I am analyzing data for my dissertation, and I have participants see initial information, make judgments, see additional information, and make the same judgments again. I don't know how or if I need to control for these initial judgments when doing analyses about the second judgments.
I understand that the first judgments cannot be covariates because they are affected by my IV/manipulations. Also, I only expect the second judgments to change for some conditions, so if I use the difference between first and second judgments, I only expect that to change for two of my four conditions.
A common way to handle comparisons between the first and second judgments would be as paired data. If condition is a between-subjects factor, then a between x within design using repeated measures ANOVA or for judgments where the scaling isn't such that you're willing to make assumptions necessary for linear models, using a generalized linear model setup that handles repeated measurements might be applicable. In SPSS for linear models, you can set up the judgments as two different variables and condition as a third, then use Analyze>General Linear Models>Repeated Measures. For generalized linear models you can use with generalized estimating equations (GEE) or mixed models, though these require a fair amount of data to be reliable. In the menus, there are Analyze>Generalized Linear Models>Generalized Estimating Equations and Analyze>Mixed Models>Generalized Linear, respectively. Each of these requires data setup for repeated measures to be in the "long" or "narrow" format, where you have a subject ID variable, a time index, the judgment variable, and the condition variable. You'd have two cases per subject, one for each time point.
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I was wondering if anyone knows how SPSS builds the interaction terms/calculates the significance for predictors behind the scenes in a GLM? From my understanding it dummy codes variables and treats the one that comes alphabetically last as the reference group.
The reason I'm asking is I have a GLM model which has 3 continuous predictors and two categorical predictors (dummy coded). When I build all the 2-way and 3-way interactions with syntax ie:
Age_Centred Age_CentredDx Age_Centredgender Age_CentredDxgender BMI_Centred BMI_CentredDx BMI_Centredgender BMI_CentredDxgender BPS_Centred BPS_CentredDx BPS_Centredgender BPS_CentredDxgender Dx Dxgender DxICV_Centred DxICV_Centredgender gender ICV_Centred ICV_Centred*gender.
vs manually creating all the variables by hand ie:
Age_Centred Age_Centred_Dx Age_Centred_gender Age_Centred_gender_Dx BMI_Centred BMI_Centred_Dx BMI_Centred_gender BMI_Centred_gender_Dx BPS_Centred BPS_Centred_Dx BPS_Centred_gender BPS_Centred_gender_Dx Dx gender_Dx ICV_Dx ICV_Centred_Dx_gender gender ICV_Centred ICV_gender.
I end up with a model which has the same intercept, overall significance, and R squared however the individual significance of the predictors changes. Refer to output below. To troubleshoot I've tried to flip the references groups when manually creating the variables but it still does not replicate the results. I've had another statistician try the same thing and ended up reaching the same point as what I did. Does it have to do with some of the parameters being redundant?
Building the terms via syntax:
Physically creating the variables by multiplying them together
All the details one might reasonably want about how GLM (and UNIANOVA, which is the same underlying code) parameterizes models, estimates parameters, and conducts hypothesis tests are available in the IBM SPSS Statistics Algorithms manual, available for download as a pdf at ftp://public.dhe.ibm.com/software/analytics/spss/documentation/statistics/26.0/en/client/Manuals/IBM_SPSS_Statistics_Algorithms.pdf. (Note that this is a large file, about 78 MB; clicking on the link starts a download.) In addition to the information in the GLM chapter, appendices F (Indicator Method) and H (Sums of Squares) are relevant, respectively, for building the design matrix and specifying linear combinations of model parameters for computing sums of squares for testing hypotheses.
In building the design matrix, categorical predictors (factors) are indeed represented by sets of indicator (0-1) variables. For a factor with k levels, k indicator variables are created, one for each observed level of the factor. The procedure does not explicitly treat the last category (sorted in ascending order, alphabetical for strings) as a reference category, though in simpler models the effect of what's done is essentially the same. If there is an intercept in the model, then the kth indicator will be redundant (linearly dependent) on the intercept and the preceding k-1 indicators. The estimation algorithm used in GLM/UNIANOVA will set the row and column in the cross-product matrix representing the redundant column in the design matrix to 0s, alias the corresponding parameter estimate to 0, and the results are similar to a reparameterization approach treating the last category as a reference category, except that you have to remember that it's there if you want to specify a linear combination of the parameters to estimate.
