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What is the best way to model compositional data problems?

Compositional data is when each example or sample is a vector that sums to 1 (or 100%). In my case, I am interested in the composition of minerals in a rock and I have sensors that tell me the sum of the minerals but not the components that make up the sum.

For example, lets say I have two minerals, $m_1$ and $m_2$, that are made up of 3 elements (like copper and other elements from the periodic table) which form a vector of length 3:

m1 = [0.1, 0.3, 0.6]
m2 = [0.6, 0.2, 0.2]

If a rock has 25% of $m_1$ and 75% of $m_2$, the sensor reading produces the sum of the two minerals (shown in the plot below):

$$0.25*m_1 + 0.75*m_2 = [0.475, 0.225, 0.3]$$

I would like to know how to model and solve the problem of unmixing a composition into its underlying components, where the sum of the elements is normalized to 100% (e.g. $0.25m_1 + 0.75m_2$ has the same composition as $0.50m_1 + 1.50m_2$).

Furthermore, my example is simplistic; in reality a composition can have more than just 2 minerals (up to 3000) and each mineral is made up of 118 elements, not just 3 (all the elements of periodic table - though many elements will be zero). The elemental composition of a mineral is assumed to be known (definition of $m_1$ and $m_2$ in the example). Also, the sensor reading is noisy - each element of the observed composition is assumed to have Gaussian noise.

What is the best way to model compositional data problems?

Compositional data is when each example or sample is a vector that sums to 1 (or 100%). In my case, I am interested in the composition of minerals in a rock and I have sensors that tell me the sum of the minerals but not the components that make up the sum.

For example, lets say I have two minerals, $m_1$ and $m_2$, that are made up of 3 elements (like copper and other elements from the periodic table) which form a vector of length 3:

m1 = [0.1, 0.3, 0.6]
m2 = [0.6, 0.2, 0.2]

If a rock has 25% of $m_1$ and 75% of $m_2$, the sensor reading produces the sum of the two minerals (shown in bottom-left subplot below):

$$ \begin{align} &0.25*m_1 + 0.75*m_2 \\ =&0.25*[0.1, 0.3, 0.6] + 0.75*[0.6, 0.2, 0.2] \\ =&[0.475, 0.225, 0.3] \end{align} $$

I would like to know how to model and solve the problem of unmixing a composition into its underlying components, where the sum of the elements is normalized to 100% (e.g. $0.25m_1 + 0.75m_2$ has the same composition as $0.50m_1 + 1.50m_2$).

Furthermore, my example is simplistic; in reality a composition can have more than just 2 minerals (up to 3000) and each mineral is made up of 118 elements, not just 3 (all the elements of periodic table - though many elements will be zero). The elemental composition of a mineral is assumed to be known (definition of $m_1$ and $m_2$ in the example). Also, the sensor reading is noisy - each element of the observed composition is assumed to have Gaussian noise.

What is the best way to model compositional data problems?

Compositional data is when each example or sample is a vector that sums to 1 (or 100%). In my case, I am interested in the composition of minerals in a rock and I have sensors that tell me the sum of the minerals but not the components that make up the sum.

For example, lets say I have two minerals, $m_1$ and $m_2$, that are made up of 3 elements (like copper and other elements from the periodic table) which form a vector of length 3:

m1 = [0.1, 0.3, 0.6]
m2 = [0.6, 0.2, 0.2]

If a rock has 25% of $m_1$ and 75% of $m_2$, the sensor reading produces the sum of the two minerals: $0.25*m_1 + 0.75*m_2$. This is plotted below:

I would like to know how to model and solve the problem of unmixing a composition into its underlying components, where the sum of the elements is normalized to 100% (e.g. $0.25m_1 + 0.75m_2$ has the same composition as $0.50m_1 + 1.50m_2$).

Furthermore, my example is simplistic; in reality a composition can have more than just 2 minerals (up to 3000) and each mineral is made up of 118 elements, not just 3 (all the elements of periodic table - though many elements will be zero). The elemental composition of a mineral is assumed to be known (definition of $m_1$ and $m_2$ in the example). Also, the sensor reading is noisy - each element of the observed composition is assumed to have Gaussian noise.

What is the best way to model compositional data problems?

Compositional data is when each example or sample is a vector that sums to 1 (or 100%). In my case, I am interested in the composition of minerals in a rock and I have sensors that tell me the sum of the minerals but not the components that make up the sum.

For example, lets say I have two minerals, $m_1$ and $m_2$, that are made up of 3 elements (like copper and other elements from the periodic table) which form a vector of length 3:

m1 = [0.1, 0.3, 0.6]
m2 = [0.6, 0.2, 0.2]

If a rock has 25% of $m_1$ and 75% of $m_2$, the sensor reading produces the sum of the two minerals (shown in the plot below):

$$0.25*m_1 + 0.75*m_2 = [0.475, 0.225, 0.3]$$

I would like to know how to model and solve the problem of unmixing a composition into its underlying components, where the sum of the elements is normalized to 100% (e.g. $0.25m_1 + 0.75m_2$ has the same composition as $0.50m_1 + 1.50m_2$).

Furthermore, my example is simplistic; in reality a composition can have more than just 2 minerals (up to 3000) and each mineral is made up of 118 elements, not just 3 (all the elements of periodic table - though many elements will be zero). The elemental composition of a mineral is assumed to be known (definition of $m_1$ and $m_2$ in the example). Also, the sensor reading is noisy - each element of the observed composition is assumed to have Gaussian noise.

What is the best way to model compositional data problems?

Compositional data is when each example or sample is a vector that sums to 1 (or 100%). In my case, I am interested in the composition of minerals in a rock and I have sensors that tell me the sum of the minerals but not the components that make up the sum.

For example, lets say I have two minerals, $m_1$ and $m_2$, that are made up of 3 elements (like copper and other elements from the periodic table) which form a vector of length 3:

m1 = [0.1, 0.3, 0.6]
m2 = [0.6, 0.2, 0.2]

If a rock has 25% of $m_1$ and 75% of $m_2$, the sensor reading produces the sum of the two minerals: $0.25*m_1 + 0.75*m_2$. This is plotted below:

I would like to know how to model and solve the problem of unmixing a composition into its underlying components, where the sum of the elements is normalized to 100% (e.g. $0.25m_1 + 0.75m_2$ has the same composition as $0.50m_1 + 1.50m_2$).

What is the best way to model compositional data problems?

Compositional data is when each example or sample is a vector that sums to 1 (or 100%). In my case, I am interested in the composition of minerals in a rock and I have sensors that tell me the sum of the minerals but not the components that make up the sum.

For example, lets say I have two minerals, $m_1$ and $m_2$, that are made up of 3 elements (like copper and other elements from the periodic table) which form a vector of length 3:

m1 = [0.1, 0.3, 0.6]
m2 = [0.6, 0.2, 0.2]

If a rock has 25% of $m_1$ and 75% of $m_2$, the sensor reading produces the sum of the two minerals: $0.25*m_1 + 0.75*m_2$. This is plotted below:

I would like to know how to model and solve the problem of unmixing a composition into its underlying components, where the sum of the elements is normalized to 100% (e.g. $0.25m_1 + 0.75m_2$ has the same composition as $0.50m_1 + 1.50m_2$).

Furthermore, my example is simplistic; in reality a composition can have more than just 2 minerals (up to 3000) and each mineral is made up of 118 elements, not just 3 (all the elements of periodic table - though many elements will be zero). The elemental composition of a mineral is assumed to be known (definition of $m_1$ and $m_2$ in the example). Also, the sensor reading is noisy - each element of the observed composition is assumed to have Gaussian noise.