How can we modify the damped random walk model to allow for the other changes seen in the temperature anomaly. We arrived at the random walk model by subtracting the anomaly from the five year average and found a standard deviation which is characteristic of the "random" variations present in the data. I wanted to compare a one-dimensional random walk with the anomaly data and tried a simple sum first but it produced permanent displacements which were not seen in the anomaly. Multiplying the random walk displacement by a damping factor produced deviations more like those found in the anomaly. We can modify the formula used by assuming the zero value represented an equilibrium value, T0, for the damped random walk. A new monthly average temperature, T', can be broken down into the equilibrium value plus a random change to the temperature fitting the normal distribution plus the damped anomaly.
To get this random walk model to produce results similar to the we would have to allow the equilibrium value to change over time. Once we have matched the changes in equilibrium over time numerically they would still have to be explained either as some external or internal change to the environment. One would have to consider sunspot activity, the melting ice pack, oscillations in the weather, greenhouse gases, etc.