If you suppress the intercept, then for the first factor entered into the model the kth indicator would not be redundant (unless the factor is preceded by an unusual covariate or set of covariates). Any subsequent factors included in the model would involve redundant parameters, as would any interactions among factors, whether or not an intercept is included. Interactions among factors are created by multiplying the 0s and 1s for each level of the factors by those for each level of the other factor. So for an interaction of two two-level factors, there are four columns generated, of which typically the last three are redundant.
Covariates are entered simply by copying the values of the variables into the design matrix. Interactions involving covariates and other covariates multiply values for the columns involved within each row, and interactions involving covariates and factors multiply covariates (or products of them) by the indicator variables for the factor(s). Usually covariate-by-covariate terms do not involve redundancies, but factor-by-covariate terms do.
To get to the specifics of what's going on with your data, I can't replicate your exact results without your data, but I am able to replicate the patterns shown if I assume you've used the binary Dx variable as a covariate and the binary gender variable as a factor in each analysis. (There seem to actually be four continuous predictors in your model rather than three, but that doesn't affect anything of importance for understanding what's going on.)
There are two aspects of the situation to be considered. One is the parameterization and how the two ways of entering the variables into the model treat the variables and whether or not they produce the same estimates of parameters. The second is how the model specification results in the Type III tests shown in the ANOVA tables.
If I'm understanding things correctly based on what you've posted here, you should find if you compare parameter estimates for the two analyses that the parameter estimates for the intercepts and the non-redundant estimates for gender ([gender=0]) are the same, and have the same standard errors. For the terms involving just covariates or products of covariates, I expect that you will find the parameter estimates to differ between the two analyses and produce different t statistics. For interactions involving gender and covariates (which is all the other variables or products created outside the procedure), I expect the estimates will be the same in magnitude and opposite in sign, with the same standard errors.
None of the estimates or tests here are wrong. The models fitted involve interaction effects. An interaction means that effect of one variable varies by the levels of the other variable(s) in the interaction, and in order to estimate the same simple effects you have to parameterize the model in the same way, at least as far as the non-redundant parameters are concerned. However, to get the Type III tests for all terms to be identical, it's not always enough to have the same parameter estimates and standard errors. Type III tests involve a concept called containment that must also be considered.
For two effects in a model, effect A is contained in effect B if:
A and B contain the same covariate terms, if any.
B contains all factor effects in A, and at least one more (with the intercept being contained in all factor-only effects).
In your original model, the intercept is included in the gender effect, gender is not included in any effects, and all the covariate main effects and two-way interactions among covariates are contained within the interactions between those terms and gender, while the three-way interactions (which include gender) are not contained within any other effects.
Type III sums of squares (not invented by SPSS, but by our friends at SAS) are based on linear combinations of parameters where a given effect is adjusted for any effects that do not contain it, and made orthogonal to any effects that contain it. The practical application of these rules is complicated (see Appendix H of the algorithms).
If you recode the gender variable to swap the 0 and 1 values, specify it as a covariate along with all the other variables, and fit the same model, you should be able to match all the non-redundant parameter estimates from the original model, along with their standard errors and t statistics. However, because the containment relationships in the original model are no longer there, the Type III tests for the terms not involving gender (which were previously contained in terms involving gender) will not match up.
The bottom line is that all results are translatable and all correct for what's being done, and that in order to make much sense out of individual terms you have to carefully focus on what's being estimated in a given parameterization, as well as the containment relationships. The difficult part gets simpler when you take seriously the fact that when variable X is involved in interaction terms, there is no single estimate of the effect of X. Any estimates are conditional one where you fix the value(s) of the terms with which X interacts.
