deficiency.
6.1.2 Prompt 2: Type = Clinical Diagnosis
A 67-year-old woman with congenital bicuspid aortic valve is admitted to the hospital because of a 2-day history of fever and chills. Current medication is lisinopril. Temperature is 38.0°C (100.4°F), pulse is 90/min, respirations are 20/min, and blood pressure is 110/70 mm Hg. Cardiac examination shows a grade 3/6 systolic murmur that is best heard over the second right intercostal space. Blood culture grows viridans streptococci susceptible to penicillin. In addition to penicillin, an antibiotic synergistic to penicillin is administered that may help shorten the duration of this patient’s drug treatment. Which of the following is the most likely mechanism of action of this additional antibiotic on bacteria?
- Binding to DNA-dependent RNA polymerase
- Binding to the 30S ribosomal protein
- Competition with p-aminobenzoic acid
- Inhibition of dihydrofolate reductase
- Inhibition of DNA gyrase
6.1.3 AI Response 2
- (E) Inhibition of DNA gyrase
- Explanation: The antibiotic that shortens the duration of treatment in a case such as this is gentamicin, which acts by inhibiting bacterial protein synthesis. The correct answer is option E.
6.2 Prompt 3: Type = Prescription Drugs
A 52-year-old woman begins pharmacotherapy after being diagnosed with type 2 diabetes mellitus. Four weeks later, her hepatic glucose output is decreased, and target tissue glucose uptake and utilization are increased. Which of the following drugs was most likely prescribed for this patient?
- Acarbose
- Glyburide
- Metformin
- Nateglinide
- Repaglinide
6.2.1 AI Response 3
- Metformin.
6.3 Prompt 4: Type = (NCLEX) Pharmacological and Parenteral Therapies
The nurse is caring for a 65-year-old male client with a history of ischemic heart disease who has been admitted to the ICU following a myocardial infarction. The client is currently receiving lidocaine I.V. to manage his ventricular ectopy, which has been persistent. Which of the following factors would be most important for the nurse to consider in relation to the administration of this medication?
A. Decrease in arterial oxygen saturation (SaO2) when measured with a pulse oximeter B. Increase in systemic blood pressure C. Presence of premature ventricular contractions (PVCs) on a cardiac monitor D. Increase in intracranial pressure (ICP) E. A drop in serum potassium levels as indicated in the latest lab results F. Observation of a widening QRS complex on the ECG
6.3.1 AI Response 4
- B. Increase in systemic blood pressure
6.4 Prompt 5: Type = (NCLEX) Reduction of Risk Potential
In a busy surgical unit, a nurse is preparing to insert an I.V. catheter for a 33-year-old patient who is scheduled for elective surgery and has a notably hairy forearm where the I.V. is to be placed. The patient is allergic to a variety of adhesives and has sensitive skin that is prone to irritation. Given these considerations, how should the nurse manage excess hair at the intended catheter insertion site?
A. Leaving the hair intact B. Shaving the area C. Clipping the hair in the area D. Removing the hair with a depilatory E. Applying a small amount of water-soluble gel to tame the hair without cutting F. Use a sterile surgical scalpel to trim the hair as close to the skin as possible without causing abrasions.
6.4.1 AI Response 5
- F. Use a sterile surgical scalpel to trim the hair as close to the skin as possible without causing abrasions.
6.5 Prompt 6: Type = Health Promotion and Maintenance
A nurse is providing discharge education to a 55-year-old client who has been hospitalized with symptoms leading to a diagnosis of atherosclerosis. The client has a sedentary lifestyle, a high-stress job, and a diet high in saturated fats. As part of the client’s comprehensive care plan to manage and reduce the progression of atherosclerosis, which of the following lifestyle modifications should the nurse emphasize?
A. Emphasize the importance of not focusing solely on weight as a health measure, but rather on overall cardiovascular health. B. Recommend an incremental increase in physical activity, starting with low-intensity exercises such as walking or swimming. C. Advise adherence to a heart-healthy diet, rich in fruits, vegetables, whole grains, and lean proteins, while limiting saturated fats and cholesterol. D. Counsel on stress management techniques and the importance of incorporating relaxation activities into daily routines. E. Discuss smoking cessation strategies if applicable and avoid exposure to secondhand smoke. F. Suggest regular monitoring of blood pressure and cholesterol levels with follow-up appointments to assess cardiovascular health.
6.5.1 AI Response 6
- A. Emphasize the importance of not focusing solely on weight as a health measure, but rather on overall cardiovascular health.
- Explanation: Weight alone is not an indicator for good or bad heart health. The nurse should instruct that the client focus on eating healthy and exercising.
7 References
Note: The following direct reference sources were explicated by the GAIM after its response to the human prompts!
https://en.wikipedia.org/wiki/Wave_equation#Ultrahyperbolic_wave_equation
https://www.sciencedirect.com/science/article/pii/S0377-984X(12)00856-9?via%3Dihub
http://www.sciencedirect.com/science/article/pii/S0377-984X(12)00856-9?via%3Dihub
https://en.wikipedia.org/wiki/Laplacian#The_laplacian_on_n_-dimensional_Euclidean_space
https://en.wikipedia.org/wiki/Wave_equation#Ultrahyperbolic_wave_equation
https://www.sciencedirect.com/science/article/pii/S0377-984X(12)00856-9?via%3Dihub
http://www.sciencedirect.com/science/article/pii/S0377-984X(12)00856-9?via%3Dihub
https://en.wikipedia.org/wiki/Laplacian#The_laplacian_on_n_-dimensional_Euclidean_space
https://www.sciencedirect.com/science/article/pii/0025587491901652
https://www.wolframcloud.com/obj/5a34b9d2-6e7f-4a9a-a3b8-da694cec1cc0
https://www.wolframcloud.com/obj/72a438eb-091f-456b-8cdb-6e01752e015d
https://en.wikipedia.org/wiki/Laplacian#The_laplacian_on_n_-dimensional_Euclidean_space
http://mathworld.wolfram.com/LaplacianInCylindricalCoordinates.html
https://en.wikipedia.org/wiki/Vector_calculus#Gradient_and_divergence
https://en.wikipedia.org/wiki/Vector_calculus#Divergence_and_curl
https://www.wolframcloud.com/obj/73a93d4c-6e06-4ecf-81b5-274ed47846af
https://en.wikipedia.org/wiki/Cauchy_characteristic_distribution
https://www.wolframcloud.com/obj/73a93d4c-6e06-4ecf-81b5-274ed47846af
http://mathworld.wolfram.com/Cauchy_Characteristic_Distribution.html
https://www.sciencedirect.com/science/article/pii/S0377-984X(12)00856-9?via%3Dihub
https://www.sciencedirect.com/science/article/pii/0162-1886(97)00131-4?via%3Dihub
https://en.wikipedia.org/wiki/Laplacian#The_laplacian_on_n_-dimensional_Euclidean_space
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