I am implementing a SARSA(lambda) model in C++ to overcome some of the limitations (the sheer amount of time and space DP models require) of DP models, which hopefully will reduce the computation time (takes quite a few hours atm for similar research) and less space will allow adding more complexion to the model.
We do have explicit transition probabilities, and they do make a difference. So how should we incorporate them in a SARSA model?
Simply select the next state according to the probabilities themselves? Apparently SARSA models don't exactly expect you to use probabilities - or perhaps I've been reading the wrong books.
PS- Is there a way of knowing if the algorithm is properly implemented? First time working with SARSA.
The fundamental difference between Dynamic Programming (DP) and Reinforcement Learning (RL) is that the first assumes that environment's dynamics is known (i.e., a model), while the latter can learn directly from data obtained from the process, in the form of a set of samples, a set of process trajectories, or a single trajectory. Because of this feature, RL methods are useful when a model is difficult or costly to construct. However, it should be notice that both approaches share the same working principles (called Generalized Policy Iteration in Sutton's book).
Given they are similar, both approaches also share some limitations, namely, the curse of dimensionality. From Busoniu's book (chapter 3 is free and probably useful for your purposes):
A central challenge in the DP and RL fields is that, in their original
form (i.e., tabular form), DP and RL algorithms cannot be implemented
for general problems. They can only be implemented when the state and
action spaces consist of a finite number of discrete elements, because
(among other reasons) they require the exact representation of value
functions or policies, which is generally impossible for state spaces
with an infinite number of elements (or too costly when the number of
states is very high).
Even when the states and actions take finitely many values, the cost
of representing value functions and policies grows exponentially with
the number of state variables (and action variables, for Q-functions).
This problem is called the curse of dimensionality, and makes the
classical DP and RL algorithms impractical when there are many state
and action variables. To cope with these problems, versions of the
classical algorithms that approximately represent value functions
and/or policies must be used. Since most problems of practical
interest have large or continuous state and action spaces,
approximation is essential in DP and RL.
In your case, it seems quite clear that you should employ some kind of function approximation. However, given that you know the transition probability matrix, you can choose a method based on DP or RL. In the case of RL, transitions are simply used to compute the next state given an action.
Whether is better to use DP or RL? Actually I don't know the answer, and the optimal method likely depends on your specific problem. Intuitively, sampling a set of states in a planned way (DP) seems more safe, but maybe a big part of your state space is irrelevant to find an optimal pocliy. In such a case, sampling a set of trajectories (RL) maybe is more effective computationally. In any case, if both methods are rightly applied, should achive a similar solution.
NOTE: when employing function approximation, the convergence properties are more fragile and it is not rare to diverge during the iteration process, especially when the approximator is non linear (such as an artificial neural network) combined with RL.
If you have access to the transition probabilities, I would suggest not to use methods based on a Q-value. This will require additional sampling in order to extract information that you already have.
It may not always be the case, but without additional information I would say that modified policy iteration is a more appropriate method for your problem.
I am reading about n-grams and I am wondering whether there is a case in practice when uni-grams would are preferred to be used over bi-grams (or higher N-grams). As I understand, the bigger N, the bigger complexity to calculate the probabilities and establish the vector space. But apart from that, are there other reasons (e.g. related to type of data)?
This boils down to data sparsity: As your n-gram length increases, the amount of times you will see any given n-gram will decrease: In the most extreme example, if you have a corpus where the maximum document length is n tokens and you are looking for an m-gram where m=n+1, you will, of course, have no data points at all because it's simply not possible to have a sequence of that length in your data set. The more sparse your data set, the worse you can model it. For this reason, despite that a higher-order n-gram model, in theory, contains more information about a word's context, it cannot easily generalize to other data sets (known as overfitting) because the number of events (i.e. n-grams) it has seen during training becomes progressively less as n increases. On the other hand, a lower-order model lacks contextual information and so may underfit your data.
For this reason, if you have a very relatively large amount of token types (i.e. the vocabulary of your text is very rich) but each of these types has a very low frequency, you may get better results with a lower-order n-gram model. Similarly, if your training data set is very small, you may do better with a lower-order n-gram model. However, assuming that you have enough data to avoid over-fitting, you then get better separability of your data with a higher-order model.
Usually, n-grams more than 1 is better as it carries more information about the context in general. However, sometimes unigrams are also calculated besides bigram and trigrams and used as fallback for them. This is usefull also, if you want high recall than precision to search unigrams, for instance, you are searching for all possible uses of verb "make".
Lets use Statistical Machine Translation as an Example:
Intuitively, the best scenario is that your model has seen the full sentence (lets say 6-grams) before and knows its translation as a whole. If this is not the case you try to divide it to smaller n-grams, keeping into consideration that the more information you know about the word surroundings, the better the translation. For example, if you want to translate "Tom Green" to German, if you have seen the bi-gram you will know it is a person name and should remain as it is but if your model never saw it, you would fall back to unigrams and translate "Tom" and "Green" separately. Thus "Green" will be translated as a color to "Grün" and so on.
Also, in search knowing more about the surrounding context makes the results more accurate.
I want to build a model that recognizes the species based on multiple indicators. The problem is, neural networks (usually) receive vectors, and my indicators are not always easily expressed in numbers. For example, one of the indicators is not only whether species performs some actions (that would be, say, '0' or '1', or anything in between, if the essence of action permits that), but sometimes, in which order are those actions performed. I want the system to be able to decide and classify species based on these indicators. There are not may classes but rather many indicators.
The amount of training data is not an issue, I can get as much as I want.
What machine learning techniques should I consider? Maybe some special kind of neural network would do? Or maybe something completely different.
If you treat a sequence of actions as a string, then using features like "an action A was performed" is akin to unigram model. If you want to account for order of actions, you should add bigrams, trigrams, etc.
That will blow up your feature space, though. For example, if you have M possible actions, then there are M (M-1) / 2 bigrams. In general, there are O(Mk) k-grams. This leads to the following issues:
The more features you have — the harder it is to apply some methods. For example, many models suffer from curse of dimensionality
The more features you have — the more data you need to capture meaningful relations.
This is just one possible approach to your problem. There may be others. For example, if you know that there's some set of parameters ϴ, that governs action-generating process in a known (at least approximately) way, you can build a separate model to infer these first, and then use ϴ as features.
The process of coming up with sensible numerical representation of your data is called feature engineering. Once you've done that, you can use any Machine Learning algorithm at your disposal.
I need to write a program that performs arithmetic (+-*/) on multiples time series of different date range (mostly from 2007-2009) and frequency (weekly, monthly, yearly...).
I came up with:
find the series with the highest freq. then fill in the other series with zeros so they have the same number of elements. then perform the operation.
How can I present the data in the most meaningful way?
Trying to think of all the possibilities
If zero can be a meaningful value for this time series (e.g. temperature in Celsius degrees), it might not be a good idea to fill all gaps with zeros (i.e. you will not be able to distinguish between the real and stub values afterwards). You might want to interpolate your time series. Basic data structure for this can be array/double linked list.
You can take several approaches:
use the finest-grained time series data (for instance, seconds) and interpolate/fill data when needed
use the coarsest-grained (for instance, years) and summarize data when needed
any middle step between the two extremes
You should always know your data, because:
in case of interpolating you have to choose the best algorithm (linear or quadratic interpolation, splines, exponential...)
in case of summarizing you have to choose an appropriate aggregation function (sum, maximum, mean...)
Once you have the same time scales for all the time series you can perform your arithmetical magick, but be aware that interpolation generates extra information, and summarization removes available information.
I've studied this problem fairly extensively. The danger of interpolation methods is that you bias various measures - especially volatility - and introduce spurious correlation. I found that Fourier interpolation mitigated this to some extent but the better approach is to go the other way: aggregate your more frequent observations to match the periodicity of your less frequent series, then compare these.